IntroductionIntroduction

Field of Fluid Mechanics can be divided into 3 branches:

• Fluid Statics: mechanics of fluids at rest• Kinematics: deals with velocities and streamlinesKinematics: deals with velocities and streamlines

w/o considering forces or energy• Fluid Dynamics: deals with the relations between• Fluid Dynamics: deals with the relations between

velocities and accelerations and forces exerted by or upon fluids in motionby or upon fluids in motion

Mechanics of fluids is extremely important in many areas of engineering and science Examples are:areas of engineering and science. Examples are:

• BiomechanicsBl d fl th h t i– Blood flow through arteries

– Flow of cerebral fluidMeteorology and Ocean Engineering• Meteorology and Ocean Engineering– Movements of air currents and water currents

Chemical Engineering• Chemical Engineering– Design of chemical processing equipment

M h i l E i i• Mechanical Engineering– Design of pumps, turbines, air-conditioning

equipment, pollution-control equipment, etc.• Civil Engineeringg g

– Transport of river sediments– Pollution of air and waterPollution of air and water– Design of piping systems

Flood control systems– Flood control systems

Wh t dWhat we do :

•Lagrangian Approach: Follow motion of some particle of the fluid and this g g pp pmust be done for all particle of the fluid

•Eulerian Approach: Follow velocity and density of fluid at particular point and this must be done for all points in space

•We write the equation of motion of a fluid under an external force

Our approach:

-Calculate the velocity of the fluid when external forces are applied

- This velocity equation is subjected to the properties of the fluid : Compressible, homogeneous etc

Euler’s equation

21 1( )v v v p v ( )

2v v p v

t

Then we derive Bernoulli’s principle

1P 21 Constant2

P v

2

Fl id ti H d t ti ilib iFluid motion -- Hydrostatic equilibrium

z

PInfinitisimal volume element at the point P

x

yp

Forces on the faces perpendicular to y axis areLet p be the pressure at the pointForces on the faces perpendicular to y-axis are

dxdzdyp

p

along the +y directiondxdzy

p

2

along the +y direction

d and dxdzdyypp

2

along the –y direction

pNet force in the y direction is dxdydz

yp

e y

ˆ

Applying the same concept along the other two directions

The Net force on the element of volume

d d d)(

dxdydzp)(

Net force per unit volume at P due to hydrostaticNet force per unit volume at P, due to hydrostatic pressure p

If there is an external force, , acting on unit mass of the element at P,

extF

,

Total force on unit volume is extFp

0 Fp

At hydrostatic equilibrium 0 extFp At hydrostatic equilibrium

If is the force of gravity on unit massextF

extFp

zext egF ˆ

zext g

egp ˆ1

)(zegp

)(zpp

dpg

dzdp

or

at equilibrium

gzp constant

Equation of motion

Local hydrodynamic variable: ( , )P r t ( , )v r t

( , )r t

For a fluid in motion in a region without sources and sinks

variable: ( , ) ( , )r t

For a fluid in motion, in a region without sources and sinks, the equation of continuity is

0 J

t

where vJ

t

For an incompressible homogeneous fluid, this reduces top g ,

0 v

The velocity field is solenoidal

This is a major difference between flow of a gas and that of a liquid

If does not have an explicit time dependence, but the fluid is inhomogeneousg

vvv 0

vv

To write the equation of motion of an element ofTo write the equation of motion of an element of fluid, using Newton’s law Force on unit volume = extFp this is equal to mass x accelerationthis is equal to mass x acceleration

pFvd

pdt

therefore 1dv F

thereforeextp F

dt

If the external force is conservative (gravity),We can write force as the gradient of a scalar potential

F Fa dissipative force due to viscosity is present a d ss pat e o ce due to scos ty s p ese twhenever a fluid moves (non-conservative)viscF

Equation of motion is in the general form

Fpvd

1

viscFpdt

viscF

is the viscous drag per unit mass

The acceleration is the total time derivative of the velocity.

dtdzv

dtdyv

dtdxv

dtv

dtvd

vdtzdtydtxdtdt

vvtv

The equation of motion (per unit volume) of the fluid is therefore

The second term is called convective derivative.

The equation of motion (per unit volume) of the fluid is therefore

v 1

pvv

tv

1

for non-viscous fluids.

From the vector identity,y,

)()()()()( ABBAABBABA

)()()()()(

writing vBA , we haveg vBA , e a e

( ) 2( ) 2 ( )v v v v v v ( ) 2( ) 2 ( )v v v v v v

1 1v

21 1( )2

v v v p vt

)(This equation is also known as Euler’s equation

Vorticityv )(

We shall write equations of motion (we shall find velocity of the fluid)For several cases of fluids:

(1) Incompressible homogenous non-viscous fluid

(2) For the Irrotational flow of a non viscous fluids

3. For Steady, irrotational flow of a homogeneous, incompressible,Non-viscous fluids

( 4) For steady, non-viscous flow (not necessarily irrotational)

(5) For Steady of a homogeneous incompressible(5) For Steady, of a homogeneous, incompressible,Non-viscous fluids (not irrotational)

alirrotationisFlow 0

alirrotationisFlow0

1

where

i th l l l l it2

where, is the local angular velocity in the fluid.

Barotropic flow of an inviscid fluid1 p This term in general can not be written as gradient

of a scalar field

If flow is incompressible or barotropic then it can be written as gradient of a scalar field. dp

gradient of a scalar field.If ρ is only function of P (barotropic) only then, ( )

( )dpf P

P

And 1( )( )

dff P P PdP P

( )

( )P f PP

For a set of equations not involving the pressure, we take curl of q g p ,the Euler’s equation

21 1v

21 1( )2

v v v p vt

Taking curl on both sides of the equation of motion, we get

v

0)(

v

tv

0)( v

t

0)(

vwithalong

t

Non-Viscous flow

0)(

vt

withalong

(1) Incompressible homogenous non-viscous fluid

0

vv

)(

v )(

vt

H did t thi ti ? S t lid

Together with appropriate boundary conditions these

How did we get this equation? See next slide :

vr

equations completely determine as a function of and t.r and t.

Equation of motion

For a fluid in motion in a region without sources and sinksFor a fluid in motion, in a region without sources and sinks, the equation of continuity is

0 J

t

where vJ

t

For an incompressible homogeneous fluid, this reduces top g ,

0 v

The velocity field is solenoidal

This is a major difference between flow of a gas and that of a liquid

(2) For the Irrotational flow of a non viscous fluids

If ti it i t ti l0v If vorticity is zero at any particular

point of time then it will remain zero forever

0 v

211 vpv forever

0

vt

2

vpt

(3) For Steady irrotational flow of a homogeneous incompressible(3) For Steady, irrotational flow of a homogeneous, incompressibleNon-viscous fluids0 v 0 v

0 v

01 2

vpv

2

t

or p 1 Cvp 2

21

a constant

In the stead state, every element arriving at A will have the b t th Li f fl d th t lisame subsequent path. Lines of flow and the streamlines are

identical.In the steady state of the homogeneous incompressible

1P d

In the steady state of the homogeneous, incompressible,Non-viscous fluids

21. . 0 .2

P drv v vdt

2 dt

21P Along the streamline21 Constant2

P v Along the streamline

or lines of flow

This is known as Bernoulli’s principleThe value of the constant may be different for different streamlines

For irrotational flow, φ is constant all over the fluid

(4) For steady, non-viscous flow (not necessarily irrotational)

00

ttv

tt

0)(0

vv and0)(0

vt

v and

211)( vpvv

2

)( p

N S d fl i li h f hNote: Steady flow implies that none of the parameters change explicitly with time. At a given point in space, the

i i i i i i ) h i l d i iquantities remain constant in time; ie) the partial derivative with respect to time is zero.

(5) For Steady, of a homogeneous, incompressible,Non isco s fl ids (not irrotational)Non-viscous fluids (not irrotational)

21( ) P

21( )2

Pv v v

0 v

Taking the dot product with velocity on both side, and therefore without invoking the condition of irrotationaltherefore without invoking the condition of irrotational

21P dr 21. . 0 .

2P drv v v

dt

This is the rate of change of φ with displacement along the di ti f tidirection of motion

Streamlines (similar to lines of force in electric field or magneic field)

A curve the tangent to the curve gives instantaneous velocityA curve, the tangent to the curve gives instantaneous velocity at that point, the streamline therefore can’t intersect one anotheranother

AV(t1)

B

( ) N

AB

LV(t2)

N

Streamlines gives instantaneous picture of the velocities at all that point.A Lines of flow is the actual path traced by small element of the moving fluid

Stream TubeA region of the moving fluid bounded on the all sides by streamlines is called a tube of flow or stream tubestreamlines is called a tube of flow or stream tube.

As streamline does not intersect each other, no fluid enters or leaves across the sides.

V

V1

V2

δA1

δA2

Wh d d d d l l h f

( ) 0v

When density do not depend explicitly on time then from continuity equation, we have

( ) 0v

( ) 0v Where V is the volume of the

b b f δA d( ) 0V

v streamtube between faces δA1 and δA2

( ). 0v dS

From divergence theorems

1 2and are the mean values of over A and Av v v 1 21 2

and are the mean values of over A and Av v v

1 21 2A = Av v

For a homogeneous, incompressible, Non-viscous fluids

A A 1 21 2

A = Av v

Thus, whenever stream tube is constricted, i.e., whereverThus, whenever stream tube is constricted, i.e., wherever streamlines gets crowded, the speed of flow of liquid islarger

For the entire collection of the stream tubes occupying the whole cross section of the passage through which the fluid flows is constant along the passagevdS is constant along the passage

A

vdS

In the case of constant density is constantA

vdSA

or 1 21 2A = Av v

1 2