govardhan.pdf

8/21/2019 govardhan.pdf

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Basic Fluid Mechanics

for Geologists

Training Course on Fluid Physicsin Geological Environments Jointly Organized

by C-MMACS and JNCASR, BangaloreJanuary 19 - 23, 2004

Raghuraman N. Govardhan

Mechanical EngineeringIndian Institute of Science, Bangalore


2/65

Outline of Lecture

• Fundamental concepts & Fluid Statics

- Fluid definition, Continuum, description and classification of fluid

motions, viscosity and other basics, Fluid statics in incompressibleand 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


3/65

Fundamental concepts

- Definition of a fluid

- Continuum

- Velocity field (streamlines)

- Thermodynamic properties (p, T, ρ)

- Viscosity

- Reynolds number

- Non-Newtonian fluids

Fluid Statics


4/65

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 intheir shape when acted upon by a force.

• Anything that flows, either liquid or gas. Some solids can alsoexhibit 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"


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Fluid – Solid : Distinction

Reaction to an applied shear

SOLID

FLUID

F

F

θθθθ(t)

F

F

θθθθ

flow

Static deformation


6/65

Fluid Definition

A fluid cannot resist a shear stressby a static deformation.

Fluid includes Liquids and Gases –

Distinction between the two comes from the effect ofcohesive molecular forces.


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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 elementalvolume

!! "

#$$%

& = →

V

mV V

δ

δ ρ δ δ *lim

*V

*V


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Density at a “point”

*V

*V 3910* mmV −≈δ

ρ

Microscopicuncertainty

Macroscopicuncertainty

!! "

#$$%

& V

m

δ

δ

!! "

#$$%

& = →

V mV V

δ δ ρ δ δ *lim


9/65

Density field

Most problems are concerned with physical

dimensions much larger than this limitingvolume

So density is essentially a point function andcan be thought of as a continuum

),,,( t z y x ρ ρ =


10/65

Velocity field

Perhaps the most importantproperty in a flow is the

velocity vector field:

),,,( t z y xV V =

wk v jui ˆˆ ++=

u, v, w are f(x,y,z,t)

(taken from www.amtec.com)

Eulerian representation


11/65

Velocity field

Lagrangian representation

Flow quantities are heredefined as functions of timeand the choice of a materialelement of fluid

),( t aV V =

awhere = location of fluid particle at t=0

Lagrangian specification describes the dynamical history of the selected fluid element


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Material derivative

4 ρ

Density variation following a fluid particle

31 ρ 2 ρ

Convectivederivative

Localderivative


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Streamline

Visual representation of a velocity or flow field: Streamlines

Streamlines are lines drawn in the flow field so that at a given instant

they are tangent to the velocity vector at every point in the flow.

Local velocity vector


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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)- Specfic heats (Cp & Cv)

Transport properties

- Viscosity (µ)- Thermal conductivity (k)

All these are functions of (x,y,z,t)


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Viscosity

Shear stress causes continuous shear deformation in a fluid.

Newtonian fluid

show a linear relation between applied shear (ττττ) and resultingstrain rate (dθ /dt)

τ ∝∝∝∝ (dθ /dt)

τ = µµµµ (dθ /dt) τ = µµµµ (du/dy)

µ = viscosity coefficient

ττττ

ττττ

θθθθ(t) u

u + du

dy


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Newtonian fluid

Viscosity coefficient (µ)

Kinematic viscosity ( ν) = µ /ρ

µ kg/(m s) ν kg/(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


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Factors affecting viscosity

LI QU I D S

• decreases with increasing

temperature, since theinteratomic forces weaken

• increases under very highpressures.

GASES

• increases with increasingtemperature, since the rate ofinteratomic collisions increases

and

• is typically independent ofpressure and density.

( ) ( )2

00

0ln T T cT T ba ++=!! "

#

$$%

&

µ

µ

7.0

00!! "

#$$%

& =!!

"

#$$%

&

T

T

µ µ


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Effect of temperature on viscosity


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Non-Newtonian fluids

Do not follow linear relationship between applied shear(ττττ) and resulting strain rate (dθ /dt)

ττττ

(dθ /dt)

Newtonian

Pseudoplastic

Dilatant

Bingham Plastic

Yieldstress

Plastic

Rheology


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Magma Viscosity

Magmatic liquid viscosity depends on:

composition (especially Si), temperature,time and pressure, each of which effectthe melt structure.

