# Basic Fluid Mechanics for Geologists

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

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Basic Fluid Mechanics
for Geologists
Training Course on Fluid Physics
in Geological Environments Jointly Organized
by C-MMACS and JNCASR, Bangalore
January 19 -23, 2004
RaghuramanN. Govardhan
Mechanical Engineering
Indian Institute of Science, Bangalore
Outline of Lecture
•Fundamental concepts& Fluid Statics
-Fluid definition, Continuum, description and classification of fluid
motions, viscosity and other basics, Fluid staticsin incompressible
and compressible fluids
•Governing equations for fluid flow &
Applications
-Integral & differential form of the governing equations, Pipe flow,
friction losses, flow measurement & Rainfall-run-off modelling
Fundamental concepts
-Definition of a fluid
-Continuum
-Velocity field (streamlines)
-Thermodynamic properties (p, T, ρ)
-Viscosity
-Reynolds number
-Non-Newtonian fluids
Fluid Statics
What is a fluid ?
Definitions of fluid on the Web:
•Any substance that FLOWs, such as a liquid or gas.
•A substance that is either a liquid or a gas
•Fluids differ from solids in that they cannot resist changes in
their shape when acted upon by a force.
•Anything that flows, either liquid or gas. Some solids can also
exhibit fluid behavior over time.
•any substance that cannot maintain its own shape
Not directly relevant:
•in cash or easily convertible to cash; "liquid (or fluid) assets"
Fluid –Solid : Distinction
Reaction to an applied shear
SOLID
FLUID
F
F
θ
θθ
θ(t)
F
F
θ
θθ
θ
flow
Static
deformation
Fluid Definition
A fluid cannot resist a shear stress
by a static deformation.
Fluid includes Liquids and Gases –
Distinction between the two comes from the effect of
cohesive molecular forces.
Fluid as a Continuum
Before defining Fluid property like density, pressure at a “point” :
Note:
-Fluids are aggregations of molecules
-Moving freely relative to each other (unlike a solid) Fluid density : mass / unit volume depends on elemental
volume

=

V
m
VV
δ
δ
ρ
δδ
*
lim
*V
δ
*V
δ
Density at a “point”
*V
δ
*V
δ
39
10*mmV

δ
ρ
Microscopic
uncertainty
Macroscopic
uncertainty

V
m
δ
δ

=

V
m
VV
δ
δ
ρ
δδ
*
lim
Density field
Most problems are concerned with physical
dimensions much larger than this limiting
volume
So density is essentially a point function and
can be thought of as a continuum
),,,(tzyx
ρ
ρ
=
Velocity field
Perhaps the most important
property in a flow is the
velocity vector field:
),,,(tzyxVV=
wkvjui
ˆ
ˆ
++=
)
u, v, w are f(x,y,z,t)
(taken from www.amtec.com)
Eulerianrepresentation
Velocity field
Lagrangianrepresentation
Flow quantities are here
defined as functions of time
and the choice of a material
element of fluid
),(taVV=
a
where = location of fluid particle at t=0
Lagrangianspecification describes the dynamical history of the selected fluid element
Material derivative
4
ρ
Density variation following a fluid particle
3
ρ
1
ρ
2
ρ
Convective
derivative
Local
derivative
Streamline
Visual representation of a velocity or flow field: Streamlines
Streamlines are lines drawn in the flow field so that at a giveninstant
they are tangent to the velocity vector at every point in the flow.
Local velocity vector
Thermodynamic properties
-Pressure (p)
-Density (ρ)
-Temperature (T)
When work, heat and energy balances are treated
-Internal energy (e)
-Enthalpy (h = u + p/
ρ)
-Entropy (s)
-Specficheats (C
p
& Cv)
Transport properties
-Viscosity (µ)
-Thermal conductivity (k)
All these are functions of (x,y,z,t)
Newtonian fluid
Viscosity coefficient (µ)
Kinematicviscosity (ν) = µ/ρ
µkg/(m s)νkg/(m s)
Air:1.8 x 10
-5
1.5 x 10
-5
Water:1.0 x 10
-3
1.0 x 10
-6
Newtonian fluid (
µ)
depends on (T, P)
•Generally variation with pressure is weak
less than 10% for 50 times increase in P for air
•Temperature has a strong effect
Factors affecting viscosity
LIQUIDS
•decreases
with increasing
temperature, since the
interatomicforces weaken
•increases under very high
pressures.
GASES
•increases
with increasing
temperature, since the rate of
interatomiccollisions increases
and
•is typically independent of
pressure and density.
()
()
2
00
0
lnTTcTTba++=

