# use of these strange equations.

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24 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

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1.
If the Navier
-
Stokes equations really are the corner stone of Fluid Mechanics shouldn't
they be inserted as part of the Fluid Mechanics foundation course.

2.
Now come I had manages to teach fluid mechanics all these years without needing to make
use of these strange equations.

Two questions occurred to me on
first seeing this diagram which
incidentally is taken from:

Fluid Mechanics Fundamentals and
Applications

Y. A.
Cengel

and J. M.
Cimbala
.

McGraw Hill

A talk by T. Swann

Mathematics From Fluids
-
1

Dr Anthony B Swann

Dept of CEE University of Auckland, Auckland, New Zealand

Int r oduct i on

Int egr at i on
over t he Regi on

I nt egr at i on ov er
t he Sur f ace

Th o u g h r a r e l y u s e d, i t i s t h e r e g i o n
i n t e g r a t i o n t h a t p r o vi d e s t h e l i n k b e t we e n
t h e s u r f a c e a n d vo l u me i n t e g r a t i o n s a n d
h e n c e p r o vi d e s t h e t h e o r e t i c a l b a s i s f o r
Ga u s s' t h e o r e m. Th e s i t u a t i o n i s
c o n ve n i e n t l y i l l u s t r a t e d b y t h e d i a g r a m:

Concl usi ons

Ma t h e ma t i c s F r o m F l u i d s

-

A review of Gauss' Theorem

The successful use Computational Fluid
Dynamic software such as Flexpde5 to
model complex flow problems requires an
appreciation of three dimensional field
theory and in particular of Gauss' theorem.

derived mathematically from a
consideration of a general three
dimensional field. The aim of this seminar
is to explore an alternative "Fluid
Mechanics" approach to teaching Gauss'
Theorem in which a consideration of the
well known principle due to Archimedes
provides the context. Once established,
the approach can be

conservation laws of mass and momentum
to fluid flows as well as to a derivation of
the well known Navier

Stokes equations.

Archimedes principle deals with the
upthrust or buoyancy force which
acts on a body immersed in a fluid.
To establish a link with Gauss'
theorem we will reinterpret it as the
resultant force acting on a body
immersed in a fluid when linearly
varying pressure field (the
hydrostatic field) exists in the fluid.

Archimedes principle states that:

When a body is placed in a fluid it
experiences an upward force which
is equal the weight of the fluid
displaced

F
1

=
-

P
δ
a
R

=

-

gz
1

a
R

z
1

Region
R

on a horizontal
plane which is the

Diagram 1 shows the force of the
water on the upper surface of the
discretised body

Total Force is obtained by
integrating over the region

Diagram 2 shows the force of the
water on the lower surface of the body

F
2

= P
δ
a
R

=

gz
2

a
R

z
2

Resultant buoyancy force

Taking the water surface as datum

z
1

Column

area =

δ
a
R

z
2

Diagram 3 shows how integration over
the region is used to find the volume
of the body.

Vertical height of the elementary volume =

z
2

-

z
1

Therefore volume of the element =

δ

= (
z
2

-

z
1
)
δ
a
R

The total volume of the body

Hence

So far so good but instead of stopping here we
consider calculation the buoyancy force by
integrating over the surface of the body. To do
this we first introduce two unit vectors

k

in the
vertical direction and
n

in the direction normal to
the surface on the fluid side. (called the
outward

unit normal
).

k

Area of surface element

δ
a
S

γ

n

Di agram 4 s hows t he hy dros t at i c pres s ure forc e
ac t i ng on a s mal l el ement of t he s urfac e of t he
body res ol ved i n t he vert i c al di rec t i on.
Int egrat i ng t hi s over t he whol e of t he s urfac e
gi ves t he buoy anc y forc e.

and henc e

Thi s es t abl i s hes an equi val enc e rel at i on
bet ween t he regi on and t he s urfac e i nt egral,
namel y:

Integration over the Volume

Di agram 5 i s s i mi l ar t o di agram 3.It s hows a s mal l
el ement of t he el ement ary c ol umn. By c ombi ni ng
an i nner i nt egrat i on w.r.t z wi t h an out er regi onal
i nt egrat i on

we c an i nt egrat e over t he ent i re
vol ume of t he body.

