The Fluid Mechanics of Microdevices—The ... - People.vcu.edu

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Oct 24, 2013 (3 years and 7 months ago)

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Mohame
d
Gad-el-Ha
k
Professor
,
Departmen
t
o
f
Aerospac
e
an
d
Mecfianica
l
Engineering
,
Universit
y
o
f
Notr
e
Dame
,
Notr
e
Dame
,
I
N
46556
,
Fello
w
ASIVI
E
E-mail
:
Mohamed.Gad-el-Hal<.1@nd.ed
u
Th
e
Flui
d
Mechanic
s
o
f
Microdevices—Th
e
Freema
n
Schola
r
Lectur
e
Manufacturing
processes
that
can
create
extremely
small
machines
have
been
devel-
oped
in
recent
years.
Microelectromechanical
systems
(MEMS)
refer
to
devices
that
have
characteristic
length
of
less
than
1
mm
but
more
than
1
micron,
that
combine
electrical
and
mechanical
components
and
that
are
fabricated
using
integrated
circuit
batch-processing
techniques.
Electrostatic,
magnetic,
pneumatic
and
thermal
actua-
tors,
motors,
valves,
gears,
and
tweezers
of
less
than
100-pm
size
have
been
fabri-
cated.
These
have
been
used
as
sensors
for
pressure,
temperature,
mass
flow,
velocity
and
sound,
as
actuators
for
linear
and
angular
motions,
and
as
simple
components
for
complex
systems
such
as
micro-heat-engines
and
micro-heat-pumps.
The
technol-
ogy
is
progressing
at
a
rate
that
far
exceeds
that
of
our
understanding
of
the
unconven-
tional
physics
involved
in
the
operation
as
well
as
the
manufacturing
of
those
minute
devices.
The
primary
objective
of
this
article
is
to
critically
review
the
status
of
our
understanding
of
fluid
flow
phenomena
particular
to
microdevices.
In
terms
of
applications,
the
paper
emphasizes
the
use
of
MEMS
as
sensors
and
actuators
for
flow
diagnosis
and
control.
Abou
t
th
e
Autho
r
Mohame
d
Gad-el-Ha
k
receive
d
hi
s
B.Sc
.
(summ
a
cu
m
laude
)
i
n
mechanica
l
engineerin
g
fro
m
Ai
n
Sham
s
Universit
y
i
n
196
6
an
d
hi
s
Ph.D
.
i
n
fluid
mechanic
s
fro
m
th
e
John
s
Hopkin
s
Universit
y
i
n
1973
.
H
e
ha
s
sinc
e
taugh
t
an
d
conducte
d
researc
h
a
t
th
e
Universit
y
o
f
Souther
n
California
,
Universit
y
o
f
Virginia
,
Institu
t
Nationa
l
Polytechniqu
e
d
e
Grenoble
,
an
d
Universit
e
d
e
Poitiers
,
an
d
ha
s
lecture
d
extensivel
y
a
t
seminar
s
i
n
th
e
Unite
d
State
s
an
d
overseas
.
Dr
.
Gad-el-Ha
k
i
s
currentl
y
Professo
r
o
f
Aerospac
e
an
d
Mechanica
l
Engineerin
g
a
t
th
e
Universit
y
o
f
Notr
e
Dame
.
Prio
r
t
o
that
,
h
e
was
a
Senio
r
Researc
h
Scientis
t
an
d
Progra
m
Manage
r
a
t
Flo
w
Researc
h
Compan
y
i
n
Seattle
,
Washington
.
Dr
.
Gad-el-Ha
k
ha
s
publishe
d
ove
r
28
0
article
s
an
d
presente
d
17
0
invite
d
lecture
s
i
n
th
e
basi
c
an
d
applie
d
researc
h
area
s
o
f
isotropi
c
turbulence
,
boundar
y
laye
r
flows,
stratifie
d
flows,
complian
t
coatings
,
unstead
y
aerodynamics
,
bi
-
ologica
l
flows,
non-Newtonia
n
fluids,
har
d
an
d
sof
t
computin
g
includin
g
geneti
c
algorithms
,
an
d
flow
control
.
H
e
i
s
th
e
autho
r
o
f
th
e
boo
k
Flow
Control,
an
d
edito
r
o
f
thre
e
Springer-Verlag'
s
book
s
Frontiers
in
Experimental
Fluid
Mechanics,
Advances
in
Fluid
Mechanics
Measurements,
an
d
Flow
Control:
Fundamen-
tals
and
Practices.
Professo
r
Gad-el-Ha
k
i
s
a
fello
w
o
f
Th
e
America
n
Societ
y
o
f
Mechanica
l
Engineers
,
a
lif
e
membe
r
o
f
th
e
America
n
Physica
l
Society
,
an
d
a
n
associat
e
fello
w
o
f
th
e
America
n
Institut
e
o
f
Aeronautic
s
an
d
Astronautics
.
H
e
ha
s
recentl
y
bee
n
inducte
d
a
s
a
n
eminen
t
enginee
r
i
n
Ta
u
Bet
a
Pi
,
a
n
honorar
y
membe
r
i
n
Sigm
a
Gamm
a
Ta
u
an
d
P
i
Ta
u
Sigma
,
an
d
a
member-at-larg
e
i
n
Sigm
a
Xi
.
Fro
m
198
8
t
o
1991
,
Dr
.
Gad-el-Ha
k
serve
d
a
s
Associat
e
Edito
r
fo
r
AIAA
Journal.
H
e
i
s
currentl
y
a
n
Associat
e
Edito
r
fo
r
Applied
Mechanics
Reviews.
I
n
1998
,
Professo
r
Gad-el-Ha
k
wa
s
name
d
th
e
Fourteent
h
ASM
E
Freema
n
Scholar
.
1
Introductio
n
How
many
times
when you
are
working
on
something
frus-
tratingly
tiny,
like
your
wife's
wrist
watch,
have
you
said
to
Contribute
d
b
y
th
e
Fluid
s
Engineerin
g
Divisio
n
fo
r
publicatio
n
i
n
th
e
JOURNA
L
O
F
FLUtD
S
ENGINEERIN
G
,
Manuscrip
t
receive
d
b
y
th
e
Fluid
s
Engineerin
g
Divisio
n
Augus
t
31
,
1998
;
revise
d
manuscrip
t
receive
d
Decembe
r
14
,
1998
.
Associat
e
Technica
l
Editor
;
D
.
P
.
Telionis
.
yourself,
'
'If
I
could
only
train
an
ant
to
do
this!''
What
I
would
like
to
suggest
is
the
possibility
of
training
an
ant
to
train
a
mite
to
do
this.
What
are
the
possibilities
of
small
but
movable
machines?
They
may
or
may
not
be
useful,
but
they
surely
would
be
fun
to
make.
(Fro
m
th
e
tal
k
"There'
s
Plent
y
o
f
Roo
m
a
t
th
e
Bottom,
"
deliv
-
ere
d
b
y
Richar
d
P
.
Feynma
n
a
t
th
e
annua
l
meetin
g
o
f
th
e
Ameri
-
ca
n
Physica
l
Society
,
Pasadena
,
California
,
2
9
Decembe
r
1959.
)
Too
l
makin
g
ha
s
alway
s
differentiate
d
ou
r
specie
s
fro
m
al
l
other
s
o
n
earth
.
Aerodynamicall
y
correc
t
woode
n
spear
s
wer
e
carve
d
b
y
archai
c
homosapien
s
clos
e
t
o
400,00
0
year
s
ago
.
Ma
n
build
s
thing
s
consisten
t
wit
h
hi
s
size
,
typicall
y
i
n
th
e
rang
e
o
f
tw
o
order
s
o
f
magnitud
e
large
r
o
r
smalle
r
tha
n
himself
,
a
s
indi
-
cate
d
i
n
Fig
.
1
.
(Thoug
h
th
e
extreme
s
o
f
length-scal
e
ar
e
outsid
e
th
e
rang
e
o
f
thi
s
figure,
man
,
a
t
slightl
y
mor
e
tha
n
10
°
m
,
amazingl
y
fits
righ
t
i
n
th
e
middl
e
o
f
th
e
smalles
t
subatomi
c
particl
e
whic
h
i
s
approximatel
y
1
0
"^"
^
m
an
d
th
e
exten
t
o
f
th
e
observabl
e
univers
e
whic
h
i
s
~1.4
2
X
10^
^
m
(1
5
billio
n
ligh
t
years)
.
A
n
egocentri
c
univers
e
indeed!
)
Bu
t
human
s
hav
e
al
-
way
s
strive
n
t
o
explore
,
build
,
an
d
contro
l
th
e
extreme
s
o
f
lengt
h
an
d
tim
e
scales
.
I
n
th
e
voyage
s
t
o
Lillipu
t
an
d
Brobding
-
na
g
o
f
Gulliver's
Travels,
Jonatha
n
Swif
t
(1727
)
speculate
s
o
n
th
e
remarkabl
e
possibilitie
s
whic
h
diminutio
n
o
r
magnificatio
n
o
f
physica
l
dimension
s
provides
.
Th
e
Grea
t
Pyrami
d
o
f
Khuf
u
wa
s
originall
y
14
7
m
hig
h
whe
n
complete
d
aroun
d
260
0
B.C.
,
whil
e
th
e
Empir
e
Stat
e
Buildin
g
constructe
d
i
n
193
1
i
s
pres
-
ently—afte
r
th
e
additio
n
o
f
a
televisio
n
antenn
a
mas
t
i
n
1950

44
9
m
high
.
A
t
th
e
othe
r
en
d
o
f
th
e
spectru
m
o
f
man-mad
e
artifacts
,
a
dim
e
i
s
slightl
y
les
s
tha
n
2
c
m
i
n
diameter
.
Watch
-
maker
s
hav
e
practice
d
th
e
ar
t
o
f
miniaturizatio
n
sinc
e
th
e
thir
-
teent
h
century
.
Th
e
inventio
n
o
f
th
e
microscop
e
i
n
th
e
seven
-
teent
h
centur
y
opene
d
th
e
wa
y
fo
r
direc
t
observatio
n
o
f
mi
-
crobe
s
an
d
plan
t
an
d
anima
l
cells
.
Smalle
r
thing
s
wer
e
man
-
mad
e
i
n
th
e
latte
r
hal
f
o
f
thi
s
century
.
Th
e
transistor—invente
d
i
n
1948—i
n
toda
y
integrate
d
circuit
s
ha
s
a
siz
e
o
f
0.2
5
micro
n
i
n
productio
n
an
d
approache
s
5
0
nanometer
s
i
n
researc
h
labora
-
tories
.
Bu
t
wha
t
abou
t
th
e
miniaturizatio
n
o
f
mechanica
l
parts

machines—envisione
d
b
y
Feynma
n
(1961
)
i
n
hi
s
legendar
y
speec
h
quote
d
above
?
Manufacturin
g
processe
s
tha
t
ca
n
creat
e
extremel
y
smal
l
ma
-
chine
s
hav
e
bee
n
develope
d
i
n
recen
t
year
s
(Angel
l
e
t
al.
,
1983
;
Journa
l
o
f
Fluid
s
Engineerin
g
Copyrigh
t
©
199
9
b
y
ASM
E
MARC
H
1999
,
Vol
.
1 2 1/
5
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Diamete
r
o
f
Eart
h
10
2
10
4
10
8
,
,
lo
S
lOl
O
,
,
10
"
10
"
i l O
"
10
"
lO
M
Astronomica
l
Uni
t
Ligli
t
Yea
r
Voyage
to
Brobdingnag
Voyage
to
Ulliput
mete
r
10-1
«
10-1
4
10-1
2
10-1
0
10-
8
'
Diamete
r
o
f
Proto
n
Ato
m
Diamete
r
Nanodevice
s
10-
*
10-
2
io
»
10
2
'
Huma
n
Hai
r
'
Ma
n
Typica
l
Man-Mad
e
Device
s
(MEMS
J
Fig
.
1
Tii
e
scal
e
o
f
things
,
i
n
meters
.
Lowe
r
scal
e
continue
s
I
n
th
e
uppe
r
ba
r
fro
m
lef
t
t
o
right
.
Gabrie
l
e
t
al
,
1988
;
1992
;
O'Connor
,
1992
;
Gravese
n
e
t
al
,
1993
;
Bryze
k
e
t
al
,
1994
;
Gabriel
,
1995
;
Hogan
,
1996
;
H
o
an
d
Tai
,
1996
;
1998
;
Tien
,
1997
;
Busch-Vishniac
,
1998
;
Amato
,
1998)
.
Electrostatic
,
magnetic
,
pneumati
c
an
d
therma
l
actua
-
tors
,
motors
,
valves
,
gear
s
an
d
tweezer
s
o
f
les
s
tha
n
10
0
/x
m
siz
e
hav
e
bee
n
fabricated
.
Thes
e
hav
e
bee
n
use
d
a
s
sensor
s
fo
r
pressure
,
temperature
,
mas
s
flow,
velocit
y
an
d
sound
,
a
s
actuator
s
fo
r
linea
r
an
d
angula
r
motions
,
an
d
a
s
simpl
e
compo
-
nent
s
fo
r
comple
x
system
s
suc
h
a
s
micro-heat-engine
s
an
d
mi
-
cro-heat-pump
s
(Lipkin
,
1993
;
Garci
a
an
d
Sniegowski
,
1993
;
1995
;
Sniegowsk
i
an
d
Garcia
,
1996
;
Epstei
n
an
d
Senturia
,
1997
;
Epstei
n
e
t
al
,
1997)
.
Th
e
technolog
y
i
s
progressin
g
a
t
a
rat
e
tha
t
fa
r
exceed
s
tha
t
o
f
ou
r
understandin
g
o
f
th
e
unconventiona
l
physic
s
involve
d
i
n
th
e
operatio
n
a
s
wel
l
a
s
th
e
manufacturin
g
o
f
thos
e
minut
e
devices
.
Th
e
presen
t
pape
r
focuse
s
o
n
on
e
aspec
t
o
f
suc
h
physics
:
fluid flow
phenomen
a
associate
d
wit
h
micro-scal
e
devices
.
I
n
term
s
o
f
applications
,
th
e
pape
r
wil
l
emphasiz
e
th
e
us
e
o
f
MEM
S
a
s
sensor
s
an
d
actuator
s
fo
r
flow
diagnosi
s
an
d
control
.
Microelectromechanica
l
system
s
(MEMS
)
refe
r
t
o
device
s
tha
t
hav
e
characteristi
c
lengt
h
o
f
les
s
tha
n
1
m
m
bu
t
mor
e
tha
n
1
micron
,
tha
t
combin
e
electrica
l
an
d
mechanica
l
component
s
an
d
tha
t
ar
e
fabricate
d
usin
g
integrate
d
circui
t
batch-processin
g
technologies
.
Curren
t
manufacturin
g
technique
s
fo
r
MEM
S
in
-
clud
e
surfac
e
silico
n
micromachining
;
bul
k
silico
n
microma
-
chining
;
lithography
,
electrodepositio
n
an
d
plasti
c
moldin
g
(or
,
i
n
it
s
origina
l
German
,
lithographi
c
galvanoformun
g
abfor
-
mung
,
LIGA)
;
an
d
electrodischarg
e
machinin
g
(EDM)
.
A
s
in
-
dicate
d
i
n
Fig
.
1
,
MEM
S
ar
e
mor
e
tha
n
fou
r
order
s
o
f
magnitud
e
large
r
tha
n
th
e
diamete
r
o
f
th
e
hydroge
n
atom
,
bu
t
abou
t
fou
r
order
s
o
f
magnitud
e
smalle
r
tha
n
th
e
traditiona
l
man-mad
e
arti
-
facts
.
Nanodevice
s
(som
e
sa
y
NEMS
)
furthe
r
pus
h
th
e
envelop
e
o
f
electromechanica
l
miniaturization
.
Despit
e
Feynman'
s
demurrin
g
regardin
g
th
e
usefulnes
s
o
f
smal
l
machines
,
MEM
S
ar
e
finding
increase
d
appUcation
s
i
n
a
variet
y
o
f
industria
l
an
d
medica
l
fields
,
wit
h
a
potentia
l
world
-
wid
e
marke
t
i
n
th
e
billion
s
o
f
dollars
.
Accelerometer
s
fo
r
auto
-
mobil
e
airbags
,
keyles
s
entr
y
systems
,
dens
e
array
s
o
f
micro
-
mirror
s
fo
r
high-definitio
n
optica
l
displays
,
scannin
g
electro
n
microscop
e
tip
s
t
o
imag
e
singl
e
atoms
,
micro-heat-exchanger
s
fo
r
coolin
g
o
f
electroni
c
circuits
,
reactor
s
fo
r
separatin
g
biologi
-
ca
l
cells
,
bloo
d
analyzer
s
an
d
pressur
e
sensor
s
fo
r
cathete
r
tip
s
ar
e
bu
t
a
fe
w
o
f
curren
t
usage
.
Microduct
s
ar
e
use
d
i
n
infrare
d
detectors
,
diod
e
lasers
,
miniatur
e
ga
s
chromatograph
s
an
d
high
-
frequenc
y
fluidic
contro
l
systems
.
Micropump
s
ar
e
use
d
fo
r
in
k
je
t
printing
,
environmenta
l
testin
g
an
d
electroni
c
cooling
.
Potentia
l
medica
l
application
s
fo
r
smal
l
pump
s
includ
e
con
-
trolle
d
deliver
y
an
d
monitorin
g
o
f
minut
e
amoun
t
o
f
medication
.
manufacturin
g
o
f
nanoliter
s
o
f
chemical
s
an
d
developmen
t
o
f
artificia
l
pancreas
.
Severa
l
ne
w
journal
s
ar
e
dedicate
d
t
o
th
e
scienc
e
an
d
technolog
y
o
f
MEMS
,
fo
r
exampl
e
lEEE/ASM
E
Journal
of
Microelectromechanical
Systems,
Journal
of
Micro-
mechanics
and
Microengineering,
and
Microscale
Thermophys-
ical
Engineering.
No
t
al
l
MEM
S
device
s
involv
e
fluid flows,
bu
t
th
e
presen
t
revie
w
wil
l
focu
s
o
n
th
e
one
s
tha
t
do
.
Microducts
,
micropumps
,
microturbine
s
an
d
microvalve
s
ar
e
example
s
o
f
smal
l
device
s
involvin
g
th
e
flow
o
f
liquid
s
an
d
gases
.
MEM
S
ca
n
als
o
b
e
relate
d
t
o
fluid flows
i
n
a
n
indirec
t
way
.
Th
e
availabilit
y
o
f
inexpensive
,
batch-processing-produce
d
microsensor
s
an
d
mi
-
croactuator
s
provide
s
opportunitie
s
fo
r
targetin
g
small-scal
e
co
-
heren
t
structure
s
i
n
macroscopi
c
turbulen
t
shea
r
flows.
Flo
w
contro
l
usin
g
MEM
S
promise
s
a
quantu
m
lea
p
i
n
contro
l
syste
m
performance
.
Th
e
presen
t
articl
e
wil
l
cove
r
bot
h
th
e
direc
t
an
d
indirec
t
aspect
s
o
f
microdevice
s
an
d
fluid flows.
Sectio
n
2
ad
-
dresse
s
th
e
questio
n
o
f
modelin
g
fluid flows
i
n
microdevices
,
an
d
Sectio
n
3
give
s
a
brie
f
overvie
w
o
f
typica
l
application
s
o
f
MEM
S
i
n
th
e
field
o
f
fluid
mechanics
.
Th
e
pape
r
b
y
Lofdah
l
an
d
Gad-el-Ha
k
(1999
)
provide
s
mor
e
detai
l
o
n
MEM
S
applica
-
tion
s
i
n
turbulenc
e
an
d
flow
control
.
Th
e
Freema
n
Scholarshi
p
i
s
bestowe
d
biennially
,
i
n
even
-
numbere
d
years
.
Th
e
Fourteent
h
Freema
n
Lectur
e
presente
d
i
n
199
8
is
,
therefore
,
th
e
las
t
o
f
it
s
kin
d
i
n
thi
s
millennium
,
an
d
th
e
topi
c
o
f
micromachine
s
i
s
perhap
s
a
fitting
en
d
t
o
a
centur
y
o
f
spectacula
r
progres
s
i
n
mechanica
l
engineerin
g
le
d
i
n
n
o
smal
l
par
t
b
y
member
s
o
f
ASM
E
International
.
2
Flui
d
Mechanic
s
Issue
s
2.
1
Prologue
.
Th
e
rapi
d
progres
s
i
n
fabricatin
g
an
d
utiliz
-
in
g
microelectromechanica
l
system
s
durin
g
th
e
las
t
decad
e
ha
s
no
t
bee
n
matche
d
b
y
correspondin
g
advance
s
i
n
ou
r
understand
-
in
g
o
f
th
e
unconventiona
l
physic
s
involve
d
i
n
th
e
operatio
n
an
d
manufactur
e
o
f
smal
l
devices
.
Providin
g
suc
h
understandin
g
i
s
crucia
l
t
o
designing
,
optimizing
,
fabricatin
g
an
d
operatin
g
improve
d
MEM
S
devices
.
Flui
d
flows
i
n
smal
l
device
s
diffe
r
fro
m
thos
e
i
n
macroscopi
c
machines
.
Th
e
operatio
n
o
f
MEMS-base
d
ducts
,
nozzles
,
valves
,
bearings
,
turbomachines
,
etc.
,
canno
t
alway
s
b
e
pre
-
dicte
d
fro
m
conventiona
l
flow
model
s
suc
h
a
s
th
e
Navier-Stoke
s
equation
s
wit
h
no-sli
p
boundar
y
conditio
n
a
t
a
fluid-solid
inter
-
face
,
a
s
routinel
y
an
d
successfull
y
applie
d
fo
r
large
r
flow
de
-
vices
.
Man
y
question
s
hav
e
bee
n
raise
d
whe
n
th
e
result
s
o
f
experiment
s
wit
h
microdevice
s
coul
d
no
t
b
e
explaine
d
vi
a
tradi
-
tiona
l
flow
modeling
.
Th
e
pressur
e
gradien
t
i
n
a
lon
g
microduc
t
wa
s
observe
d
t
o
b
e
non-constan
t
an
d
th
e
measure
d
flowrate
wa
s
6
/
Vol
.
121
,
MARC
H
199
9
Transaction
s
o
f
th
e
ASM
E
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
higher than that predicted from the conventional continuum
flow model. Load capacities of microbearings were diminished
and electric currents needed to move micromotors were extraor-
dinarily high. The dynamic response of micromachined acceler-
ometers operating at atmospheric conditions was observed to
be over-damped.
In the early stages of development of this exciting new field,
the objective was to build MEMS devices as productively as
possible. Microsensors were reading something, but not many
researchers seemed to know exactly what. Microactuators were
moving, but conventional modeling could not precisely predict
their motion. After a decade of unprecedented progress in
MEMS technology, perhaps the time is now ripe to take stock,
slow down a bit and answer the many questions that arose. The
ultimate aim of this long-term exercise is to achieve rational-
design capability for useful microdevices and to be able to
characterize definitively and with as little empiricism as possible
the operations of microsensors and microactuators.
In dealing with fluid flow through microdevices, one is faced
with the question of which model to use, which boundary condi-
tion to apply and how to proceed to obtain solutions to the
problem at hand. Obviously surface effects dominate in small
devices. The surface-to-volume ratio for a machine with a char-
acteristic length of I m is I m"', while that for a MEMS device
having a size of 1 //m is 10' m~'. The million-fold increase in
surface area relative to the mass of the minute device substan-
tially affects the transport of mass, momentum and energy
through the surface. The small length-scale of microdevices
may invalidate the continuum approximation altogether. Slip
flow, thermal creep, rarefaction, viscous dissipation, compress-
ibility, intermolecular forces and other unconventional effects
may have to be taken into account, preferably using only first
principles such as conservation of mass, Newton's second law,
conservation of energy, etc.
In this section, we discuss continuum as well as molecular-
based flow models, and the choices to be made. Computing
typical Reynolds, Mach and Knudsen numbers for the flow
through a particular device is a good start to characterize the
flow. For gases, microfluid mechanics has been studied by incor-
porating slip boundary conditions, thermal creep, viscous dissi-
pation as well as compressibility effects into the continuum
equations of motion. Molecular-based models have also been
attempted for certain ranges of the operating parameters. Use
is made of the well-developed kinetic theory of gases, embodied
in the Boltzmann equation, and direct simulation methods such
as Monte Carlo. Microfluid mechanics of liquids is more com-
plicated. The molecules are much more closely packed at nor-
mal pressures and temperatures, and the attractive or cohesive
potential between the liquid molecules as well as between the
liquid and solid ones plays a dominant role if the characteristic
length of the flow is sufficiently small. In cases when the tradi-
tional continuum model fails to provide accurate predictions or
postdictions, expensive molecular dynamics simulations seem
to be the only first-principle approach available to rationally
characterize liquid flows in microdevices. Such simulations are
not yet feasible for realistic flow extent or number of molecules.
As a consequence, the microfluid mechanics of liquids is much
less developed than that for gases.
2.2 Fluid Modeling. There are basically two ways of
modeling a flow field. Either as the fluid really is, a collection
of molecules, or as a continuum where the matter is assumed
continuous and indefinitely divisible. The former modeling
is subdivided into deterministic methods and probabilistic
ones, while in the latter approach the velocity, density, pres-
sure, etc., are defined at every point in space and time, and
conservation of mass, energy and momentum lead to a set of
nonlinear partial differential equations (Euler, Navier-
Stokes, Burnett, etc.). Fluid modeling classification is de-
picted schematically in Fig. 2.
Fig. 2 Molecular and continuum flow models
The continuum model, embodied in the Navier-Stokes equa-
tions, is applicable to numerous flow situations. The model
ignores the molecular nature of gases and liquids and regards
the fluid as a continuous medium describable in terms of the
spatial and temporal variations of density, velocity, pressure,
temperature and other macroscopic flow quantities. For dilute
gas flows near equilibrium, the Navier-Stokes equations are
derivable from the molecularly-based Boltzmann equation, but
can also be derived independently of that for both liquids and
gases. In the case of direct derivation, some empiricism is neces-
sary to close the resulting indeterminate set of equations. The
continuum model is easier to handle mathematically (and is
also more familiar to most fluid dynamists) than the alternative
molecular models. Continuum models should therefore be used
as long as they are applicable. Thus, careful considerations of
the validity of the Navier-Stokes equations and the like are in
order.
Basically, the continuum model leads to fairly accurate pre-
dictions as long as local properties such as density and velocity
can be defined as averages over elements large compared with
the microscopic structure of the fluid but small enough in com-
parison with the scale of the macroscopic phenomena to permit
the use of differential calculus to describe them. Additionally,
the flow must not be too far from thermodynamic equilibrium.
The former condition is almost always satisfied, but it is the
latter which usually restricts the validity of the continuum equa-
tions. As will be seen in Section 2.3, the continuum flow equa-
tions do not form a determinate set. The shear stress and heat
flux must be expressed in terms of lower-order macroscopic
quantities such as velocity and temperature, and the simplest
(i.e., linear) relations are valid only when the flow is near
thermodynamic equilibrium. Worse yet, the traditional no-slip
boundary condition at a solid-fluid interface breaks down even
before the linear stress-strain relation becomes invalid.
To be more specific, we temporarily restrict the discussion
to gases where the concept of mean free path is well defined.
Liquids are more problematic and we defer their discussion to
Section 2.7. For gases, the mean free path £ is the average
distance traveled by molecules between collisions. For an ideal
gas modeled as rigid spheres, the mean free path is related to
temperature T and pressure p as follows
I: =
1 kT
\/27 f2
TTpa
(1)
where n is the number density (number of molecules per unit
volume), a is the molecular diameter, and k is the Boltzmann
constant.
The continuum model is valid when X' is much smaller than
a characteristic flow dimension L. As this condition is violated,
the flow is no longer near equilibrium and the linear relation
Journal of Fluids Engineering MARCH 1999, Vol. 1 2 1/7
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
between stress and rate of strain and the no-slip velocity condi-
tion are no longer valid. Similarly, the linear relation between
heat flux and temperature gradient and the no-jump temperature
condition at a solid-fluid interface are no longer accurate when
£ is not much smaller than L.
The length-scale L can be some overall dimension of the
flow, but a more precise choice is the scale of the gradient of
a macroscopic quantity, as for example the density p.
(2)
dp
dy
i:n= 0.0001 0.001 0,01
Continuum flow
(ordinary density levels)
Transition regime
(noderately raretkd)
Slip-flow regime
(Bllghtly rarcHed)
Fig. 3 Knudsen number regimes
Free-molecule flow
(highly ranried)
The ratio between the mean free path and the characteristic
length is known as the Knudsen number
Kn
£
L
(3)
and generally the traditional continuum approach is valid, albeit
with modified boundary conditions, as long as Kn < 0.1.
There are two more important dimensionless parameters in
fluid mechanics, and the Knudsen number can be expressed in
terms of those two. The Reynolds number is the ratio of inertial
forces to viscous ones
Re = ^ ^
(4)
where v„ is a characteristic velocity, and v is the kinematic
viscosity of the fluid. The Mach number is the ratio of flow
velocity to the speed of sound
Ma (5)
The Mach number is a dynamic measure of fluid compressibility
and may be considered as the ratio of inertial forces to elastic
ones. From the kinetic theory of gases, the mean free path is
related to the viscosity as follows
V = - = -£v,„
p 2
where p, is the dynamic viscosity, and v„ is the mean molecular
speed which is somewhat higher than the sound speed Uo,
Try
a„
(7)
where y is the specific heat ratio (i.e. the isentropic exponent).
Combining Equations ( 3) - ( 7), we reach the required relation
/TTT Ma
Kn = , /
2 Re
(8)
In boundary layers, the relevant length-scale is the shear-
layer thickness 6, and for laminar flows
(9)
Kn
8
L
~
1
VRe
Ma _
Re« ~
Ma
\/Re
(10)
where Re^ is the Reynolds number based on the freestream
velocity v„ and the boundary layer thickness 8, and Re is based
on D„ and the streamwise length-scale L.
Rarefied gas flows are in general encountered in flows in
small geometries such as MEMS devices and in low-pressure
applications such as high-altitude flying and high-vacuum gad-
gets. The local value of Knudsen number in a particular flow
determines the degree of rarefaction and the degree of validity
of the continuum model. The different Knudsen number regimes
are determined empirically and are therefore only approximate
for a particular flow geometry. The pioneering experiments in
rarefied gas dynamics were conducted by Knudsen in 1909. In
the limit of zero Knudsen number, the transport terms in the
continuum momentum and energy equations are negligible and
the Navier-Stokes equations then reduce to the inviscid Euler
equations. Both heat conduction and viscous diffusion and dissi-
pation are negligible, and the flow is then approximately isen-
tropic (i.e., adiabatic and reversible) from the continuum view-
point while the equivalent molecular viewpoint is that the veloc-
ity distribution function is everywhere of the local equilibrium
or Maxwellian form. As Kn increases, rarefaction effects be-
come more important, and eventually the continuum approach
breaks down altogether. The different Knudsen number regimes
are depicted in Fig. 3, and can be summarized as follows
Euler equations (neglect molecular diffusion):
Kn -» 0 (Re ^ «))
Navier-Stokes equations with no-slip boundary conditions:
Kn s 10"'
Navier-Stokes equations with slip boundary conditions:
10"' < Kn s 10-'
^ ' Transition regime:
10"
Free-molecule flow:
Kn < 10
Kn > 10
We will return to those regimes in the following subsections.
As an example, consider air at standard temperature (T =
288 K) and pressure {p = 1.01 X 10^ N/m^). A cube one
micron to the side contains 2.54 X 10' molecules separated by
an average distance of 0.0034 micron. The gas is considered
dilute if the ratio of this distance to the molecular diameter
exceeds 7, and in the present example this ratio is 9, barely
satisfying the dilute gas assumption. The mean free path com-
puted from Eq. (1) is X' = 0.065 pm. A microdevice with
characteristic length of 1 pm would have Kn = 0.065, which
is in the slip-flow regime. At lower pressures, the Knudsen
number increases. For example, if the pressure is 0.1 atm and
the temperature remains the same, Kn = 0.65 for the same 1-
pm device, and the flow is then in the transition regime. There
would still be over 2 million molecules in the same one-micron
cube, and the average distance between them would be 0.0074
pm. The same device at 100 km altitude would have Kn = 3
X lO'', well into the free-molecule flow regime. Knudsen num-
ber for the flow of a light gas like helium is about 3 times larger
than that for air flow at otherwise the same conditions.
Consider a long microchannel where the entrance pressure is
atmospheric and the exit conditions are near vacuum. As air
goes down the duct, the pressure and density decrease while
the velocity, Mach number and Knudsen number increase. The
pressure drops to overcome viscous forces in the channel. If
isothermal conditions prevail, density also drops and conserva-
8 / Vol. 121, MARCH 1999 Transactions of the ASME
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
tio
n
o
f
mas
s
require
s
th
e
flow
t
o
accelerat
e
dow
n
th
e
constant
-
are
a
tube
.
(Mor
e
likel
y
th
e
flow
wil
l
b
e
somewher
e
i
n
betwee
n
isotherma
l
an
d
adiabatic
,
Fann
o
flow.
I
n
tha
t
cas
e
bot
h
densit
y
an
d
temperatur
e
decreas
e
downstream
,
th
e
forme
r
no
t
a
s
fas
t
a
s
i
n
th
e
isotherma
l
case
.
Non
e
o
f
tha
t
change
s
th
e
qualitativ
e
argument
s
mad
e
i
n
th
e
example.
)
Th
e
fluid
acceleratio
n
i
n
tur
n
affect
s
th
e
pressur
e
gradient
,
resultin
g
i
n
a
nonlinea
r
pressur
e
dro
p
alon
g
th
e
channel
.
Th
e
Mac
h
numbe
r
increase
s
dow
n
th
e
tube
,
limite
d
onl
y
b
y
choked-flo
w
conditio
n
(M
a
=1 )
.
Addi
-
tionally
,
th
e
norma
l
componen
t
o
f
velocit
y
i
s
n
o
longe
r
zero
.
Wit
h
lowe
r
density
,
th
e
mea
n
fre
e
pat
h
increase
s
an
d
K
n
corre
-
spondingl
y
increases
.
Al
l
flow
regime
s
depicte
d
i
n
Fig
.
3
ma
y
occu
r
i
n
th
e
sam
e
tube
:
continuu
m
wit
h
no-sli
p
boundar
y
condi
-
tions
,
slip-flo
w
regime
,
transitio
n
regim
e
an
d
free-molecul
e
flow.
Th
e
ai
r
flow
ma
y
als
o
chang
e
fro
m
incompressibl
e
t
o
compressibl
e
a
s
i
t
move
s
dow
n
th
e
microduct
.
A
simila
r
sce
-
nari
o
ma
y
tak
e
plac
e
i
f
th
e
entranc
e
pressur
e
is
,
say
,
5
atm
,
whil
e
th
e
exi
t
i
s
atmospheric
.
Thi
s
deceivingl
y
simpl
e
duc
t
flow
ma
y
i
n
fac
t
manifes
t
ever
y
singl
e
complexit
y
discusse
d
i
n
thi
s
section
.
I
n
th
e
followin
g
si
x
subsections
,
w
e
discus
s
i
n
tur
n
th
e
Na
-
vier-Stoke
s
equations
,
compressibilit
y
effects
,
boundar
y
condi
-
tions
,
molecular-base
d
models
,
liqui
d
flows
an
d
surfac
e
phe
-
nomena
.
2.
3
Continuu
m
Model
.
W
e
recal
l
i
n
thi
s
subsectio
n
th
e
traditiona
l
conservatio
n
relation
s
i
n
fluid
mechanics
.
N
o
deriva
-
tio
n
i
s
give
n
her
e
an
d
th
e
reade
r
i
s
referre
d
t
o
an
y
advance
d
textboo
k
i
n
fluid
mechanics
,
e.g.
,
Batchelo
r
(1967)
,
Landa
u
an
d
Lifshit
z
(1987)
,
Sherma
n
(1990)
,
Kund
u
(1990)
,
an
d
Panto
n
(1996)
.
I
n
here
,
instead
,
w
e
emphasiz
e
th
e
precis
e
assumption
s
neede
d
t
o
obtai
n
a
particula
r
for
m
o
f
thos
e
equations
.
A
contin
-
uu
m
fluid
implie
s
tha
t
th
e
derivative
s
o
f
al
l
th
e
dependen
t
vari
-
able
s
exis
t
i
n
som
e
reasonabl
e
sense
.
I
n
othe
r
words
,
loca
l
propertie
s
suc
h
a
s
densit
y
an
d
velocit
y
ar
e
define
d
a
s
average
s
ove
r
element
s
larg
e
compare
d
wit
h
th
e
microscopi
c
structur
e
o
f
th
e
fluid
bu
t
smal
l
enoug
h
i
n
compariso
n
wit
h
th
e
scal
e
o
f
th
e
macroscopi
c
phenomen
a
t
o
permi
t
th
e
us
e
o
f
differentia
l
calculu
s
t
o
describ
e
them
.
A
s
mentione
d
earlier
,
suc
h
condition
s
ar
e
almos
t
alway
s
met
.
Fo
r
suc
h
fluids,
an
d
assumin
g
th
e
law
s
o
f
non-relativisti
c
mechanic
s
hold
,
th
e
conservatio
n
o
f
mass
,
momentu
m
an
d
energ
y
ca
n
b
e
expresse
d
a
t
ever
y
poin
t
i
n
spac
e
an
d
tim
e
a
s
a
se
t
o
f
partia
l
differentia
l
equation
s
a
s
follow
s
dt
9
,
.
dxu
0
+
Ilk
at
axt
de
dxt
+
PS.
de
dt
"
dxk
dxt
+
ffi
dui
dxt
(11
)
(12
)
(13
)
wher
e
p
i
s
th
e
fluid
density
,
«
*
i
s
a
n
instantaneou
s
velocit
y
componen
t
(u,
v,
w),
a^i
i
s
th
e
second-orde
r
stres
s
tenso
r
(surfac
e
forc
e
pe
r
uni
t
area)
,
an
d
gi
i
s
th
e
bod
y
forc
e
pe
r
uni
t
mass
,
e
i
s
th
e
interna
l
energy
,
an
d
qt
i
s
th
e
su
m
o
f
hea
t
flux
vector
s
du
e
t
o
conductio
n
an
d
radiation
.
Th
e
independen
t
variable
s
ar
e
tim
e
t
an
d
th
e
thre
e
spatia
l
coordinate
s
Xi
,
X2
an
d
x^
ox
{x,y,
z).
Equation
s
(11)
,
(12)
,
an
d
(13
)
constitut
e
5
differentia
l
equa
-
tion
s
fo
r
th
e
1
7
unknown
s
p,
u,,
at,,
e
,
an
d
qt-
Absen
t
an
y
bod
y
couples
,
th
e
stres
s
tenso
r
i
s
symmetri
c
havin
g
onl
y
si
x
independen
t
components
,
whic
h
reduce
s
th
e
numbe
r
o
f
un
-
known
s
t
o
14
.
Obviously
,
th
e
continuu
m
flow
equation
s
d
o
not
for
m
a
determinat
e
set
.
T
o
clos
e
th
e
conservatio
n
equations
,
relatio
n
betwee
n
th
e
stres
s
tenso
r
an
d
deformatio
n
rate
,
relatio
n
betwee
n
th
e
hea
t
flux
vecto
r
an
d
th
e
temperatur
e
field
an
d
ap
-
propriat
e
equation
s
o
f
stat
e
relatin
g
th
e
differen
t
thermodynami
c
propertie
s
ar
e
needed
.
Th
e
stress-rat
e
o
f
strai
n
relatio
n
an
d
th
e
hea
t
flux-temperature
relatio
n
ar
e
approximatel
y
linea
r
i
f
th
e
flow
i
s
no
t
to
o
fa
r
fro
m
thermodynami
c
equilibrium
.
Thi
s
i
s
a
phenomenologica
l
resul
t
but
ca
n
b
e
rigorousl
y
derive
d
fro
m
th
e
Boltzman
n
equatio
n
fo
r
a
dilut
e
ga
s
assumin
g
th
e
flow
i
s
nea
r
equilibriu
m
(se
e
Sectio
n
2.6)
.
Fo
r
a
Newtonian
,
isotropic
,
Fou
-
rier
,
idea
l
gas
,
fo
r
example
,
thos
e
relation
s
rea
d
J
.
/
dui
duk
<7(
,
=
-p6ki
+
f^[-^
+
T-
\
oxk
aXi
dT
+
\i
duj
dxj
6k,
(14
)
qi
=
-K
h
Hea
t
flu
x
du
e
t
o
radiatio
n
(15
)
OXi
de
=
c„dT
an
d
p
=
pW
(16
)
wher
e
p
i
s
th
e
thermodynami
c
pressure
,
p
an
d
\
ar
e
th
e
first
an
d
secon
d
coefficient
s
o
f
viscosity
,
respectively
,
6n
i
s
th
e
uni
t
second-orde
r
tenso
r
(Kronecke
r
delta)
,
K
i
s
th
e
therma
l
conduc
-
tivity
,
r i
s
th
e
temperatur
e
field,
c