µ
µ
7.0
00

=

T
T
µ
µ
Effect of temperature on viscosity
Non-Newtonian fluids
Do not follow linear relationship between applied shear

ττ
τ)and resulting strain rate (dθ/dt)
τ
ττ
τ
(dθ/dt)
Newtonian
Pseudoplastic
Dilatant
Bingham Plastic
Yield
stress
Plastic
Rheology
Magma Viscosity
Magmaticliquid viscosity depends on:
composition (especially Si), temperature,
time and pressure, each of which effect
the melt structure. It is possible to estimate the viscosity of a magmatic
liquid at temperatures well above liquidus
temperatures (that is, temperatures at which only
liquid is present) from chemical compositions and
empirical extrapolation of experimental data. The
range of temperatures of naturally flowing magmas,
however, is near or within the crystallization
interval, where stress-strain relationships are not
linear (that is, they are crystal-liquid mixtures and
show Bingham behavior). Under such conditions,
the only way to predict viscosities is by analogy with
similar compositions investigated experimentally.
Information source and for further reading:
http://www.geo.ua.edu/volcanology/lecture_notes_files/controls_on_magma_viscos.html
Magma viscosity
Silica composition
The strong dependence of viscosity of molten
silicatesonSicontent can be illustrated by those of
various Na-Si-O compounds:
0.24:1:4
1.52:1:3
281:1:2.5
1010
0:1:2
(poise)
Na:Si:O
The decrease in viscosity can be attributed to a reduction
in the proportion of framework silica tetrahedral, and
therefore strong Si-O bonds in the magma.
Temperature
Temperatures of erupting magmas normally fallbetween700° and 1200°C; lower values, observed in partly
crystallized lavas, probably correspond to the limiting conditions under which magmas flow. Magmas do not
crystallize instantaneously, but over an interval of temperature.
Temperature has a strong influence on viscosity: as temperature increases viscosity decreases, an effect particularly
evident in the behavior of lava flows. As lavas flow away from their source or vent, they lose heat by radiation and
conduction, so that their viscosity steadily increases.Forexample:
a) measured viscosity of a Mauna Loa flowincreased2-fold over a 12-mile distance from vent;
b) measured viscosity of a small flow fromMt. Etna increased 375-fold in a distance of about 1500 feet.The decrease in viscosity can be attributed toanincrease in distance between cationsand anions, and therefore, a decreaseinSi-O bond
strength.
Magma viscosity
Time
At temperatures below the beginning of crystallization viscosityalso increases with time. If magma is
undisturbed at a constant temperature, its viscosity may continue to increase for many hours before it reaches a
steady value. The viscosity increases with time results partly an increasing proportion of crystals (which raise the
effective magma viscositybytheir interference in melt flow), and partly from increasing orderingand
polymerizing (linking) of the framework tetrahedra.
Pressure
The effect of pressure is relatively unknown, but viscosity appears to decrease with increasing
pressure at least at temperatures above the liquidus. As pressure increases at constant temperature,
the rate at which viscosity decreases is less in basaltic magma than that in andesiticmagma. The
viscosity decrease may be related to a change in the coordination number of Al from 4 to 6 in the
melt, thereby reducing the number of framework-forming tetrahedra.
•Bubble content
•Crystal content
Fluid density
Liquids
Water ~ 1000 kg/m
3
-Density in liquids is nearly constant
-water density increases by 1% if the pressure is increased by afactor of 220 !
-for a temperature increase of 100 K, density decreases by 5%
-Magma -Magma densities range from about 2200 kg/m
3
to 2800 kg/m
3, illustrating a
close density-melt composition relationship.
Magma density decreases with increasing temperature and gas content. These densities
increase a few percent between liquid and crystalline states.
Gases
Air ~ 1.2 kg/m
3
-Density is highly variable
-ideal gas law : p =
ρRT(perfect gas law)
-real gas:at low temperatures & high pressure –intermolecular forces
become important
Reynolds number
Dimensionless parameter correlating viscous behaviour
forcesViscous
forcesInertial
=
Low Re:
-Viscous forces dominate
-Flow is “Laminar”
-flow structure is characterized by smooth motion in laminaeor layers
High Re:
-Viscous forces are very small
-Flow is “Turbulent”
-flow structure is characterized by random three-dimensional motions
of fluid particles
νµ
ρ
VLVL
==Re
Low & High Reynolds number
Low Re
ViscousLaminar
High Re
Inertial
Turbulent
νµ
ρ
VLVL
==Re
Reynolds pipe experiment
Laminar
Transition : Re ~ 2000 Turbulent
Low & High Reynolds number
Low Re
ViscousLaminar
High Re
Inertial
Turbulent
Classification of flows
Continuum
Fluid Mechanics
Inviscid
µ= 0
Turbulent
High Re
Laminar
Low Re
Viscous
Compressible
Incompressible
Fluid statics
Fluids by definition cannot resist shear
⇒in fluids at rest there can be no shear
Only stresses are normal pressure forces
Net force in x-direction
dzdyp
z
x
y
dzdydx
x
p
dF
x

