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

(τ

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

Consider steady flow of water

through a bend,

Mass:

Steady

2211

AuAu=

Momentum

Steady

)(

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

gradient

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

R: radius, D: diameter

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”

Hydraulic radius

Open channel

Rh

= A/P

P =“wetted perimeter”

Hydraulic radius

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

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