MOTION OF FLUID ELEMENT (KINEMATICS)

Before formulating the effects of forces on fluid motion (dynamics), first we consider the motion (kinematics) of a fluid in a flow field. When a fluid element moves in a flow field, it may under go translation, linear deformation, rotation, and angular deformation as a consequence of spatial variations in the velocity.

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Figure. Pictorial representation of the components of fluid motion in a flow field.

Rate of Translation

Tranlation in unit time is equal to velocity,

Acceleration of a Fluid Particle in a Velocity Field

Figure. Motion of a particle in a flow field.

Consider a particle moving in a velocity field. At time t, the particle is at a position x,

y, z and has a velocity .

At time t+dt, the particle has moved to a new position, with coordinates x+dx, y+dy, z+dz, and has a velocity given by

The change in the velocity of the particle moving from location to

is given by

Dividing both sides by dt, the total acceleration of the particle is obtained as

Since

then

),,,( tzyxVVtp

),,,( dttdzzdyydxxVVdttp

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Vdy

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t

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y

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Vda

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ktzyxwjtzyxvitzyxutzyxVp

,,,,,,,,,,,,

Acceleration of a fluid particle in a velocity field requires a special derivative, it is

denoted by the symbol .

Thus,

This derivation is called the substantial, the material or particle derivative.

The significance of the terms,

The convective acceleration may be written as a single vector expression using the vector gradient operator, .

Thus,

It is possible to express the above equation in terms of three scalar equations as

The components of acceleration in cylindrical coordinates may be obtained by utilizing the appropriate expression for the vector operator . Thus

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t

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Example: The velocity field for a fluid flow is given by

Determine:

a) the acceleration vector,

b) the acceleration of the fluid particle at point P(1,2,3) and at time t = 1 sec.

kztjxyıxtzyxV

32),,,( 2

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To be completed in class

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FLUID ROTATION Definition: The rotation of a fluid particle is defined as the average angular velocity of any two mutually perpendicular line elements of particle in each corrdinate plane. Hence a particle may rotate about three coordinate axes. Thus, in general, rotation of a fluid element can be expressed as: kjı zyx

u v

A A

B

B

2

y

y

uu

2

y

y

uu

2

x

x

vv

2

x

x

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b

B

A’ A

B’

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x

Figure. Rectangular fluid particle with two instantaneous perpendicular line AA’ and BB’.

By definition, the rotation of fluid element about z-axis can be written as

dt

d

dt

dBBAAz

ba

2

1

2

1

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1

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By considering the rotation of pairs of perpendicular lines in the yz and xz planes, one can show that

Then

We recognize the term in the square brackets as

Then, in vector notation, we can write

The factor of ½ can be eliminated in above equation by defining a quantity called the vorticity, , to be twice the rotation,

The vorticity is the measure of the rotation of a fluid element as it moves in the flow field.

In cylindrical coordinates the vorticity is

x

w

z

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1 1z r z rr z

V rVV V V VV e e e

r z z r r r

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Circulation

The circulation, , is defined as the line integral of the tangential velocity component about a closed curve fixed in the flow,

where, is an elemental vector, of the length ds, tangent to curve; a positive sense corresponds to a counterclockwise path of integration around the curve.

For the closed curve Oacb,

Thus, the circulation around a closed contour is equal to the total vorticity enclosed within it.

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2

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A

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C

)(2

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Example: Consider flow fields with purely tangential motion (circular streamlines): Vr = 0 and V = f(r). Evaluate the rotation, vorticity, and circulation for rigid-body rotation, and “a forced vortex”. Show that it is possible to choose f(r) so that the flow is irrotational; to produce “a free vortex”.

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To be completed in class

Angular Deformation of a Fluid Element

Definition: Angular deformation of a fluid element involves changes in the angle between two mutually perpendicular lines in the fluid.

dt

d

dt

d

dt

d ba

x

v

t

xtxxv

t

x

tdt

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dt

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Figure. Angular deformation of a fluid element in a two dimensional flow field.

From the definition, the rate of angular deformation of fluid element can be expressed as

The shear stress is related to the rate of angular deformation through the fluid viscosity. For one-dimensional Newtonian laminar flow the shear stress is given by

Consequently, the rate of the angular deformation in the xy plane is obtained as

Rate of the angular deformation in the yz plane z

v

y

w

Rate of the angular deformation in the zx plane z

u

x

w

y

uyx

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Linear Deformation

x

y

x

y

x

y

u u+(du/dx)x

v

Definition: Rate of linear deformation of a fluid element is defined as the change in the unit length in unit time in each coordinate direction.

x+(du/dx)xt

Rate of linear deformation in x-dir:

x

u

xt

xtxxux

)/(

Rate of linear deformation in y-dir: y

v

Rate of linear deformation in z-dir: z

w

Similarly in y- and z-directions,

Change in the length of the sides of the fluid element may produce change in volume of the element. The rate of local instantaneous volume dilatation is given by

Volume dilation rate Vz

w

y

v

x

u

MOMENTUM EQUATION

To derive the differential form of momentum equation, we shall apply Newton’s second law to an infinitesimal fluid particle of mass dm.

Newton’s second law for a finite system is given by

where the linear momentum, , of the system is given by

Then for an infinitesimal system of mass dm, Newton’s second law is written as

systemdt

PdF

P

)(systemmass

system dmVP

t

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y

Vv

x

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t

V

z

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y

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VDdmFd

system

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Forces Acting on a Fluid Particle

The forces acting on a fluid element may be classified as body forces and surface forces. Surface forces include both normal forces and tangential (shear) forces. Surface force acting on a fluid element can be expressed in terms of stresses.

Stresses acting on a differential fluid element in the x-direction are shown in the figure.

Figure. Stresses in the x direction on an element of fluid.

To obtain the net surface force in the x-direction, , we must sum the forces in the x direction.

xSdF

dxdydz

zdxdy

dz

z

dxdzdy

ydxdz

dy

y

dydzdx

xdydz

dx

xdF

zxzx

zxzx

yxyx

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22

22

22

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dxdydzzyx

dF zxyxxxSx

dxdydzzyx

gdFdFdF zxyxxxxSBx xx

dxdydzzyx

gdFdFdF

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gdFdFdF

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x

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t

v

zyxg

zyyyxyy

z

ww

y

wv

x

wu

t

w

zyxg zzyzxz

z

These three equations are the differential equations of motion for any fluid satisfying the continuum assumption. Before the equations can be used to solve problems, suitable expressions for the stresses must be obtained in terms of the velocity and pressure fields. Since the relation between stress and velocity is different for Newtonian and non-Newtonian fluids, to express the stresses in terms of velocity and pressure, we need to identify the type of fluid.

Newtonian Fluid: Navier-Stokes Equations

For a Newtonian fluid the viscous stress is proportional to the rate of shearing strain (angular deformation rate). The stresses may be expressed in terms of velocity gradients and fluid properties in rectangular coordinates as follows:

where p is the local thermodynamic pressure.

If these expressions are introduced into the differential equations of motion, we obtain

y

u

x

vyxxy

z

v

y

wzyyz

x

w

z

uxzzx

x

uVpxx

2

3

2

y

vVpyy

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3

2

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2

3

2

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These equations of motion are called the Navier-Stokes equations. The equations are greatly simplified when applied to incompressible flow with constant viscosity. Under these conditions the equations reduce to

The Navier-Stokes equations in cylindrical coordinates, for constant density and viscosity, are given in the textbook.

For the case of frictionless flow ( = 0) the equations of motion reduce to Euler’s equation,

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