−=
dzdydx
x
p
p)(

+
Hydrostatic equation
dzdydx
z
p
k
y
p
j
x
p
iFd
pressure

+

+

−=
ˆ
ˆˆ
()
p
dzdydx
Fd
fd
pressure
pressure
∇−==
gfd
gravity
ρ
=
For equilibrium the pressure gradient force has to be balanced by the body forces
(like gravity)
per unit volume
0=+
pressuregravity
fdfd
gp
ρ
=∇
g
z
p
ρ
−=

0=

y
p
0=

x
p
zgp∆−=∆
ρ
if incompressible,ρ=constant, then
Atmosphere
For the purpose of calculating the pressure and density of the atmosphere, we can
regard air as a perfect gas obeying the perfect gas law equation.
Substituting the
perfect gas law into the differential equation of force balance,and integrating, we find an
expression for the pressure:
where p
0
is the atmospheric pressure at the earth's surface, z=0. The density
ρ
may be
found readily by dividing equation by RT(z).
Note that the atmospheric absolute temperature T(z) must be known as a function
of altitude in order to evaluate the integral.
Atmosphere
Fluid statics-atmosphere
Can determine pressure, density as functions of altitude from the “hydrostatic equation”.
International Standard Atmosphere
Created by ICAO (International Civil Aviation Organization)
The ISA is a "model" of the atmosphere, designed to allow for standardized comparison
of conditions on a given day.
Based on the International Standard Atmosphere:fordry air (ICAO 1964):
1. At mean sea level pressure=101325 Pa, temp=15 deg C
Atmosphere -pressure
Linearly varying temperature
Constant temperature region
Standard Atmosphere
SECOND SESSION
Outline of Lecture
•Fundamental concepts& Fluid Statics
-Fluid definition, Continuum, description and classification of
fluid motions, viscosity and other basics, Fluid staticsin
incompressible and compressible fluids
•Governing equations for fluid flow &
Applications
-Integral & differential form of the governing equations,
-Pipe flow, friction losses, flow measurement
& Rainfall-run-off modelling
Approach
Fluid flow analysis:
•Control volume, or large-scale
•Differential, or small-scale
Flow must satisfy the three basic laws of mechanics:
•Conservation of mass (continuity)
•Conservation of Linear momentum (Newton’s second law)
•Conservation of energy (first law of thermodynamics)
System
All the laws of mechanics are written for a system, which is defined as an
arbitrary quantity of mass of fixed identity.
Mass:
(dmsys/dt) = 0
Momentum:
F = m (dV/dt)
Energy:
dQ/dt–dW/dt= dE/dt
Difficult to follow a fluid of fixed identity. Easier to look ata specific region ….
Control Volume
Write the basic laws for a specific region:
Consider a fixed Control Volume:
Let B =any property (mass, momentum, energy)
β= B per unit mass = dB/dm
dAnudV
dt
d
dt
dB
CSCV
sys
)(⋅+

=
∫∫∫∫∫
ρββρ
Flux out of
the CV
Increase within
CV
n
u
Integral form
Mass:
B = m
β= dm/dm =1
From system, (dm
sys/dt) = 0
dAnudV
dt
d
dt
dm
CSCV
sys
)(⋅+

=
∫∫∫∫∫
ρρ
0)(=⋅+

∫∫∫∫∫
dAnudV
dt
d
CSCV
ρρ
Integral form
Momentum:From system,
umB=
u=
β
dAnuudVu
dt
d
F
CSCV
)(⋅+