i
s
th
e
specifi
c
hea
t
a
t
constan
t
volume
,
an
d
"R
i
s
th
e
ga
s
constan
t
whic
h
i
s
give
n
b
y
th
e
Boltz
-
man
n
constan
t
divide
d
b
y
th
e
mas
s
o
f
a
n
individua
l
molecul
e
(k
=
m'R).
(Newtonia
n
implie
s
a
linea
r
relatio
n
betwee
n
th
e
stres
s
tenso
r
an
d
th
e
symmetri
c
par
t
o
f
th
e
deformatio
n
tenso
r
(rat
e
o
f
strai
n
tensor)
.
Th
e
isotrop
y
assumptio
n
reduce
s
th
e
8
1
constant
s
o
f
proportionalit
y
i
n
tha
t
linea
r
relatio
n
t
o
tw
o
con
-
stants
.
Fourie
r
fluid
i
s
tha
t
fo
r
whic
h
th
e
conductio
n
par
t
o
f
th
e
hea
t
flux
vecto
r
i
s
linearl
y
relate
d
t
o
th
e
temperatur
e
gradient
,
an
d
agai
n
isotrop
y
implie
s
tha
t
th
e
constan
t
o
f
proportionalit
y
i
n
thi
s
relatio
n
i
s
a
singl
e
scalar.
)
Th
e
Stokes
'
hypothesi
s
relate
s
th
e
first
an
d
secon
d
coefficient
s
o
f
viscosit
y
thu
s
\
+
|;t
i
=
0
,
althoug
h
th
e
validit
y
o
f
thi
s
assumptio
n
fo
r
othe
r
tha
n
dilute
,
monatomi
c
gase
s
ha
s
occasionall
y
bee
n
questione
d
(Gad-el
-
Hak
,
1995)
.
Wit
h
th
e
abov
e
constitutiv
e
relation
s
an
d
neglect
-
in
g
radiativ
e
hea
t
transfe
r
(
a
reasonabl
e
assumptio
n
whe
n
deal
-
in
g
wit
h
lo
w
t
o
moderat
e
temperature
s
sinc
e
th
e
radiativ
e
hea
t
flux
i
s
proportiona
l
t
o
T")
,
Equation
s
( I I )
,
(12)
,
an
d
(13)
,
respectively
,
rea
d
9p
9
,
.
^
ot
axk
(17
)
/
dui
dUi
dt
dxi
dp
d
dXi
oxi
.
dUi
dut\
duj
M
T.
'
+
°i!i^

dXk
dXi
I
dxj
(18
)
fdT
dT
"'"'
'
¥
+
"
'
&
.
d
(
dT\
dut
^
,,„
,
dXk
\
dXkl
oXk
Th
e
thre
e
component
s
o
f
th
e
vecto
r
equatio
n
(18
)
ar
e
th
e
Na
-
vier-Stoke
s
equation
s
expressin
g
th
e
conservatio
n
o
f
momen
-
tu
m
fo
r
a
I>Jewtonia
n
fluid.
I
n
th
e
therma
l
energ
y
equatio
n
(19)
,
4>
i
s
th
e
alway
s
positiv
e
(a
s
require
d
b
y
th
e
Secon
d
La
w
o
f
thermodynamics
)
dissipatio
n
functio
n
expressin
g
th
e
irrevers
-
ibl
e
convfTsio
n
o
f
mechanica
l
energ
y
t
o
interna
l
energ
y
a
s
a
resul
t
o
f
th
e
deformatio
n
o
f
a
fluid
element
.
Th
e
secon
d
ter
m
o
n
th
e
right-han
d
sid
e
o
f
(19
)
i
s
th
e
reversibl
e
wor
k
don
e
(pe
r
uni
t
time
)
b
y
th
e
pressur
e
a
s
th
e
volum
e
o
f
a
fluid
materia
l
elemen
t
changes
.
Fo
r
a
Newtonian
,
isotropi
c
fluid,
th
e
viscou
s
dissipatio
n
rat
e
i
s
give
n
b
y
I
dui
dut
dxt
dXi
^
\
+\
dUj
dxi
(20
)
Ther
e
ar
e
no
w
si
x
unknowns
,
p,
«,
,
p
an
d
T,
an
d
th
e
five
couple
d
equation
s
(17)
,
(18)
,
an
d
(19
)
plu
s
th
e
equatio
n
o
f
Journa
l
o
f
Fluid
s
Engineerin
g
MARC
H
1999
,
Vol
.
1 2 1/
9
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
stat
e
relatin
g
pressure
,
densit
y
an
d
temperature
.
Thes
e
si
x
equa
-
tion
s
togethe
r
wit
h
sufficien
t
numbe
r
o
f
initia
l
an
d
boundar
y
condition
s
constitut
e
a
well-posed
,
albei
t
formidable
,
problem
.
Th
e
syste
m
o
f
equation
s
( I 7) - ( 19
)
i
s
a
n
excellen
t
mode
l
fo
r
th
e
lamina
r
o
r
turbulen
t
flow
o
f
mos
t
fluids
suc
h
a
s
ai
r
an
d
wate
r
unde
r
man
y
circumstances
,
includin
g
high-spee
d
ga
s
flows
fo
r
whic
h
th
e
shoc
k
wave
s
ar
e
thic
k
relativ
e
t
o
th
e
mea
n
fre
e
pat
h
o
f
th
e
molecules
.
(Thi
s
conditio
n
i
s
me
t
i
f
th
e
shoc
k
Mac
h
numbe
r
i
s
les
s
tha
n
2.
)
Considerabl
e
simplificatio
n
i
s
achieve
d
i
f
th
e
flow
i
s
assume
d
incompressible
,
usuall
y
a
reasonabl
e
assumptio
n
provide
d
tha
t
th
e
characteristi
c
flow
spee
d
i
s
les
s
tha
n
0.
3
o
f
th
e
spee
d
o
f
sound
.
(Althoug
h
a
s
wil
l
b
e
demonstrate
d
i
n
th
e
followin
g
sub
-
section
,
ther
e
ar
e
circumstance
s
whe
n
eve
n
a
low-Mach-numbe
r
flow
shoul
d
b
e
treate
d
a
s
compressible.
)
Th
e
incompressibilit
y
assumptio
n
i
s
readil
y
satisfie
d
fo
r
almos
t
al
l
liqui
d
flows
an
d
man
y
ga
s
flows.
I
n
suc
h
cases
,
th
e
densit
y
i
s
assume
d
eithe
r
a
constan
t
o
r
a
give
n
functio
n
o
f
temperatur
e
(o
r
specie
s
concen
-
tration)
.
(Withi
n
th
e
so-calle
d
Boussines
q
approximation
,
den
-
sit
y
variation
s
hav
e
negligibl
e
effec
t
o
n
inerti
a
bu
t
ar
e
retaine
d
i
n
th
e
buoyanc
y
terms
.
Th
e
incompressibl
e
continuit
y
equatio
n
i
s
therefor
e
used.
)
Th
e
governin
g
equation
s
fo
r
suc
h
flow
ar
e
dut
dxt
0
(21
)
dui
dui
h
Uk
dt
dx,,
2.
4
Compressibility
.
Th
e
issu
e
o
f
whethe
r
t
o
conside
r
th
e
continuu
m
flow
compressibl
e
o
r
incompressibl
e
seem
s
t
o
b
e
rathe
r
straightforward
,
bu
t
i
s
i
n
fac
t
ful
l
o
f
potentia
l
pitfalls
.
I
f
th
e
loca
l
Mac
h
numbe
r
i
s
les
s
tha
n
0.3
,
the
n
th
e
flow
o
f
a
compressibl
e
fluid
lik
e
ai
r
can—accordin
g
t
o
th
e
conventiona
l
wisdom—b
e
treate
d
a
s
incompressible
.
Bu
t
th
e
well-know
n
M
a
<
0.
3
criterio
n
i
s
onl
y
a
necessar
y
no
t
a
sufficien
t
on
e
t
o
allo
w
treatmen
t
o
f
th
e
flow
a
s
approximatel
y
incompressible
.
I
n
othe
r
words
,
ther
e
ar
e
situation
s
wher
e
th
e
Mac
h
numbe
r
ca
n
b
e
exceedingl
y
smal
l
whil
e
th
e
flow
i
s
compressible
.
A
s
i
s
wel
l
documente
d
i
n
hea
t
transfe
r
textbooks
,
stron
g
wal
l
heatin
g
o
r
coolin
g
ma
y
caus
e
th
e
densit
y
t
o
chang
e
sufficientl
y
an
d
th
e
incompressibl
e
approximatio
n
t
o
brea
k
down
,
eve
n
a
t
lo
w
speeds
.
Les
s
know
n
i
s
th
e
situatio
n
encountere
d
i
n
som
e
micro
-
device
s
wher
e
th
e
pressur
e
ma
y
strongl
y
chang
e
du
e
t
o
viscou
s
effect
s
eve
n
thoug
h
th
e
speed
s
ma
y
not
b
e
hig
h
enoug
h
fo
r
th
e
Mac
h
numbe
r
t
o
g
o
abov
e
th
e
traditiona
l
threshol
d
o
f
0.3
.
Correspondin
g
t
o
th
e
pressur
e
change
s
woul
d
b
e
stron
g
densit
y
change
s
tha
t
mus
t
b
e
take
n
int
o
accoun
t
whe
n
writin
g
th
e
con
-
tinuu
m
equation
s
o
f
motion
.
I
n
thi
s
section
,
w
e
systematicall
y
explai
n
al
l
situation
s
relevan
t
t
o
MEM
S
wher
e
compressibilit
y
effect
s
mus
t
b
e
considered
.
(Tw
o
othe
r
situation
s
wher
e
com
-
pressibilit
y
effect
s
mus
t
als
o
b
e
considere
d
ar
e
length-scale
s
comparabl
e
t
o
th
e
scal
e
heigh
t
o
f
th
e
atmospher
e
an
d
rapidl
y
varyin
g
flows
a
s
i
n
soun
d
propagatio
n
(se
e
Lighthill
,
1963)
.
Neithe
r
o
f
thes
e
situation
s
i
s
likel
y
t
o
b
e
encountere
d
i
n
micro
-
devices.
)
Le
t
u
s
rewrit
e
th
e
ful
l
continuit
y
equatio
n
(11
)
a
s
follow
s
dp
d
dxi
dxt
M
/
dUi
dui
l
^
fdT
dT\
d
I
1
-
u,
=
dxt
J
dxk
''•\-^^
K—\
+
dxj
+
Pgi
(22
)
(23
)
wher
e
</>i„com
p
i
s
th
e
incompressibl
e
limi
t
o
f
Eq
.
(20)
.
Thes
e
ar
e
no
w
five
equation
s
fo
r
th
e
five
dependen
t
variable
s
w,
,
p
an
d
T.
Not
e
tha
t
th
e
left-han
d
sid
e
o
f
Eq
.
(23
)
ha
s
th
e
specifi
c
hea
t
a
t
constan
t
pressur
e
c,,
an
d
no
t
c„
.
I
t
i
s
th
e
convectio
n
o
f
enthalpy—an
d
not
interna
l
energy—tha
t
i
s
balance
d
b
y
hea
t
conductio
n
an
d
viscou
s
dissipation
.
Thi
s
i
s
th
e
correc
t
incom
-
pressible-flo
w
limit—o
f
a
compressibl
e
fluid—as
discusse
d
i
n
detai
l
i
n
Sectio
n
10.
9
o
f
Panto
n
(1996)
;
a
subtl
e
poin
t
perhap
s
bu
t
on
e
tha
t
i
s
frequentl
y
misinterprete
d
i
n
textbooks
.
Th
e
sys
-
te
m
o
f
equation
s
( 21) - ( 23
)
i
s
couple
d
i
f
eithe
r
th
e
viscosit
y
o
r
densit
y
depend
s
o
n
temperature
,
otherwis
e
th
e
energ
y
equa
-
tio
n
i
s
uncouple
d
fro
m
th
e
continuit
y
an
d
momentu
m
equation
s
an
d
ca
n
therefor
e
b
e
solve
d
after
th
e
velocit
y
an
d
pressur
e
fields
ar
e
determined
.
Fo
r
bot
h
th
e
compressibl
e
an
d
th
e
incompressibl
e
equation
s
o
f
motion
,
th
e
transpor
t
term
s
ar
e
neglecte
d
awa
y
fro
m
soli
d
wall
s
i
n
th
e
limi
t
o
f
infinit
e
Reynold
s
numbe
r
(K
n
-
>
0)
.
Th
e
fluid
i
s
the
n
approximate
d
a
s
invisci
d
an
d
non-conducting
,
an
d
th
e
correspondin
g
equation
s
rea
d
(fo
r
th
e
compressibl
e
case
)

+
TT-
iPUt)
=
0
at
axk
duj
Ik
+
Uk
dUi
dxt
dp
oxi
(24
)
(25
)
Dt
dxi
(27
)
wher
e
D/Dt
i
s
th
e
substantia
l
derivativ
e
(d/dt
+
Ukdldx^),
expressin
g
change
s
followin
g
a
fluid
element
.
Th
e
prope
r
crite
-
rio
n
fo
r
th
e
incompressibl
e
approximatio
n
t
o
hol
d
i
s
tha
t
(l/p){Dp/Dt)
i
s
vanishingl
y
small
.
I
n
othe
r
words
,
i
f
densit
y
change
s
followin
g
a
fluid
particl
e
ar
e
small
,
th
e
flow
i
s
approxi
-
matel
y
incompressible
.
Densit
y
ma
y
chang
e
arbitraril
y
fro
m
on
e
particl
e
t
o
anothe
r
withou
t
violatin
g
th
e
incompressibl
e
flow
assumption
.
Thi
s
i
s
th
e
cas
e
fo
r
exampl
e
i
n
th
e
stratifie
d
atmospher
e
an
d
ocean
,
wher
e
th
e
variable-density/temperature
/
salinit
y
flow
i
s
ofte
n
treate
d
a
s
incompressible
.
Fro
m
th
e
stat
e
principl
e
o
f
thermodynamics
,
w
e
ca
n
expres
s
th
e
densit
y
change
s
o
f
a
simpl
e
syste
m
i
n
term
s
o
f
change
s
i
n
pressur
e
an
d
temperature
.
p
=
pip,
T)
Usin
g
th
e
chai
n
rul
e
o
f
calculus
,
1
Dp
Dp
DT
=
a
a

Dt
Dt
Dt
(28
)
(29
)
wher
e
a
an
d
(5
are
,
respectively
,
th
e
isotherma
l
compressibilit
y
coefficien
t
an
d
th
e
bul
k
expansio
n
coefficient—tw
o
thermody
-
nami
c
variable
s
tha
t
characteriz
e
th
e
fluid
susceptibilit
y
t
o
chang
e
o
f
volume—whic
h
ar
e
define
d
b
y
th
e
followin
g
rela
-
tion
s
a{p.
T ).i
^
p
dp
(30
)
pc„
(dT
dT
\
-—
+
Ut

\
dt
dx.
dut
dxt
(26
)
0(P.
T)
-
pdf
(31
)
Th
e
Eule
r
equatio
n
(25
)
ca
n
b
e
integrate
d
alon
g
a
streamlin
e
an
d
th
e
resultin
g
Bernoulli'
s
equatio
n
provide
s
a
direc
t
relatio
n
betwee
n
th
e
velocit
y
an
d
pressure
.
Fo
r
idea
l
gases
,
a
=
Up,
an
d
/
3
=
l/T.
Note
,
however
,
tha
t
i
n
th
e
followin
g
argument
s
i
t
wil
l
not
b
e
necessar
y
t
o
invok
e
th
e
idea
l
ga
s
assumption
.
1
0
/
Vol
.
121
,
MARC
H
199
9
Transaction
s
o
f
thi
e
ASiVI
E
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Th
e
flow
mus
t
b
e
treate
d
a
s
compressibl
e
i
f
pressur
e
and/o
r
temperatur
e
change
s
ar
e
sufficientl
y
strong
.
Equatio
n
(29
)
mus
t
o
f
cours
e
b
e
properl
y
nondimensionalize
d
befor
e
decidin
g
whethe
r
a
ter
m
i
s
larg
e
o
r
small
.
I
n
here
,
w
e
follo
w
closel
y
th
e
procedur
e
detaile
d
i
n
Panto
n
(1996)
.
Conside
r
first
th
e
cas
e
o
f
adiabati
c
walls
.
Densit
y
i
s
normal
-
ize
d
wit
h
a
referenc
e
valu
e
p„
,
velocitie
s
wit
h
a
referenc
e
spee
d
v„,
spatia
l
coordinates
,
an
d
tim
e
with
,
respectively
,
L
an
d
Llv„,
th
e
isotherma
l
compressibilit
y
coefficien
t
an
d
bul
k
expansio
n
coefficien
t
wit
h
referenc
e
value
s
a„
an
d
/?„
.
Th
e
pressur
e
i
s
nondimensionalize
d
wit
h
th
e
inertia
l
pressure-scal
e
p„vl.
Thi
s
scal
e
i
s
twic
e
th
e
dynami
c
pressure
,
i.e.
,
th
e
pressur
e
chang
e
a
s
a
n
invisci
d
fluid
movin
g
a
t
th
e
referenc
e
spee
d
i
s
brough
t
t
o
rest
.
Temperatur
e
change
s
fo
r
th
e
cas
e
o
f
adiabati
c
wall
s
resul
t
fro
m
th
e
irreversibl
e
conversio
n
o
f
mechanica
l
energ
y
int
o
in
-
terna
l
energ
y
vi
a
viscou
s
dissipation
.
Temperatur
e
i
s
therefor
e
nondimensionalize
d
a
s
follow
s
r
-
T„
T
(32
)
pressur
e
bee
n
nondimensionalize
d
usin
g
th
e
viscou
s
scal
e
(Pi,v,J
L)
instea
d
o
f
th
e
inertia
l
on
e
(p„i)o)
,
th
e
revise
d
equatio
n
(33
)
woul
d
hav
e
Re^
'
appearin
g
explicitl
y
i
n
th
e
first
ter
m
i
n
th
e
right-han
d
side
,
accentuatin
g
th
e
importanc
e
o
f
thi
s
ter
m
whe
n
viscou
s
force
s
dominate
.
A
simila
r
resul
t
ca
n
b
e
gleane
d
whe
n
th
e
Mac
h
numbe
r
i
s
interprete
d
a
s
follow
s
Ma
^
=
vl
al
dp
p„vl
dp
Po
dp
Ap
A/
9
po
Ap
—- r
^
=

(34
)
Ap
Po
wher
e
s
i
s
th
e
entropy
.
Again
,
th
e
abov
e
equatio
n
assume
s
tha
t
pressur
e
change
s
ar
e
inviscid
,
an
d
therefor
e
smal
l
Mac
h
numbe
r
mean
s
negligibl
e
pressur
e
an
d
densit
y
changes
.
I
n
a
flow
domi
-
nate
d
b
y
viscou
s
effects—suc
h
a
s
tha
t
insid
e
a
microduct

densit
y
change
s
ma
y
b
e
significan
t
eve
n
i
n
th
e
limi
t
o
f
zer
o
Mac
h
number
.
Identica
l
argument
s
ca
n
b
e
mad
e
i
n
th
e
cas
e
o
f
isotherma
l
walls
.
Her
e
stron
g
temperatur
e
change
s
ma
y
b
e
th
e
resul
t
o
f
wal
l
heatin
g
o
r
cooling
,
eve
n
i
f
viscou
s
dissipatio
n
i
s
negligible
.
Th
e
prope
r
temperatur
e
scal
e
i
n
thi
s
cas
e
i
s
give
n
i
n
term
s
o
f
th
e
wal
l
temperatur
e
T„,
an
d
th
e
referenc
e
temperatur
e
T„
a
s
follow
s
wher
e
T„
i
s
a
referenc
e
temperature
,
p„
,
K„
,
an
d
c,
,
are
,
respec
-
tively
,
referenc
e
viscosity
,
therma
l
conductivit
y
an
d
specifi
c
hea
t
a
t
constan
t
pressure
,
an
d
P
r
i
s
th
e
referenc
e
Prandt
l
num
-
ber
,
ip„Ci,J/K„.
I
n
th
e
presen
t
formulation
,
th
e
scalin
g
use
d
fo
r
pressur
e
i
s
base
d
o
n
th
e
Bernoulli'
s
equation
,
an
d
therefor
e
neglect
s
vis
-
cou
s
effects
.
Thi
s
particula
r
scalin
g
guarantee
s
tha
t
th
e
pressur
e
ter
m
i
n
th
e
momentu
m
equatio
n
wil
l
b
e
o
f
th
e
sam
e
orde
r
a
s
th
e
inerti
a
term
.
Th
e
temperatur
e
scalin
g
assume
s
tha
t
th
e
conduction
,
convectio
n
an
d
dissipatio
n
term
s
i
n
th
e
energ
y
equatio
n
hav
e
th
e
sam
e
orde
r
o
f
magnitude
.
Th
e
resultin
g
di
-
mensionles
s
for
m
o
f
Eq
.
(29
)
read
s
Dp*
p-*
Dt*
y„
M
a