Al s o s i nc e

and t he t ot al

and henc e t hat

or as a vol ume i nt egral

Abstract

ABS Oc t o b e r 2 00 7

Thi s res ul t i s at t he very heart of our us e of
c ont rol vol umes

i n t he s t udy fl ui d dy nami c s of
fl ows. Thi s i s bec aus e i t s hows t hat, i n
anal y s i ng

t he dy nami c s of a fl ow, we c an, i n
many c as es, c ons i der onl y t he fl ow c ondi t i ons
t hat ex i s t at t he s urfac e of t he c ont rol vol ume
and c an i gnore t hos e t hat ex i s t wi t hi n.

pres s ure c hange c an be found by i nt egrat i ng
t he vert i c al gradi ent of t he pres s ure from t he
bot t om t o t he t op of t he c ol umn i t fol l ows t hat

whi c h reduc es t o

N.B.
-
v e b e c a u s e a p o s i t i v e b u o y a n c y f o r c e r e q u i r e s a n e g a t i v e g r a d i e n t

Bonaventura
Cavalieri

born 1598, Mi l an [ It al y ]

di ed Nov. 30, 1647, Bol ogna, Papal St at es

Caval i eri, i n 1629, devel oped hi s
met hod of
i ndi vi s i bl es
, a means of det ermi ni ng t he s i z e of
geomet ri c fi gures s i mi l ar t o t he met hods of i nt egral
c al c ul us.

z
1

z
2

z

Thi s l eads t o t he doubl e i nt egral ex pres s i on

Further Work

Mathematics From Fluids
-

2

Dr Anthony B Swann

Dept of CEE University of Auckland, Auckland, New Zealand

Int r oduct i on

Int egr at i on over t he Regi on

Int egr at i on over t he Sur f ace

A g a i n we s e e t h a t t h e r e g i o n a l i n t e g r a t i o n
p r o vi d e s t h e l i n k b e t we e n t h e s u r f a c e a n d
vo l u me i n t e g r a t i o n s a n d h e n c e i t i s a n o t h e r
e x a mp l e o f Ga u s s' t h e o r e m a t wo r k. Th e
s i t u a t i o n i n t h i s c a s e i s:

Concl usi ons

Ap p l i c a t i o n t o t h e ma ss c o n se r v a t i o n
l a w o f F l u i d Me c h a n i c s

T h e a p p l i c a t i o n o f c o n se r v a t i o n l a w s t o
f l u i d f l o w s r e q u i r e s t h e u se o f a c o n t r o l
v o l u me t o d e f i n e t h e e x t e n t o f t h e b o d y
o f f l u i d u n d e r c o n si d e r a t i o n. S u c h a
c o n t r o l v o l u me c a n b e c o n c e p t u a l i se d
b y f i r st c o n si d e r i n g Ar c h i me d e s's
su b me r g e d b o d y a n d t h e n r e mo v i n g i t s
su b st a n c e. T h i s l e a v e s t h e e sse n c e o f
t h e b o d y i n mu c h t h e sa me w a y t h a t t h e
r e mo v a l o f t h e "Ch e sh i r e c a t" l e a v e s
o n l y t h e "Gr i n". Be i n g o n l y e sse n c e, t h e
f l o w i s n o t a f f e c t e d b y t h e p r e se n c e o f
t h e C.V. We c a n h o w e v e r u se i t t o
d e t e r mi n e w h e t h e r t h e C.V. h a s a n e t
i n f l o w o r a n e t o u t f l o w o r w h e t h e r t h e
f l o w i s b a l a n c e d.

T h e c o n t i n u i t y l a w o f i n c o mp r e s s i b l e f l u i d
f l o w e f f e c t i v e l y s a y s t h a t i n t h e a b s e n c e o f
a n y s o u r c e s o r s i n k s, a n y n e t i n f l o w t o t h e
C.V. t h r o u g h o n e p a r t o f i t s s u r f a c e mu s t b e
j u s t b a l a n c e d b y a n e t o u t f l o w t h r o u g h
a n o t h e r p a r t o f i t s s u r f a c e. S o i f we f i r s t
d i v i d e t h e s u r f a c e i n t o s ma l l e l e me n t s t h e n
mu l t i p l y
t h e a r e a o f e a c h e l e me n t b y t h e
n o r ma l c o mp o n e n t o f t h e f l o w v e l o c i t y
wh e r e t h e f l o w c r o s s e s t h e e l e me n t a n d
f i n a l l y s u m t h e r e s u l t s
f o r a l l t h e e l e me n t s
t h e n
t h e r e s u l t s h o u l d b e z e r o. T h i s c o n c e p t
f o r ms t h e b a s i s o f a n i mp o r t a n t t h e o r e m
c a l l e d t h e "
Di v e r g e n c e T h e o r e m".
I t i s i n
e f f e c t Ga u s s' T h e o r e m a p p l i e d t o t h e
c o n t i n u i t y p r o b l e m
.

Di a g r a m 6 i s b a s e d o n d i a g r a m 3 e x c e p t t h a t
b o d y i s n o w a n i n s u b s t a n t i a l
s h a p e ( t h e C.V.).