=
∫∫∫∫∫

ρρ
dAnuudVu
dt
d
dt
umd
F
CSCV
sys
)(
)(
⋅⋅+

⋅==
∫∫∫∫∫

ρρ
dAnuudVu
dt
d
dt
umd
CSCV
sys
)(
)(
⋅⋅+

⋅=
∫∫∫∫∫
ρρ
Integral form
Energy:
From system,
EB=
e=
β
e = einternal
+ ekinetic
+ epotential
+ eelectrostatic
dAnudV
dt
d
dt
dW
dt
dQ
CSCV
)(⋅+

=−
∫∫∫∫∫
βρβρ
dAnudV
dt
d
dt
Ed
dt
dW
dt
dQ
CSCV
sys
)(
)(
⋅+

==−
∫∫∫∫∫
βρβρ
dAnudV
dt
d
dt
Ed
CSCV
sys
)(
)(
⋅+

=
∫∫∫∫∫
βρβρ
Control Volume Analysis
0=⋅nu
0)(=⋅+

∫∫∫∫∫
dAnudV
dt
d
CSCV
ρρ
dAnuudVu
dt
d
F
CSCV
)(⋅+

=
∫∫∫∫∫

ρρ
Control Volume
0=⋅nu
1
u
2
u
through a bend,
Mass:
2211
AuAu=
Momentum
)(
2
2
21
2
1
AuAuF
y
ρρ
+−=

0=

x
F
y
x
Differential form
0)(=⋅+

∫∫∫∫∫
dAnudV
dt
d
CSCV
ρρ
0)(=

⋅∇+

∫∫∫
dVu
t
CV
ρ
ρ
0)(=⋅∇+

u
t
ρ
ρ
Can be written in the form:
Mass
Valid for any volume V, possible only if:
0)(=⋅∇+u
D
t
D
ρ
ρ
)(
ρ
ρ
ρ
∇⋅+

=u
t
Dt
D
Integral form :
(or)
Differential form
τρρ
⋅∇+∇−=

∇⋅+

pguu
t
u
gravitational
force
Pressure
viscous
force
Momentum
If we assume Newtonian fluid
upguu
t
u
2
∇+∇−=

∇⋅+

µρρ

⋅∇∇+∇+∇−=

∇⋅+

)(
3
1
2
uupguu
t
u
µρρ
ρ
= constant
Navier-Stokes equation
Differential form
heat
conduction
Viscous
Dissipation
Energy
steady motion of a frictionless non-conducting fluid
B = constant
Bernoulli equation
(for material fluid
element)
Bernoulli equation
Commonly used form in pipe flows (in terms of head):
pumpturbinefriction
hhhz
g
V
g
p
z
g
V
g
p
−++

++=

++
2
2
22
1
2
11
22
ρρ
1
2
Flow measurement
Flow measurement
Fox & McDonald
Pipe flow –Major loss
Major losses: Frictional losses in piping system
P1
P2
L: pipe length τ w: wall shear stress
Consider a laminar, fully developedcircular pipe flow
p
P+dp
τ w
[()]()(),
,
ppdpRRdx
dp
R
dx
ppp
h
g
L
D
f
L
D
V
g
w
w
L
w
−+=
−=
=

==
F
H
I
K
=
F
H
I
K
F
H
G
I
K
J
π
τ
π
τ
γγ
τ
ρ
2
12
2
2
2
4
2
Pressure force balances frictional force
integrate from 1 to 2
where f is defined as frictional factor characterizing
pressure loss due to pipe wall shear stress

=

=
g
V
D
L
f
D
L
g
h
w
L
2
4
2

ρ
ρρ
ρ
τ
ττ
τ

=

=
g
V
D
L
f
D
L
g
h
w
L
2
4
2

ρ
ρρ
ρ
τ
ττ
τ
When the pipe flow is laminar,it can beshown (not here) that
by recognizing that as Reynolds number
Therefore, frictional factor is a function of the Reynolds number
Similarly, for a turbulent flow, f=function of Reynolds number also
. Another parameter that influences the friction is the surface
roughness as relativeto the pipe diameter
D
Such that
D
Pipe frictional factor is a function of pipe Reynolds
number and the relative roughness of pipe.
This relation is sketched in the Moody diagram as shown in the following page.
The diagram shows f as a function of the Reynolds number (Re), with a series of
parametric curves related to the relative roughness
D
f
VD
VD
f
fF
fF
==
=
=
=
F
H
I
K
F
H
I
K
64
64
µ
ρ
ρ
µ
ε
ε
ε
,Re,
Re
,
(Re)
.
Re,:
.