^
a
Dp*
Pr
B/3*
DT*
Dt*
Dt*
(33
)
wher
e
th
e
superscrip
t
*
indicate
s
a
nondimensiona
l
quantity
,
M
a
i
s
th
e
referenc
e
Mac
h
number
,
an
d
A
an
d
B
ar
e
dimensionles
s
constant
s
define
d
by
A
=
a„p„CpJ„,
an
d
B
=
/3„T„.
I
f
th
e
scalin
g
i
s
properl
y
chosen
,
th
e
term
s
havin
g
th
e
*
superscrip
t
i
n
th
e
right-han
d
sid
e
shoul
d
b
e
o
f
orde
r
one
,
an
d
th
e
relativ
e
importanc
e
o
f
suc
h
term
s
i
n
th
e
equation
s
o
f
motio
n
i
s
deter
-
mine
d
b
y
th
e
magnitud
e
o
f
th
e
dimensionles
s
parameter(s
)
ap
-
pearin
g
t
o
thei
r
left
,
e.g
.
Ma
,
Pr
,
etc
.
Therefore
,
a
s
Ma
^
-^
0
,
temperatur
e
change
s
du
e
t
o
viscou
s
dissipatio
n
ar
e
neglecte
d
(unles
s
P
r
i
s
ver
y
large
,
a
s
fo
r
exampl
e
i
n
th
e
cas
e
o
f
highl
y
viscou
s
polymer
s
an
d
oils)
.
Withi
n
th
e
sam
e
orde
r
o
f
approxi
-
mation
,
al
l
thermodynami
c
propertie
s
o
f
th
e
fluid
ar
e
assume
d
constant
.
Pressur
e
change
s
ar
e
als
o
neglecte
d
i
n
th
e
limi
t
o
f
zer
o
Mac
h
number
.
Hence
,
fo
r
M
a
<
0.
3
(i.e
.
Ma
^
<
0.09)
,
densit
y
change
s
followin
g
a
fluid
particl
e
ca
n
b
e
neglecte
d
an
d
th
e
flow
ca
n
the
n
b
e
approximate
d
a
s
incompressible
.
(Wit
h
a
n
erro
r
o
f
abou
t
10
%
a
t
M
a
=
0.3
,
4
%
a
t
M
a
=
0.2
,
1
%
a
t
M
a
=
0.1
,
an
d
s
o
on.
)
However
,
ther
e
i
s
a
cavea
t
i
n
thi
s
argument
.
Pressur
e
change
s
du
e
t
o
inerti
a
ca
n
indee
d
b
e
neglecte
d
a
t
smal
l
Mac
h
number
s
an
d
thi
s
i
s
consisten
t
wit
h
th
e
wa
y
w
e
nondimension
-
alize
d
th
e
pressur
e
ter
m
above
.
If
,
o
n
th
e
othe
r
hand
,
pressur
e
change
s
ar
e
mostl
y
du
e
t
o
viscou
s
effects
,
a
s
i
s
th
e
cas
e
fo
r
exampl
e
i
n
a
lon
g
duc
t
o
r
a
ga
s
bearing
,
pressur
e
change
s
ma
y
b
e
significan
t
eve
n
a
t
lo
w
speed
s
(lo
w
Ma)
.
I
n
tha
t
cas
e
th
e
ter
m
Dp
*l
Dt
*
i
n
Eq
.
(33
)
i
s
n
o
longe
r
o
f
orde
r
one
,
an
d
ma
y
b
e
larg
e
regardles
s
o
f
th
e
valu
e
o
f
Ma
.
Densit
y
the
n
ma
y
chang
e
significantl
y
an
d
th
e
flow
mus
t
b
e
treate
d
a
s
compressible
.
Ha
d
(35
)
wher
e
f
i
s
th
e
ne
w
dimensionles
s
temperature
.
Th
e
nondimen
-
siona
l
form
o
f
Eq
.
(29
)
no
w
read
s
'
''Pl=y„M,^a*^-p*B(^"'
p*
D0
Dt*
DT
Dt*
(36
)
Her
e
w
e
notic
e
tha
t
th
e
temperatur
e
ter
m
i
s
differen
t
fro
m
tha
t
i
n
Eq
.
(33)
.
M
a
i
s
n
o
longe
r
appearin
g
i
n
thi
s
term
,
an
d
stron
g
temperatur
e
changes
,
i.e.
,
larg
e
{T„
-
T„)/T„,
ma
y
caus
e
stron
g
densit
y
change
s
regardles
s
o
f
th
e
valu
e
o
f
th
e
Mac
h
number
.
Additionally
,
th
e
thermodynami
c
propertie
s
o
f
th
e
fluid
ar
e
no
t
constan
t
but
depen
d
o
n
temperature
,
an
d
a
s
a
result
,
th
e
continu
-
ity
,
momentu
m
an
d
energ
y
equation
s
ar
e
al
l
coupled
.
Th
e
pres
-
sur
e
ter
m
i
n
Eq
.
(36)
,
o
n
th
e
othe
r
hand
,
i
s
exacd
y
a
s
i
t
was
i
n
th
e
adiabati
c
cas
e
an
d
th
e
sam
e
argument
s
mad
e
befor
e
apply
:
th
e
flow
shoul
d
b
e
considere
d
compressibl
e
i
f
M
a
>
0.3
,
o
r
i
f
pressur
e
change
s
du
e
t
o
viscou
s
force
s
ar
e
sufficientl
y
large
.
Experiment
s
i
n
gaseou
s
microduct
s
confir
m
th
e
abov
e
argu
-
ments
.
Fo
r
bot
h
low
-
an
d
high-Mach-numbe
r
flows,
pressur
e
gradient
s
i
n
lon
g
microchannel
s
ar
e
non-constant
,
consisten
t
wit
h
th
e
compressibl
e
flow
equations
.
Suc
h
experiment
s
wer
e
conducte
d
by
,
amon
g
others
,
Prud'homm
e
e
t
al
.
(1986)
,
Pfahle
r
e
t
al
.
(1991)
,
va
n
de
n
Ber
g
e
t
al
.
(1993)
,
Li
u
e
t
al
.
(1993
;
1995)
,
Pon
g
e
t
al
.
(1994)
,
Harle
y
e
t
al
.
(1995)
,
Pieko
s
an
d
Breue
r
(1996)
,
Arkili
c
(1997)
,
an
d
Arkili
c
e
t
al
.
(1995
;
1997a
;
1997b)
.
Sampl
e
result
s
wi
U
b
e
presente
d
i
n
th
e
followin
g
sub
-
section
.
Ther
e
i
s
on
e
las
t
scenari
o
i
n
whic
h
significan
t
pressur
e
an
d
densit
y
change
s
ma
y
tak
e
plac
e
withou
t
viscou
s
o
r
inertia
l
ef
-
fects
.
Tha
t
i
s
th
e
cas
e
o
f
quasi-stati
c
compression/expansio
n
o
f
a
ga
s
in
,
fo
r
example
,
a
piston-cylinde
r
arrangement
.
Th
e
re
-
sultin
g
compressibilit
y
effect
s
are
,
however
,
compressibilit
y
o
f
th
e
fluid
an
d
no
t
o
f
th
e
flow.
2.
5
Boundar
y
Conditions
.
Th
e
equation
s
o
f
motio
n
de
-
scribe
d
i
n
Sectio
n
2.
3
requir
e
a
certai
n
numbe
r
o
f
initia
l
an
d
boundar
y
condition
s
fo
r
prope
r
mathematica
l
formulatio
n
o
f
flow
problems
.
I
n
thi
s
subsection
,
w
e
describ
e
th
e
boundar
y
condition
s
a
t
a
fluid-solid
interface
.
Boundar
y
condition
s
i
n
th
e
invisci
d
flow
theor
y
pertai
n
onl
y
t
o
th
e
velocit
y
componen
t
norma
l
t
o
a
soli
d
surface
.
Th
e
highes
t
spatia
l
derivativ
e
o
f
ve
-
locit
y
i
n
th
e
invisci
d
equation
s
o
f
motio
n
i
s
first-order,
an
d
onl
y
on
e
velocit
y
boundar
y
conditio
n
a
t
th
e
surfac
e
i
s
admissible
.
Journa
l
o
f
Fluid
s
Engineerin
g
MARC
H
1999
,
Vol
.
1 2 1/1
1
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
The normal velocity component at a fluid-solid interface is spec-
ified, and no statement can be made regarding the tangential
velocity component. The normal-velocity condition simply
states that a fluid-particle path cannot go through an imperme-
able wall. Real fluids are of course viscous and the correspond-
ing momentum equation has second-order derivatives of veloc-
ity, thus requiring an additional boundary condition on the ve-
locity component tangential to a solid surface.
Traditionally, the no-slip condition at a fluid-solid interface
is enforced in the momentum equation and an analogous no-
temperature-jump condition is applied in the energy equation.
The notion underlying the no-slip/no-jump condition is that
within the fluid there cannot be any finite discontinuities of
velocity/temperature. Those would involve infinite velocity/
temperature gradients and so produce infinite viscous stress/
heat flux that would destroy the discontinuity in infinitesimal
time. The interaction between a fluid particle and a wall is
similar to that between neighboring fluid particles, and therefore
no discontinuities are allowed at the fluid-solid interface either.
In other words, the fluid velocity must be zero relative to the
surface and the fluid temperature must equal to that of the
surface. But strictly speaking those two boundary conditions
are valid only if the fluid flow adjacent to the surface is in
thermodynamic equilibrium. This requires an infinitely high
frequency of collisions between the fluid and the solid surface.
In practice, the no-slip/no-jump condition leads to fairly accu-
rate predictions as long as Kn < 0.001 (for gases). Beyond
that, the collision frequency is simply not high enough to ensure
equilibrium and a certain degree of tangential-velocity slip and
temperature jump must be allowed. This is a case frequently
encountered in MEMS flows, and we develop the appropriate
relations in this subsection.
For both liquids and gases, the linear Navier boundary condi-
tion empirically relates the tangential velocity slip at the wall
AM I „ to the local shear
AML =
= L,
du
(37)
where L, is the constant slip length, and duldy\„ is the strain
rate computed at the wall. In most practical situations, the slip
length is so small that the no-slip condition holds. In MEMS
applications, however, that may not be the case. Once again we
defer the discussion of liquids to Section 2.7, and focus for now
on gases.
Assuming isothermal conditions prevail, the above slip rela-
tion has been rigorously derived by Maxwell (1879) from con-
siderations of the kinetic theory of dilute, monatomic gases.
Gas molecules, modeled as rigid spheres, continuously strike
and reflect from a solid surface, just as they continuously collide
with each other. For an idealized perfectly smooth (at the molec-
ular scale) wall, the incident angle exactly equals the reflected
angle and the molecules conserve their tangential momentum
and thus exert no shear on the wall. This is termed specular
reflection and results in perfect slip at the wall. For an extremely
rough wall, on the other hand, the molecules reflect at some
random angle uncorrelated with their entry angle. This perfectly
diffuse reflection results in zero tangential-momentum for the
reflected fluid molecules to be balanced by a finite slip velocity
in order to account for the shear stress transmitted to the wall.
A force balance near the wall leads to the following expression
for the slip velocity
^wall
dy
(38)
where £ is the mean free path. The right-hand side can be
considered as the first term in an infinite Taylor series, sufficient
if the mean free path is relatively small enough. The equation
above states that significant slip occurs only if the mean velocity
of the molecules varies appreciably over a distance of one mean
free path. This is the case, for example, in vacuum applications
and/or flow in microdevices. The number of collisions between
the fluid molecules and the solid in those cases is not large
enough for even an approximate flow equilibrium to be estab-
lished. Furthermore, additional (nonlinear) terms in the Taylor
series would be needed as £ increases and the flow is further
removed from the equihbrium state.
For real walls some molecules reflect diffusively and some
reflect specularly. In other words, a portion of the momentum
of the incident molecules is lost to the wall and a (typically
smaller) portion is retained by the reflected molecules. The
tangential-momentum-accommodatio n coefficient cr„ is defined
as the fraction of molecules reflected diffusively. This coeffi-
cient depends on the fluid, the solid and the surface finish,
and has been determined experimentally to be between 0.2-0.8
(Thomas and Lord, 1974; Seidl and Steiheil, 1974; Porodnov
et al, 1974; Arkilic et al., 1997b; Arkilic, 1997), the lower
limit being for exceptionally smooth surfaces while the upper
limit is typical of most practical surfaces. The final expression
derived by Maxwell for an isothermal wall reads
"wall
(T„ du
dy
(39)
For cr„ = 0, the slip velocity is unbounded, while for cr„ = 1,
Eq. (39) reverts to (38).
Similar arguments were made for the temperature-jump
boundary condition by von Smoluchowski (1898). For an ideal
gas flow in the presence of wall-normal and tangential tempera-
ture gradients, the complete slip-flow and temperature-jump
boundary conditions read
_2- a„ 1 3 P r ( y - - l ), ,
Mgas M„a|i — / ''"»' + 7 (~?;t)n.
(40)
_ 2 — a„ / du\ 3 fj, I dT
Twall —
a-,-
(jj
UT
OT
2 ( 7 - 1)
( r + 1)
2y
( 7 + 1)
P%
2'RTpp
i-qy).
£_
Pr
dy).
(41)
where x and y are the streamwise and normal coordinates, p
and n are respectively the fluid density and viscosity, /? is the
gas constant, Tg^ is the temperature of the gas adjacent to the
wall, r„aii is the wall temperature, r „ is the shear stress at the
wall, Pr is the Prandtl number, 7 is the specific heat ratio, and
(qx)w and {qy\y are, respectively, the tangential and normal heat
flux at the wall.
The tangential-momentum-accommodatio n coefficient a„ and
the thermal-accommodation coefficient ar are given by, respec-
tively,
T. -7-
(42)
dEi — dE„
where the subscripts i, r, and w stand for, respectively, incident,
reflected and solid wall conditions, T is a tangential momentum
flux, and dE is an energy flux.
Ti
Ti
dEi
- Tr
- T,„
— dEr
12 / Vol. 121, MARCH 1999
Transactions of the ASME
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Th
e
secon
d
ter
m
i
n
th
e
right-han
d
sid
e
o
f
Eq
.
(40
)
i
s
th
e
thermal
creep
whic
h
generate
s
sli
p
velocit
y
i
n
th
e
fluid
opposit
e
t
o
th
e
directio
n
o
f
th
e
tangentia
l
hea
t
flux,
i.e.
,
flow
i
n
th
e
directio
n
o
f
increasin
g
temperature
.
A
t
sufficientl
y
hig
h
Knud
-
se
n
numbers
,
streamwis
e
temperatur
e
gradien
t
i
n
a
condui
t
lead
s
t
o
a
measurabl
e
pressur
e
gradien
t
alon
g
th
e
tube
.
Thi
s
ma
y
b
e
th
e
cas
e
i
n
vacuu
m
application
s
an
d
MEM
S
devices
.
Therma
l
cree
p
i
s
th
e
basi
s
fo
r
th
e
so-calle
d
Knudse
n
pump—
a
devic
e
wit
h
n
o
movin
g
parts—i
n
whic
h
rarefie
d
ga
s
i
s
haule
d
fro
m
on
e
col
d
chambe
r
t
o
a
ho
t
one
.
(Th
e
terminolog
y
Knudsen
pump
ha
s
bee
n
use
d
by
,
fo
r
example
,
Varg
o
an
d
Munt
z
(1996)
,
but
accordin
g
t
o
Loe
b
(1961)
,
th
e
origina
l
experiment
s
demon
-
stratin
g
suc
h
pum
p
wer
e
carrie
d
ou
t
b
y
Osborn
e
Reynolds.
)
Clearly
,
suc
h
pum
p
perform
s
bes
t
a
t
hig
h
Knudse
n
numbers
,
an
d
i
s
typicall
y
designe
d
t
o
operat
e
i
n
th
e
free-molecul
e
flow
regime
.
I
n
dimensionles
s
form
,
Eqs
.
(40
)
an
d
(41)
,
respectively
,
rea
d
I*

^ K
n
c

du*
dy*
^wal
l
2
-
a

vm
£'
+

3
!
(d^u
w
/d^u
[dy'
(48
)
Attempt
s
t
o
implemen
t
th
e
abov
e
sli
p
conditio
n
i
n
numerica
l
simulation
s
ar
e
rathe
r
difficult
.
Second-orde
r
an
d
highe
r
deriva
-
tive
s
o
f
velocit
y
canno
t
b
e
compute
d
accuratel
y
nea
r
th
e
wall
.
Base
d
o
n
asymptoti
c
analysis
,
Besko
k
(1996
)
an
d
Besko
k
an
d
Karniadaki
s
(1994
;
1998
)
propose
d
th
e
followin
g
alternativ
e
higher-orde
r
boundar
y
conditio
n
fo
r
th
e
tangentia
l
velocity
,
in
-
cludin
g
th
e
therma
l
cree
p
term
.
»ga
s
K
n
I
-
bKn
du*
dy*
3
( 7 - 1
)
Kn^R
e
27
r
y
E
c
dT*
dx*
(49
)
3
(
y
-
1
)
Kn'R
e
-
*
ga
s
-
*
wa
l
27
r
a-j-
7
E
c
dT*
dx*
a-r
ly
(
r
+
1
)
K
n
/
dT*
P
r
\dy*
(44
)
(45
)
wher
e
th
e
superscrip
t
*
indicate
s
dimensionles
s
quantity
,
K
n
i
s
th
e
Knudse
n
number
.
R
e
i
s
th
e
Reynold
s
number
,
an
d
E
c
i
s
th
e
Ecker
t
numbe
r
define
d
b
y
E
c
=
CpA
T
=
(
7
l ) ^ M a
'
A
T
(46
)
wher
e
v„
i
s
a
referenc
e
velocity
,
A
T
=
(Tga
s
-To),
an
d
To
i
s
a
referenc
e
temperature
.
Not
e
tha
t
ver
y
lo
w
value
s
o
f
0-

an
d
er
r
lea
d
t
o
substantia
l
velocit
y
sli
p
an
d
temperatur
e
jum
p
eve
n
fo
r
flows
wit
h
smal
l
Knudse
n
number
.
Th
e
first
ter
m
i
n
th
e
right-han
d
sid
e
o
f
Eq
.
(44
)
i
s
first-order
i
n
Knudse
n
number
,
whil
e
th
e
therma
l
cree
p
ter
m
i
s
second
-
order
,
meanin
g
tha
t
th
e
cree
p
phenomeno
n
i
s
potentiall
y
sig
-
nifican
t
a
t
larg
e
value
s
o
f
th
e
Knudse
n
number
.
Equatio
n
(45
)
i
s
first-order
i
n
Kn
.
Usin
g
Eqs
.
(8
)
an
d
(46)
,
th
e
therma
l
cree
p
ter
m
i
n
Eq
.
(44
)
ca
n
b
e
rewritte
n
i
n
term
s
o
f
A
T
an
d
Reynold
s
number
.
Thus
,
I*
«ga
s

K
n
CTl
,
du*
dy*
3
Ar j
_
4
To
R
e
dT*
dx*
(47
)
I
t
i
s
clea
r
tha
t
larg
e
temperatur
e
change
s
alon
g
th
e
surfac
e
o
r
lo
w
Reynold
s
number
s
lea
d
t
o
significan
t
therma
l
creep
.
Th
e
continuu
m
Navier-Stoke
s
equation
s
wit
h
no-slip/no
-
temperatur
e
jum
p
boundar
y
condition
s
ar
e
vali
d
a
s
lon
g
a
s
th
e
Knudse
n
numbe
r
doe
s
no
t
excee
d
0.001
.
First-orde
r
slip/tem
-
perature-jum
p
boundar
y
condition
s
shoul
d
b
e
applie
d
t
o
th
e
Navier-Stoke
s
equation
s
i
n
th
e
rang
e
o
f
0.00
1
<
K
n
<
0.1
.
Th
e
transitio
n
regim
e
span
s
th
e
rang
e
o
f
0.
1
<
K
n
<
10
,
an
d
second-orde
r
o
r
highe
r
slip/temperature-jum
p
boundar
y
condi
-
tion
s
ar
e
applicabl
e
there
.
Note
,
however
,
tha
t
th
e
Navier-Stoke
s
equation
s
ar
e
first-order
accurat
e
i
n
K
n
a
s
wil
l
b
e
show
n
i
n
Sectio
n
2.6
,
an
d
ar
e
themselve
s
no
t
vali
d
i
n
th
e
transitio
n
re
-
gime
.
Eithe
r
higher-orde
r
continuu
m
equations
,
e.g.
,
Burnet
t
equations
,
shoul
d
b
e
use
d
ther
e
o
r
molecula
r
modelin
g
shoul
d
b
e
invoked
,
abandonin
g
th
e
continuu
m
approac
h
altogether
.
Fo
r
isotherma
l
walls
,
Besko
k
(1994
)
derive
d
a
higher-orde
r
slip-velocit
y
conditio
n
a
s
follow
s
wher
e
fo
i
s
a
high-orde
r
sli
p
coefficien
t
determine
d
fro
m
th
e
presumabl
y
know
n
no-sli
p
solution
,
thu
s
avoidin
g
th
e
computa
-
tiona
l
difficultie
s
mentione
d
above
.
I
f
thi
s
high-orde
r
sli
p
coef
-
ficient
i
s
chose
n
d&b
=
ul/2ul,
wher
e
th
e
prim
e
denote
s
deriva
-
tiv
e
wit
h
respec
t
t
o
y
an
d
th
e
velocit
y
i
s
compute
d
fro
m
th
e
no
-
sli
p
Navier-Stoke
s
equations
,
Eq
.
(49
)
become
s
second-orde
r
accurat
e
i
n
Knudse
n
number
.
Beskok'
s
procedur
e
ca
n
b
e
ex
-
tende
d
t
o
third
-
an
d
higher-order
s
fo
r
bot
h
th
e
slip-velocit
y
an
d
therma
l
cree
p
terms
.
Simila
r
argument
s
ca
n
b
e
applie
d
t
o
th
e
temperature-jum
p
boundar
y
condition
,
an
d
th
e
resultin
g
Taylo
r
serie
s
read
s
i
n
dimensionles
s
for
m
(Beskok
,
1996)
,
'
ga
s
^
wal
l
Uj
cJr
ly
(
r
+
1
)
_1
_
P
r
K
n
dT*
dy*
Kn
^
/
d'^T*
2
!
\dy
(50
)
Again
,
th
e
difficultie
s
associate
d
wit
h
computin
g
second
-
an
d
higher-orde
r
derivative
s
o
f
temperatur
e
ar
e
alleviate
d
usin
g
a
n
identica
l
procedur
e
t
o
tha
t
utilize
d
fo
r
th
e
tangentia
l
velocit
y
boundar
y
condition
.
Severa
l
experiment
s
i
n
low-pressur
e
macroduct
s
o
r
i
n
micro
-
duct
s
confir
m
th
e
necessit
y
o
f
applyin
g
sli
p
boundar
y
conditio
n
a
t
sufficientl
y
larg
e
Knudse
n
numbers
.
Amon
g
the
m
ar
e
thos
e
conducte
d
b
y
Knudse
n
(1909)
,
Pfable
r
a
t
al
.
(1991)
,
Tiso
n
(1993)
,
Li
u
e
t
al
.
(1993
;
1995)
,
Pon
g
e
t
al
.
(1994)
,
Arkili
c
e
t
al
.
(1995)
,
Harle
y
e
t
al
.
(1995)
,
an
d
Shi
h
e
t
al
.
(1995
;
1996)
.
Th
e
experiment
s
ar
e
complemente
d
b
y
th
e
numerica
l
simula
-
tion
s
carrie
d
ou
t
b
y
Besko
k
(1994
;
1996)
,
Besko
k
an
d
Karnia
-
daki
s
(1994
;
1998)
,
an
d
Besko
k
e
t
al
.
(1996)
.
Her
e
w
e
presen
t
selecte
d
example
s
o
f
th
e
experimenta
l
an
d
numerica
l
results
.
Tiso
n
(1993
)
conducte
d
pip
e
flow
experiment
s
a
t
ver
y
lo
w
pressures
.
Hi
s
pip
e
ha
s
a
diamete
r
o
f
2
m
m
an
d
a
length-to
-
diamete
r
rati
o
o
f
200
.
Bot
h
inle
t
an
d
outle
t
pressure
s
wer
e
varie
d
t
o
yiel
d
Knudse
n
numbe
r
i
n
th
e
rang
e
o
f
K
n
=
0-200
.
Figur
e
4
show
s
th
e
variatio
n
o
f
mas
s
flowrate
a
s
a
functio
n
o
f
(p
?

pi),
wher
e
p
,
i
s
th
e
inle
t
pressur
e
an
d
po
i
s
th
e
outle
t
pressure
.
(Th
e
origina
l
dat
a
i
n
thi
s
figure
wer
e
acquire
d
b
y
S
.
A
.
Tiso
n
an
d
plotte
d
b
y
Besko
k
e
t
al
.
(1996).
)
Th
e
pressur
e
dro
p
i
n
thi
s
rarefie
d
pip
e
flow
i
s
nonlinear
,
characteristi
c
o
f
low-Reynolds
-
number
,
compressibl
e
flows.
Thre
e
distinc
t
flow
regime
s
ar
e
identified
:
(1
)
sli
p
flow
regime
,
0
<
K
n
<
0.6
;
(2
)
transitio
n
regime
,
0.
6
<
K
n
<
17
,
wher
e
th
e
mas
s
flowrate
i
s
almos
t
constan
t
a
s
th
e
pressur
e
changes
;
an
d
(3
)
free-molecul
e
flow,
K
n
>
17
.
Not
e
tha
t
th
e
demarkatio
n
betwee
n
thes
e
thre
e
re
-
Journa
l
o
f
Fluid
s
Engineerin
g
MARC
H
1999
,
Vol
.
12
1
/
1
3
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
60
0
40
0
20
0
s
:
10
0
e
.
6
0
%
4
0

S
2
0
1
0
8
6
4
1
20
0
-
A
1
>Kn
A.
>
A
m
l
A
A
ll>Kn>
0.
6
1
7
A
A
0.
6
>Kn>
0.
0
-
-
0.
1
1
0
10
0
100
0
10'
'
Fig
.
4
Variatio
n
o
f
mas
s
flowrat
e
a
s
a
functio
n
o
f
{pf
-
pi).
Origina
l
dat
a
acquire
d
b
y
S
.
A
.
Tiso
n
an
d
plotte
d
b
y
Beslco
k
e
t
al
.
(1996)
.
gime
s
i
s
slightl
y
differen
t
fro
m
tha
t
mentione
d
i
n
Sectio
n
2.2
.
A
s
stated
,
th
e
differen
t
Knudse
n
numbe
r
regime
s
ar
e
determine
d
empiricall
y
an
d
ar
e
therefor
e
onl
y
approximat
e
fo
r
a
particula
r
flow
geometry
.
Shi
h
e
t
al
.
(1995
)
conducte
d
thei
r
experiment
s
i
n
a
micro
-
channe
l
usin
g
heliu
m
a
s
a
fluid.
Th
e
inle
t
pressur
e
varie
d
bu
t
th
e
duc
t
exi
t
was
atmospheric
.
Microsensor
s
wher
e
fabricate
d
in-sit
u
alon
g
thei
r
MEM
S
channe
l
t
o
measur
e
th
e
pressure
.
Figur
e
5
show
s
thei
r
measure
d
mas
s
flowrate
versu
s
th
e
inle
t
pressure
.
Th
e
dat
a
ar
e
compare
d
t
o
th
e
no-sli
p
solutio
n
an
d
th
e
sli
p
solutio
n
usin
g
thre
e
differen
t
value
s
o
f
th
e
tangential
-
momentum-accommodatio
n
coefficient
,
0.8
,
0.
9
an
d
1.0
.
Th
e
agreemen
t
i
s
reasonabl
e
wit
h
th
e
cas
e
(T

=
1.0
,
indicatin
g
per
-
hap
s
tha
t
th
e
channe
l
use
d
b
y
Shi
h
e
t
al
.
wa
s
quit
e
roug
h
o
n
th
e
molecula
r
scale
.
I
n
a
secon
d
experimen
t
(Shi
h
e
t
al.
,
1996)
,
nitrou
s
oxid
e
wa
s
use
d
a
s
th
e
fluid
.
Th
e
squar
e
o
f
th
e
pressur
e
distributio
n
alon
g
th
e
channe
l
i
s
plotte
d
i
n
Fig
.
6
fo
r
five
differ
-
en
t
inle
t
pressures
.
Th
e
experimenta
l
dat
a
(symbols
)
compar
e
wel
l
wit
h
th
e
theoretica
l
prediction
s
(soli
d
lines)
.
Again
,
th
e
nonlinea
r
pressur
e
dro
p
show
n
indicate
s
tha
t
th
e
ga
s
flow
i
s
compressible
.
Arkili
c
(1997
)
provide
d
a
n
elegan
t
analysi
s
o
f
th
e
compress
-
ible
,
rarefie
d
flow
i
n
a
microchannel
.
Th
e
result
s
o
f
hi
s
theor
y
ar
e
compare
d
t
o
th
e
experiment
s
o
f
Pon
g
e
t
al
.
(1994
)
i
n
Fig
.
7
.
Th
e
dotte
d
lin
e
i
s
th
e
incompressibl
e
flow
solution
,
wher
e
th
e
pressur
e
i
s
predicte
d
t
o
dro
p
linearl
y
wit
h
streamwis
e
dis
-
8

7
&
6
a
^
4
3
-
2
-
-
-

Dat
a
^y
No-sli
p
solutio
n
y"^
y^
"
Sli
p
solutio
n
a
^
=
0.
9
y''
^
*
/
Sli
p
solutio
n
c,
=
0.
8
y
^'^y^
"
y
yw<
A^
..
.
y
^y^
y
'^
M
4
r
•••••••-'"
X
•,/

-<iii
r
-'-'"
"
."^m
•••''
1
1
1 1
1
1
1
1
0
I
S
2
0
2
5
Inle
t
Pressur
e
[psig
]
3
0
3
5
Inle
t
Pressur
e
0
8.
4
psi
g

12.
1
psi
g
A
15.
5
psi
g
X
19.
9
psi
g
0
23.
0
psi
g
100
0
200
0
300
0
Channe
l
Lengt
h
()ini
)
400
0
Fig
.
6
Pressur
e
distributio
n
o
f
nitrou
s
oxid
e
i
n
a
microduct
.
Fro
m
Shi
h
etal
.
(1996)
.
tance
.
Th
e
dashe
d
lin
e
i
s
th
e
compressibl
e
flow
solutio
n
tha
t
neglect
s
rarefactio
n
effect
s
(assume
s
K
n
=
0)
.
Finally
,
th
e
soli
d
Un
e
i
s
th
e
theoretica
l
resul
t
tha
t
take
s
int
o
accoun
t
bot
h
compressibilit
y
an
d
rarefactio
n
vi
a
slip-flo
w
boundar
y
conditio
n
compute
d
a
t
th
e
exi
t
Knudse
n
numbe
r
o
f
K
n
=
0.06
.
Tha
t
theor
y
compare
s
mos
t
favorabl
y
wit
h
th
e
experimenta
l
data
.
I
n
th
e
compressibl
e
flow
throug
h
th
e
constant-are
a
duct
,
densit
y
de
-
crease
s
an
d
thu
s
velocit
y
increase
s
i
n
th
e
streamwis
e
direction
.
A
s
a
result
,
th
e
pressur
e
distributio
n
i
s
nonlinea
r
wit
h
negativ
e
curvature
.
A
moderat
e
Knudse
n
numbe
r
(i.e
.
moderat
e
slip
)
actuall
y
diminishes
,
albei
t
rathe
r
weakly
,
thi
s
curvature
.
Thus
,
compressibilit
y
an
d
rarefactio
n
effect
s
lea
d
t
o
opposin
g
trends
,
a
s
pointe
d
ou
t
b
y
Besko
k
e
t
al
.
(1996)
.
2.
6
Molecular-Base
d
Models
.
I
n
th
e
continuu
m
model
s
discusse
d
i
n
Sectio
n
2.3
,
th
e
macroscopi
c
fluid
propertie
s
ar
e
th
e
dependen
t
variable
s
whil
e
th
e
independen
t
variable
s
ar
e
th
e
thre
e
spatia
l
coordinate
s
an
d
time
.
Th
e
molecula
r
model
s
recogniz
e
th
e
fluid
a
s
a
myria
d
o
f
discret
e
particles
:
molecules
,
atoms
,
ion
s
an
d
electrons
.
Th
e
goa
l
her
e
i
s
t
o
determin
e
th
e
position
,
velocit
y
an
d
stat
e
o
f
al
l
particle
s
a
t
al
l
times
.
Th
e
molecula
r
approac
h
i
s
eithe
r
deterministi
c
o
r
probabilisti
c
(refe
r
t
o
Fig
.
2)
.
Provide
d
tha
t
ther
e
i
s
a
sufficien
t
numbe
r
o
f
micro
-
scopi
c
particle
s
withi
n
th
e
smalles
t
significan
t
volum
e
o
f
a
flow,
th
e
macroscopi
c
propertie
s
a
t
an
y
locatio
n
i
n
th
e
flow
ca
n
the
n
b
e
compute
d
fro
m
th
e
discrete-particl
e
informatio
n
b
y
a
suitabl
e
averagin
g
o
r
weighte
d
averagin
g
process
.
Th
e
presen
t
subsec
-
tio
n
discusse
s
molecular-base
d
model
s
an
d
thei
r
relatio
n
t
o
th
e
continuu
m
model
s
previousl
y
considered
.
2.
8
1
2.
4
1.
6
I
1.
2
0.
8
«
Pon
g
e
t
al
.
(1994
)
-
-
Outle
t
Knudse
n
numbe
r
=
0.
0

