Fluid Flow Equations

7/31/2019 Fluid Flow Equations

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Fluid Flow Equations


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Equation of Continuity

Law of conservation of mass

(1)


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])()([ xxx uuzytzyx

])()([yyy

vvxz

])()([zzz

wwzy

The mass balance is given as:

w

zv

yu

xt

By dividing the entire equation by x y z and taking the

limit as x, y, and z go to zero, and then using the

definitions of the partial derivatives, we get

This is the equation of continuity, which describes the time

rate of change of the fluid density at a fixed point in space.

(2)


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This equation can be written more concisely by usingvector notation as follows

).( Vt

Rate of increase

of mass per unit

volume

Net rate of addition of

mass per unit volume

by convection

(3)


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Recalling Eq.(2) and open all terms on right side as perproduct rule for derivatives we have

w

zv

yu

xt

(2)

zw

z

w

yv

y

v

xu

x

u

t

zw

yv

xu

z

w

y

v

x

u

t

z

w

y

v

x

u

zw

yv

xu

t

).( VDt

D

(4)


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Where is the Substantial derivative or the

derivative following the motion.

This is the rate of density change that would be noted byan observer moving downstream at the velocity of thefluid.

Note that in Eq. (3) is the rate of change observedfrom a fixed point.

).( VDt

D

(4)

Dt

D

t


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Equation of MotionTo get the equation of motion we write a momentumbalance over the volume element x yz in Figure of theform

(5)


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Fluid is flowing through all six faces of the volume

element in any arbitrary direction. First consider the x-component of each term in Eq.(5).

Momentum enters and leaves the volume element partlybyConvection from flow of bulk fluid, and partly by

Viscous action as the result of velocity gradients.

The rate at which the x componentof momentum enters the face at x by

Convection is:

zyuux )(

The rate at which the x componentof momentum leaves the face atx+x byConvection is:

zyuu xx )(


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SimilarlyThe rate at which the x component

of momentum enters the face atybyConvection is:

zxvu y )(

The rate at which the x component

of momentum leaves the face aty+ybyConvection is:

zxvu yy )(

AndThe rate at which the x component

of momentum enters the face at z byConvection is:

yxwu z )(

The rate at which the x component

of momentum leaves the face atz+z b Convection is:

yxwu zz )(


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The net convective flow into volume element is:

xxx uuuuzy )()( yyy vuvuzx )()(

zzz wuwuyx )()( (6)


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Similarly momentum along x component throughmolecular transport.

The rate at which the x component ofmomentum enters the face at x byMolecular transport is:

zyxxx )(

The rate at which the x component of

momentum leaves the face at x+x byMolecular transport is:

zyxxxx

)(

The rate at which the x componentof momentum enters the face atyby

Molecular transport is:

zxyyx )(

The rate at which the x componentof momentum leaves the face at

y+ybyMolecular transport is:

zxyyyx )(


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And

The rate at which the x componentof momentum enters the face at z by

Molecular transport is:

yxzzx )(

The rate at which the x componentof momentum leaves the face atz+z byMolecular transport is:

yxzzzx )(


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Summing up these six contributions gives the net flow ofx momentum into volume element by viscous action:

xxxxxxxzy )()(

zzzxzzx

yx

)()(

yyyxyyxzx )()(

(7)

xxnormal stress on the x face, and yx and zx are the x-directed tangential stresses, or shear stresses,

on the y and z faces resulting from viscous forces, respectively.

Similar expressions may be written for momentum flow in

the y and z directions.

The shear stresses result from the deformation of the volumeelement; the normal stress is related principally to thechange in and the dilation of the element.

xu

/


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In most cases the important forces acting on the systemarise from the fluid pressure p and gravitational force

per unit mass g. The resultant of the forces in the xdirection is:

zyxgppzy xxxx )( (8)

Finally, the rate of accumulationof x component within theelement is:

t

uzyx

(9)


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Substituting the expressions from Eqs.(6), (7), (8) and(9) and putting in Eq. (5) we have:

zyxgppzy xxxx )(

t

uzyx

(10)

xxx uuuuzy )()( yyy vuvuzx )()(

zzz

wuwuyx

)()( xxxxxxx

zy )()(

zzzxzzx

yx

)()( yyyxyyxzx )()(


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Dividing Eq.(10) byxyz, taking the limit as x, yand z approaches to zero, given the x component of the

equation of motion:

(11)

xgxp

zxyxxx

zyxwu

zvu

yuu

xu

t

Above equation may be rearranged with the help of theequation of continuity to give:

x

zxyxxx gzyxx

p

Dt

Du

(12)


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Similarly equations may be derived for the y and zcomponents. Adding the three components vectorially

gives:

(13) ].[pDt

VD

The stress at any point depends on the velocity gradientsand the rheological properties of the fluid.


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Navier-Stokes Equations

For a fluid of constant density and viscosity, theequations of motion, known as the Navier-Stokesequations are

xgx

p

z

u

y

u

x

u

z

uwy

uvx

uut

u

2

2

2

2

2

2

yg

y

p

z

v

y

v

x

v

z

vw

y

vv

x

vu

t

v

2

2

2

2

2

2

zgz

p

z

w

y

w

x

w

z

ww

y

wv

x

wu

t

w

2

2

2

2

2

2


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Fluid Flow Equations

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