=
D
Ff
ε
Re,
D
ε
Pipe flow
Losses in Pipe Flows Major Losses: due to friction, significant head loss is associated with the straight portions of
pipe flows. This loss can be calculated using the Moody chart.
Minor Losses: Additional components (valves, bends, tees, contractions, etc)in
pipe flows also contribute to the total head loss of the system.Their contributions
are generally termed minor losses.
The head losses and pressure drops can be characterized by usingthe loss coefficient,
KL
, which is defined as
One of the example of minor losses is the entrance flow loss. Atypical flow pattern
for flow entering a sharp-edged entrance is shown in the following page. A vena
contractaregion is formed at the inlet because the fluid can not turn a sharp corner.
Flow separation and associated viscous effects will tend to decrease the flow energy;
the phenomenon is fairly complicated. To simplify the analysis,a head loss and the
associated loss coefficient are used in the extended Bernoulli’sequation to take into
consideration this effect as described in the next page.
K
h
Vg
p
pKV
L
L
L
V
===
22
1
2
2
2
1
2
/
,

ρ
ρ
so that
Minor Loss
V2
V3
V1
gh
K
zzg
K
VVppp
g
V
Khz
g
Vp
hz
g
Vp
L
L
LLL
+
=−
+
=≈==
=++=−++

1
2
)(2(
1
1
,0,
2
,
22
:Equation sBernoulli' Extended
313131
2
3
3
2
33
1
2
11
γγ
(1/2)ρV2
2
(1/2)ρV3
2
KL
(1/2)ρV3
2
p→p∞
gz
V
p
ρ
ρ
++
2
2
Open channel flow
Rh
= A/P
A= cross-sectional area
P =“wetted perimeter”
Open channel
Rh
= A/P
P =“wetted perimeter”
Open Channel
Rainfall/Runoff Relationships
Depending on the nature of precipitation, soil type,
moisture history, etc., an ever-varying portion of
the precipitation becomes runoff, moving via
overland flow into stream channels
•these stormflowevents are typically recorded as
hydrographs of discharge, or stream height
(stage) vs. time
•A hydrograph is a plot of discharge vs. time at any
point of interest in a watershed, usually its outlet.
Hydrographs are the ultimate measure of a
watershed's response to precipitation events
•for any storm, the initial precipitation does not
contribute directly to flow at the outlet, instead it
is stored or absorbed. This is termed the initial
abstraction (Fig. 2
), precipitation that falls before
the storm hydrograph begins.
•direct runoff is that portion of the precipitation
that moves directly into the channel, appearing in
the hydrograph
•losses represent storage of precipitation upstream
from the outlet after the storm hydrograph
begins. Often lumped with abstraction.
•excess precipitation runs off, and forms the storm
hydrograph
GEOS 4310/5310 Lecture Notes, Fall 2002
Dr. T. Brikowski, UTD
Rainfall/Runoff
Idealized model: HortonianOverland Flow
•when precipitation exceeds infiltration capacity of soil,
Hortonianoverland flow results
•infiltration rate declines exponentially as soil saturates
•Horton model (1940) assumes uniform infiltration capacity for
watershed
Overland Flow (OF) actually unimportant in most watersheds
(studies performed in 1960's) •often only 10% of a watershed regularly supplies OF during a
storm event
•in those areas, often only 10-30% of precip. becomes OF
•vegetation also absorbs much precip.
•well-vegetated watersheds in humid climate rarely show OF
•arid zones (sparse vegetation) during high-intensity rainfall
will show HortonianOF
Rainfall/Runoff
Best model: variable source area
•interflow (subsurface stormflow) is prime contributor to streamflow
•OF is important near streams, where slopes become saturated by
interflow
•return flow (emergence of interflow) also important near streams
BaseflowCharacteristics
storm hydrograph has two
contributions
•``Fast'' response:
overland flow, interflow,
etc. direct runoff
•Baseflow: discharge of
groundwater flow to
stream
hydrograph separation helps
distinguish these
components
Gaining/losing stream
Flash flood prediction
Starting from precipitation …
…. Storm hydrograph
actual discharge volume flow rate (Q) and
height (d) in discharge channel
Outline of Lecture
•Fundamental concepts& Fluid Statics
-Fluid definition, Continuum, description and classification of fluid
motions, viscosity and other basics, Fluid staticsin incompressible
and compressible fluids
•Governing equations for fluid flow &
Applications
-Control Volume analysis using basic laws of Fluid Mechanics,
Pipe flow, friction losses, flow measurement & Rainfall-run-off-
modelling