Outle
t
Knudse
n
numbe
r
=
0.0
6
Incompressibl
e
flo
w
solutio
n
Fig
.
5
Mas
s
flowrat
e
versu
s
inle
t
pressur
e
i
n
a
microchannel
.
Fro
m
Shi
h
etal
.
(1995)
.
0
0.
2
0.
4
0.
6
0.
8
1
Non-Dimensiona
l
Positio
n
(x
)
Fig
.
7
Pressur
e
distributio
n
i
n
a
lon
g
microchannel
.
Th
e
symbol
s
ar
e
experimenta
l
dat
a
whil
e
th
e
soli
d
line
s
ar
e
differen
t
theoretica
l
predic
-
tions
.
Fro
m
Arkili
c
(1997)
.
1
4
/
Vol
.
121
,
MARC
H
199
9
Transaction
s
o
f
th
e
ASM
E
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
The most fundamental of the molecular models is a determin-
istic one. The motion of the molecules are governed by the
laws of classical mechanics, although, at the expense of greatly
complicating the problem, the laws of quantum mechanics can
also be considered in special circumstances. The modern molec-
ular dynamics computer simulations (MD) have been pioneered
by Alder and Wainwright (1957; 1958; 1970) and reviewed by
Ciccotti and Hoover (1986), Allen and Tildesley (1987), Haile
(1993), and Koplik and Banavar (1995). The simulation begins
with a set of A' molecules in a region of space, each assigned
a random velocity corresponding to a Boltzmann distribution
at the temperature of interest. The interaction between the parti-
cles is prescribed typically in the form of a two-body potential
energy and the time evolution of the molecular positions is
determined by integrating Newton's equations of motion. Be-
cause MD is based on the most basic set of equations, it is valid
in principle for any flow extent and any range of parameters.
The method is straightforward in principle but there are two
hurdles: choosing a proper and convenient potential for particu-
lar fluid and solid combinations, and the colossal computer
resources required to simulate a reasonable flow field extent.
For purists, the former difficulty is a sticky one. There is no
totally rational methodology by which a convenient potential
can be chosen. Part of the art of MD is to pick an appropriate
potential and validate the simulation results with experiments
or other analytical/computational results. A commonly used
potential between two molecules is the generalized Lennard-
Jones 6-12 potential, to be used in Section 2.7 and further
discussed in Section 2.8.
The second difficulty, and by far the most serious limitation
of molecular dynamics simulations, is the number of molecules
A' that can realistically be modeled on a digital computer. Since
the computation of an element of trajectory for any particular
molecule requires consideration of all other molecules as poten-
tial collision partners, the amount of computation required by
the MD method is proportional to A'^. Some saving in computer
time can be achieved by cutting off the weak tail of the potential
(see Fig. 12) at, say, r,. = 2.5a, and shifting the potential by a
linear term in r so that the force goes smoothly to zero at the
cutoff. As a result, only nearby molecules are treated as potential
collision partners, and the computation time for A' molecules
no longer scales with A'^.
The state of the art of molecular dynamics simulations in the
1990s is such that with a few hours of CPU time, general
purpose supercomputers can handle around 10,000 molecules.
At enormous expense, the fastest parallel machine available can
simulate around 1 million particles. Because of the extreme
diminution of molecular scales, the above translates into regions
of liquid flow of about 0.01 fxm (100 A) in Unear size, over
time intervals of around 0.001 (US, just enough for continuum
behavior to set in, for simple molecules. To simulate 1 s of
real time for complex molecular interactions, e.g., including
vibration modes, reorientation of polymer molecules, collision
of colloidal particles, etc., requires unrealistic CPU time mea-
sured in thousands of years.
MD simulations are highly inefficient for dilute gases where
the molecular interactions are infrequent. The simulations are
more suited for dense gases and liquids. Clearly, molecular
dynamics simulations are reserved for situations where the con-
tinuum approach or the statistical methods are inadequate to
compute from first principles important flow quantities. Slip
boundary conditions for liquid flows in extremely small devices
is such a case as will be discussed in Section 2.7.
An alternative to the deterministic molecular dynamics is the
statistical approach where the goal is to compute the probability
of finding a molecule at a particular position and state. If the
appropriate conservation equation can be solved for the proba-
bility distribution, important statistical properties such as the
mean number, momentum or energy of the molecules within
an element of volume can be computed from a simple weighted
averaging. In a practical problem, it is such average quantities
that concern us rather than the detail for every single molecule.
Clearly, however, the accuracy of computing average quantities,
via the statistical approach, improves as the number of mole-
cules in the sampled volume increases. The kinetic theory of
dilute gases is well advanced, but that for dense gases and
liquids is much less so due to the extreme complexity of having
to include multiple collisions and intermolecular forces in the
theoretical formulation. The statistical approach is well covered
in books such as those by Kennard (1938), Hirschfelder et
al. (1954), Schaaf and Chambre (1961), Vincenti and Kruger
(1965), Kogan (1969), Chapman and Cowling (1970), Cercig-
nani (1988), and Bird (1994), and review articles such as those
by Kogan (1973), Muntz (1989), and Oran et al. (1998).
In the statistical approach, the fraction of molecules in a
given location and state is the sole dependent variable. The
independent variables for monatomic molecules are time, the
three spatial coordinates and the three components of molecular
velocity. Those describe a six-dimensional phase space. (The
evolution equation of the probability distribution is considered,
hence time is the 7th independent variable.) For diatomic or
polyatomic molecules, the dimension of phase space is in-
creased by the number of internal degrees of freedom. Orienta-
tion adds an extra dimension for molecules which are not spheri-
cally symmetric. Finally, for mixtures of gases, separate proba-
bility distribution functions are required for each species.
Clearly, the complexity of the approach increases dramatically
as the dimension of phase space increases. The simplest prob-
lems are, for example, those for steady, one-dimensional flow
of a simple monatomic gas.
To simplify the problem we restrict the discussion here to
monatomic gases having no internal degrees of freedom. Fur-
thermore, the fluid is restricted to dilute gases and molecular
chaos is assumed. The former restriction requires the average
distance between molecules 6 to be an order of magnitude larger
than their diameter a. That will almost guarantee that all colli-
sions between molecules are binary collisions, avoiding the
complexity of modeling multiple encounters. (Dissociation and
ionization phenomena involve triple collisions and therefore
require separate treatment.) The molecular chaos restriction im-
proves the accuracy of computing the macroscopic quantities
from the microscopic information. In essence, the volume over
which averages are computed has to have sufficient number
of molecules to reduce statistical errors. It can be shown that
computing macroscopic flow properties by averaging over a
number of molecules will result in statistical fluctuations with
a standard deviation of approximately 0.1 % if one million mole-
cules are used and around 3% if one thousand molecules are
used. The molecular chaos limit requires the length-scale L for
the averaging process to be at least 100 times the average dis-
tance between molecules (i.e., typical averaging over at least
one million molecules).
Figure 8, adapted from Bird (1994), shows the limits of
vahdity of the dilute gas approximation (S/a > 7), the contin-
uum approach (Kn < 0.1, as discussed previously in Section
2.2), and the neglect of statistical fluctuations {LIS > 100).
Using a molecular diameter of cr = 4 X 10 "'" m as an example,
the three limits are conveniently expressed as functions of the
normalized gas density p/ p„ or number density n/n„, where the
reference densities p„ and n„ are computed at standard condi-
tions. All three limits are straight lines in the log-log plot of L
versus p/p„, as depicted in Figure 8. Note the shaded triangular
wedge inside which both the Boltzmann and Navier-Stokes
equations are valid. Additionally, the lines describing the three
limits very nearly intersect at a single point. As a consequence,
the continuum breakdown limit always lies between the dilute
gas limit and the limit for molecular chaos. As density or charac-
teristic dimension is reduced in a dilute gas, the Navier-Stokes
model breaks down before the level of statistical fluctuations
becomes significant. In a dense gas, on the other hand, signifi-
Journal of Fluids Engineering
MARCH 1999, Vol. 121 / 15
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10,00
0
10
^
10-
^
10-
2
Densit
y
rati
o
/i/«
o
o
r
p/p
^
Fig
.
8
Effectiv
e
limit
s
o
f
differen
t
flo
w
models
.
Fro
m
Bir
d
(1994)
.
can
t
fluctuation
s
ma
y
b
e
presen
t
eve
n
whe
n
th
e
Navier-Stoke
s
mode
l
i
s
stil
l
valid
.
Th
e
startin
g
poin
t
i
n
statistica
l
mechanic
s
i
s
th
e
Liouvill
e
equatio
n
whic
h
expresse
s
th
e
conservatio
n
o
f
th
e
A'-particl
e
distributio
n
functio
n
i
n
6A^-dimensiona
l
phas
e
spac
e
(thre
e
posi
-
tion
s
an
d
thre
e
velocitie
s
fo
r
eac
h
molecul
e
o
f
a
monatomi
c
ga
s
wit
h
n
o
interna
l
degree
s
o
f
freedom)
,
wher
e
N
i
s
th
e
numbe
r
o
f
particle
s
unde
r
consideration
.
Considerin
g
onl
y
externa
l
force
s
whic
h
d
o
no
t
depen
d
o
n
th
e
velocit
y
o
f
th
e
molecule
s
(thi
s
exclude
s
Lorent
z
forces
,
fo
r
example)
,
th
e
Liouvill
e
equa
-
tio
n
fo
r
a
syste
m
o
f
N
mas
s
point
s
read
s
-7-+
lik--^+
i F
r
0
(51
)
wher
e
9
i
s
th
e
probabilit
y
o
f
finding a
molecul
e
a
t
a
particula
r
poin
t
i
n
phas
e
space
,
t
i
s
time
,
^i,
i
s
th
e
three-dimensiona
l
veloc
-
it
y
vecto
r
fo
r
th
e
fcth
molecule
,
x
^
i
s
th
e
three-dimensiona
l
positio
n
vecto
r
fo
r
th
e
A:t
h
molecule
,
an
d
F
i
s
th
e
externa
l
forc
e
vector
.
Not
e
tha
t
th
e
do
t
produc
t
i
n
th
e
abov
e
equatio
n
i
s
carrie
d
out
ove
r
eac
h
o
f
th
e
thre
e
component
s
o
f
th
e
vector
s
^
,
x
,
an
d
F
,
an
d
tha
t
th
e
summatio
n
i
s
ove
r
al
l
molecules
.
Obviousl
y
suc
h
a
n
equatio
n
i
s
no
t
tractabl
e
fo
r
realisti
c
numbe
r
o
f
particles
.
A
hierarch
y
o
f
reduce
d
distributio
n
function
s
ma
y
b
e
ob
-
taine
d
b
y
repeate
d
integratio
n
o
f
th
e
Liouvill
e
equatio
n
above
.
Th
e
final
equatio
n
i
n
th
e
hierarch
y
i
s
fo
r
th
e
singl
e
particl
e
distributio
n
whic
h
als
o
involve
s
th
e
two-particl
e
distributio
n
function
.
Assumin
g
molecula
r
chaos
,
tha
t
final
equatio
n
be
-
come
s
a
close
d
on
e
(i.e.
,
on
e
equatio
n
i
n
on
e
unknown)
,
an
d
i
s
know
n
a
s
th
e
Boltzman
n
equation
,
th
e
fundamenta
l
relatio
n
o
f
th
e
kineti
c
theor
y
o
f
gases
.
Tha
t
final
equatio
n
i
n
th
e
hierarch
y
i
s
th
e
onl
y
on
e
whic
h
carrie
s
an
y
hop
e
o
f
obtainin
g
analytica
l
solutions
.
A
simple
r
direc
t
derivatio
n
o
f
th
e
Boltzman
n
equatio
n
i
s
provide
d
b
y
Bir
d
(1994)
.
Fo
r
monatomi
c
ga
s
molecule
s
i
n
binar
y
collisions
,
th
e
integro-differentia
l
Boltzman
n
equatio
n
read
s
;

=
1
,
2
,
3
(52
)
wher
e
nf
i
s
th
e
produc
t
o
f
th
e
numbe
r
densit
y
an
d
th
e
normal
-
ize
d
velocit
y
distributio
n
functio
n
(dn/n
=/d ^ )
,
Xj
an
d
^j
are
,
respectively
,
th
e
coordinate
s
an
d
speed
s
o
f
a
molecul
e
(consti
-
tuting
,
togethe
r
wit
h
time
,
th
e
seve
n
independen
t
variable
s
o
f
th
e
single-dependent-variabl
e
equation)
,
Fj
i
s
a
know
n
externa
l
force
,
an
d
/(/
,
/
*
)
i
s
th
e
nonlinea
r
collisio
n
integra
l
tha
t
de
-
scribe
s
th
e
ne
t
effec
t
o
f
populatin
g
an
d
depopulatin
g
collision
s
o
n
th
e
distributio
n
function
.
Th
e
collisio
n
integra
l
i
s
th
e
sourc
e
o
f
difficult
y
i
n
obtainin
g
analytica
l
solution
s
t
o
th
e
Boltzman
n
equation
,
an
d
i
s
give
n
b
y
J(fJ*)
rtCO
/l 47
i
J-0
0
J
o
n\f*ft
-
ffi)l.cTdn(d^h
(53
)
wher
e
th
e
superscrip
t
*
indicate
s
post-collisio
n
values
,
/and/
i
represen
t
tw
o
differen
t
molecules
,
^,
.
i
s
th
e
relativ
e
spee
d
be
-
twee
n
tw
o
molecules
,
a
i
s
th
e
molecula
r
cross-section
,
fi
i
s
th
e
soli
d
angle
,
an
d
d^
=
d^id^2d£,3,.
Onc
e
a
solutio
n
fo
r
/
i
s
obtained
,
macroscopi
c
quantitie
s
suc
h
a
s
density
,
velocity
,
temperature
,
etc.
,
ca
n
b
e
compute
d
fro
m
th
e
appropriat
e
weighte
d
integra
l
o
f
th
e
distributio
n
func
-
tion
.
Fo
r
example
.
p
=
mn
=
m
I
(nf)d^
Ui
=
/
'^kT
J
6
M
mec.
M
(54
)
(55
)
(56
)
I
f
th
e
Boltzman
n
equatio
n
i
s
nondimensionalize
d
wit
h
a
char
-
acteristi
c
lengt
h
L
an
d
characteristi
c
spee
d
[2(k/m)T]"^,
wher
e
k
i
s
th
e
Boltzman
n
constant
,
m
i
s
th
e
molecula
r
mass
,
an
d
T
i
s
temperature
,
th
e
invers
e
Knudse
n
numbe
r
appear
s
ex
-
pliciti
y
i
n
th
e
right-han
d
sid
e
o
f
th
e
equatio
n
a
s
follow
s
at
^'
dx,
'a
l
Kn^'-
^
''
j
=
1
,
2
,
3
(57
)
wher
e
th
e
superscrip
t
represent
s
a
dimensionles
s
variable
,
and
/
i
s
nondimensionalize
d
usin
g
a
referenc
e
numbe
r
densit
y
Th
e
five
conservatio
n
equation
s
fo
r
th
e
transpor
t
o
f
mass
,
momentum
,
an
d
energ
y
ca
n
b
e
derive
d
b
y
multiplyin
g
th
e
Boltz
-
man
n
equatio
n
abov
e
by
,
respectively
,
th
e
molecula
r
mass
,
mo
-
mentu
m
an
d
energy
,
the
n
integratin
g
ove
r
al
l
possibl
e
molecula
r
velocities
.
Subjec
t
t
o
th
e
restriction
s
o
f
dilut
e
ga
s
an
d
molecula
r
chao
s
state
d
earlier
,
th
e
Boltzman
n
equatio
n
i
s
vali
d
fo
r
al
l
range
s
o
f
Knudse
n
numbe
r
fro
m
0
t
o

.
Analytica
l
solution
s
t
o
thi
s
equatio
n
fo
r
arbitrar
y
geometrie
s
ar
e
difficul
t
mostl
y
becaus
e
o
f
th
e
nonlinearit
y
o
f
th
e
collisio
n
integral
.
Simpl
e
model
s
o
f
thi
s
integra
l
hav
e
bee
n
propose
d
t
o
facilitat
e
analyti
-
ca
l
solutions
;
see
,
fo
r
example
,
Bhatnaga
r
e
t
al
.
(1954)
.
Ther
e
ar
e
tw
o
importan
t
asymptote
s
t
o
Eq
.
(57)
.
First
,
a
s
K
n
-
*
00
,
molecula
r
collision
s
becom
e
unimportant
.
Thi
s
i
s
th
e
free-molecul
e
flow
regim
e
depicte
d
i
n
Fig
.
3
fo
r
K
n
>
10
,
wher
e
th
e
onl
y
importan
t
collisio
n
i
s
tha
t
betwee
n
a
ga
s
molecul
e
an
d
th
e
soli
d
surfac
e
o
f
a
n
obstacl
e
o
r
a
conduit
.
Analytica
l
solution
s
ar
e
the
n
possibl
e
fo
r
simpl
e
geometries
,
an
d
numerica
l
simula
-
tion
s
fo
r
complicate
d
geometrie
s
ar
e
straightforwar
d
onc
e
th
e
surface-reflectio
n
characteristic
s
ar
e
accuratel
y
modeled
.
Sec
-
ond
,
a
s
K
n
-
>
0
,
collision
s
becom
e
importan
t
an
d
th
e
flow
approache
s
th
e
continuu
m
regim
e
o
f
conventiona
l
fluid
dynam
-
ics
.
Th
e
Secon
d
La
w
specifie
s
a
tendenc
y
fo
r
thermodynami
c
system
s
t
o
rever
t
t
o
equilibriu
m
state
,
smoothin
g
ou
t
an
y
discon
-
tinuities
i
n
macroscopi
c
flow
quantities
.
Th
e
numbe
r
o
f
molecu
-
la
r
collision
s
i
n
th
e
limi
t
K
n
-
>
0
i
s
s
o
larg
e
tha
t
th
e
flow
approache
s
th
e
equilibriu
m
stat
e
i
n
a
tim
e
shor
t
compare
d
t
o
th
e
macroscopi
c
time-scale
.
Fo
r
example
,
fo
r
ai
r
a
t
standar
d
condition
s
(T
-
28
8
K;p=l
atm)
,
eac
h
molecul
e
experiences
,
o
n
th
e
average
,
1
0
collision
s
pe
r
nanosecon
d
an
d
travel
s
1
6
/
Vol
.
121
,
MARC
H
199
9
Transaction
s
o
f
th
e
ASM
E
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1 micron in the same time period. Such a molecule has already
forgotten its previous state after 1 ns. In a particular flow field,
if the macroscopic quantities vary little over a distance of 1 fjxa
or over a time interval of 1 ns, the flow of STP air is near
equilibrium.
At Kn = 0, the velocity distribution function is everywhere
of the local equilibrium or Maxwellian form:
/
0) _
«„
^ -''^ e x p [ - ( e - M)']
(58)
where | and u are, respectively, the dimensionless speeds of a
molecule and of the flow. In this Knudsen number limit, the
velocity distribution of each element of the fluid instantaneously
adjusts to the equilibrium thermodynamic state appropriate to
the local macroscopic properties as this molecule moves through
the flow field. From the continuum viewpoint, the flow is isen-
tropic and heat conduction and viscous diffusion and dissipation
vanish from the continuum conservation relations.
The Chapman-Enskog theory attempts to solve the Boltz-
mann equation by considering a small perturbation of / from
the equilibrium Maxwellian form. For small Knudsen numbers,
the distribution function can be expanded in terms of Kn in the
form of a power series
/ = /«" + Kn/<" + Kn ^"' +
(59)
By substituting the above series in the Boltzmann equation (57)
and equating terms of equal order, the following recurrent set
of integral equations result:
/(/"",/'")
dt
dxj
d^J
(60)
The first integral is nonlinear and its solution is the local Max-
wellian distribution, Eq. (58). The distribution functions/'',
f^\ etc., each satisfies an inhomogeneous linear equation whose
solution leads to the transport terms needed to close the contin-
uum equations appropriate to the particular level of approxima-
tion. The continuum stress tensor and heat flux vector can be
written in terms of the distribution function, which in turn can
be specified in terms of the macroscopic velocity and tempera-
ture and their derivatives (Kogan, 1973). The zeroth-order
equation yields the Euler equations, the first-order equation re-
sults in the linear transport terms of the Navier-Stokes equa-
tions, the second-order equation gives the nonlinear transport
terms of the Burnett equations, and so on. Keep in mind, how-
ever, that the Boltzmann equation as developed in this subsec-
tion is for a monatomic gas. This excludes the all important air
which is composed largely of diatomic nitrogen and oxygen.
As discussed in Sections 2.2, 2.3, and 2.5, the Navier-Stokes
equations can and should be used up to a Knudsen number of
0.1. Beyond that, the transition flow regime commences (0.1
< Kn < 10). In this flow regime, the molecular mean free path
for a gas becomes significant relative to a characteristic distance
for important flow-property changes to take place. The Burnett
equations can be used to obtain analytical/numerical solutions
for at least a portion of the transition regime for a monatomic
gas, although their complexity have precluded much results for
reaUstic geometries. There is also a certain degree of uncertainty
about the proper boundary conditions to use with the continuum
Burnett equations, and experimental validation of the results
have been very scarce. Additionally, as the gas flow further
departs from equilibrium, the bulk viscosity (=X + 3^, where
\ is the second coefficient of viscosity) is no longer zero, and
the Stokes' hypothesis no longer holds (see Gad-el-Hak, 1995,
for an interesting summary of the issue of bulk viscosity).
In the transition regime, the molecularly-based Boltzmann
equation cannot easily be solved either, unless the nonlinear
collision integral is simplified. So, clearly the transition regime
is one of dire need of alternative methods of solution. MD
simulations as mentioned earlier are not suited for dilute gases.
The best approach for the transition regime right now is the
direct simulation Monte Carlo (DSMC) method developed by
Bird (1963; 1965; 1976; 1978; 1994) and briefly described
below. Some recent reviews of DSMC include those by Muntz
(1989), Cheng (1993), Cheng and Emmanuel (1995), and
Oran et al. (1998). The mechanics as well as the history of the
DSMC approach and its ancestors are well described in the
book by Bird (1994).
UnUke molecular dynamics simulations, DSMC is a statisti-
cal computational approach to solving rarefied gas problems.
Both approaches treat the gas as discrete particles. Subject to
the dilute gas and molecular chaos assumptions, the direct simu-
lation Monte Carlo method is valid for all ranges of Knudsen
number, although it becomes quite expensive for Kn < 0.1.
Fortunately, this is the continuum regime where the Navier-
Stokes equations can be used analytically or computationally.
DSMC is therefore ideal for the transition regime (0.1 < Kn
< 10), where the Boltzmann equation is difficult to solve. The
Monte Carlo method is, like its name sake, a random number
strategy based directly on the physics of the individual molecu-
lar interactions. The idea is to track a large number of randomly
selected, statistically representative particles, and to use their
motions and interactions to modify their positions and states.
The primary approximation of the direct simulation Monte
Carlo method is to uncouple the molecular motions and the
intermolecular collisions over small time intervals. A significant
advantage of this approximation is that the amount of computa-
tion required is proportional to A^, in contrast to N^ for molecular
dynamics simulations. In essence, particle motions are modeled
deterministically while collisions are treated probabilistically,
each simulated molecule representing a large number of actual
molecules. Typical computer runs of DSMC in the 1990s in-
volve tens of millions of intermolecular collisions and fluid-
solid interactions.
The DSMC computation is started from some initial condition
and followed in small time steps that can be related to physical
time. Colliding pairs of molecules in a small geometric cell in
physical space are randomly selected after each computational
time step. Complex physics such as radiation, chemical reac-
tions and species concentrations can be included in the simula-
tions without the necessity of nonequilibrium thermodynamic
assumptions that commonly afflict nonequilibrium continuum-
flow calculations. DSMC is more computationally intensive
than classical continuum simulations, and should therefore be
used only when the continuum approach is not feasible.
The DSMC technique is explicit and time marching, and
therefore always produces unsteady flow simulations. For mac-
roscopically steady flows, Monte Carlo simulation proceeds un-
til a steady flow is established, within a desired accuracy, at
sufficiently large time. The macroscopic flow quantities are then
the time average of all values calculated after reaching the
steady state. For macroscopically unsteady flows, ensemble av-
eraging of many independent Monte Carlo simulations is carried
out to obtain the final results within a prescribed statistical
accuracy.
2.7 Liquid Flows. From the continuum point of view,
liquids and gases are both fluids obeying the same equations of
motion. For incompressible flows, for example, the Reynolds
number is the primary dimensionless parameter that determines
the nature of the flow field. True, water, for example, has density
and viscosity that are, respectively, three and two orders of
magnitude higher than those for air, but if the Reynolds number
and geometry are matched, liquid and gas flows should be iden-
tical. (Barring phenomena unique to liquids such as cavitation,
free surface flows, etc.) For MEMS applications, however, we
anticipate the possibility of non-equilibrium flow conditions and
Journal of Fluids Engineering
MARCH 1999, Vol. 121 / 17
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th
e
consequen
t
invalidit
y
o
f
th
e
Navier-Stoke
s
equation
s
an
d
th
e
no-sli
p
boundar
y
conditions
.
Suc
h
circumstance
s
ca
n
bes
t
b
e
researche
d
usin
g
th
e
molecula
r
approach
.
Thi
s
wa
s
discusse
d
fo
r
gase
s
i
n
Sectio
n
2.6
,
an
d
th
e
correspondin
g
argument
s
fo
r
liquid
s
wil
l
b
e
give
n
i
n
th
e
presen
t
subsection
.
Th
e
literatur
e
o
n
non-Newtonia
n
fluids
i
n
genera
l
an
d
polymer
s
i
n
particula
r
i
s
vas
t
(fo
r
example
,
th
e
bibliographi
c
surve
y
b
y
Nadolin
k
an
d
Haigh
,
1995
,
cite
s
ove
r
4,90
0
reference
s
o
n
polyme
r
dra
g
reduc
-
tio
n
alone
)
an
d
provide
s
a
ric
h
sourc
e
o
f
informatio
n
o
n
th
e
molecula
r
approac
h
fo
r
liqui
d
flows.
Solids
,
liquid
s
an
d
gase
s
ar
e
distinguishe
d
merel
y
b
y
th
e
degre
e
o
f
proximit
y
an
d
th
e
intensit
y
o
f
motion
s
o
f
thei
r
constit
-
uen
t
molecules
.
I
n
solids
,
th
e
molecule
s
ar
e
packe
d
closel
y
an
d
confined
,
eac
h
hemme
d
i
n
b
y
it
s
neighbor
s
(Chapma
n
an
d
CowUng
,
1970)
.
Onl
y
rarel
y
woul
d
on
e
soli
d
molecul
e
sli
p
fro
m
it
s
neighbor
s
t
o
joi
n
a
ne
w
set
.
A
s
th
e
soli
d
i
s
heated
,
molecula
r
motio
n
become
s
mor
e
violen
t
an
d
a
sligh
t
therma
l
expansio
n
take
s
place
.
A
t
a
certai
n
temperatur
e
tha
t
depend
s
o
n
ambien
t
pressure
,
sufficientl
y
intens
e
motio
n
o
f
th
e
molecule
s
enable
s
the
m
t
o
pas
s
freel
y
fro
m
on
e
se
t
o
f
neighbor
s
t
o
another
.
Th
e
molecule
s
ar
e
n
o
longe
r
confine
d
bu
t
ar
e
nevertheles
s
stil
l
closel
y
packed
,
an
d
th
e
substanc
e
i
s
no
w
considere
d
a
liquid
.
Furthe
r
heatin
g
o
f
th
e
matte
r
eventuall
y
release
s
th
e
molecule
s
altogether
,
allowin
g
the
m
t
o
brea
k
th
e
bond
s
o
f
thei
r
mutua
l
attractions
.
Unlik
e
solid
s
an
d
liquids
,
th
e
resultin
g
ga
s
expand
s
t
o
fill
an
y
volum
e
availabl
e
t
o
it
.
Unlik
e
solids
,
bot
h
liquid
s
an
d
gase
s
canno
t
resis
t
finite
shea
r
forc
e
withou
t
continuou
s
deformation
;
tha
t
i
s
th
e
definitio
n
o
f
a
fluid
medium
.
I
n
contras
t
t
o
th
e
reversible
,
elastic
,
stati
c
defor
-
matio
n
o
f
a
soUd
,
th
e
continuou
s
deformatio
n
o
f
a
fluid
resultin
g
fro
m
th
e
applicatio
n
o
f
a
shea
r
stres
s
result
s
i
n
a
n
irreversibl
e
wor
k
tha
t
eventuall
y
become
s
rando
m
therma
l
motio
n
o
f
th
e
molecules
;
tha
t
i
s
viscou
s
dissipation
.
Ther
e
ar
e
aroun
d
25
-
millio
n
molecule
s
o
f
ST
P
ai
r
i
n
a
one-micro
n
cube
.
Th
e
sam
e
cub
e
woul
d
contai
n
aroun
d
34-billio
n
molecule
s
o
f
water
.
So
,
liqui
d
flows
ar
e
continuu
m
eve
n
i
n
extremel
y
smal
l
device
s
throug
h
whic
h
ga
s
flows
woul
d
not
.
Th
e
averag
e
distanc
e
be
-
twee
n
molecule
s
i
n
th
e
ga
s
exampl
e
i
s
on
e
orde
r
o
f
magnitud
e
highe
r
tha
n
th
e
diamete
r
o
f
it
s
molecules
,
whil
e
tha
t
fo
r
th
e
liqui
d
phas
e
approache
s
th
e
molecula
r
diameter
.
A
s
a
result
,
liquid
s
ar
e
almos
t
incompressible
.
Thei
r
isotherma
l
compress
-
ibilit
y
coefficien
t
a
an
d
bul
k
expansio
n
coefficien
t
/
3
ar
e
muc
h
smalle
r
compare
d
t
o
thos
e
fo
r
gases
.
Fo
r
water
,
fo
r
example
,
a
hundred-fol
d
increas
e
i
n
pressur
e
lead
s
t
o
les
s
tha
n
0.5
%
de
-
creas
e
i
n
volume
.
Soun
d
speed
s
throug
h
liquid
s
ar
e
als
o
hig
h
relativ
e
t
o
thos
e
fo
r
gases
,
an
d
a
s
a
resul
t
mos
t
liqui
d
flows
ar
e
incompressible
.
Th
e
exceptio
n
bein
g
propagatio
n
o
f
ultra-high
-
frequenc
y
soun
d
wave
s
an
d
cavitatio
n
phenomena
.
(Not
e
tha
t
w
e
distinguis
h
betwee
n
a
fluid
an
d
a
flow
bein
g
compressible
/
incompressible
.
Fo
r
example
,
th
e
flow
o
f
th
e
highl
y
compress
-
ibl
e
ai
r
ca
n
b
e
eithe
r
compressibl
e
o
r
incompressible.
)
Th
e
mechanis
m
b
y
whic
h
liquid
s
transpor
t
mass
,
momentu
m
an
d
energ
y
mus
t
b
e
ver
y
differen
t
fro
m
tha
t
fo
r
gases
.
I
n
dilut
e
gases
,
intermolecula
r
force
s
pla
y
n
o
rol
e
an
d
th
e
molecule
s
spen
d
mos
t
o
f
thei
r
tim
e
i
n
fre
e
flight
betwee
n
brie
f
colUsion
s
a
t
whic
h
instance
s
th
e
molecules
'
directio
n
an
d
spee
d
abruptl
y
change
.
Th
e
rando
m
molecula
r
motion
s
ar
e
responsibl
e
fo
r
gas
-
eou
s
transpor
t
processes
.
I
n
liquids
,
o
n
th
e
othe
r
hand
,
th
e
mole
-
cule
s
ar
e
closel
y
packe
d
thoug
h
no
t
fixed
i
n
on
e
position
.
I
n
essence
,
th
e
liqui
d
molecule
s
ar
e
alway
s
i
n
a
collision
state
.
Applyin
g
a
shea
r
forc
e
mus
t
creat
e
a
velocit
y
gradien
t
s
o
tha
t
th
e
molecule
s
mov
e
relativ
e
t
o
on
e
another
,
ad
infinitum
a
s
lon
g
a
s
th
e
stres
s
i
s
applied
.
Fo
r
liquids
,
momentu
m
transpor
t
du
e
t
o
th
e
rando
m
molecula
r
motio
n
i
s
negligibl
e
compare
d
t
o
tha
t
du
e
t
o
th
e
intermolecula
r
forces
.
Th
e
strainin
g
betwee
n
liqui
d
molecule
s
cause
s
som
e
t
o
separat
e
fro
m
thei
r
origina
l
neighbors
,
bringin
g
the
m
int
o
th
e
forc
e
field
o
f
ne
w
molecules
.
Acros
s
th
e
plan
e
o
f
th
e
shea
r
stress
,
th
e
su
m
o
f
al
l
intermolecula
r
force
s
must
,
o
n
th
e
average
,
balanc
e
th
e
impose
d
shear
.
Liquid
s
a
t
res
t
transmi
t
onl
y
norma
l
force
,
bu
t
whe
n
a
velocit
y
gradien
t
occurs
,
th
e
ne
t
intermolecula
r
forc
e
woul
d
hav
e
a
tangentia
l
component
.
Th
e
incompressibl
e
Navier-Stoke
s
equation
s
describ
e
liqui
d
flows
unde
r
mos
t
circumstances
.
Liquids
,
however
,
d
o
no
t
hav
e
a
we
U
advance
d
molecular-base
d
theor
y
a
s
tha
t
fo
r
dilut
e
gases
.
Th
e
concep
t
o
f
mea
n
fre
e
pat
h
i
s
no
t
ver
y
usefu
l
fo
r
liquid
s
an
d
th
e
condition
s
unde
r
whic
h
a
liqui
d
flow
fail
s
t
o
b
e
i
n
quasi
-
equilibriu
m
stat
e
ar
e
no
t
wel
l
defined
.
Ther
e
i
s
n
o
Knudse
n
numbe
r
fo
r
liqui
d
flows
t
o
guid
e
u
s
throug
h
th
e
maze
.
W
e
d
o
not
know
,
fro
m
first
principles
,
th
e
condition
s
unde
r
whic
h
th
e
no-sli
p
boundar
y
conditio
n
become
s
inaccurate
,
o
r
th
e
poin
t
a
t
whic
h
th
e
(stress)-(rat
e
o
f
strain
)
relatio
n
o
r
th
e
(hea
t
flux)-
(temperatur
e
gradient
)
relatio
n
fail
s
t
o
b
e
linear
.
Certai
n
empiri
-
ca
l
observation
s
indicat
e
tha
t
thos
e
simpl
e
relation
s
tha
t
w
e
tak
e
fo
r
grante
d
occasionall
y
fai
l
t
o
accuratel
y
mode
l
liqui
d
flows.
Fo
r
example
,
i
t
ha
s
bee
n
show
n
i
n
rheologica
l
studie
s
(Loos
e
an
d
Hess
,
1989
)
tha
t
non-Newtonia
n
behavio
r
commence
s
whe
n
th
e
strai
n
rat
e
approximatel
y
exceed
s
twic
e
th
e
molecula
r
frequency-scal
e
y
=

