Pratik Vadher - Fluid Mechanics

Shantilal Shah Engineering College

Production Engineering

Sem.- 4 th.Group- 22

Velocity Potential & Potential Flow, Relation between Stream

function & Velocity Potential

by: Pratik Vadher 130430125113

Guided by : Prof. Vinay A Parikh

Prof. Prashant V Sartanpara Prof. Nital P Nirmal

DIFFERENCES BETWEEN f and y

1. Flow field variables are found by: Differentiating f in the same direction as velocities Differentiating y in direction normal to velocities

2. Potential function f applies for irrotational flow only3. Stream function y applies for rotational or irrotational flows

4. Potential function f applies for 2D flows [f(x,y) or f(r,q)] and 3D flows [f(x,y,z) or f(r,q, f)]

5. Stream function y applies for 2D y(x,y) or y(r, q) flows only

6. Stream lines (y =constant) and equipotential lines (f =constant) are mutually perpendicular Slope of a line with y =constant is the negative reciprocal of the

slope of a line with f =constant

Velocity Potential Function

It is defined as a scalar function of space & time such that its negative derivative w.r.t any direction gives the fluid velocity in that direction. It is defined by f (Phi). Mathematically, the velocity, potential is defined as f = f (x,y,z) for steady flow such that

Where u, v and w are the components of velocity in x, y & z directions respectively.

The velocity components in cylindrical polar co-ordinates in terms of velocity potential function are given by

Where Ur = Velocity component in radial direction& Uq = Velocity component in tangential direction

The continuity equation for an incompressible steady flow is

Substituting the values of u, v & w from above equation, we get

For two-dimensional case, above equation reduces to

If any value of (f Phi) that satisfies the Laplace equation, will correspond to some case of fluid flow.Properties of the Potential function. The rotational components are given by

Conti..

Substituting the values of u, v and w from equation in the above rotational components, we get

If f is a continuous function,

Then equ.

Therefore

When rotational components are zero, the flow is called irrotational. Hence the properties of the potential function are :

1. If velocity potential ( ) f exits, the flow should be irrotational.

2. If velocity potential ( ) f satisfies the Laplace equ. It represents the

possible steady incompressible irrotational flow.

Stream FunctionIt is defined as the scalar function of space &

time, such that it’s partial derivative w.r.t any direction gives the velocity component at right angles to that direction. It is denoted as y (Psi) and defined only for two-dimensional flow. Mathematically, for steady flow it is defined as y = f(x,y) such that

And

The velocity components in cylindrical polar co-ordinates in terms of stream function are given as

Where Ur = radial velocity and Uq= tangential velocity.

The continuity equation for two dimensional flow is

Substituting the values of u and v from above equation, we get

Hence existence of y means a possible case of fluid flow. The flow may be rotational or irrotational.The rotational component Wz is

Given by

Substituting the values of u and v from equation in the above rotational component, we get

For irrotational flow, Wz = 0. Hence above equation becomes as

Which is Laplace equation for .y

The Properties of Stream Function ( )y are :

1. If stream function ( ) y exists, it is possible case of fluid flow which may be rotational or irrotational.

2. If stream function ( ) y satisfies the Laplace equation, it is a possible case of an irrotational flow.

Relation between Stream Function & Velocity Potential Functions

We have,

From stream function equation we have,

Thus, we have

Hence