Th e mo vi n g f l u i d
, n o w
p a s s e s
t h r o u g h
i t
u n h i n d e r e d.
Th e mo vi n g f l u i d c o n s t i t u t e s a 3
-
D
flow field described by lines
called
streamlines
which are tangential to the velocity vector at
every point
along their length.

The flow through
this
element

Q
u

is given by
:

Diagram 7 shows flow through a single

a
r

on the upper
control surface.

To get the total flow crossing the surface we
must include the flows across the two sets of
risers. Diagram 8 shows all three
integration
regions
.

The result for the three regions is:

Next we calculate the flow by integrating over
the surface of the C.V.

Diagram 9 shows a patch of the surface above a
n

is the unit normal to the patch
and
V

is the
local velocity
vector.

and hence

This establishes an equivalence relation
between the region and the surface
integrals,
namely:

Integration over the Volume

Diagram 10 is similar to diagram 6 except that it
shows a small element of the elementary column.
Now since integrating the gradient of
V
z

over the
height of the column gives the total change in
V
z

across the column we can use it to substitute for
the expression under the regional integral in the
previous result. This leads to a double integral
with the inner integration being performed over
the entire height of the column.

The expressions for the other directions are.

Abstract

ABS January 2008

Now we can add the three expressions to get
the net outflow from the U.C.V. The result is:

Gauss

born 1777, Braunschweig, [Germany]

died 1855, Göttingen, Hanover, {Germany]

Carl Friedrich Gauss, in 1813, developed the
divergence theorem (also called Gauss' Theorem)
which states that "The net flux of any vector field
through any closed surface is equal to the divergence
of the field integrated throughout the volume enclosed
by the surface".

z

The resulting expression for the z
-
direction velocity
component is:

Control Surface

Region

Control Volume

V
z

V
x

V
y

Area =
δ
a
R

The net outflow flow through all the treads is
obtained by integrating over the
entire region for
both upper and lower control surfaces and
subtracting thus
:

V

Rate of flow through this
surface element

V

Surface
Patch

n

Region
R
z

Region
R
x

Region
R
y

z
2

z
1

We may note that the expression

is called the divergence of the vector field
div(V)
and hence that

The divergence of the velocity field is a
measure of the overall rate at which fluid is
either approaching or retreating from a point in
the flow.

Mathematics From Fluids
-

3

Dr Anthony B Swann

Dept of CEE University of Auckland, Auckland, New Zealand

Introduction

Integration over the
Surface

Volume Integration

In this poster we have obtained the ingredients
required to apply the momentum principle to the
C.V. The result for say the z
-
direction in volume
integral form is:

Conclusions

Application to the linear momentum
conservation law of Fluid Mechanics

The linear momentum conservation law
is simply an extension of the law of
continuity with the linear momentum
vector replacing the velocity vector.
The U.C.V. is used as before to control
the calculation. In all other respects the
derivation is the same

Poster 1 introduced Gauss, theorem
as it applies to forces acting at the
surface of a body (real or imagined)
while poster 2 applied Gauss' theorem
a quantity (namely volume) which the
field is convecting through an
imaginary body (referred to as the
Control Volume
). In this poster we will
use Gauss' theorem to apply the
momentum principle to the C.V. using
the previous applications as templates.
As momentum is convected by the flow
we will use the surface integral from
poster 2 diagram 9 as our template and
simply replace volume by momentum.
Thus:

To handle the forces active on the control
surface we turn to our poster 1 template.
Further, since gravity is a body force it is
logical to consider the volume integral from
our template which is:

Becomes:

However, for bodies on or near the earths
surface, the gradient of the gravity field will be
sensibly constant, directed in the vertical and
equal to
g

times the mass of the body
.
Thus
we can substitute
ρ
g
for
-
(
P
)

t o g e t:

A c c o r d i n g t o Ne wt o n, V i s c o u s f o r c e s i n a f l u i d
c a n b e c h a r a c t e r i s e d a s "
T h e r e s i s t a n c e wh i c h
a r i s e s f r o m t h e l a c k o f s l i p p e r i n e s s o r i g i n a t i n g i n
a f l u i d wh i c h, o t h e r t h i n g s b e i n g e q u a l, i s
p r o p o r t i o n a l t o t h e v e l o c i t y b y wh i c h t h e p a r t s o f
t h e f l u i d a r e b e i n g s e p a r a t e d f r o m e a c h o t h e r
.
"In other words, viscous stress is proportional to
the rate of strain of the fluid.

The rate at which fluid particles are separating
from each other at the tread surface is simply

V
z
/

z

or in other words
(
V
z
)

k
.