>
2r -
'
dy
wher
e
th
e
molecula
r
time-scal
e
T
i
s
give
n
b
y
(61
)
(62
)
wher
e
m
i
s
th
e
molecula
r
mass
,
an
d
a
an
d
e
ar
e
respectivel
y
th
e
characteristi
c
length
-
an
d
energy-scal
e
fo
r
th
e
molecules
.
Fo
r
ordinar
y
liquid
s
suc
h
a
s
water
,
thi
s
time-scal
e
i
s
extremel
y
smal
l
an
d
th
e
threshol
d
shea
r
rat
e
fo
r
th
e
onse
t
o
f
non-Newton
-
ia
n
behavio
r
i
s
therefor
e
extraordinaril
y
high
.
Fo
r
high-molecu
-
lar-weigh
t
polymers
,
o
n
th
e
othe
r
hand
,
m
an
d
a
ar
e
bot
h
man
y
order
s
o
f
magnitud
e
highe
r
tha
n
thei
r
respectiv
e
value
s
fo
r
wa
-
ter
,
an
d
th
e
linea
r
stress-strai
n
relatio
n
break
s
dow
n
a
t
realisti
c
value
s
o
f
th
e
shea
r
rate
.
Th
e
movin
g
contac
t
lin
e
whe
n
a
liqui
d
spread
s
o
n
a
soli
d
substrat
e
i
s
a
n
exampl
e
wher
e
sli
p
flow
mus
t
b
e
allowe
d
t
o
avoi
d
singula
r
o
r
unrealisti
c
behavio
r
i
n
th
e
Navier-Stoke
s
solution
s
(Dussa
n
an
d
Davis
,
1974
;
Dussan
,
1976
;
1979
;
Thompso
n
an
d
Robbins
,
1989)
.
Othe
r
example
s
wher
e
sUp-flo
w
mus
t
b
e
admit
-
te
d
includ
e
come
r
flows
(Moffatt
,
1964
;
Kopli
k
an
d
Banavar
,
1995
)
an
d
extrusio
n
o
f
polyme
r
melt
s
fro
m
capillar
y
tube
s
(Pearso
n
an
d
Petrie
,
1968
;
Richardson
,
1973
;
Den
,
1990)
.
Existin
g
experimenta
l
result
s
o
f
liqui
d
flow
i
n
microdevice
s
ar
e
contradictory
.
Thi
s
i
s
no
t
surprisin
g
give
n
th
e
difficult
y
o
f
suc
h
experiment
s
an
d
th
e
lac
k
o
f
a
guidin
g
rationa
l
theory
.
Pfahle
r
e
t
al
.
(1990
;
1991)
,
Pfahle
r
(1992)
,
an
d
Ba
u
(1994
)
summariz
e
th
e
relevan
t
literature
.
Fo
r
small-length-scal
e
flows,
a
phenomenologica
l
approac
h
fo
r
analyzin
g
th
e
dat
a
i
s
t
o
defin
e
a
n
apparen
t
viscosit
y
//

calculate
d
s
o
tha
t
i
f
i
t
wer
e
use
d
i
n
th
e
traditiona
l
no-sli
p
Navier-Stoke
s
equation
s
instea
d
o
f
th
e
fluid
viscosit
y
//
,
th
e
result
s
woul
d
b
e
i
n
agreemen
t
wit
h
experimenta
l
observations
.
Israelachvi
H
(1986
)
an
d
Ge
e
e
t
al
.
(1990
)
foun
d
tha
t
jU

=
/
/
fo
r
thin-fil
m
flows
a
s
lon
g
a
s
th
e
film
thicknes
s
exceed
s
1
0
molecula
r
layer
s
( «
5
nm)
.
Fo
r
thinne
r
films,
/u

depend
s
o
n
th
e
numbe
r
o
f
molecula
r
layer
s
an
d
ca
n
b
e
a
s
muc
h
a
s
10
'
time
s
large
r
tha
n
//
.
Cha
n
an
d
Horn'
s
(1985
)
result
s
ar
e
somewha
t
different
;
th
e
apparen
t
viscosit
y
deviate
s
fro
m
th
e
fluid
viscosit
y
fo
r
films
thinne
r
tha
n
5
0
nm
.
I
n
polar-liqui
d
flows
throug
h
capillaries
,
Migu
n
an
d
Prokhor
-
enk
o
(1987
)
repor
t
tha
t
ji,,
increase
s
fo
r
tube
s
smalle
r
tha
n
1
micro
n
i
n
diameter
.
I
n
contrast
,
Deby
e
an
d
Clelan
d
(1959
)
repor
t
/x

smalle
r
tha
n
/.
t
fo
r
paraffi
n
flow
i
n
porou
s
glas
s
wit
h
averag
e
por
e
siz
e
severa
l
time
s
large
r
tha
n
th
e
molecula
r
length
-
scale
.
Experimentin
g
wit
h
microchannel
s
rangin
g
i
n
depth
s
fro
m
0.
5
micro
n
t
o
5
0
microns
,
Pfahle
r
e
t
al
.
(1991
)
foun
d
tha
t
lia
i
s
consistentl
y
smalle
r
tha
n
fj,
fo
r
bot
h
liqui
d
(isopropy
l
alcohol
;
silicon
e
oil
)
an
d
ga
s
(nitrogen
;
helium
)
flows
i
n
micro
-
1
8
/
Vol
.
121
,
MARC
H
199
9
Transaction
s
o
f
th
e
ASM
E
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Fig
.
9
Velocit
y
profile
s
i
n
a
Couett
e
flo
w
geometr
y
a
t
differen
t
Interfacia
l
parameters
.
Al
l
thre
e
profile
s
ar
e
fo
r
U
=
O-
T
'
,
an
d
h
=
24.570-
.
Th
e
dashe
d
lin
e
i
s
th
e
no-sll
p
Couette-flo
w
solution
.
Fro
m
Thompso
n
an
d
Troia
n
(1997)
.
channels
.
Fo
r
liquids
,
th
e
apparen
t
viscosit
y
decrease
s
wit
h
de
-
creasin
g
channe
l
depth
.
Othe
r
researcher
s
usin
g
smal
l
capillar
-
ie
s
repor
t
tha
t
/x

i
s
abou
t
th
e
sam
e
a
s
fj,
(Anderso
n
an
d
Quinn
,
1972
;
Tukerman
n
an
d
Pease
,
1981
;
1982
;
Tuckermann
,
1984
;
Guvenc
,
1985
;
Nakagaw
a
e
t
al.
,
1990)
.
Th
e
abov
e
contradictor
y
result
s
poin
t
t
o
th
e
nee
d
fo
r
replac
-
in
g
phenomenologica
l
model
s
b
y
first-principles
ones
.
Th
e
lac
k
o
f
molecular-base
d
theor
y
o
f
liquids—despit
e
extensiv
e
re
-
searc
h
b
y
th
e
rheolog
y
an
d
polyme
r
communities—leave
s
mo
-
lecula
r
dynamic
s
simulation
s
a
s
th
e
neares
t
weapo
n
t
o
first-
principle
s
arsenal
.
M
D
simulation
s
offe
r
a
uniqu
e
approac
h
t
o
checkin
g
th
e
validit
y
o
f
th
e
traditiona
l
continuu
m
assumptions
.
However
,
a
s
wa
s
pointe
d
out
i
n
Sectio
n
2.6
,
suc
h
simulation
s
ar
e
limite
d
t
o
exceedingl
y
minut
e
flow
extent
.
Thompso
n
an
d
Troia
n
(1997
)
provid
e
molecula
r
dynamic
s
simulation
s
t
o
quantif
y
th
e
slip-flo
w
boundar
y
conditio
n
depen
-
denc
e
o
n
shea
r
rate
.
Recal
l
th
e
linea
r
Navie
r
boundar
y
conditio
n
introduce
d
i
n
Sectio
n
2.5
,
A
I
-
_
r
^
"
A
M
I


Uf\md
~
Mwal
l

A
s
TT
"
dy
(63
)
wher
e
L
,
i
s
th
e
constan
t
sli
p
length
,
an
d
duldy
\

,
i
s
th
e
strai
n
rat
e
compute
d
a
t
th
e
wall
.
Th
e
goa
l
o
f
Thompso
n
an
d
Troian'
s
simulation
s
wa
s
t
o
determin
e
th
e
degre
e
o
f
sli
p
a
t
a
solid-liqui
d
interfac
e
a
s
th
e
interfacia
l
parameter
s
an
d
th
e
shea
r
rat
e
change
.
I
n
thei
r
simulations
,
a
simpl
e
Uqui
d
underwen
t
plana
r
shea
r
i
n
a
Couett
e
cel
l
a
s
show
n
i
n
Fig
.
9
.
Th
e
typica
l
cel
l
measure
d
12.5
1
X
7.2
2
X
h,
i
n
unit
s
o
f
molecula
r
length-scal
e
a,
wher
e
th
e
channe
l
dept
h
h
varie
d
i
n
th
e
rang
e
o
f
16.7l
a

lA.Sla,
an
d
th
e
correspondin
g
numbe
r
o
f
molecule
s
simulate
d
range
d
fro
m
1,15
2
t
o
1,728
.
Th
e
liqui
d
i
s
treate
d
a
s
a
n
isotherma
l
ensembl
e
o
f
spherica
l
molecules
.
A
shifte
d
Lennart-Jone
s
6-1
2
potentia
l
i
s
use
d
t
o
mode
l
intermolecula
r
interactions
,
wit
h
energy
-
an
d
length-scale
s
e
an
d
a,
an
d
cut-of
f
distanc
e
r
,
=
2.2(7
:
V{r)
=
4e
Th
e
truncate
d
potentia
l
i
s
se
t
t
o
zer
o
fo
r
r
>
r^.
(64
)
Th
e
fluid-solid
interactio
n
i
s
als
o
modele
d
wit
h
a
truncate
d
Lennard-Jone
s
potential
,
wit
h
energy
-
an
d
length-scale
s
e"'^an
d
ff""^,
an
d
cut-of
f
distanc
e
r,.
.
Th
e
equilibriu
m
stat
e
o
f
th
e
fluid
i
s
a
well-define
d
liqui
d
phas
e
characterize
d
b
y
numbe
r
densit
y
n
=
O.Slcr"
'
an
d
temperatur
e
T
=
l.le/fe
,
wher
e
k
i
s
th
e
Boltz
-
man
n
constant
.
Th
e
steady-stat
e
velocit
y
profile
s
resultin
g
fro
m
Thompso
n
an
d
Troian'
s
(1997
)
M
D
simulation
s
ar
e
depicte
d
i
n
Fig
.
9
fo
r
differen
t
value
s
o
f
th
e
interfacia
l
parameter
s
e'"^
,
a''^
an
d
«""
.
Thos
e
parameters
,
show
n
i
n
unit
s
o
f
th
e
correspondin
g
fluid
parameter
s
e
,
a
an
d
n,
characterize
,
respectively
,
th
e
strengt
h
o
f
th
e
liquid-soli
d
coupling
,
th
e
therma
l
roughnes
s
o
f
th
e
inter
-
fac
e
an
d
th
e
commensurabilit
y
o
f
wal
l
an
d
liqui
d
densities
.
Th
e
macroscopi
c
velocit
y
profile
s
recove
r
th
e
expecte
d
flow
behavio
r
fro
m
continuu
m
hydrodynamic
s
wit
h
boundar
y
condi
-
tion
s
involvin
g
varyin
g
degree
s
o
f
slip
.
Not
e
tha
t
whe
n
sli
p
exists
,
th
e
shea
r
rat
e
y
n
o
longe
r
equal
s
U/h.
Th
e
degre
e
o
f
sli
p
increase
s
(i.e
.
th
e
amoun
t
o
f
momentu
m
transfe
r
a
t
th
e
wall-flui
d
interfac
e
decreases
)
a
s
th
e
relativ
e
wal
l
densit
y
n"
increase
s
o
r
th
e
strengt
h
o
f
th
e
wall-flui
d
couplin
g
a"^
de
-
creases
;
i
n
othe
r
word
s
whe
n
th
e
relativ
e
surfac
e
energ
y
corru
-
gatio
n
o
f
th
e
wal
l
decreases
.
Conversely
,
th
e
corrugatio
n
i
s
maximize
d
whe
n
th
e
wal
l
an
d
fluid
densitie
s
ar
e
commensurat
e
an
d
th
e
strengt
h
o
f
th
e
wall-flui
d
couplin
g
i
s
large
.
I
n
thi
s
case
,
th
e
liqui
d
feel
s
th
e
corrugation
s
i
n
th
e
surfac
e
energ
y
o
f
th
e
soU
d
owin
g
t
o
th
e
atomi
c
close-packing
.
Consequently
,
ther
e
i
s
efficien
t
momentu
m
transfe
r
an
d
th
e
no-sli
p
conditio
n
applies
,
o
r
i
n
extrem
e
cases
,
a
'stick
'
boundar
y
conditio
n
take
s
hold
.
Variation
s
o
f
th
e
sli
p
lengt
h
L^
an
d
viscosit
y
fj.
a
s
function
s
o
f
shea
r
rat
e
y
ar
e
show
n
i
n
part
s
(a
)
an
d
(b)
o
f
Fig
.
10
,
fo
r
five
differen
t
set
s
o
f
interfacia
l
parameters
.
Fo
r
Couett
e
flow,
th
e
sli
p
lengt
h
i
s
compute
d
fro
m
it
s
definition
,
L
,
=
Au\„/y
=
(U/y
-
h)/2.
Th
e
sli
p
length
,
viscosit
y
an
d
shea
r
rat
e
ar
e
normalize
d
i
n
th
e
figure
usin
g
th
e
respectiv
e
molecula
r
scale
s
fo
r
lengt
h
a,
viscosit
y
era
*'
,
an
d
invers
e
tim
e
T
^'
.
Th
e
viscos
-
it
y
o
f
th
e
fluid
i
s
constan
t
ove
r
th
e
entir
e
rang
e
o
f
shea
r
rate
s
(Fig
.
W(b)),
indicatin
g
Newtonia
n
behavior
.
A
s
indicate
d
ear
-
lier
,
non-Newtonia
n
behavio
r
i
s
expecte
d
fo
r
y
>
2
r
"'
,
wel
l
abov
e
th
e
shea
r
rate
s
use
d
i
n
Thompso
n
an
d
Troian'
s
simula
-
tions
.
40
-
3
0
>-l
"
2
0
1
0
%
3
U
J
0
I
I
I
1
I
I
I — I — I — I — I — I — I — I

(a
)
0
0.
6

0.
1

0.
6
D
0.
4

0.
2
1.
0
1.
0
0.7
5
0.7
5
0.7
5
n
"
1
1
4
4
4
a
D
D
D
«
D

a
D
°
.
/
:
*

*
-
,•
"


• •"• •

,^
n
O
o
o
O
o
O
O
O
O
-
I
1
1
1
1 1
1
1
1
1
1
1
1
-
(b
)
;
|-li-"|-ff
e
i-flt-ani-^—NW
-
J
I
I
U
0.00
1
0.0
1
0.
1
1.
0
Fig
.
1
0
Variatio
n
o
f
sli
p
lengt
h
an
d
viscosit
y
a
s
function
s
o
f
shea
r
rate
.
Fro
m
Thompso
n
an
d
Troia
n
(1997)
.
Journa
l
o
f
Fluid
s
Engineerin
g
MARC
H
1999
,
Vol
.
12
1
/
1
9
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
A
t
lo
w
shea
r
rates
,
th
e
sli
p
lengt
h
behavio
r
i
s
consisten
t
wit
h
th
e
Navie
r
model
,
i.e.
,
i
s
independen
t
o
f
th
e
shea
r
rate
.
It
s
limitin
g
valu
e
L"
range
s
fro
m
0
t
o
~\la
fo
r
th
e
rang
e
o
f
interfacia
l
parameter
s
chose
n
(Fig
.
10(a))
.
I
n
general
,
th
e
amoun
t
o
f
sli
p
increase
s
wit
h
decreasin
g
surfac
e
energ
y
corru
-
gation
.
Mos
t
interestingly
,
a
t
hig
h
shea
r
rate
s
th
e
Navie
r
condi
-
tio
n
break
s
dow
n
a
s
th
e
sli
p
lengt
h
increase
s
rapidl
y
wit
h
y.
Th
e
critica
l
shear-rat
e
valu
e
fo
r
th
e
sli
p
lengt
h
t
o
diverge
,
jc,
decrease
s
a
s
th
e
surfac
e
energ
y
corrugatio
n
decreases
.
Surpris
-
ingly
,
th
e
boundar
y
conditio
n
i
s
nonlinea
r
eve
n
thoug
h
th
e
liq
-
ui
d
i
s
stil
l
Newtonian
.
I
n
dilut
e
gases
,
a
s
discusse
d
i
n
Sectio
n
2.6
,
th
e
linea
r
sli
p
conditio
n
an
d
th
e
Navier-Stoke
s
equations
,
wit
h
thei
r
Unea
r
stress-strai
n
relation
,
ar
e
bot
h
vali
d
t
o
th
e
sam
e
orde
r
o
f
approximatio
n
i
n
Knudse
n
number
.
I
n
othe
r
words
,
deviatio
n
fro
m
linearit
y
i
s
expecte
d
t
o
tak
e
plac
e
a
t
th
e
sam
e
valu
e
o
f
K
n
=
0.1
.
I
n
liquids
,
i
n
contrast
,
th
e
sli
p
lengt
h
appear
s
t
o
becom
e
nonUnea
r
an
d
t
o
diverg
e
a
t
a
critica
l
valu
e
o
f
shea
r
rat
e
wel
l
belo
w
th
e
shea
r
rat
e
a
t
whic
h
th
e
linea
r
stress-strai
n
relatio
n
fails
.
Moreover
,
th
e
boundar
y
conditio
n
deviatio
n
fro
m
linearit
y
i
s
no
t
gradua
l
bu
t
i
s
rathe
r
catastrophic
.
Th
e
critica
l
valu
e
o
f
shea
r
rat
e
y^
signal
s
th
e
poin
t
a
t
whic
h
th
e
soli
d
ca
n
n
o
longe
r
impar
t
momentu
m
t
o
th
e
liquid
.
Thi
s
mean
s
tha
t
th
e
sam
e
liqui
d
molecule
s
sheare
d
agains
t
differen
t
substrate
s
wil
l
experienc
e
varyin
g
amount
s
o
f
sli
p
an
d
vic
e
versa
.
Base
d
o
n
th
e
abov
e
results
,
Thompso
n
an
d
Troia
n
(1997
)
sugges
t
a
universa
l
boundar
y
conditio
n
a
t
a
solid-liqui
d
inter
-
face
.
Scalin
g
th
e
sli
p
lengt
h
L
,
b
y
it
s
asymptoti
c
limitin
g
valu
e
L"
an
d
th
e
shea
r
rat
e
y
b
y
it
s
critica
l
valu
e
y,,
,
collapse
s
th
e
dat
a
i
n
th
e
singl
e
curv
e
show
n
i
n
Figur
e
11
.
Th
e
dat
a
point
s
ar
e
wel
l
describe
d
b
y
th
e
relatio
n
L,
=
L
°
1
-
y
(65
)
Th
e
nonlinea
r
behavio
r
clos
e
t
o
a
critica
l
shea
r
rat
e
suggest
s
tha
t
th
e
boundar
y
conditio
n
ca
n
significantl
y
affec
t
flow
behavio
r
a
t
macroscopi
c
distance
s
fro
m
th
e
wall
.
Experiment
s
wit
h
poly
-
mer
s
confir
m
thi
s
observatio
n
(Atwoo
d
an
d
Schwalter
,
1989)
.
Th
e
rapi
d
chang
e
i
n
th
e
sli
p
lengt
h
suggest
s
tha
t
fo
r
flows
i
n
th
e
vicinit
y
o
f
y^,
smal
l
change
s
i
n
surfac
e
propertie
s
ca
n
lea
d
t
o
larg
e
fluctuations
i
n
th
e
apparen
t
boundar
y
condition
.
Thomp
-
so
n
an
d
Troia
n
(1997
)
conclud
e
tha
t
th
e
Navie
r
sli
p
conditio
n
i
s
bu
t
th
e
low-shear-rat
e
limi
t
o
f
a
mor
e
generalize
d
universa
l
relationshi
p
whic
h
i
s
nonlinea
r
an
d
divergent
.
Thei
r
relatio
n
provide
s
a
mechanis
m
fo
r
relievin
g
th
e
stres
s
singularit
y
i
n
spreadin
g
contac
t
line
s
an
d
corne
r
flows,
a
s
i
t
naturall
y
allow
s
fo
r
varyin
g
degree
s
o
f
sh
p
o
n
approac
h
t
o
region
s
o
f
highe
r
rat
e
o
f
strain
.
T
o
plac
e
th
e
abov
e
result
s
i
n
physica
l
terms
,
conside
r
wate
r
a
t
a
temperatur
e
o
f
7
=
28
8
K
.
(Wate
r
molecule
s
ar
e
comple
x
ones
,
formin
g
directional
,
short-rang
e
covalen
t
bonds
.
Thu
s
re
-
quirin
g
a
mor
e
comple
x
potentia
l
tha
n
th
e
Lennard-Jone
s
t
o
describ
e
th
e
intermolecula
r
interactions
.
Fo
r
th
e
purpos
e
o
f
th
e
qualitativ
e
exampl
e
describe
d
here
,
however
,
w
e
us
e
th
e
compu
-
tationa
l
result
s
o
f
Thompso
n
an
d
Troia
n
(1997
)
wh
o
employe
d
th
e
L-
J
potential.
)
Th
e
energy-scal
e
i
n
th
e
Lennard-Jone
s
poten
-
tia
l
i
s
the
n
e
=
3.6
2
X
10"^
'
J
.
Fo
r
water
,
m
=
2.9
9
X
10"^
'
kg
,
a
=
2.8
9
X
10"'
"
m
,
an
d
a
t
standar
d
temperatur
e
n
=
3.3
5
X
10^*
*
molecules/m
\
Th
e
molecula
r
time-scal
e
ca
n
thu
s
b
e
computed
,
T
=
[ma^/e]"
^
=
8.3
1
X
10"'
^
s
.
Fo
r
th
e
thir
d
cas
e
depicte
d
i
n
Fig
.
1
1
(th
e
ope
n
squares)
,
y^r
=
0.1
,
an
d
th
e
critica
l
shea
r
rat
e
a
t
whic
h
th
e
sli
p
conditio
n
diverge
s
i
s
thu
s
t
c
=
1.
2
X
10
"
s""'
.
Suc
h
a
n
enormou
s
rat
e
o
f
.strai
n
ma
y
b
e
foun
d
i
n
extremel
y
smal
l
device
s
havin
g
extremel
y
hig
h
speeds
.
(Not
e
howeve
r
tha
t
y,
.
fo
r
high-molecular-weigh
t
polymer
s
woul
d
b
e
man
y
order
s
o
f
magnitud
e
smalle
r
tha
n
th
e
valu
e
develope
d
her
e
fo
r
water.
)
O
n
th
e
othe
r
hand
,
th
e
condition
s
t
o
achiev
e
a
measurabl
e
sli
p
o
f
17o
-
(th
e
soli
d
circle
s
i
n
Fig
.
10
)
ar
e
not
difficul
t
t
o
encounte
r
i
n
microdevices
:
densit
y
o
f
soli
d
5
4
3
2
-
1
0
-
-
-
-
~
-
1
1
1
/.7
c

1.
9
*
4.
5
a
8.
2

16.
8
w
-
,
,
1
1
I
"
Ye
t
0.3
6
0.1
4
0.1
0
0.0
6
- • • - » »
Q
1
'
1
1
1
1
1
1
- * - « T i a —^"
^
1
1
1
1
1
1
1
\
1
f
t
+
^
/
1
1
1
-
-
-
-
-
~
1
0.0
1
0.
1
Y/Y
e
1.
0
Fig
.
1
1
Universa
l
relatio
n
o
f
sli
p
lengt
h
a
s
a
functio
n
o
f
shea
r
rate
.
Fro
m
Thompso
n
an
d
Troia
n
(1997)
.
fou
r
time
s
tha
t
o
f
liquid
,
an
d
energy-scal
e
fo
r
wall-flui
d
interac
-
tio
n
tha
t
i
s
on
e
fifth
o
f
energy-scal
e
fo
r
liquid
.
Th
e
limitin
g
valu
e
o
f
sli
p
lengt
h
i
s
independen
t
o
f
th
e
shea
r
rat
e
an
d
ca
n
b
e
compute
d
fo
r
wate
r
a
s

=
11
a
=
4.9
1
X
10"
^
m
.
Conside
r
a
wate
r
microbearin
g
havin
g
a
shaf
t
diainete
r
o
f
10
0
fxm
an
d
rotatio
n
rat
e
o
f
20,00
0
rp
m
an
d
a
minimu
m
ga
p
of
h
=
I
fim.
I
n
thi
s
case
,
U
=
O.l
m/
s
an
d
th
e
no-sli
p
shea
r
rat
e
i
s
U/h
=
10
'
s"'
.
Whe
n
sli
p
occur
s
a
t
th
e
limitin
g
valu
e
jus
t
computed
,
th
e
shea
r
rat
e
an
d
th
e
wal
l
slip-velocit
y
ar
e
compute
d
a
s
follow
s
U
=
9.9
0
X
lO'^
s
h
+
2L1
AM|,

=
t
4
=
4.8
7
X
10"
"
m/
s
(66
)
(67
)
A
s
a
resul
t
o
f
th
e
Navie
r
slip
,
th
e
shea
r
rat
e
i
s
reduce
d
b
y
1
%
fro
m
it
s
no-sli
p
value
,
an
d
th
e
sli
p
velocit
y
a
t
th
e
wal
l
i
s
abou
t
0.5
%
o
f
U,
smal
l
bu
t
no
t
insignificant
.
2.
8
Surfac
e
Phenomena
.
A
s
mentione
d
i
n
Sectio
n
2.1
,
th
e
surface-to-volum
e
rati
o
fo
r
a
machin
e
wit
h
a
characteristi
c
lengt
h
o
f
1
m
i
s
1
m"'
,
whil
e
tha
t
fo
r
a
MEM
S
devic
e
havin
g
a
siz
e
o
f
1
yu
m
i
s
lO*
"
m
'
.
Th
e
million-fol
d
increas
e
i
n
surfac
e
are
a
relativ
e
t
o
th
e
mas
s
o
f
th
e
minut
e
devic
e
substantiall
y
affect
s
th
e
transpor
t
o
f
mass
,
momentu
m
an
d
energ
y
throug
h
th
e
surface
.
Obviousl
y
surfac
e
effect
s
dominat
e
i
n
smal
l
de
-
vices
.
Th
e
surfac
e
boundar
y
condition
s
i
n
MEM
S
flows
hav
e
alread
y
bee
n
discusse
d
i
n
Section
s
2.
5
an
d
2.7
.
I
n
microdevices
,
i
t
ha
s
bee
n
show
n
tha
t
i
t
i
s
possibl
e
t
o
hav
e
measurabl
e
slip
-
velocit
y
an
d
temperatur
e
jum
p
a
t
a
solid-flui
d
interface
.
I
n
thi
s
subsection
,
w
e
illustrat
e
othe
r
ramification
s
o
f
th
e
larg
e
surface
-
to-volum
e
rati
o
uniqu
e
t
o
MEMS
,
an
d
provid
e
a
molecula
r
viewpoin
t
t
o
surfac
e
forces
.
I
n
microdevices
,
bot
h
radiativ
e
an
d
convectiv
e
hea
t
loss/gai
n
ar
e
enhance
d
b
y
th
e
hug
e
surface-to-volum
e
ratio
.
Conside
r
a
devic
e
havin
g
a
characteristi
c
lengt
h
Lj
.
Us
e
o
f
th
e
lumpe
d
capacitanc
e
metho
d
t
o
comput
e
th
e
rat
e
o
f
convectiv
e
hea
t
trans
-
fer
,
fo
r
example
,
i
s
justifie
d
i
f
th
e
Bio
t
numbe
r
(
=
hLjK,,
wher
e
h
i
s
th
e
convectiv
e
hea
t
transfe
r
coefficien
t
o
f
th
e
fluid
an
d
K
J
i
s
th
e
therma
l
conductivit
y
o
f
th
e
solid
)
i
s
les
s
tha
n
0.1
.
Smal
l
L,
implie
s
smal
l
Bio
t
number
,
an
d
a
nearl
y
unifor
m
temperatur
e
withi
n
th
e
solid
.
Withi
n
thi
s
approximation
,
th
e
rat
e
a
t
whic
h
hea
t
i
s
los
t
t
o
th
e
surroundin
g
fluid
i
s
give
n
b
y
PSLU
dt
-hLl{T,
-
To,)
(68
)
wher
e
p,
an
d
c,
ar
e
respectivel
y
th
e
densit
y
an
d
specifi
c
hea
t
o
f
th
e
solid
,
T,
i
s
it
s
(uniform
)
temperature
,
an
d
Too
i
s
th
e
ambi
-
en
t
fluid
temperature
.
Solutio
n
o
f
th
e
abov
e
equatio
n
i
s
trivial
.
2
0
/
Vol
.
121
,
MARC
H
199
9
Transaction
s
o
f
th
e
ASM
E
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
an
d
th
e
temperatur
e
o
f
a
ho
t
surfac
e
drop
s
exponentiall
y
wit
h
tim
e
fro
m
a
n
initia
l
temperatur
e
T,
,
TAt)
Ti
=
ex
p
wher
e
th
e
tim
e
constan
t
?J
i
s
give
n
b
y
^
--
(69
)
(70
)
Fo
r
smal
l
devices
,
th
e
tim
e
i
t
take
s
th
e
soli
d
t
o
coo
l
dow
n
i
s
proportionall
y
small
.
Clearly
,
th
e
million-fol
d
increas
e
i
n
sur
-
face-to-volum
e
rati
o
implie
s
a
proportiona
l
increas
e
i
n
th
e
rat
e
a
t
whic
h
hea
t
escapes
.
Identica
l
scalin
g
argument
s
ca
n
b
e
mad
e
regardin
g
mas
s
transfer
.
Anothe
r
effec
t
o
f
th
e
diminishe
d
scal
e
i
s
th
e
increase
d
impor
-
tanc
e
o
f
surfac
e
force
s
an
d
th
e
wanin
g
importanc
e
o
f
bod
y
forces
.
Base
d
o
n
biologica
l
studies
.
Wen
t
(1968
)
conclude
s
tha
t
th
e
demarkatio
n
length-scal
e
i
s
aroun
d
1
mm
.
Belo
w
that
,
surfac
e
force
s
dominat
e
ove
r
gravitationa
l
forces
.
A
10-m
m
piec
e
o
f
pape
r
wil
l
fal
l
dow
n
whe
n
gentl
y
place
d
o
n
a
smooth
,
vertica
l
wall
,
whil
e
a
0.1-m
m
piec
e
wil
l
stick
.
Tr
y
it
!
Stiction
i
s
a
majo
r
proble
m
i
n
MEM
S
applications
.
Certai
n
structure
s
suc
h
a
s
long
,
thi
n
polysilico
n
beam
s
an
d
large
,
thi
n
com
b
drive
s
hav
e
a
propensit
y
t
o
stic
k
t
o
thei
r
substrate
s
an
d
thu
s
fai
l
t
o
perfor
m
a
s
designe
d
(Mastrangel
o
an
d
Hsu
,
1992
;
Tan
g
e
t
al.
,
1989)
.
Conventiona
l
dr
y
frictio
n
betwee
n
tw
o
solid
s
i
n
relativ
e
mo
-
tio
n
i
s
proportiona
l
t
o
th
e
norma
l
forc
e
whic
h
i
s
usuall
y
a
com
-
ponen
t
o
f
th
e
movin
g
devic
e
weight
.
Th
e
frictio
n
i
s
independen
t
o
f
th
e
contact-surfac
e
are
a
becaus
e
th
e
va
n
de
r
Waal
s
cohesiv
e
force
s
ar
e
negligibl
e
relativ
e
t
o
th
e
weigh
t
o
f
th
e
macroscopi
c
device
.
I
n
MEM
S
applications
,
th
e
cohesiv
e
intermolecula
r
force
s
betwee
n
tw
o
surface
s
ar
e
significan
t
an
d
th
e
stictio
n
i
s
independen
t
o
f
th
e
devic
e
mas
s
bu
t
i
s
proportiona
l
t
o
it
s
surfac
e
area
.
Th
e
first
micromoto
r
di
d
not
move—despit
e
larg
e
electri
c
curren
t
throug
h
it—unti
l
th
e
contac
t
are
a
betwee
n
th
e
100
-
micro
n
roto
r
an
d
th
e
substrat
e
was
reduce
d
significantl
y
b
y
placin
g
dimple
s
o
n
th
e
rotor'
s
surfac
e
(Fa
n
e
t
al.
,
1988
;
1989
;
Ta
i
andMuller
,
1989)
.
On
e
las
t
exampl
e
o
f
surfac
e
effect
s
tha
t
t
o
m
y
knowledg
e
ha
s
not
bee
n
investigate
d
fo
r
microflow
s
i
s
th
e
adsorbe
d
laye
r
i
n
gaseou
s
wall-bounde
d
flows.
I
t
i
s
wel
l
know
n
(see
,
fo
r
example
,
Brunauer
,
1944
;
Lighthill
,
1963
)
tha
t
whe
n
a
ga
s
flows
i
n
a
duct
,
th
e
ga
s
molecule
s
ar
e
attracte
d
t
o
th
e
soli
d
surfac
e
b
y
th
e
va
n
de
r
Waal
s
an
d
othe
r
force
s
o
f
cohesion
.
Th
e
potentia
l
en
-
erg
y
o
f
th
e
ga
s
molecule
s
drop
s
o
n
reachin
g
th
e
surface
.
Th
e
adsorbe
d
laye
r
partake
s
th
e
therma
l
vibration
s
o
f
th
e
solid
,
an
d
th
e
ga
s
molecule
s
ca
n
onl
y
escap
e
whe
n
thei
r
energ
y
exceed
s
th
e
potentia
l
energ
y
minimum
.
I
n
equilibrium
,
a
t
leas
t
par
t
o
f
th
e
soli
d
woul
d
b
e
covere
d
b
y
a
monomolecula
r
laye
r
o
f
ad
-
sorbe
d
ga
s
molecules
.
Molecula
r
specie
s
wit
h
significan
t
partia
l
pressure—relativ
e
t
o
thei
r
vapo
r
pressure—ma
y
locall
y
for
m
layer
s
tw
o
o
r
mor
e
molecule
s
thick
.
Consider
,
fo
r
example
,
th
e
flow
o
f
a
mixtur
e
o
f
dr
y
ai
r
an
d
wate
r
vapo
r
a
t
STP
.
Th
e
energ
y
o
f
adsorptio
n
o
f
wate
r
i
s
muc
h
large
r
tha
n
tha
t
fo
r
nitroge
n
an
d
oxygen
,
makin
g
i
t
mor
e
difficul
t
fo
r
wate
r
molecule
s
t
o
escap
e
th
e
potentia
l
energ
y
trap
.
I
t
follow
s
tha
t
th
e
lif
e
tim
e
o
f
wate
r
molecule
s
i
n
th
e
adsorbe
d
laye
r
significantly
exceed
s
tha
t
fo
r
th
e
ai
r
molecule
s
(b
y
60,00
0
folds
,
i
n
fact
)
and
,
a
s
a
result
,
th
e
thi
n
surfac
e
laye
r
woul
d
b
e
mostl
y
water
.
Fo
r
example
,
i
f
th
e
proportio
n
o
f
wate
r
vapo
r
i
n
th
e
ambien
t
ai
r
i
s
1:1,00
0
(i.e.
,
ver
y
lo
w
humidit
y
level)
,
th
e
rati
o
o
f
wate
r
t
o
ai
r
i
n
th
e
ad
-
sorlje
d
laye
r
woul
d
b
e
60:1
.
Microscopi
c
roughnes
s
o
f
th
e
soli
d
surfac
e
cause
s
partia
l
condensatio
n
o
f
th
e
wate
r
alon
g
portion
s
havin
g
sufficientl
y
stron
g
concav
e
curvature
.
So
,
surface
s
ex
-
pose
d
t
o
non-dr
y
ai
r
flows
ar
e
mainl
y
liqui
d
wate
r
surfaces
.
I
n
mos
t
applications
,
thi
s
thi
n
adsorbe
d
laye
r
ha
s
littl
e
effec
t
o
n
th
e
flow
dynamics
,
despit
e
th
e
fac
t
tha
t
th
e
densit
y
an
d
viscosit
y
o
f
liqui
d
wate
r
ar
e
fa
r
greate
r
tha
n
thos
e
fo
r
air
.
I
n
MEM
S
applications
,
however
,
th
e
laye
r
thicknes
s
ma
y
no
t
b
e
a
n
insig
-
nifican
t
portio
n
o
f
th
e
characteristi
c
flow
dimensio
n
an
d
th
e
wate
r
laye
r
ma
y
hav
e
a
measurabl
e
effec
t
o
n
th
e
ga
s
flow. A
hybri
d
approac
h
o
f
molecula
r
dynamic
s
an
d
continuu
m
flow
simulation
s
o
r
MD-Mont
e
Carl
o
simulation
s
ma
y
b
e
use
d
t
o
investigat
e
thi
s
issue
.
I
t
shoul
d
b
e
note
d
tha
t
quit
e
recently
,
Majumda
r
an
d
Mezi
c
(1998
;
1999
)
hav
e
studie
d
th
e
stabilit
y
an
d
ruptur
e
int
o
droplet
s
o
f
thi
n
liqui
d
films
o
n
soli
d
surfaces
.
The
y
poin
t
out
tha
t
th
e
fre
e
energ
y
o
f
a
liqui
d
film
consist
s
o
f
a
surfac
e
tensio
n
compo
-
nen
t
a
s
wel
l
a
s
highl
y
nonlinea
r
volumetri
c
intermolecula
r
force
s
resultin
g
fro
m
va
n
de
r
Waals
,
electrostatic
,
hydratio
n
an
d
elasti
c
strai
n
interactions
.
Fo
r
wate
r
films
o
n
hydrophili
c
surface
s
suc
h
a
s
silic
a
an
d
mica
,
Majumda
r
an
d
Mezi
c
(1998
)
estimat
e
th
e
equilibriu
m
film
thicknes
s
t
o
b
e
abou
t
0.
5
n
m
(
2
monolayers
)
fo
r
a
wid
e
rang
e
o
f
ambient-ai
r
relativ
e
humidities
.
Th
e
equilibriu
m
thicknes
s
grow
s
ver
y
sharply
,
however
,
a
s
th
e
relativ
e
humidit
y
approache
s
100%
.
Majumda
r
an
d
Mezic'
s
(1998
;
1999
)
result
s
ope
n
man
y
questions
.
Wha
t
ar
e
th
e
stabilit
y
characteristic
s
o
f
thei
r
wate
r
film
i
n
th
e
presenc
e
o
f
ai
r
flow
abov
e
it
?
Woul
d
thi
s
wate
r
film
affec
t
th
e
accommodatio
n
coefficien
t
fo
r
microduc
t
ai
r
flow?
I
n
a
moder
n
Winchester-typ
e
har
d
disk
,
th
e
driv
e
mechanis
m
ha
s
a
read/writ
e
hea
d
tha
t
floats
5
0
n
m
abov
e
th
e
surfac
e
o
f
th
e
spinnin
g
platter
.
Th
e
hea
d
an
d
platte
r
togethe
r
wit
h
th
e
ai
r
laye
r
i
n
betwee
n
for
m
a
slide
r
bearing
.
Woul
d
th
e
compute
r
performanc
e
b
e
affecte
d
adversel
y
b
y
th
e
hig
h
relativ
e
humidit
y
o
n
a
particula
r
da
y
whe
n
th
e
adsorbe
d
wate
r
film
i
s
n
o
longe
r
'thin'
?
I
f
a
microduc
t
haul
s
liqui
d
water
,
woul
d
th
e
wate
r
film
adsorbe
d
b
y
th
e
soli
d
wall
s
influenc
e
th
e
effectiv
e
viscosit
y
o
f
th
e
wate
r
flow?
Electrostati
c
force
s
ca
n
exten
d
t
o
almos
t
1
micro
n
(th
e
Deby
e
length)
,
an
d
tha
t
lengt
h
i
s
know
n
t
o
b
e
highl
y
pH-dependent
.
Woul
d
th
e
wate
r
flow
b
e
influence
d
b
y
th
e
surfac
e
an
d
liqui
d
chemistry
?
Woul
d
thi
s
explai
n
th
e
contra
-
dictor
y
experimenta
l
result
s
o
f
liqui
d
flows
i
n
microduct
s
dis
-
cusse
d
i
n
Sectio
n
2.7
?
Th
e
fe
w
example
s
abov
e
illustrat
e
th
e
importanc
e
o
f
surfac
e
effect
s
i
n
smal
l
devices
.
Fro
m
th
e
continuu
m
viewpoint
,
force
s
a
t
a
solid-flui
d
interfac
e
ar
e
th
e
limi
t
o
f
pressur
e
an
d
viscou
s
force
s
actin
g
o
n
a
paralle
l
elementar
y
are
a
displace
d
int
o
th
e
fluid,
whe
n
th
e
displacemen
t
distanc
e
i
s
allowe
d
t
o
ten
d
t
o
zero
.
Fro
m
th
e
molecula
r
poin
t
o
f
view
,
al
l
macroscopi
c
surfac
e
force
s
ar
e
ultimatel
y
trace
d
t
o
intermolecula
r
forces
,
whic
h
sub
-
jec
t
i
s
extensivel
y
covere
d
i
n
th
e
boo
k
b
y
Israelachvill
i
(1991
)
an
d
reference
s
therein
.
Her
e
w
e
provid
e
a
ver
y
brie
f
introduc
-
tio
n
t
o
th
e
molecula
r
viewpoint
.
Th
e
fou
r
force
s
i
n
natur
e
ar
e
(1
)
th
e
stron
g
an
d
(2
)
wea
k
force
s
describin
g
th
e
interaction
s
betwee
n
neutrons
,
protons
,
electrons
,
etc.
;
(3
)
th
e
electromag
-
neti
c
force
s
betwee
n
atom
s
an
d
molecules
;
an
d
(4
)
gravitationa
l
force
s
betwee
n
masses
.
Th
e
rang
e
o
f
actio
n
o
f
th
e
first
tw
o
force
s
i
s
aroun
d
10"
'
nm
,
an
d
henc
e
neithe
r
concern
s
u
s
overl
y
i
n
MEM
S
applications
.
Th
e
electromagneti
c
force
s
ar
e
effectiv
e
ove
r
a
muc
h
large
r
thoug
h
stil
l
smal
l
distanc
e
o
n
th
e
orde
r
o
f
th
e
interatomi
c
separation
s
(0.1-0.
2
nm)
.
Effect
s
ove
r
longe
r
range—severa
l
order
s
o
f
magnitud
e
longer—ca
n
an
d
d
o
ris
e
fro
m
th
e
shor
t
rang
e
intermolecula
r
forces
.
Fo
r
example
,
th
e
ris
e
o
f
liqui
d
colum
n
i
n
capillarie
s
an
d
th
e
actio
n
o
f
detergen
t
molecule
s
i
n
removin
g
oil
y
dir
t
fro
m
fabri
c
ar
e
th
e
resul
t
o
f
intermolecula
r
interactions
.
Gravitationa
l
force
s
deca
y
wit
h
th
e
distanc
e
t
o
secon
d
power
,
whil
e
intermolecula
r
force
s
deca
y
muc
h
quicker
,
typicall
y
wit
h
th
e
sevent
h
power
.
Cohesiv
e
force
s
ar
e
therefor
e
negligibl
e
onc
e
th
e
distanc
e
betwee
n
mole
-
cule
s
exceed
s
fe
w
molecula
r
diameters
,
whil
e
massiv
e
bodie
s
lik
e
star
s
an
d
planet
s
ar
e
stil
l
strongl
y
interacting
,
vi
a
gravity
,
ove
r
astronomica
l
distances
.
Electromagneti
c
force
s
ar
e
th
e
sourc
e
o
f
al
l
intermolecula
r
interaction
s
an
d
th
e
cohesiv
e
force
s
holdin
g
atom
s
an
d
mole
-
cule
s
togethe
r
i
n
solid
s
an
d
liquids
.
The
y
ca
n
b
e
classifie
d
int
o
(1
)
purel
y
electrostati
c
arisin
g
fro
m
th
e
Coulom
b
forc
e
betwee
n
Journa
l
o
f
Fluid
s
Engineerin
g
MARC
H
1999
,
Vol
.
12
1
/
2
1
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Potential energy
Force field
Fig. 12 Typical Lennard-Jones 6-12 potential and the intermolecular
force field resulting from It. Only a small portion of the potential function
is shown for clarity.
charges, interactions between charges, permanent dipoles, quad-
rupoles, etc.; (2) polarization forces arising from the dipole
moments induced in atoms and molecules by the electric field
of nearby charges and permanent dipoles; and (3) quantum
mechanical forces that give rise to covalent or chemical bonding
and to repulsive steric or exchange interactions that balance the
attractive forces at very short distances. The Hellman-Feynman
theorem of quantum mechanics states that once the spatial distri-
bution of the electron clouds has been determined by solving the
appropriate Schrodinger equation, intermolecular forces may be
calculated on the basis of classical electrostatics, in effect reduc-
ing all intermolecular forces to Coulombic forces. Note, how-
ever, that intermolecular forces exist even when the molecules
are totally neutral. Solutions of the Schrodinger equation for
general atoms and molecules are not easy of course, and alterna-
tive modeUng are sought to represent intermolecular forces. The
van der Waals attractive forces are usually represented with a
potential that varies as the inverse-sixth power of distance, while
the repulsive forces are represented with either a power or an
exponential potential.
A commonly used potential between two molecules is the
generalized Lennard-Jones (L-J 6-12) pair potential given by
Vy(r) = 4e
-d.
(71)
where Vg is the potential energy between two particles i and
j, r is the distance between the two molecules, e and a are,
respectively, characteristic energy and length-scales, and Cij and
dij are parameters to be chosen for the particular fluid and soUd
combinations under consideration. The first term in the right-
hand side is the strong repulsive force that is felt when two
molecules are at extremely close range comparable to the molec-
ular length-scale. That short-range repulsion prevents overlap
of the molecules in physical space. The second term is the
weaker, van der Waals attractive force that commences when
the molecules are sufficiently close (several times a). That
negative part of the potential represents the attractive polariza-
tion interaction of neutral, spherically symmetric particles. The
power of 6 associated with this term is derivable from quantum
mechanics considerations, while the power of the repulsive part
of the potential is found empirically. The Lennard-Jones poten-
tial is zero at very large distances, has a weak negative peak
at r sUghtly larger than a, is zero at r = cr, and is infinite as
The force field resulting from this potential is given by
P , , dVij 48e
Fijir) = - ^ =
or a
(72)
for c = <5? = 1. The minimum potential V^^^ = - e, corresponds to
the equilibrium position (zero force) and occurs at r = \.\1a.
The attractive van der Waals contribution to the minimum po-
tential is - 2e, while the repulsive energy contribution is -he.
Thus the inverse 12th power repulsive force term decreases the
strength of the binding energy at equilibrium by 50%.
The L-J potential is commonly used in molecular dynamics
simulations to model intermolecular interactions between dense
gas or liquid molecules and between fluid and solid molecules.
As mentioned in Section 2.7, such potential is not accurate
for complex substances such as water whose molecules form
directional covalent bonds. As a result, MD simulations for
water are much more involved.
3 Typical Fluid Applications
3.1 Prologue. The physics of fluid flows in microdevices
was covered in Section 2. In this section, we provide a number
of examples of useful applications of MEMS devices in fluid
mechanics. The list is by no means exhaustive, but includes the
use of MEMS-based sensors and actuators for flow diagnosis
and control, a recently developed viscous micropump/microtur -
bine, and analysis of a journal microbearing. The paper by
LOfdahl and Gad-el-Hak (1999) offers more detail on some of
the topics covered in this section.
3.2 Turbulence Measurements. Microelectromechani -
cal systems offer great opportunities for better flow diagnosis
and control, particularly for turbulent flows. The batch pro-
cessing fabrication of microdevices makes it possible to produce
large number of identical transducers within extremely tight
tolerance. Microsensors and microactuators are small, inexpen-
sive, combine electronic and mechanical parts, have low energy
consumption and can be distributed over a wide area. In this
subsection we discuss the advantages of using MEMS-based
sensors for turbulence measurements, and in the following sub-
section the issue of flow control will be addressed.
Turbulence remains largely an enigma, analytically unap-
proachable yet practically very important. For a turbulent flow,
the dependent variables are random functions of space and time,
and no straightforward method exists for analytically obtaining
stochastic solutions to the governing nonlinear, partial differen-
tial equations. The statistical approach to solving the Navier-
Stokes equations sets a more modest aim of solving for the
average flow quantities rather than the instantaneous ones. But
as a result of the nonlinearity of the governing equations, this
approach always leads to more unknowns than equations (the
closure problem), and solutions based on first principles are
again not possible. Turbulence, therefore, is a conundrum that
appears to yield its secrets only to physical and numerical exper-
iments, provided that the wide band of relevant scales is fully
resolved—a far-from-trivial task particularly at high Reynolds
numbers.
A turbulent flow field is composed of a hierarchy of eddies
having a broad range of time- and length-scales. The largest
eddies have a spatial extension of approximately the same size
as the width of the flow field, while the smallest eddies are of
the size where viscous effects become dominant and energy is
transferred from kinetic into internal. The ratio of the smallest
length-scale—the Kolmogorov microscale r\—to the largest
scale / is related to the turbulence Reynolds number as follows
»?
Re"
(73)
A typical L-J 6-12 potential and force field are shown in Fig. 12,
Similar expressions can be written for time- and velocity-scales
(see, for example, Tennekes and Lumley, 1972). Not only does
a sensor have to be sufficiently small to resolve the smallest
eddies, but multi-sensors distributed over a large volume are
22 / Vol. 121, MARCH 1999
Transactions of the ASME
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needed to detect any flow structures at the largest scale. Clearly,
the problem worsens as the Reynolds number increases.
In wall-bounded flows, the shear-layer thickness provides a
measure of the largest eddies in the flow. The smallest scale is
the viscous wall unit. Viscous forces dominate over inertia in
the near-wall region. The characteristic scales there are obtained
from the magnitude of the mean vorticity in the region and its
viscous diffusion away from the wall. Thus, the viscous time-
scale, t^, is given by the inverse of the mean wall vorticity
t„
dU
dy
(74)
where U is the mean streamwise velocity. The viscous length-
scale, /„, is determined by the characteristic distance by which
the (spanwise) vorticity is diffused from the wall, and is thus
given by
l„
vt^
(75)
where v is the kinematic viscosity. The wall velocity-scale (so-
called friction velocity, u-!) follows directly from the time and
length-scales
Ur = —
dy
(76)
where T„ is the mean shear stress at the wall, and p is the fluid
density. A wall unit implies scaling with the viscous scales, and
the usual ( ) * notation is used; for example, y'^ = y/l„ =
yUr/v- In the wall region, the characteristic length for the large
eddies is y itself, while the Kolmogorov scale is related to the
distance from the wall y as follows
V
(77)
where K is the von Karman constant (i«0.41). As j;^ changes
in the range of 1-5 (the extent of the viscous sublayer), r]
changes from 0.8 to 1.2 wall units.
It is clear from the above that the spatial and temporal resolu-
tions for any probe to be used to resolve high-Reynolds-number
turbulent flows are extremely tight. For example, both the Kol-
mogorov scale and the viscous length-scale change from few
microns at the typical field Reynolds number—based on the
momentum thickness—of 10^, to a couple of hundred microns
at the typical laboratory Reynolds number of 10^ MEMS sen-
sors for pressure, velocity, temperature and shear stress are at
least one order of magnitude smaller than conventional sensors
(Ho and Tai, 1996; Lofdahl et al, 1996). Their small size
improves both the spatial and temporal resolutions of the mea-
surements, typically few microns and few microseconds, respec-
tively. For example, a micro-hot-wire (called hot point) has
very small thermal inertia and the diaphragm of a micro-pres-
sure-transducer has correspondingly fast dynamic response.
Moreover, the microsensors' extreme miniaturization and low
energy consumption make them ideal for monitoring the flow
state without appreciably affecting it. Lastly, literally hundreds
of microsensors can be fabricated on the same silicon chip
at a reasonable cost, making them well suited for distributed
measurements. The UCLA/Caltech team (see, for example, Ho
and Tai, 1996; 1998, and references therein) has been very
effective in developing many MEMS-based sensors and actua-
tors for turbulence diagnosis and control.
3.3 Flow Control. Due to their small size, fast response,
low unit-cost and energy consumption and an ability to combine
mechanical and electronic components, MEMS-based sensors
Fig. 13 Classification of flow control strategies
and actuators are presently the best candidate for reactive con-
trol of turbulent flows where distributed arrays of sensing and
actuation elements are required. In this subsection we offer a
brief introduction to targeted flow control. More detail are found
in the review papers by Gad-el-Hak (1989; 1994; 1996), Wil-
kinson (1990), and Moin and Bewley (1994).
The ability to actively or passively manipulate a flow field
to effect a desired change is of immense technological impor-
tance, and this undoubtedly accounts for the fact that the subject
is more hotly pursued by scientists and engineers than any other
topic in fluid mechanics. The potential benefits of realizing
efficient flow control systems range from saving billions of
dollars in annual fuel cost for land, air and sea vehicles to
achieving economically/environmentally more competitive in-
dustrial processes involving fluid flows. Flow control can be
used to achieve transition delay/advance, separation postpone-
ment/provocation, lift enhancement, drag reduction, turbulence
augmentation/suppression or noise reduction.
33.1 Classification Schemes. There are different classi-
fication schemes for flow control methods. One is to consider
whether the technique is applied at the wall or away from it.
Surface parameters that can influence the flow include
roughness, shape, curvature, rigid-wall motion, compliance,
temperature and porosity. Heating and cooling of the surface
can influence the flow via the resulting viscosity and density
gradients. Mass transfer can take place through a porous wall
or a wall with slots. Suction and injection of primary fluid can
have significant effects on the flow field, influencing particularly
the shape of the velocity profile near the wall and thus the
boundary layer susceptibility to transition and separation. Dif-
ferent additives, such as polymers, surfactants, micro-bubbles,
droplets, particles, dust or fibers, can also be injected through
the surface in water or air wall-bounded flows. Control devices
located away from the surface can also be beneficial. Large-
eddy breakup devices (also called outer-layer devices, or
OLDs), acoustic waves bombarding a shear layer from outside,
additives introduced in the middle of a shear layer, manipulation
of freestream turbulence levels and spectra, gust, and magneto-
and electro-hydrodynamic body forces are examples of flow
control strategies applied away from the wall.
A second scheme for classifying flow control methods con-
siders energy expenditure and the control loop involved. As
shown in the schematic in Fig. 13, a control device can be
Journal of Fluids Engineering
MARCH 1999, Vol. 121 / 23
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Controlled
variable
(a)
Measured
variable
T
Power
A"""""" ""I A Controlled
variable
Feedforward
signal
(b)
T
Power
Measured/controlled
variable
Feedback
signal
(c)
Vtt
(jSensov)
Fig. 14 Different control loops for active flow control, ( a) Predeter -
mined, open-loop control; ( b) reactive, feedforward, open-loop control;
(c) reactive, feedbaci<, closed-loop controi.
passive, requiring no auxiliary power, or active, requiring en-
ergy expenditure. As for the action of passive devices, some
prefer to use the term flow management rather than flow control
(Fiedler and Fernholz, 1990), reserving the latter terminology
for dynamic processes. Active control—always requiring actua-
tors—is further divided into predetermined or reactive. Prede-
termined control includes the application of steady or unsteady
energy input without regard to the particular state of the flow.
The control loop in this case is open as shown in Figure 14(a),
and no sensors are required. Reactive control is a special class
of active control where the control input is continuously ad-
justed based on measurements of some kind. The control loop
in this case can either be an open, feedforward one (Fig. 14(b))
or a closed, feedback loop (Fig. 14(c)). Classical control theory
deals, for the most part, with reactive control.
The distinction between feedforward and feedback is particu-
larly important when dealing with the control of flow structures
which convect over stationary sensors and actuators. In feedfor-
ward control, the measured variable and the controlled variable
differ. For example, the pressure or velocity can be sensed at
an upstream location, and the resulting signal is used together
with an appropriate control law to trigger an actuator which in
turn influences the velocity at a downstream position. Feedback
control, on the other hand, necessitates that the controlled vari-
able be measured, fed back and compared with a reference
input. Reactive feedback control is further classified into four
categories: Adaptive, physical model-based, dynamical sys-
tems-based and optimal control (Moin and Bewley, 1994).
A yet another classification scheme is to consider whether
the control technique directly modifies the shape of the instanta-
neous/mean velocity profile or selectively influence the small
dissipative eddies. An inspection of the Navier-Stokes equations
written at the surface indicates that the spanwise and streamwise
vorticity fluxes at the wall can be changed, either instantane-
ously or in the mean, via wall motion/compliance, suction/
injection, streamwise or spanwise pressure-gradient (respec-
tively), or normal viscosity-gradient. (Note that streamwise
vorticity exists only if the velocity field is three-dimensional.
instantaneously or in the mean.) These vorticity fluxes deter-
mine the fullness of the corresponding velocity profiles. For
example, suction (or downward wall motion), favorable pres-
sure-gradient or lower wall-viscosity results in vorticity flux
away from the wall, making the surface a source of spanwise
and streamwise vorticity. The corresponding fuller velocity pro-
files have negative curvature at the wall and are more resistant
to transition and to separation but are associated with higher
skin-friction drag. Conversely, an inflectional velocity profile
can be produced by injection (or upward wall motion), adverse
pressure-gradient or higher wall-viscosity. Such profile is more
susceptible to transition and to separation and is associated with
lower, even negative, skin friction. Note that many techniques
are available to effect a wall viscosity-gradient; for example
surface heating/cooling, film boiling, cavitation, sublimation,
chemical reaction, wall injection of lower/higher viscosity fluid
and the presence of shear thinning/thickening additive.
Flow control devices can alternatively target certain scales
of motion rather than globally changing the velocity profile.
Polymers, riblets and LEBUs, for example, appear to selectively
damp only the small dissipative eddies in turbulent wall-
bounded flows. These eddies are responsible for the (instanta-
neous) inflectional profile and the secondary instability in the
buffer zone, and their suppression leads to increased scales, a
delay in the reduction of the (mean) velocity-profile slope and
consequent thickening of the wall region. In the buffer zone,
the scales of the dissipative and energy containing eddies are
roughly the same and, hence, the energy containing eddies will
also be suppressed resulting in reduced Reynolds stress produc-
tion, momentum transport and skin friction.
Considering the extreme complexity of the turbulence prob-
lem in general and the unattainability of first-principles analyti-
cal solutions in particular, it is not surprising that controlling a
turbulent flow remains a challenging task, mired in empiricism
and unfulfilled promises and aspirations. Brute force suppres-
sion, or taming, of turbulence via active, energy-consuming
control strategies is always possible, but the penalty for doing
so often exceeds any potential benefits. The artifice is to achieve
a desired effect with minimum energy expenditure. This is
where the concept of reactive control and the use of microsen-
sors/microactuators come into the domain of this paper.
3.3.2 Control of Turbulence. Numerous methods of flow
control have already been successfully implemented in practical
engineering devices. Yet, very few of tlie classical strategies
are effective in controlling free-shear or wall-bounded turbulent
flows. Serious limitations exist for some familiar control tech-
niques when applied to certain turbulent flow situations. For
example, in attempting to reduce the skin-friction drag of a
body having a turbulent boundary layer using global suction,
the penalty associated with the control device often exceeds the
saving derived from its use. What is needed is a way to reduce
this penalty to achieve a more efficient control.
Flow control is most effective when applied near the transi-
tion or separation points; in other words, near the critical flow
regimes where flow instabilities magnify quickly. Therefore,
delaying/advancing laminar-to-turbulence transition and pre-
venting/provoking separation are relatively easier tasks to ac-
complish. To reduce the skin-friction drag in a non-separating
turbulent boundary layer, where the mean flow is quite stable,
is a more challenging problem. Yet, even a modest reduction
in the fluid resistance to the motion of, for example, the world-
wide commercial air fleet is translated into fuel savings esti-
mated to be in the billions of dollars. Newer ideas for turbulent
flow control focus on the direct onslaught on coherent struc-
tures. Spurred by the recent developments in chaos control,
microfabrication and soft computing tools, reactive control of
turbulent flows is now in the realm of the possible for future
practical devices.
24 / Vol. 121, MARCH 1999 Transactions of the ASME
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Suc
h
futuristi
c
system
s
ar
e
envisage
d
a
s
consistin
g
o
f
a
larg
e
numbe
r
o
f
intelligent
,
interactive
,
microfabricate
d
wal
l
sensor
s
an
d
actuator
s
arrange
d
i
n
a
checkerboar
d
patter
n
an
d
targete
d
toward
s
specifi
c
organize
d
structure
s
tha
t
occu
r
randoml
y
withi
n
th
e
boundar
y
layer
.
Sensor
s
detec
t
oncomin
g
coheren
t
structures
,
an
d
adaptiv
e
controller
s
proces
s
th
e
sensor
s
informa
-
tio
n
an
d
provid
e
contro
l
signal
s
t
o
th
e
actuator
s
whic
h
i
n
tur
n
attemp
t
t
o
favorabl
y
modulat
e
th
e
quasi-periodi
c
events
.
Finit
e
numbe
r
o
f
wal
l
sensor
s
perceiv
e
onl
y
partia
l
informatio
n
abou
t
th
e
entir
e
flow field
above
.
However
,
a
low-dimensiona
l
dynam
-
ica
l
mode
l
o
f
th
e
near-wal
l
regio
n
use
d
i
n
a
Kalma
n
filter
ca
n
mak
e
th
e
mos
t
o
f
th
e
partia
l
informatio
n
fro
m
th
e
sensors
.
Con
-
ceptuall
y
al
l
o
f
tha
t
i
s
not
to
o
difficult
,
but
i
n
practic
e
th
e
complexit
y
o
f
suc
h
a
contro
l
syste
m
i
s
dauntin
g
an
d
muc
h
researc
h
an
d
developmen
t
wor
k
stil
l
remain
.
Targete
d
contro
l
implie
s
sensin
g
an
d
reactin
g
t
o
a
particula
r
quasi-periodi
c
structur
e
i
n
th
e
boundar
y
layer
.
Th
e
wal
l
seem
s
t
o
b
e
th
e
logica
l
plac
e
fo
r
suc
h
reactiv
e
control
,
becaus
e
o
f
th
e
relativ
e
eas
e
o
f
placin
g
somethin
g
i
n
there
,
th
e
sensitivit
y
o
f
th
e
flow
i
n
genera
l
t
o
surfac
e
perturbation
s
an
d
th
e
proximit
y
an
d
therefor
e
accessibilit
y
t
o
th
e
dynamicall
y
al
l
importan
t
near
-
wal
l
coheren
t
events
.
3.3.3
Targeted
Control.
A
s
discusse
d
above
,
successfu
l
technique
s
t
o
reduc
e
th
e
ski
n
frictio
n
i
n
a
turbulen
t
flow,
suc
h
a
s
polymers
,
particle
s
o
r
riblets
,
appea
r
t
o
ac
t
indirectl
y
throug
h
loca
l
interactio
n
wit
h
discret
e
turbulen
t
structures
,
particularl
y
small-scal
e
eddies
,
withi
n
th
e
flow.
Commo
n
characteristic
s
o
f
al
l
thes
e
method
s
ar
e
increase
d
losse
s
i
n
th
e
near-wal
l
region
,
thickenin
g
o
f
th
e
buffe
r
layer
,
an
d
lowere
d
productio
n
o
f
Reyn
-
old
s
shea
r
stres
s
(Bandyopadhyay
,
1986)
.
Method
s
tha
t
ac
t
directl
y
o
n
th
e
mea
n
flow,
suc
h
a
s
suctio
n
o
r
lowerin
g
o
f
near
-
wal
l
viscosity
,
als
o
lea
d
t
o
inhibitio
n
o
f
Reynold
s
stress
.
How
-
ever
,
ski
n
frictio
n
i
s
increase
d
whe
n
an
y
o
f
thes
e
velocity
-
profil
e
modifier
s
i
s
applie
d
globally
.
Coul
d
thes
e
seemingl
y
inefficien
t
techniques
,
e.g.
,
globa
l
suc
-
tion
,
b
e
use
d
mor
e
sparingl
y
an
d
b
e
optimize
d
t
o
reduc
e
thei
r
associate
d
penalty
?
I
t
appear
s
tha
t
th
e
mor
e
successfu
l
drag
-
reductio
n
methods
,
e.g.
,
polymers
,
ac
t
selectivel
y
o
n
particula
r
scale
s
o
f
motio
n
an
d
ar
e
though
t
t
o
b
e
associate
d
wit
h
stabiliza
-
tio
n
o
f
th
e
secondar
y
instabilities
.
I
t
i
s
als
o
clea
r
tha
t
energ
y
i
s
waste
d
whe
n
suctio
n
o
r
heating/coolin
g
i
s
use
d
t
o
suppres
s
th
e
turbulenc
e
throughou
t
th
e
boundar
y
laye
r
whe
n
th
e
mai
n
inter
-
es
t
i
s
t
o
affec
t
a
near-wal
l
phenomenon
.
On
e
ponders
,
wha
t
woul
d
becom
e
o
f
wal
l
turbulenc
e
i
f
specifi
c
coheren
t
structure
s
ar
e
t
o
b
e
targeted
,
b
y
th
e
operato
r
throug
h
a
reactiv
e
contro
l
scheme
,
fo
r
modification
?
Th
e
myria
d
o
f
organize
d
structure
s
presen
t
i
n
al
l
shea
r
flows
ar
e
instantaneousl
y
identifiable
,
quasi
-
periodi
c
motion
s
(Cantwell
,
1981
;
Robinson
,
1991)
.
Burstin
g
event
s
i
n
wall-bounde
d
flows,
fo
r
example
,
ar
e
bot
h
intermitten
t
an
d
rando
m
i
n
spac
e
a
s
wel
l
a
s
time
.
Th
e
rando
m
aspect
s
o
f
thes
e
event
s
reduc
e
th
e
effectivenes
s
o
f
a
predetermine
d
activ
e
contro
l
strategy
.
I
f
suc
h
structure
s
ar
e
nonintrusivel
y
detecte
d
an
d
altered
,
o
n
th
e
othe
r
hand
,
ne
t
performanc
e
gai
n
migh
t
b
e
achieved
.
I
t
seem
s
clear
,
however
,
tha
t
tempora
l
phasin
g
a
s
wel
l
a
s
spatia
l
selectivit
y
woul
d
b
e
require
d
t
o
achiev
e
prope
r
contro
l
targete
d
toward
s
rando
m
events
.
A
nonreactiv
e
versio
n
o
f
th
e
abov
e
ide
a
i
s
th
e
.^elective
suc-
tion
technique
whic
h
combine
s
suctio
n
t
o
achiev
e
a
n
asymptoti
c
turbulen
t
boundar
y
laye
r
an
d
longitudina
l
riblet
s
t
o
fix
th
e
loca
-
tio
n
o
f
low-spee
d
streak
s
(Gad-el-Ha
k
an
d
Blackwelder
,
1989)
.
Althoug
h
fa
r
fro
m
indicatin
g
ne
t
dra
g
reduction
,
th
e
availabl
e
result
s
ar
e
encouragin
g
an
d
furthe
r
optimizatio
n
i
s
needed
.
Whe
n
implemente
d
vi
a
a
n
arra
y
o
f
reactiv
e
contro
l
loops
,
th
e
selectiv
e
suctio
n
metho
d
i
s
potentiall
y
capabl
e
o
f
skin-frictio
n
reductio
n
tha
t
approache
s
60%
.
3.3.4
Required
Characteristics.
Th
e
randomnes
s
o
f
th
e
burstin
g
event
s
necessitate
s
tempora
l
phasin
g
a
s
wel
l
a
s
spatia
l
selectivit
y
t
o
effec
t
selectiv
e
(targeted
)
control
.
Practica
l
appli
-
cation
s
o
f
method
s
targete
d
a
t
controllin
g
a
particula
r
turbulen
t
structur
e
t
o
achiev
e
a
prescibe
d
goa
l
woul
d
therefor
e
requir
e
implementin
g
a
larg
e
numbe
r
o
f
surfac
e
sensors/actuator
s
to
-
gethe
r
wit
h
appropriat
e
contro
l
algorithms
.
Tha
t
strateg
y
fo
r
controllin
g
wall-bounde
d
turbulen
t
flows
ha
s
bee
n
advocate
d
by
,
amon
g
other
s
an
d
i
n
chronologica
l
order
,
Gad-el-Ha
k
an
d
Blackwelde
r
(1987
;
1989)
,
Blackwelde
r
an
d
Gad-el-Ha
k
(1990)
,
Lumle
y
(1991
;
1996)
,
Cho
i
etal
.
(1992
;
1994)
,
Reyn
-
old
s
(1993)
,
Jacobso
n
an
d
Reynold
s
(1993
;
1995
;
1998)
,
Moi
n
an
d
Bewle
y
(1994)
,
Gad-el-Ha
k
(1994
;
1996
;
1998)
,
McMi
-
chae
l
(1996)
,
Mehregan
y
e
t
al
.
(1996)
,
an
d
Lumle
y
an
d
Blos
-
se
y
(1998)
.
Specia
l
mentio
n
shoul
d
als
o
b
e
mad
e
o
f
th
e
UCLA
/
Caltec
h
tea
m
wh
o
ha
s
bee
n
ver
y
effectiv
e
i
n
developin
g
man
y
MEMS-base
d
sensor
s
an
d
actuator
s
fo
r
turbulenc
e
diagnosi
s
an
d
control
.
Thei
r
lis
t
o
f
publication
s
i
n
th
e
field
i
s
rathe
r
long
,
bu
t
see
,
fo
r
example
.
H
o
an
d
Ta
i
(1996)
,
Tsa
o
e
t
al
.
(1997)
,
H
o
e
t
al
.
(1997)
,
H
o
an
d
Ta
i
(1998)
,
an
d
reference
s
therein
.
I
t
i
s
instructiv
e
t
o
estimat
e
som
e
representativ
e
characteristic
s
o
f
th
e
require
d
arra
y
o
f
sensors/actuators
.
Conside
r
a
typica
l
commercia
l
aircraf
t
cruisin
g
a
t
a
spee
d
o
f
U^
=
30
0
m/
s
an
d
a
t
a
n
altitud
e
o
f
1
0
km
.
Th
e
densit
y
an
d
kinemati
c
viscosit
y
o
f
ai
r
an
d
th
e
uni
t
Reynold
s
numbe
r
i
n
thi
s
cas
e
are
,
respectively
,
p
=
0.
4
kg/m
^
V
=
2,X
10"
'
mVs
,
an
d
R
e
=
lO'/m
.
Assum
e
furthe
r
tha
t
th
e
portio
n
o
f
fuselag
e
t
o
b
e
controlle
d
ha
s
a
turbu
-
len
t
boundar
y
laye
r
characteristic
s
whic
h
ar
e
identica
l
t
o
thos
e
fo
r
a
zero-pressure-gradien
t
flat
plat
e
a
t
a
distanc
e
o
f
1
m
fro
m
th
e
leadin
g
edge
.
I
n
thi
s
case
,
th
e
skin-frictio
n
coefficien
t
an
d
th
e
frictio
n
velocit
y
are
,
respectively
,
C
/
=
0.00
3
an
d
u^
=
11.6
2
m/s
.
(Not
e
tha
t
th
e
ski
n
frictio
n
decrease
s
a
s
th
e
distanc
e
fro
m
th
e
leadin
g
edg
e
increases
.
I
t
i
s
als
o
strongl
y
affecte
d
b
y
suc
h
thing
s
a
s
th
e
externall
y
impose
d
pressur
e
gradient
.
Therefore
,
th
e
estimate
s
provide
d
i
n
her
e
ar
e
fo
r
illustratio
n
purpose
s
only.
)
A
t
thi
s
location
,
on
e
viscou
s
wal
l
uni
t
i
s
onl
y
i//Ur
=
2.
6
microns
.
I
n
orde
r
fo
r
th
e
surfac
e
arra
y
o
f
sensors
/
actuator
s
t
o
b
e
hydraulicall
y
smooth
,
i
t
shoul
d
not
protrud
e
beyon
d
th
e
viscou
s
sublayer
,
or
Sv/Ur
=
1
3
fira.
Wall-spee
d
streak
s
ar
e
th
e
mos
t
visible
,
reliabl
e
an
d
detect
-
abl
e
indicator
s
o
f
th
e
preburs
t
turbulenc
e
productio
n
process
.
Th
e
detectio
n
criterio
n
i
s
simpl
y
lo
w
velocit
y
nea
r
th
e
wall
,
an
d
th
e
actuato
r
respons
e
shoul
d
hs
t
o
accelerat
e
(o
r
t
o
remove
)
th
e
low-spee
d
regio
n
befor
e
i
t
break
s
down
.
Loca
l
wal
l
motion
,
tangentia
l
injection
,
suctio
n
o
r
heatin
g
triggere
d
o
n
sense
d
wall
-
pressur
e
o
r
wall-shea
r
stres
s
coul
d
b
e
use
d
t
o
caus
e
loca
l
accel
-
eratio
n
o
f
near-wal
l
fluid.
Th
e
recen
t
numerica
l
experiment
s
o
f
Berkoo
z
e
t
al
.
(1993
)
indicat
e
tha
t
effectiv
e
contro
l
o
f
burstin
g
pai
r
o
f
roll
s
ma
y
b
e
achieve
d
b
y
usin
g
th
e
equivalen
t
o
f
tw
o
wall-mounte
d
shea
r
sensors
.
I
f
th
e
goa
l
i
s
t
o
stabiliz
e
o
r
t
o
eliminat
e
al
l
low-spee
d
streak
s
i
n
th
e
boundar
y
layer
,
a
reasonabl
e
estimat
e
fo
r
th
e
spanwis
e
an
d
streamwis
e
distance
s
betwee
n
individua
l
element
s
o
f
a
checkerboar
d
arra
y
is
,
respectively
,
10
0
an
d
100
0
wal
l
unit
s
o
r
26
0
//
m
an
d
260
0
//m
,
fo
r
ou
r
particula
r
example
.
(Not
e
tha
t
10
0
an
d
100
0
wal
l
unit
s
ar
e
equa
l
to
,
respectively
,
th
e
averag
e
spanwis
e
wavelengt
h
betwee
n
tw
o
adjacen
t
streak
s
an
d
th
e
averag
e
streamwis
e
exten
t
fo
r
a
typica
l
low-spee
d
region
.
On
e
ca
n
argu
e
tha
t
thos
e
estimate
s
ar
e
to
o
conservative
:
onc
e
a
regio
n
i
s
relaminarized
,
i
t
woul
d
perhap
s
sta
y
a
s
suc
h
fo
r
quit
e
a
whil
e
a
s
th
e
flow
convect
s
downstream
.
Th
e
nex
t
ro
w
o
f
sensors/actuator
s
ma
y
therefor
e
b
e
relegate
d
t
o
a
downstrea
m
locatio
n
wel
l
beyon
d
100
0
wal
l
units.
)
A
reasonabl
e
siz
e
fo
r
eac
h
elemen
t
i
s
probabl
y
one-tent
h
o
f
th
e
spanwis
e
separation
,
o
r
2
6
^m
.
A
(
1
m
X
1
m
)
portio
n
o
f
th
e
surfac
e
woul
d
hav
e
t
o
b
e
covere
d
wit
h
abou
t
n
=
1.
5
millio
n
elements
.
Thi
s
i
s
a
colossa
l
number
,
but
th
e
densit
y
o
f
sensors/actuator
s
ca
n
b
e
considerabl
y
reduce
d
i
f
w
e
moderat
e
ou
r
goa
l
o
f
targetin
g
ever
y
singl
e
burstin
g
even
t
(an
d
als
o
i
f
les
s
conservativ
e
assumption
s
ar
e
used)
.
I
t
i
s
wel
l
know
n
tha
t
no
t
ever
y
low-spee
d
strea
k
lead
s
t
o
a
burst
.
O
n
th
e
average
,
a
particula
r
senso
r
woul
d
detec
t
a
n
incipi
-
en
t
burstin
g
even
t
ever
y
wall-uni
t
interva
l
o
f
P^
=
Pul/u
=
Journa
l
o
f
Fluid
s
Engineerin
g
MARC
H
1999
,
Vol
.
12
1
/
2
5
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
250, or F = 56 /is. The corresponding dimensionless and dimen-
sional frequencies aief* = 0.004 and /= 18 kHz, respectively.
At different distances from the leading edge and in the presence
of nonzero pressure-gradient, the sensors/actuators array would
have different characteristics, but the corresponding numbers
would still be in the same ballpark as estimated in here.
As a second example, consider an underwater vehicle moving
at a speed of U„ = 10 m/s. Despite the relatively low speed,
the unit Reynolds number is still the same as estimated above
for the air case. Re = 10''/m, due to the much lower kinematic
viscosity of water. At one meter from the leading edge of an
imaginary flat plate towed in water at the same speed, the fric-
tion velocity is only Ur = 0.39 m/s, but the wall unit is still the
same as in the aircraft example, u/Ur = 2.6 ^m. The density
of required sensors/actuators array is the same as computed for
the aircraft example, n = 1.5 X 10'' elements/m^. The antici-
pated average frequency of sensing a bursting event is, however,
much lower at / = 600 Hz.
Similar calculations have also been made by Gad-el-Hak
(1993; 1994), Reynolds (1993), and Wadsworth et al. (1993).
Their results agree closely with the estimates made here for
typical field requirements. In either the airplane or the subma-
rine case, the actuator's response need not be too large. As will
be shown in Section 3.3.5, wall displacement on the order of
10 wall units (26 ij,m in both examples), suction coefficient of
about 0.0006, or surface cooling/heating on the order of 40°C/
2°C (in the first/second example, respectively) should be suffi-
cient to stabilize the turbulent flow.
As computed in the two examples above, both the required
size for a sensor/actuator element and the average frequency
at which an element would be activated are within the presently
known capabilities of microfabrication technology. The number
of elements needed per unit area is, however, alarmingly large.
The unit cost of manufacturing a programmable sensor/actuator
element would have to come down dramatically, perhaps match-
ing the unit cost of a conventional transistor, before the idea
advocated in here would become practical.
An additional consideration to the size, amplitude, and fre-
quency response is the energy consumed by each sensor/actua-
tor element. Total energy consumption by the entire control
system obviously has to be low enough to achieve net savings.
Consider the following calculations for the aircraft example.
One meter from the leading edge, the skin-friction drag to be
reduced is approximately 54 N/m^. Engine power needed to
overcome this retarding force per unit area is 16 kW/m^, or
lO'* yuW/sensor. If a 60% drag-reduction is achieved, this energy
consumption is reduced to 4320 ytiW/sensor. This number will
increase by the amount of energy consumption of a sensor/
actuator unit, but hopefully not back to the uncontrolled levels.
The voltage across a sensor is typically in the range of V =
O.I-l V, and its resistance in the range of R = 0.1-1 Mtt.
This means a power consumption by a typical sensor in the
range of f = V^/R = 0.1-10 fj,W, well below the anticipated
power savings due to reduced drag. For a single actuator in the
form of a spring-loaded diaphragm with a spring constant of
k = 100 N/m, oscillating up and down at the bursting frequency
of / = 18 kHz, with an amplitude of y = 26 microns, the power
consumption is f = {l)ky^f = 600 /.tW/actuator. If suction is
used instead, C, = 0.0006, and assuming a pressure difference
of Ap = 10"* N/m^ across the suction holes/slots, the corre-
sponding power consumption for a single actuator is f =
CgU„Ap/n = 1200 /:/W/actuator. It is clear then that when the
power penalty for the sensor/actuator is added to the lower-
level drag, a net saving is still achievable. The corresponding
actuator power penalties for the submarine example are even
smaller (f = 20 /.tW/actuator, for the wall motion actuator, and
f = 40 ytiW/actuator, for the suction actuator), and larger sav-
ings are therefore possible.
3.3.5 Microdevices for Flow Control. MEMS integrates
electronics and mechanical components and can therefore exe-
cute sense-decision-actuation on a monolithic level. Microsen-
sors/microactuators would be ideal for the reactive flow control
concept advocated in the present subsection. Methods of flow
control targeted toward specific coherent structures involve non-
intrusive detection and subsequent modulation of events that
occur randomly in space and time. To achieve proper targeted
control of these quasi-periodic vortical events, temporal phasing
as well as spatial selectivity are required. Practical implementa-
tion of such an idea necessitates the use of a large number of
intelligent, communicative wall sensors and actuators arranged
in a checkerboard pattern. Section 3.3.4 provided estimates for
the number, characteristics and energy consumption of such
elements required to modulate the turbulent boundary layer
which develops along a typical commercial aircraft or nuclear
submarine. An upper-bound number to achieve total turbulence
suppression is about one million sensors/actuators per square
meter of the surface, although as argued earlier the actual num-
ber needed to achieve effective control could perhaps be one
or two orders of magnitude below that.
The sensors would be expected to measure the amplitude,
location, and phase or frequency of the signals impressed upon
the wall by incipient bursting events. Instantaneous wall-pres-
sure or wall-shear stress can be sensed, for example. The normal
or in-plane motion of a minute membrane is proportional to the
respective point force of primary interest. For measuring wall
pressure, microphone-like devices respond to the motion of a
vibrating surface membrane or an internal elastomer. Several
types are available including variable-capacitance (condenser
or electret), ultrasonic, optical (e.g., optical-fiber and diode-
laser), and piezoelectric devices (see, for example, Lofdahl et
al., 1993; 1994). A potentially useful technique for our purposes
has been tried at MIT (Warkentin et al., 1987; Young et al.,
1988; Haritonidis et al., 1990a; 1990b). An array of extremely
small (0.2 mm in diameter) laser-powered microphones
(termed picophones) was machined in silicon using integrated
circuit fabrication techniques, and was used for field measure-
ment of the instantaneous surface pressure in a turbulent bound-
ary layer. The wall-shear stress, though smaller and therefore
more difficult to measure than pressure, provides a more reliable
signature of the near-wall events.
Actuators are expected to produce a desired change in the
targeted coherent structures. The local acceleration action
needed to stabilize an incipient bursting event can be in the form
of adaptive wall, transpiration or waU heat transfer. TraveUng
surface waves can be used to modify a locally convecting pres-
sure gradient such that the wall motion follows that of the
coherent event causing the pressure change. Surface motion in
the form of a Gaussian hill with height y* = 0[1O] should be
sufficient to suppress typical incipient bursts (Lumley, 1991;
Carlson and Lumley, 1996). Such time-dependent alteration in
wall geometry can be generated by driving a flexible skin using
an array of piezoelectric devices (dilate or contract depending
on the polarity of current passing through them), electromag-
netic actuators, magnetoelastic ribbons (made of nonlinear ma-
terials that change their stiffness in the presence of varying
magnetic fields), or Terfenol-d rods (a novel metal composite,
developed at Grumman Corporation, which changes its length
when subjected to a magnetic field). Note should also be made
of other exotic materials that can be used for actuation. For
example, electrorheological fluids (Halsey and Martin, 1993)
instantly solidify when exposed to an electric field, and may
thus be useful for the present application. Recently constructed
microactuators specifically designed for flow control include
those by Wiltse and Glezer (1993), James et al. (1994), Jacob-
son and Reynolds (1993; 1995; 1998), and Vargo and Muntz
(1996).
Suction/injection at many discrete points can be achieved by
simply connecting a large number of minute streamwise slots.
26 / Vol. 121, MARCH 1999
Transactions of the ASME
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
arrange
d
i
n
a
checkerboar
d
pattern
,
t
o
a
low-pressure/high
-
pressur
e
reservoi
r
locate
d
underneat
h
th
e
workin
g
surface
.
Th
e
transpiratio
n
throug
h
eac
h
individua
l
slo
t
i
s
turne
d
o
n
an
d
of
f
usin
g
a
correspondin
g
numbe
r
o
f
independentl
y
controlle
d
mi
-
crovalves
.
Alternatively
,
positive-displacemen
t
o
r
rotar
y
micro
-
pump
s
(see
,
fo
r
example
.
Se
n
e
t
al
,
1996
;
Sharatchandr
a
e
t
al.
,
1997
)
ca
n
b
e
use
d
fo
r
blowin
g
o
r
suckin
g
fluid
throug
h
smal
l
holes/slits
.
Base
d
o
n
th
e
result
s
o
f
Gad-el-Ha
k
an
d
Blackwelde
r
(1989)
,
equivalen
t
suctio
n
coefficient
s
o
f
abou
t
0.000
6
shoul
d
b
e
sufficien
t
t
o
stabiliz
e
th
e
near-wal
l
region
.
Assumin
g
tha
t
th
e
skin-frictio
n
coefficien
t
i
n
th
e
uncontrolle
d
boundar
y
laye
r
i
s
Cf
=
0.003
,
an
d
assumin
g
furthe
r
tha
t
th
e
suctio
n
use
d
i
s
sufficien
t
t
o
establis
h
a
n
asymptoti
c
boundar
y
laye
r
(dSg/dx
=
0
,
wher
e
5g
i
s
th
e
momentu
m
thickness)
,
th
e
ski
n
frictio
n
i
n
th
e
reactivel
y
controlle
d
cas
e
i
s
the
n
Cf
=
Q
+
2C
,
=
0.0012
,
o
r
40
%
o
f
th
e
origina
l
value
.
Th
e
ne
t
benefi
t
would
,
o
f
course
,
b
e
reduce
d
b
y
th
e
energ
y
expenditur
e
o
f
th
e
suctio
n
pum
p
(o
r
micropumps
)
a
s
wel
l
a
s
th
e
arra
y
o
f
microsensor
s
an
d
micro
-
valves
.
Finally
,
i
f
th
e
burstin
g
event
s
ar
e
t
o
b
e
eliminate
d
b
y
lowerin
g
th
e
near-wal
l
viscosity
,
direc
t
electric-resistanc
e
heatin
g
ca
n
b
e
use
d
i
n
liqui
d
flows
an
d
thermoelectri
c
device
s
base
d
o
n
th
e
Peltie
r
effec
t
ca
n
b
e
use
d
fo
r
coolin
g
i
n
th
e
cas
e
o
f
gaseou
s
boundar
y
layers
.
Th
e
absolut
e
viscosit
y
o
f
wate
r
a
t
20°
C
de
-
crease
s
b
y
approximatel
y
2
%
fo
r
eac
h

C
ris
e
i
n
temperature
,
whil
e
fo
r
room-temperatur
e
air
,
/i
t
decrease
s
b
y
approximatel
y
0.2
%
fo
r
eac
h

C
dro
p
i
n
temperature
.
Th
e
streamwis
e
momen
-
tu
m
equatio
n
writte
n
a
t
th
e
wal
l
ca
n
b
e
use
d
t
o
sho
w
tha
t
a
suctio
n
coefficien
t
o
f
0.000
6
ha
s
approximatel
y
th
e
sam
e
effec
t
o
n
th
e
wall-curvatur
e
o
f
th
e
instantaneou
s
velocit
y
profil
e
a
s
a
surfac
e
heatin
g
o
f

C
i
n
wate
r
o
r
a
surfac
e
coolin
g
o
f
40°
C
i
n
ai
r
(Liepman
n
an
d
Nosenchuck
,
1982
;
Liepman
n
e
t
al.
,
1982)
.
Sensor
s
an
d
actuator
s
o
f
th
e
type
s
discusse
d
i
n
thi
s
sectio
n
ca
n
b
e
combine
d
o
n
individua
l
electroni
c
chip
s
usin
g
microfa
-
bricatio
n
technology
.
Th
e
chip
s
ca
n
b
e
interconnecte
d
i
n
a
com
-
municatio
n
networ
k
tha
t
i
s
controlle
d
b
y
a
massivel
y
paralle
l
compute
r
o
r
a
self-learnin
g
neura
l
network
,
perhap
s
eac
h
sen
-
sor/actuato
r
uni
t
communicatin
g
onl
y
wit
h
it
s
immediat
e
neigh
-
bors
.
I
n
othe
r
words
,
i
t
ma
y
no
t
b
e
necessar
y
fo
r
on
e
sensor
/
actuato
r
t
o
exchang
e
signal
s
wit
h
anothe
r
fa
r
awa
y
unit
.
Factor
s
t
o
b
e
considere
d
i
n
a
n
eventua
l
field
applicatio
n
o
f
chip
s
pro
-
duce
d
usin
g
microfabricatio
n
processe
s
includ
e
sensitivit
y
o
f
sensors
,
sufficienc
y
an
d
frequenc
y
respons
e
o
f
actuators
'
action
,
fabricatio
n
o
f
larg
e
array
s
a
t
affordabl
e
prices
,
survivabilit
y
i
n
th
e
hostil
e
field
environment
,
an
d
energ
y
require
d
t
o
powe
r
th
e
sensors/actuators
.
A
s
argue
d
b
y
Gad-el-Ha
k
(1994)
,
sensor
/
actuato
r
chip
s
currentl
y
produce
d
ar
e
smal
l
enoug
h
fo
r
typica
l
field
application
,
an
d
the
y
ca
n
b
e
programme
d
t
o
provid
e
a
sufficientl
y
large/fas
t
actio
n
i
n
respons
e
t
o
a
certai
n
senso
r
outpu
t
(se
e
als
o
Jacobso
n
an
d
Reynolds
,
1995)
.
Presen
t
proto
-
type
s
are
,
however
,
stil
l
quit
e
expensiv
e
a
s
wel
l
a
s
delicate
.
Bu
t
s
o
wa
s
th
e
transisto
r
whe
n
first
introduced
!
I
t
i
s
hope
d
tha
t
th
e
uni
t
pric
e
o
f
futur
e
sensor/actuato
r
element
s
woul
d
follo
w
th
e
sam
e
dramati
c
trend
s
witnesse
d
i
n
cas
e
o
f
th
e
simpl
e
transisto
r
an
d
eve
n
th
e
muc
h
mor
e
comple
x
integrate
d
circuit
.
Th
e
pric
e
anticipate
d
b
y
Texa
s
Instrument
s
fo
r
a
n
arra
y
o
f
0.5-
2
million
,
individuall
y
actuate
d
mirror
s
use
d
i
n
high-definitio
n
optica
l
dis
-
play
s
hint
s
tha
t
th
e
technolog
y
i
s
wel
l
i
n
it
s
wa
y
t
o
mass
-
produc
e
phenomenall
y
inexpensiv
e
microsensor
s
an
d
microac
-
tuators
.
Additionally
,
curren
t
automotiv
e
application
s
ar
e
a
rig
-
orou
s
provin
g
groun
d
fo
r
MEMS
:
under-the-hoo
d
sensor
s
ca
n
alread
y
withstan
d
hars
h
condition
s
suc
h
a
s
intens
e
heat
,
shock
,
continua
l
vibration
,
corrosiv
e
gase
s
an
d
electromagneti
c
fields.
3.
4
Micropumps
.
Ther
e
hav
e
bee
n
severa
l
studie
s
o
f
mi
-
crofabricate
d
pumps
.
Som
e
o
f
the
m
us
e
non-mechanica
l
effects
.
Th
e
Knudse
n
pum
p
mentione
d
i
n
Sectio
n
2.
5
use
s
th
e
thermal
-
cree
p
effec
t
t
o
mov
e
rarefie
d
gase
s
fro
m
on
e
chambe
r
t
o
an
-
other
.
Ion-dra
g
i
s
use
d
i
n
electrohydrodynami
c
pump
s
(Bar
t
e
t
al
,
1990
;
Richte
r
e
t
al.
,
1991
;
Fuh
r
e
t
al.
,
1992)
;
thes
e
rel
y
o
n
y
B
K
2a::
co
i
^
2h
Fig
.
1
5
Schemati
c
o
f
micropum
p
deveiope
d
b
y
Se
n
e
t
ai
.
(1996
)
th
e
electrica
l
propertie
s
o
f
th
e
fluid
an
d
ar
e
thu
s
not
suitabl
e
fo
r
man
y
applications
.
Valveles
s
pumpin
g
b
y
ultrasoun
d
ha
s
als
o
bee
n
propose
d
(Morone
y
e
t
al.
,
1991)
,
bu
t
produce
s
ver
y
littl
e
pressur
e
difference
.
Mechanica
l
pump
s
base
d
o
n
conventiona
l
centrifuga
l
o
r
axia
l
turbomachiner
y
wil
l
no
t
wor
k
a
t
micromachin
e
scale
s
wher
e
th
e
Reynold
s
number
s
ar
e
typicall
y
small
,
o
n
th
e
orde
r
o
f
1
o
r
less
.
Centrifuga
l
force
s
ar
e
negligibl
e
and
,
furthermore
,
th
e
Kutt
a
conditio
n
throug
h
whic
h
lif
t
i
s
normall
y
generate
d
i
s
invali
d
whe
n
inertia
l
force
s
ar
e
vanishingl
y
small
.
I
n
genera
l
ther
e
ar
e
thre
e
way
s
i
n
whic
h
mechanica
l
micropump
s
ca
n
work
:
1
.
Positive-displacemen
t
pumps
.
Thes
e
ar
e
mechanica
l
pump
s
wit
h
a
membran
e
o
r
diaphrag
m
actuate
d
i
n
a
reciprocatin
g
mod
e
an
d
wit
h
unidirectiona
l
inle
t
an
d
out
-
le
t
valves
.
The
y
wor
k
o
n
th
e
sam
e
physica
l
principl
e
a
s
thei
r
large
r
cousins
.
Micropump
s
wit
h
piezoelectri
c
actuator
s
hav
e
bee
n
fabricate
d
(Va
n
Linte
l
e
t
al
,
1988
;
Esash
i
e
t
al
,
1989
;
Smits
,
1990)
.
Othe
r
actuators
,
suc
h
a
s
thermopneumatic
,
electrostatic
,
electromagneti
c
o
r
bi
-
metallic
,
ca
n
b
e
use
d
(Piste
r
e
t
al.
,
1990
;
Dorin
g
e
t
al
,
1992
;
Gabrie
l
e
t
al.
,
1992)
.
Thes
e
exceedingl
y
minut
e
positive-displacemen
t
pump
s
requir
e
eve
n
smalle
r
valves
,
seal
s
an
d
mechanisms
,
a
not-too-trivia
l
microma
-
nufacturin
g
challenge
.
I
n
additio
n
ther
e
ar
e
long-ter
m
problem
s
associate
d
wit
h
wea
r
o
r
cloggin
g
an
d
conse
-
quen
t
leakin
g
aroun
d
valves
.
Th
e
pumpin
g
capacit
y
o
f
thes
e
pump
s
i
s
als
o
limite
d
b
y
th
e
smal
l
displacemen
t
an
d
frequenc
y
involved
.
Gea
r
pump
s
ar
e
a
differen
t
kin
d
o
f
positive-displacemen
t
device
.
2
.
Continuous
,
parallel-axi
s
rotar
y
pumps
.
A
screw-type
,
three-dimensiona
l
devic
e
fo
r
lo
w
Reynold
s
number
s
wa
s
propose
d
b
y
Taylo
r
(1972
)
fo
r
propulsio
n
purpose
s
an
d
show
n
i
n
hi
s
semina
l
film.
I
t
ha
s
a
n
axi
s
o
f
rotatio
n
paralle
l
t
o
th
e
flow
directio
n
implyin
g
tha
t
th
e
powerin
g
moto
r
mus
t
b
e
submerge
d
i
n
th
e
flow,
th
e
flow
turne
d
throug
h
a
n
angle
,
o
r
tha
t
complicate
d
gearin
g
woul
d
b
e
needed
.
3
.
Continuous
,
transverse-axi
s
rotar
y
pumps
.
Thi
s
i
s
th
e
clas
s
o
f
machine
s
tha
t
wa
s
recentl
y
develope
d
b
y
Se
n
e
t
al
.
(1996)
.
The
y
hav
e
show
n
tha
t
a
rotatin
g
body
,
asymmetricall
y
place
d
withi
n
a
duct
,
wil
l
produc
e
a
ne
t
flow
du
e
t
o
viscou
s
action
.
Th
e
axi
s
o
f
rotatio
n
ca
n
b
e
perpendicula
r
t
o
th
e
flow
directio
n
an
d
th
e
cylinde
r
ca
n
thu
s
b
e
easil
y
powere
d
fro
m
outsid
e
a
duct
.
A
relate
d
viscous-flo
w
pum
p
wa
s
designe
d
b
y
Ode
U
an
d
Kovasz
-
na
y
(1971
)
fo
r
a
wate
r
channe
l
wit
h
densit
y
stratification
.
However
,
thei
r
desig
n
operate
s
a
t
a
muc
h
highe
r
Reyn
-
old
s
numbe
r
an
d
i
s
to
o
complicate
d
fo
r
microfabrication
.
A
s
evidence
d
fro
m
th
e
thir
d
ite
m
above
,
i
t
i
s
possibl
e
t
o
generat
e
axia
l
fluid
motio
n
i
n
ope
n
channel
s
throug
h
th
e
rotatio
n
o
f
a
cylinde
r
i
n
a
viscou
s
fluid
medium
.
Odel
l
an
d
Kovaszna
y
(1971
)
studie
d
a
pum
p
base
d
o
n
thi
s
principl
e
a
t
hig
h
Reynold
s
numbers
.
Se
n
e
t
al
.
(1996
)
carrie
d
ou
t
a
n
experimenta
l
stud
y
o
f
a
differen
t
versio
n
o
f
suc
h
a
pump
.
Th
e
nove
l
viscou
s
pump
.
Journa
l
o
f
Fluid
s
Engineerin
g
MARC
H
1999
,
Vol
.
12
1
/
2
7
Downloaded 29 Feb 2008 to 218.199.24.116. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
shown schematically in Fig. 15, consists simply of a transverse-
axis cylindrical rotor eccentrically placed in a channel, so that
the differential viscous resistance between the small and large
gaps causes a net flow along the duct. The Reynolds numbers
involved in Sen et al.'s work were low (0.01 < Re = lujo^lv
< 10, where w is the radian velocity of the rotor, and a is its
radius), typical of microscale devices, but achieved using a
macroscale rotor and a very viscous fluid. The bulk velocities
obtained were as high as 10% of the surface speed of the rotating
cylinder. Sen et al. (1996) have also tried cylinders with square
and rectangular cross-sections, but the circular cylinder deliv-
ered the best pumping performance.
A finite-element solution for low-Reynolds-number, uniform
flow past a rotating cylinder near an impermeable plane bound-
ary has already been obtained by Liang and Liou (1995). How-
ever, a detailed two-dimensional Navier-Stokes simulations of
the pump described above have been carried out by Sharatchan-
dra et al. (1997), who extended the operating range of Re
beyond 100. The effects of varying the channel height H and
the rotor eccentricity e have been studied. It was demonstrated
that an optimum plate spacing exists and that the induced flow
increases monotonically with eccentricity; the maximum flow-
rate being achieved with the rotor in contact with a channel
wall. Both the experimental results of Sen et al. (1996) and the
2-D numerical simulations of Sharatchandra et al. (1997) have
verified that, at Re < 10, the pump characteristics are linear and
therefore kinematically reversible. Sharatchandra et al. (1997;
1998a) also investigated the effects of slip flow on the pump
performance as well as the thermal aspects of the viscous de-
vice. Wall slip does reduce the traction at the rotor surface and
thus lowers the performance of the pump somewhat. However,
the slip effects appear to be significant only for Knudsen num-
bers greater than 0.1, which is encouraging from the point of
view of microscale applications.
In an actual implementation of the micropump, several practi-
cal obstacles need to be considered. Among those are the larger
stiction and seal design associated with rotational motion of
microscale devices. Both the rotor and the channel have a finite,
in fact rather small, width. DeCourtye et al. (1998) numerically
investigated the viscous micropump performance as the width
of the channel W becomes exceedingly small. The bulk flow
generated by the pump decreased as a result of the additional
resistance to the flow caused by the side walls. However, effec-
tive pumping was still observed with extremely narrow chan-
nels. Finally, Shartchandra et al. (1998b) used a genetic algo-
rithm to determine the optimum waif shape to maximize the
micropump performance. Their genetic algorithm uncovered
shapes that were nonintuitive but yielded vastly superior pump
performance.
Though most of the micropump discussion above is of flow
in the steady state, it should be possible to give the eccentric
cylinder a finite number of turns or even a portion of a turn
to displace a prescribed minute volume of fluid. Numerical
computations will easily show the order of magnitude of the
volume discharged and the errors induced by acceleration at
the beginning of the rotation and deceleration at the end. Such
system can be used for microdosage delivery in medical applica-
tions.
3.5 Microturbines. DeCourtye et al. (1998) have de-
scribed the possible utilization of the inverse micropump device
(Section 3.4) as a turbine. The most interesting application of
such a microturbine would be as a microsensor for measuring
exceedingly small flowrates on the order of nanoliter/s (i.e.,
microflow metering for medical and other applications).
The viscous pump described in Section 3.4 operates best at
low Reynolds numbers and should therefore be kinematically
reversible in the creeping-flow regime. A microturbine based
on the same principle should, therefore, lead to a net torque in
the presence of a prescribed bulk velocity. The results of three-
0.07
0.06
0.05
0.04
(0M)(2a)
V 0.03
0.02
0.01
0
0.02
0.06
0.1
U(2a)
W= "j/^
y^.e
0.14
0.
Fig. 16 Turbine rotation as a function of the bulic veiocity in tlie cPiannei.
From DeCourtye et al. (1998).
dimensional numerical simulations of the envisioned microtur-
bine are summarized in this subsection. The Reynolds number
for the turbine problem is defined in terms of the bulk velocity,
since the rotor surface speed is unknown in this case,
Re
U{la)
(78)
where U is the prescribed bulk velocity in the channel, a is the
rotor radius, and v is the kinematic viscosity of the fluid.
Figure 16 shows the dimensionless rotor speed as a function
of the bulk velocity, for two dimensionless channel widths
W = 00 and W = 0.6. In these simulations, the dimensionless
channel depth is H = 2.5 and the rotor eccentricity is e/e,„ax =
0.9. The relation is linear as was the case for the pump problem.
The slope of the lines is 0.37 for the 2-D turbine and 0.33 for
the narrow channel with W = 0.6. This means that the induced
rotor speed is, respectively, 0.37 and 0.33 of the bulk velocity
in the channel. (The rotor speed can never, of course, exceed
the fluid velocity even if there is no load on the turbine. Without
load, the integral of the viscous shear stress over the entire
surface area of the rotor is exactly zero, and the turbine achieves
its highest albeit finite rpm.) For the pump, the corresponding
numbers were 11.11 for the 2-D case and 100 for the 3-D case.
Although it appears that the side walls have bigger influence
on the pump performance, it should be noted that in the turbine
case a vastly higher pressure drop is required in the 3-D duct
to yield the same bulk velocity as that in the 2-D duct (dimen-
sionless pressure drop of Ap* = ApAa^lpv^ = —29 versus
Ap* = - 1.5).
The turbine characteristics are defined by the relation between
the shaft speed and the applied load. A turbine load results in
a moment on the shaft, which at steady state balances the torque
due to viscous stresses. At a fixed bulk velocity, the rotor speed
is determined for different loads on the turbine. Again, the
turbine characteristics are linear in the Stokes (creeping) flow
regime, but the side walls have weaker, though still adverse,
effect on the device performance as compared to the pump case.
For a given bulk velocity, the rotor speed drops linearly as the
external load on the turbine increases. At large enough loads,
the rotor will not spin, and maximum rotation is achieved when
the turbine is subjected to zero load.
At present it is difficult to measure flowrates on the order of
10^'^ mVs (1 nanoliter/s). One possible way is to directly
collect the effluent over time. This is useful for calibration but
is not practical for on-line flow measurement. Another is to use
heat transfer from a wire or film to determine the local flowrate
as in a thermal anemometer. Heat transfer from slowly moving
fluids is mainly by conduction so that temperature gradients can
be large. This is undesirable for biological and other fluids
easily damaged by heat. The viscous mechanism that has been
28 / Vol. 121, MARCH 1999
Transactions of the ASME
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proposed and verified for pumping may be turned around and
used for measuring. As demonstrated in this subsection, a freely
rotating cylinder eccentrically placed in a duct will rotate at a
rate proportional to the flowrate due to a turbine effect. In fact
other geometries such as a freely rotating sphere in a cylindrical
tube should also behave similarly. The calibration constant,
which depends on system parameters such as geometry and
bearing friction, should be determined computationally to ascer-
tain the practical viability of such a microflow meter. Geome-
tries that are simplest to fabricate should be explored and studied
in detail.
3.6 Microbearings. Many of the micromachines use ro-
tating shafts and other moving parts which carry a load and
need fluid bearings for support, most of them operating with
air or water as the lubricating fluid. The fluid mechanics of
these bearings are very different compared to that of their larger
cousins. Their study falls in the area of microfluid mechanics,
an emerging discipline which has been greatly stimulated by
its applications to micromachines and which is the subject of
this paper.
Macroscale journal bearings develop their load-bearing ca-
pacity from large pressure differences which are a consequence
of the presence of a viscous fluid, an eccentricity between the
shaft and its housing, a large surface speed of the shaft, and a
small clearance to diameter ratio. Several closed-form solutions
of the no-slip flow in a macrobearing have been developed.
Wannier (1950) used modified Cartesian coordinates to find
an exact solution to the biharmonic equation governing two-
dimensional journal bearings in the no-slip, creeping flow re-
gime. Kamal (1966) and Ashino and Yoshida (1975) worked
in bipolar coordinates; they assumed a general form for the
streamfunction with several constants which were determined
using the boundary conditions. Though all these methods work
if there is no slip, they cannot be readily adapted to slip flow.
The basic reason is that the flow pattern changes if there is slip
at the walls and the assumed form of the solution is no longer
valid.
Microbearings are different in the following aspects: (1) be-
ing so small, it is difficult to manufacture them with a clearance
that is much smaller than the diameter of the shaft; (2) because
of the small shaft size, its surface speed, at normal rotational
speeds, is also small; and (3) air bearings in particular may be
small enough for non-continuum effects to become important.
(The microturbomachines being developed presently at MIT
operate at shaft rotational speeds on the order of 1 million rpm,
and are therefore operating at different flow regime from that
considered here.) For these reasons the hydrodynamics of lubri-
cation is very different at microscales. The lubrication approxi-
mation that is normally used is no longer directly applicable
and other effects come into play. From an analytical point of
view there are three consequences of the above: fluid inertia is
negligible, slip flow may be important for air and other gases,
and relative shaft clearance need not be small.
In a recent study, Maureau et al. (1997) analyzed microbear-
ings represented as an eccentric cylinder rotating in a stationary
housing. The flow Reynolds number is assumed small, the clear-
ance between shaft and housing is not small relative to the
overall bearing dimensions, and there is slip at the walls due
to nonequilibrium effects. The two-dimensional governing
equations are written in terms of the streamfunction in bipolar
coordinates. Following the method of Jeffery (1920), Maureau
et al. (1997) succeeded in obtaining an exact infinite-series
solution of the Navier-Stokes equations for the specified geome-
try and flow conditions. In contrast to macrobearings and due
to the large clearance, flow in a microbearing is characterized
by the possibility of a recirculation zone which strongly affects
the velocity and pressure fields. For high values of the eccentric-
ity and low slip factors the flow develop a recirculation region,
as shown in the streamlines plot in Fig. 17.
Fig. 17 Effect of slip factor and eccentricity on tlie microbearing stream-
lines. From top to bottom, eccentricity changes as c = 0.2,0.5, 0.8. From
left to right, slip factor changes as S = (2 - <r^l(r)Kn = 0, 0.1,0.5. From
Maureau etal. (1997).
From the infinite-series solution the frictional torque and the
load-bearing capacity can be determined. The results show that
both are similarly affected by the eccentricity and the slip factor:
they increase with the former and decrease with the latter. For
a given load, there is a corresponding eccentricity which gener-
ates a force sufficient to separate shaft from housing (i.e. suffi-
cient to prevent solid-to-solid contact). As the load changes the
rotational center of the shaft shifts a distance necessary for the
forces to balance. It is interesting to note that for a weight that
is vertically downwards, the equilibrium displacement of the
center of the shaft is in the horizontal direction. This can lead
to complicated rotor dynamics governed by mechanical inertia,
viscous damping and pressure forces. A study of this dynamics
may of interest. Real microbearings have finite shaft lengths,
and end walls and other three-dimensional effects influence
the bearing characteristics. Numerical simulations of the three-
dimensional problem can readily be carried out and may also
be of interest to the designers of microbearings. Other potential
research includes determination of a criterion for onset of cavita-
tion in liquid bearings. From the results of these studies, infor-
mation related to load, rotational speed and geometry can be
generated that would be useful for the designer.
Finally, Piekos et al. (1997) have used full Navier-Stokes
computations to study the stability of ultra-high-speed, gas mi-
crobearings. They conclude that it is possible—despit e signifi-
cant design constraints—to attain stability for specific bearings
to be used with the MIT microturbomachines (Epstein and Sent-
uria, 1997; Epstein et al., 1997), which incidentally operate at
much higher Reynolds numbers (and rpm) than the micro-
pumps/microturbines/microbearing s considered thus far in this
and the previous two subsections. According to Piekos et al.
(1997), high-speed bearings are more robust than low-speed
ones due to their reduced running eccentricities and the large
loads required to maintain them.
4 Concluding Remarks
The forty-year-old vision of Richard Feynman of building
minute machines is now a reality. Microelectromechanical sys-
tems have witnessed explosive growth during the last decade
and are finding increased applications in a variety of industrial
and medical fields. The physics of fluid flows in microdevices
Journal of Fluids Engineering
MARCH 1999, Vol. 121 / 29
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an
d
som
e
representativ
e
apphcation
s
hav
e
bee
n
explore
d
i
n
thi
s
paper
.
Whil
e
w
e
no
w
kno
w
a
lo
t
mor
e
tha
n
w
e
di
d
jus
t
fe
w
year
s
ago
,
muc
h
physic
s
remain
s
t
o
b
e
explore
d
s
o
tha
t
rationa
l
tool
s
ca
n
b
e
develope
d
fo
r
th
e
design
,
fabricatio
n
an
d
operatio
n
o
f
MEM
S
devices
.
Th
e
traditiona
l
Navier-Stoke
s
mode
l
o
f
fluid flows
wit
h
no
-
sli
p
boundar
y
condition
s
work
s
onl
y
fo
r
a
certai
n
rang
e
o
f
th
e
governin
g
parameters
.
Thi
s
mode
l
basicall
y
demand
s
tw
o
con
-
ditions
.
(1
)
Th
e
fluid
i
s
a
continuum
,
whic
h
i
s
almos
t
alway
s
satisfie
d
a
s
ther
e
ar
e
usuall
y
mor
e
tha
n
1
millio
n
molecule
s
i
n
th
e
smalles
t
volum
e
i
n
whic
h
appreciabl
e
macroscopi
c
change
s
tak
e
place
.
Thi
s
i
s
th
e
molecula
r
chao
s
restriction
.
(2
)
Th
e
flow
i
s
no
t
to
o
fa
r
fro
m
thermodynami
c
equilibrium
,
whic
h
i
s
satisfie
d
i
f
ther
e
i
s
sufficien
t
numbe
r
o
f
molecula
r
encounter
s
durin
g
a
tim
e
perio
d
smal
l
compare
d
t
o
th
e
smalles
t
time-scal
e
fo
r
flow
changes
.
Durin
g
thi
s
tim
e
perio
d
th
e
averag
e
molecul
e
woul
d
hav
e
move
d
a
distanc
e
smal
l
compare
d
t
o
th
e
smalles
t
flow
length-scale
.
Fo
r
gases
,
th
e
Knudse
n
numbe
r
determine
s
th
e
degre
e
o
f
rarefactio
n
an
d
th
e
applicabilit
y
o
f
traditiona
l
flow
models
.
A
s
K
n
-
*
0
,
th
e
time
-
an
d
length-scale
s
o
f
molecula
r
encounter
s
ar
e
vanishingl
y
smal
l
compare
d
t
o
thos
e
fo
r
th
e
flow,
an
d
th
e
velocit
y
distributio
n
o
f
eac
h
elemen
t
o
f
th
e
fluid
instantaneousl
y
adjust
s
t
o
th
e
equilibriu
m
thermodynami
c
stat
e
appropriat
e
t
o
th
e
loca
l
macroscopi
c
propertie
s
a
s
thi
s
molecul
e
move
s
throug
h
th
e
flow field.
Fro
m
th
e
continuu
m
viewpoint
,
th
e
flow
i
s
isen
-
tropi
c
an
d
hea
t
conductio
n
an
d
viscou
s
diffusio
n
an
d
dissipatio
n
vanis
h
fro
m
th
e
continuu
m
conservatio
n
relations
,
leadin
g
t
o
th
e
Eule
r
equation
s
o
f
motion
.
A
t
smal
l
bu
t
finite
Kn
,
th
e
Na
-
vier-Stoke
s
equation
s
describ
e
near-equilibrium
,
continuu
m
flows.
Sli
p
flow
mus
t
b
e
take
n
int
o
accoun
t
fo
r
Kn
>
0.001
.
Th
e
sli
p
boundar
y
conditio
n
i
s
a
t
first
linea
r
i
n
Knudse
n
numbe
r
the
n
nonlinea
r
effect
s
tak
e
ove
r
beyon
d
a
Knudse
n
numbe
r
o
f
0.1
.
A
t
th
e
sam
e
transitio
n
regime
,
i.e.
,
0.
1
<
K
n
<
10
,
th
e
linea
r
(stress)-(rat
e
o
f
strain
)
an
d
(hea
t
flux)-(temperature
gradient
)
relations—neede
d
t
o
clos
e
th
e
Navier-Stoke
s
equa
-
tions—als
o
brea
k
down
,
an
d
alternativ
e
continuu
m
equation
s
(e.g.
,
Burnett
)
o
r
molecular-base
d
model
s
mus
t
b
e
invoked
.
I
n
th
e
transitio
n
regime
,
provide
d
tha
t
th
e
dilut
e
ga
s
an
d
molecula
r
chao
s
assumption
s
hold
,
solution
s
t
o
th
e
difficul
t
Boltzman
n
equatio
n
ar
e
sought
,
bu
t
physica
l
simulation
s
suc
h
a
s
Mont
e
Carl
o
method
s
ar
e
mor
e
readil
y
execute
d
i
n
thi
s
rang
e
o
f
Knud
-
se
n
number
.
I
n
th
e
free-molecul
e
flow
regime
,
i.e.
,
K
n
>
10
,
th
e
nonlinea
r
collisio
n
integra
l
i
s
negligibl
e
an
d
th
e
Boltzman
n
equatio
n
i
s
drasticall
y
simplified
.
Analytica
l
solution
s
ar
e
possi
-
bl
e
i
n
thi
s
cas
e
fo
r
simpl
e
geometrie
s
an
d
numerica
l
integratio
n
o
f
th
e
Boltzman
n
equatio
n
i
s
straightforwar
d
fo
r
arbitrar
y
geom
-
etries
,
provide
d
tha
t
th
e
surface-reflectio
n
characteristic
s
ar
e
accuratel
y
modeled
.
Gaseou
s
flows
ar
e
ofte
n
compressibl
e
i
n
microdevice
s
eve
n
a
t
lo
w
Mac
h
numbers
.
Viscou
s
effect
s
ca
n
caus
e
sufficien
t
pres
-
sur
e
dro
p
an
d
densit
y
change
s
fo
r
th
e
flow
t
o
b
e
treate
d
a
s
compressible
.
I
n
a
long
,
constant-are
a
microduct
,
al
l
Knudse
n
numbe
r
regime
s
ma
y
b
e
encountere
d
an
d
th
e
degre
e
o
f
rarefac
-
tio
n
increase
s
alon
g
th
e
tube
.
Th
e
pressur
e
dro
p
i
s
nonlinea
r
an
d
th
e
Mac
h
numbe
r
increase
s
downstream
,
limite
d
onl
y
b
y
choked-flo
w
condition
.
Simila
r
deviatio
n
an
d
breakdow
n
o
f
th
e
traditiona
l
Navier
-
Stoke
s
equation
s
occu
r
fo
r
liquid
s
a
s
well
,
bu
t
ther
e
th
e
situatio
n
i
s
mor
e
murky
.
Existin
g
experiment
s
ar
e
contradictory
.
Ther
e
i
s
n
o
kineti
c
theor
y
o
f
liquids
,
an
d
first-principles
predictio
n
method
s
ar
e
scarce
.
Molecula
r
dynamic
s
simulation
s
ca
n
b
e
used
,
bu
t
the
y
ar
e
limite
d
t
o
extremel
y
smal
l
flow
extents
.
Nev
-
ertheless
,
measurabl
e
sli
p
i
s
predicte
d
fro
m
M
D
simulation
s
a
t
realisti
c
shea
r
rate
s
i
n
microdevices
.
MEM
S
ar
e
finding
increase
d
application
s
i
n
th
e
diagnosi
s
an
d
contro
l
o
f
turbulen
t
flows.
Th
e
us
e
o
f
microsensor
s
an
d
microactuator
s
promise
s
a
quantu
m
lea
p
i
n
th
e
performanc
e
o
f
reactiv
e
flow
contro
l
systems
,
an
d
i
s
no
w
i
n
th
e
real
m
o
f
th
e
possibl
e
fo
r
futur
e
practica
l
devices
.
Simple
,
viscous-base
d
micropump
s
ca
n
b
e
utilize
d
fo
r
microdosag
e
delivery
,
an
d
microturbine
s
ca
n
b
e
use
d
fo
r
measurin
g
flowrates
i
n
th
e
nanoliter/
s
range
.
Bot
h
o
f
thes
e
ca
n
b
e
o
f
valu
e
i
n
severa
l
medica
l
applications
.
Muc
h
nontraditiona
l
physic
s
i
s
stil
l
t
o
b
e
learne
d
an
d
man
y
excitin
g
application
s
o
f
microdevice
s
ar
e
ye
t
t
o
b
e
discovered
.
Th
e
futur
e
i
s
brigh
t
fo
r
thi
s
emergin
g
field
o
f
scienc
e
an
d
tech
-
nolog
y
an
d
member
s
o
f
th
e
America
n
Societ
y
o
f
Mechanica
l
Engineer
s
shoul
d
b
e
i
n
th
e
forefron
t
o
f
thi
s
progress
.
Richar
d
Feynma
n
wa
s
righ
t
abou
t
th
e
possibilit
y
o
f
buildin
g
.mite-siz
e
machines
,
bu
t
wa
s
somewha
t
cautiou
s
i
n
forecastin
g
tha
t
suc
h
machines
,
whil
e
"woul
d
b
e
fu
n
t
o
make,
"
ma
y
o
r
ma
y
not
b
e
useful
.
Acknowledgment
s
M
y
origina
l
involvemen
t
wit
h
microdevice
s
wa
s
performe
d
unde
r
a
contrac
t
fro
m
th
e
Nationa
l
Scienc
e
Foundation
,
unde
r
th
e
Smal
l
Grant
s
fo
r
Explorator
y
Researc
h
initiativ
e
(SGE
R
Gran
t
no
.
CTS-95-21612)
.
Th
e
technica
l
monitor
s
wer
e
Rober
t
Powel
l
an
d
Roge
r
Arndt
.
I
woul
d
lik
e
t
o
expres
s
m
y
sincer
e
appreciatio
n
t
o
th
e
member
s
o
f
th
e
ASM
E
Freema
n
Schola
r
committee
,
Richar
d
Bajura
,
Clayto
n
Crow
e
an
d
Michae
l
Billet
,
fo
r
thei
r
confidenc
e
i
n
m
y
abiUt
y
t
o
delive
r
thi
s
treatise
.
I
a
m
ver
y
gratefu
l
t
o
Chih-Min
g
Ho
,
Fazl
e
Hussai
n
an
d
Georg
e
Kar
-
niadaki
s
fo
r
thei
r
continuou
s
suppor
t
an
d
encouragemen
t
throug
h
th
e
years
.
Hai
m
Bau
,
A
M
Beskok
,
Kennet
h
Breuer
,
Stuar
t
Jacobso
n
an
d
Sandr
a
Troia
n
provide
d
invaluabl
e
hel
p
i
n
preparin
g
thi
s
manuscript
,
includin
g
sharin
g
som
e
o
f
thei
r
re
-
port
s
an
d
papers
.
Finally
,
m
y
sincer
e
gratitud
e
t
o
m
y
frien
d
an
d
colleagu
e
Mihi
r
Sen
,
fo
r
al
l
th
e
grea
t
time
s
w
e
hav
e
ha
d
whil
e
workin
g
togethe
r
o
n
smal
l
device
s
an
d
bi
g
ideas
.
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s
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B
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i
n
Channels
,
Pipe
s
an
d
Duct
s
a
t
Micro
-
an
d
Nano-Scales,
"
Microscale
Thermophysicsl
Engi-
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t
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an
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an
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s
i
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