It is possible to estimate the viscosity of a magmaticliquid at temperatures well above liquidustemperatures (that is, temperatures at which onlyliquid is present) from chemical compositions andempirical extrapolation of experimental data. Therange of temperatures of naturally flowing magmas,however, is near or within the crystallizationinterval, where stress-strain relationships are notlinear (that is, they are crystal-liquid mixtures andshow Bingham behavior). Under such conditions,the only way to predict viscosities is by analogy withsimilar compositions investigated experimentally.

Information source and for further reading:http://www.geo.ua.edu/volcanology/lecture_notes_files/controls_on_magma_viscos.html


21/65

Magma viscosity

Silica composition

The strong dependence of viscosity of molten

silicateson Si content can be illustrated by those of

various Na-Si-O compounds:

0.24:1:4

1.52:1:3

281:1:2.5

10100:1:2

(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 fallbetween 700° 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.For example:

a) measured viscosity of a Mauna Loa flowincreased 2-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 toan increase in distance between cations and anions, and therefore, a decreasein Si-O bond

strength.


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Magma viscosity

Time

At temperatures below the beginning of crystallization viscosity also increases with time. If magma is

undisturbed at a constant temperature, its viscosity may continue to increase for many hours before it reaches asteady value. The viscosity increases with time results partly an increasing proportion of crystals (which raise the

effective magma viscosityby their 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 andesitic magma. 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


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Fluid density

Liquids

Water ~ 1000 kg/m3

- Density in liquids is nearly constant- water density increases by 1% if the pressure is increased by a factor of 220 !- for a temperature increase of 100 K, density decreases by 5%

- Magma - Magma densities range from about 2200 kg/m3 to 2800 kg/m3, illustrating a

close density-melt composition relationship.Magma density decreases with increasing temperature and gas content. These densitiesincrease a few percent between liquid and crystalline states.

Gases

Air ~ 1.2 kg/m3

- Density is highly variable

- ideal gas law : p = ρRT (perfect gas law)

- real gas: at low temperatures & high pressure – intermolecular forcesbecome important


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Reynolds number

Dimensionless parameter correlating viscous behaviour

forcesViscous forces Inertial =

Low Re:

- Viscous forces dominate- Flow is “Laminar”

- flow structure is characterized by smooth motion in laminae or layers

High Re:

- Viscous forces are very small- Flow is “Turbulent”

- flow structure is characterized by random three-dimensional motions

of fluid particles

ν µ ρ VLVL ==Re


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Low & High Reynolds number

Low Re

Viscous

Laminar

High Re

Inertial

Turbulent

ν µ

VLVL==Re


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Reynolds pipe experiment

Laminar

Transition : Re ~ 2000

Turbulent


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Low & High Reynolds number

Low ReViscous

Laminar

High ReInertial

Turbulent


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Classification of flows

Continuum

Fluid Mechanics

Inviscid

µ = 0

Turbulent

High Re

Laminar

Low Re

Viscous

Compressible Incompressible


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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

dz dy p

z

x

y

dz dydx x

pdF x

∂

∂−=

dz dydx x

p p )( ∂

∂+


30/65

Hydrostatic equation

dz dydx z

pk

y

p j

x

pi F d pressure !!

"

#$$%

&

∂

∂+

∂

∂+

∂

∂−= ˆˆˆ

( ) pdz dydx

F d f d

pressure

pressure ∇−==

g f d gravity ρ =

For equilibrium the pressure gradient force has to be balanced by the body forces

(like gravity)

per unit volume

0=+ pressure gravity f d f d

g p ρ =∇

g z

p ρ −=

∂

∂0=

∂

∂

y

p0=

∂

∂

x

p

z g p ∆−=∆if in co m p r e s si b le , ρ=constant, then


31/65

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 p0 is the atmospheric pressure at the earth's surface, z=0. The density ρ may befound 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.


32/65

Atmosphere


33/65

Fluid statics - atmosphere

Can determine pressure, density as functions of altitude from the “hydrostatic equation”.

I n t e r n a t i o n a l St a n d a r d A t m o sp h e r e

Created by ICAO (International Civil Aviation Organization)

The ISA is a "model" of the atmosphere, designed to allow for standardized comparisonof conditions on a given day.

Based on the International Standard Atmosphere:for dry air (ICAO 1964):

1. At mean sea level pressure=101325 Pa, temp=15 deg C


34/65

Atmosphere - pressure

Linearly varying temperature

Constant temperature region


35/65

Standard Atmosphere


36/65

SECOND SESSION


37/65

Outline of Lecture


- Fluid definition, Continuum, description and classification offluid motions, viscosity and other basics, Fluid statics inincompressible 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


38/65

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)


39/65

System

All the laws of mechanics are written for a system, which is defined as anarbitrary 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 at a specific region ….