The viscous
force on the step is obtained by multiplying by

a
R

and the coefficient of
proportionality

..

Of course in this case other
forces will be acting on the risers.

Again the double integration can be replaced by
an integration over the volume of the C.V.
giving:

Integration over the
Volume

Diagram 11 is similar to diagram 10 except that
the velocity vectors must be replaced by the
component gradient vectors since these give the
magnitude and direction of the viscous stress
components. This lets us assess the total for the
C.V. For instance, for the
z
-
direction component,
of the gradient we have :

Since these are all z
-
direction components of
the viscous force the total component will be

Abstract

ABS January 2008

Similarly for the x and y direction components

and

When integrated over the entire surface, these
expressions give the change of momentum of
the component in each direction. Further, by
the momentum principle this change will be
equal to the sum of all external forces resolved
in that direction. thus:

Where

M
x
is the x
-
direction
component of the momentum
convected through the surface
patch.

Similarly:

Gravity Force

Expanding this expression we get:

However, our template also tells us that :

Which implies that for instance for the x
-
direction momentum :

But conservation of mass requires that the
divergence term be zero leaving only the 2
nd

term whence:

as the gravity force acting on the fluid inside the
C.V., the other components being zero.

A
more general result is obtained by letting
X
,
Y

and
Z

be the component of the gravity force per
unit mass in the
x, y

and
z

directions
respectively. This gives three components of
the gravity force namely:

Pressure Force

The pressure force comes directly from the poster 1
template except that we need to consider each
coordinate direction separately. For instance, for
the
x
-
direction we can write:

Viscous Force

Which is simply:

Newton

Born 1642,
at Woolsthorpe in Lincolnshire (UK)

died 1727, Kensington, London

In Book II of his
Principia, Newton laid the
foundations of a theory of fluid mechanics which
included the effects of viscosity on a flow.

Region

z

z
2

z
1

(V
z2

)
z
2

G
(
V
z
1
)
z
1

Area element
da
R

or:

Similar results for the other velocity
component scalar fields leads to the final
result :

The Constitutive equations

Mathematics From Fluids
-

4

Dr Anthony B Swann

Dept of CEE University of Auckland, Auckland, New Zealand

Introduction

RHS Terms

The Navier
-
Stokes equations are derived from
the volume integral form of the momentum
principle equations applied to an elementary
oblong control volume.

This is in contrast with the conventional method
which is to use the surface integral form of the
equations and integrate over a piecewise
continuous control surface chosen to make the
integration process as straight forward as
possible.

The practical difference is that the Navier
-
Stokes
solution will give information about velocities,
pressures etc everywhere in the flow field. In
other words to “model” the flow. Conventional
momentum principle solutions yield results only
bulk results which apply to the whole control
volume.

The constitutive equations allow us the use
velocity gradient information from the Navies
-
Stokes solution to find the distribution of normal
and shear forces throughout the field.

Conclusions

We might for instance write the formula for a
control volume of infinitesimal volume
d

we
can then omit the volume integrals thus:

Part 2. The constitutive equations relate the
forces acting on a fluid particle to the state of
stress generated inside the fluid. To derive
the constitutive equations first consider the
net forces acting on the surface of an oblong
fluid particle at point
P
:

Now because, for an incompressible flow,

μ

div(
V
)
is zero we may add it to the LHS of the
equation The result is

Then integrating both sides of each equation
w.r.t
. its denominator gives the constitutive
equations:

Combining terms having a common denominator
we get :

Abstract

ABS January 2008

Associating the terms in brackets, the equation
reduces to

the other six constitutive equations can be easily
generated by the cyclic interchange of suffices
technique.

The first term on the RHS which is the gravity
force term becomes:

The above can now be expanded in partial
differential form. In this case the LHS
becomes:

The second or pressure term becomes:

While the third or viscosity term becomes: :

Back substitution then gives the well known
Navier
-
Stokes equation (see background):

If we now equate the expressions for say

F
x we get
:

Navier, Claude
-
Louis

Part 1. Versions of the basic momentum
principle formula
:

P

Borne 1785 in Dijon France died 1836 in Paris

Although he made great contributions relating
to elasticity and stress in solids, His major
contribution however remains the Navier
-
Stokes

equations(1822), central to Fluid Mechanics.

In part 1, the formula embodying the
application of the momentum principle
to the Universal Control Volume is
presented in different forms .

Part 2 investigates the implications of
the formula for the gravity, pressure
and viscous forces which must act on a
fluid particle having the same shape as
the Control Volume used previously.
The result is a set of equations known
collectively
as
The
Constitutive
Equations

From the diagram

Thus: