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PRESENTED BY:

1NAVIER'S-STOKES EQUATION

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` Fora fluid of constant density and constant

viscosity we used equationof motion known as

Naviers-stokes equation.` In vector form:

gVP

Dt

DV!

2V

2NAVIER'S-STOKES EQUATION

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` Considera volume element,now the momentun

balance onthe element as

` Rate of accumulation = (rate of momentum in)-(rate of momentum out)+(forces actingonthe

system)

` eqn-(1)

` The velocityof fluid In cartesian form

kwjviuV !

3NAVIER'S-STOKES EQUATION

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` Considerthe fluid flowsthoughthe six facesofthe

element.First we considered that

` The fluid flowingthroughthe x-direction:

` Momentum enters at x =

` Momentum leaves at x+x =

` Similarly fory-direction:

`

Momentum enters aty =` Momentum leaves aty+y =

zyuux

(()(V

zyuuxx ((()(V

zxuu y (()(V

zxuuyy

(((

)(V

4NAVIER'S-STOKES EQUATION

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` Forz-direction:

` Momentum enters at z =` Momentum leaves at z+z =

` Now the net convective flow throughthe element:

eqn-(2)

yxuuz

(()(V

yxuuzz

((()(V

])()[(

])()[(])()[(

zzz

yyyxxx

uuuuyx

uuuuzxuuuuzy

(

((

((

((((

5NAVIER'S-STOKES EQUATION

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` Now the normal stress actingonthe element:

`

` Normal Stress actingonthe x-face:

` Normal stress at x =

` Normal stress at x+x =

` Similarlynormal stress actingony-face:

` Normal stress aty =

`

Normal stress aty+y =

zyxxx

(()(X

zyxxxx

(((

X

zxyyx

((X

zxyyyx (((X

6NAVIER'S-STOKES EQUATION

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` Forthe z-face:

` Normal stress at z =` Normal stress at z+z =

` The netnormal stress actingonthe element:

`

eqn-(3)

` The shearstressresult from the deformationof

the volume element.

yxz

zxX

yxzzzx

X

][

][][

zzzxzzx

yyyyxxxxxxxx

yx

yxzxzy

(

((

((

((((

7NAVIER'S-STOKES EQUATION

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` The fores arise due tothe system is fluid pressure

p and the gravitational force perunit massg.

` The resultantofthe forces inthe x-direction:

` eqn-(4)` Finallythe rate of accumulationof x momentum

withinthe element isx yz (u/ t)

` Substitute all the above expression inthe equationno. 1 and resulting equation divide

zyxgppzyxxxx

V

8NAVIER'S-STOKES EQUATION

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` byxyz.takingthe limit asx,y, and z

appoachesto zerogivesthe x componentofthe

equationof motion:

` eqn-(5)

xzxyxxxg

x

p

ttt

wu

t

vu

t

uu

t

u

t

VXXX

VVVV

x

x

x

x

x

x

x

x

x

x

x

x

x

x!

x

x

)(

)(

9NAVIER'S-STOKES EQUATION

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` The equation isrearranged withthe help of

equationof continuitytogive:

`

eqn-(6)` Similarequation may be derived forthe y and z

components. Addingthe three equation vectorially

gives

` eqn-(7)

xzxyxxx gzyxt

p

Dt

DuVXXXV

x

x

x

x

x

x

x

x! )(

gpDt

DV VXV ! ][

10NAVIER'S-STOKES EQUATION

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` The stress at any point is dependsonthe velocity

gradient and the rheological propertiesofthe fluids.Forthe newtonian fluids,the x-direction component

ofthe stresstensorare

` eqn-(8)

eqn-(9)

eqn-(10)

z

w

y

v

x

uV

x

v

y

u

Vx

u

yxxy

xx

x

x

x

x

x

x!

x

x

x

x!!

x

x!

)(

))(3

2(2

QXX

OQQX

11NAVIER'S-STOKES EQUATION

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` Where k= bulk viscosity

` The general equationof motion fora newtonian

fluid with varying and viscosity are exemplified by

the following equation forthe x direction:

`

` eqn(11)

xg

z

u

x

w

zx

v

y

u

y

Vx

u

x

p

Dt

Du

VQ

OQQV

xx

xx

xx

xx

xx

xx

xx

xx

!

][][

])3

2(2[

12NAVIER'S-STOKES EQUATION

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z

y

gVz

w

z

z

v

y

w

yz

v

x

w

xz

p

Dt

Dw

gz

v

y

w

z

Vy

v

y

y

u

x

v

xy

p

Dt

Dv

VOQQ

QQV

VQ

OQQ

QV

xx

xx

x

x

x

x

x

x

x

x

x

x

x

x

x

x!

xx

xx

xx

xx

xx

xx

xx

xx

xx

!

]3

22[

][][

][

]3

22[

][

13NAVIER'S-STOKES EQUATION

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` Fora fluid of constant density and constant viscosity

the equations (5) becomes

`

` eqn(14)

` eqn(15)

xx

xx

xx

xx

z

uw

y

uv

x

uu

t

uV

xg

x

p

z

u

y

u

x

u VQ xx

xx

xx

xx! )(

2

2

2

2

2

2

x

x

x

x

x

x

x

x

z

u

w

u

v

u

ut

u

V

gp

z

uuuVQ

x

x

x

x

x

x

x

x!

2

2

2

2

2

2

14NAVIER'S-STOKES EQUATION

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x

x

x

x

x

x

x

x

z

uw

y

uv

x

uu

t

wV

zg

z

p

z

w

y

w

x

wVQ

x

x

x

x

x

x

x

x!

2

2

2

2

2

2

gVpDt

DVVQV ! 2

15NAVIER'S-STOKES EQUATION

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` Unitoperationof chemical engineering-MacCabe

smith and harriott

NAVIER'S-STOKES EQUATION 16