40/65

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

CS CV

sys)( ⋅+!

! "

#$$%

& = ((((( ρ β β ρ

Flux out ofthe CV

Increase withinCV

nu


41/65

Integral form

Mass:

B = mβ = dm/dm =1

From system, (dmsys /dt) = 0

dAnudV

dt

d

dt

dm

CS CV

sys)( ⋅+!

!

"

#$$

%

& = ((((( ρ ρ

0)( =⋅+!! "

#$$%

& ((((( dAnudV dt

d

CS CV

ρ ρ

f


42/65

Integral form

Momentum:

From system,

um B =

u=

dAnuudV udt

d F

CS CV

)( ⋅+!! "

#$$%

& = ((((() ρ ρ

dAnuudV udt

d

dt

umd F

CS CV

sys)(

)(⋅⋅+!

! "

#$$%

& ⋅== ((((() ρ ρ

dAnuudV udt

d

dt

umd

CS CV

sys)(

)(⋅⋅+!

! "

#$$%

& ⋅= ((((( ρ ρ

I l f


43/65

Integral form

Energy:

From system,

E B =

e=

e = einternal + ekinetic + epotential + eelectrostatic

dAnudV dt

d

dt

dW

dt

dQ

CS CV

)( ⋅+!! "

#$$%

& =− ((((( β ρ β ρ

dAnudV dt

d

dt

E d

dt

dW

dt

dQ

CS CV

sys)(

)(⋅+!

! "

#$$%

& ==− ((((( β ρ β ρ

dAnudV dt

d

dt

E d

CS CV

sys)(

)(⋅+!

! "

#$$%

& = ((((( β ρ β ρ

Control Volume Analysis


44/65

Control Volume Analysis

0=⋅nu

0)( =⋅+!! "

#$$%

& ((((( dAnudV dt

d

CS CV

ρ ρ

dAnuudV udt d F

CS CV

)( ⋅+!! "

#$$%

& = ((((() ρ ρ

Control Volume

0=⋅nu

1u 2u

Consider steady flow of waterthrough a bend,

Mass:

Steady

2211 Au Au =

Momentum

Steady

)( 22

21

2

1 Au Au F y ρ ρ +−=) 0=) x F

y

x

Diff ti l f


45/65

Differential form

0)( =⋅+!!

"

#$$

%

& ((((( dAnudV

dt

d

CS CV

ρ ρ

0)( =! "

#$%

& ⋅∇+

∂

∂((( dV ut 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 Dt D ρ

)( ρ ρ ρ ∇⋅+∂∂= u

t Dt D

Integral form :

(or)

Diff ti l f


46/65

Differential form

τ ρ ρ ⋅∇+∇−=

*+

,

-.

/∇⋅+

∂

∂ p g uu

t

u

gravitational

force

Pressure

gradient

viscous

force

Momentum

If we assume Newtonian fluid

u p g uut u 2∇+∇−=*+

,-./ ∇⋅+∂∂ µ ρ ρ

!

"

#$

%

& ⋅∇∇+∇+∇−=*

+

,-.

/∇⋅+

∂

∂)(

3

12 uu p g uu

t

u µ ρ ρ

ρ = constant

Navier-Stokes equation

Differential form


47/65

Differential form

heatconduction

ViscousDissipation

Energy

steady motion of a frictionless non- conducting fluid

B = constant

Bernoulli equation

(for material fluidelement)

Bernoulli equation


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Bernoulli equation

Commonly used form in pipe flows (in terms of head):

pumpturbine friction hhh z g

V g

p z g

V g

p −++!! "

#$$%

& ++=!!

"

#$$%

& ++ 2

2

221

2

11

22 ρ ρ

1

2

Flow measurement


49/65

Flow measurement

Flow measurement


50/65

Flow measurement

Fox & McDonald

Pipe flow – Major loss


51/65

Pipe flow – Major loss

Major losses: Frictional losses in piping system

P1

P2R: radius, D: diameter

L: pipe lengthτw: wall shear stress

Consider a laminar, fully developed circular pipe flow

p P+dp

τw[ ( )]( ) ( ) ,

,

p p dp R R dx

dp R

dx

p p ph

g

L

D f

L

D

V

g

w

w

Lw

− + =

− =

= −

= = FHIK=

FH

IKFHG

IKJ

π τ π

τ

γ γ

τ

ρ

2

1 2

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

Lf

D

L

g h w L

2

4 2

ρ ρρ ρ

τ ττ τ

!!

"

#$$

%

& !

"

#$

%

& =!

"

#$

%

& !!

"

#$$

%

& =

g

V

D

Lf

D

L

g

h w L

2

4 2

ρ ρρ ρ

τ ττ τ

Pipe flow


52/65

W hen the pipe flow is laminar, it can be shown (not here) that

by recognizing that as R eynolds num ber

Therefore, frictional factor is a function of the Reynolds num ber

Similarly, for a turbulent flow , f = function of Reynolds number also

. A nother parameter that influences the friction is the surface

roughness as relativeto the pipe diam eterD

Su ch thatD

P ipe frictional factor is a function of p ipe R eynolds

num ber and the relative roughness of pipe.

Th is relation is sketched in the M oody d iagram as show n in the follow ing page.

Th e diagram show s f as a function of the Reynolds num ber (R e), w ith a series of

param etric curves related to the relative roughness D

f VD

VD

f

f F

f F

= =

=

=

= FHIK

FHIK

64

64

µ

ρ

ρ

µ

ε

ε

ε

, R e ,

Re,

(Re)

.

R e, :

.

! "

#$%

& =

D F f

ε Re,

D

ε

Pipe flow


53/65

Losses in Pipe Flows


54/65

p

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 using the loss coefficient,

K L, which is defined as

One of the example of minor losses is the entrance flow loss. A typical flow pattern

for flow entering a sharp-edged entrance is shown in the following page. A venacontracta region 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’s equation to take intoconsideration this effect as described in the next page.

K h

V g

p p K V

L

L

LV = = =

2 2

12

2

2 12/,

∆∆

ρ ρ so that

Minor Loss


55/65

V2 V3

V1

gh K

z z g K

V V p p p

g

V K h z

g

V ph z

g

V p

L L

L L L

+=−

+=≈==

=++=−++

∞1

2)(2(

11,0,

2,

22:EquationsBernoulli'Extended

313131

2

33

2

331

2

11

γ γ

(1/2)ρV22 (1/2)ρV3

2

K L(1/2)ρV32

p→ p∞

gz V

p ρ ρ

++

2

2

Open channel flow


56/65

Open channel flow

Rh = A/P

A= cross-sectional area

P =“wetted perimeter”

Hydraulic radius

Open channel


57/65

Open channel

Rh = A/P

P =“wetted perimeter”

Hydraulic radius

Open Channel


58/65

Open Channel

Rainfall/Runoff Relationships


59/65

Rainfall/Runoff Relationships

Depending on the nature of precipitation, soil type,moisture history, etc., an ever-varying portion ofthe precipitation becomes runoff, moving viaoverland flow into stream channels

• these stormflow events are typically recorded ashydrographs of discharge, or stream height(stage) vs. time

• A hydrograph is a plot of discharge vs. time at anypoint of interest in a watershed, usually its outlet.Hydrographs are the ultimate measure of awatershed's response to precipitation events

• for any storm, the initial precipitation does notcontribute directly to flow at the outlet, instead itis stored or absorbed. This is termed the initialabstraction (Fig. 2), precipitation that falls beforethe storm hydrograph begins.

• direct runoff is that portion of the precipitationthat moves directly into the channel, appearing inthe hydrograph

• losses represent storage of precipitation upstreamfrom the outlet after the storm hydrographbegins. Often lumped with abstraction.

• excess precipitation runs off, and forms the stormhydrograph

GEOS 4310/5310 Lecture Notes, Fall 2002Dr. T. Brikowski, UTD

Rainfall/Runoff


60/65

/

Idealized model: Hortonian Overland Flow• when precipitation exceeds infiltration capacity of soil,

Hortonian overland 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 Hortonian OF

Rainfall/Runoff


61/65

/

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

Baseflow Characteristics


62/65

storm hydrograph has twocontributions

• ``Fast'' response:overland flow, interflow,etc. direct runoff

• Baseflow: discharge ofgroundwater flow tostream

hydrograph separation helps

distinguish thesecomponents

Gaining/losing stream


63/65

g/ g

Flash flood prediction


64/65

p

Starting from precipitation …

…. Storm hydrograph

actual discharge volume flow rate (Q) andheight (d) in discharge channel

Outline of Lecture


65/65


- Fluid definition, Continuum, description and classification of fluid

motions, viscosity and other basics, Fluid statics in incompressibleand 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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