STREAM FUNCTION &

VELOCITY POTENTIAL CLASS NOTES 

Stream function ‘ψ’ and velocity potential ‘φ’

are arbitrary fictitious parameters. These

do not exist in actual practice. Fluid flow is

a complex phenomenon. Thus these parameters help to

understand complexity of fluid dynamics in a easy manner.

STREAM FUNCTION

It is the volume flux in the flow direction. Volume flux is volume flow rate per meter length normal to the flow direction. Volume per meter becomes two dimensional quantities. Therefore stream function is also 2- dimensional. It is represented by the symbol ‘ψ’ which is pronounced as ‘ psi ’. Since there is no flow normal to a stream line, a streamline can be represented by a ψ= constant parameter.

 STREAM FUNCTION RELATIONS

(i) u = -dψ / dy, velocity in x direction is negative derivative of stream function with respect to ‘y’
(ii) v = dψ/ dx,  velocity in y direction is positive derivative of stream function with respect to ‘x’

These give rise to slope of the stream function, dy/dx = –v/u

PROPERTIES OF THE STREAM FUNCTION

If the stream function (ψ) exists, the flow is there and the continuity equation is automatically satisfied. If stream
function satisfies Laplace Equation, It represents the possible case of ir-rotational flow.

PROVE THAT STREAM FUNCTION ( ψ =C) IS CONSTANT FOR A STREAM LINE

Proof
Assume  ψ = constant
Then dψ =0
We know ψ is  2-D function
d ψ= (∂ψ/∂x)dx + (∂ψ/∂y) dy
Thus dψ = -vdx + u dy
But d ψ=0 for ψ to be constant
-vdx + u dy =0
Then dy/dx = v/u
This equation gives dy/dx from a triangle which is formed with u as horizontal and v as vertical i.e. v is perpendicular to u.
Stream line is a line, which is everywhere tangent to the velocity vector at a given instant. There can be no flow across the stream line. Therefore velocity in perpendicular direction is zero. Therefore dψ=0 is true for a streamline. Thus ψ is constant for a streamline.
Finally    u = -dψ/dy,   velocity in x direction is negative derivative of stream function with respect to ‘y’
v = dψ/dx,    velocity in y direction is positive derivative of stream function with respect to ‘x’

VELOCITY POTENTIAL

Velocity Potential φ is a Scalar Function of space and time co-ordinates such that its NEGATIVE derivative with respect to any direction give the fluid velocity in that direction. Thus φ is a 3-D function.
φ= f(x, y, z, t)

For a steady flow

φ= (x, y, z)

If u, v, w are the components of velocity in x, y and z directions, then by definition

u= – (∂φ/∂x)

v= – (∂φ/∂y)

w= – (∂φ/∂z)
Then ∂u/∂y = ∂^{2 φ}/∂x ∂y
And ∂v/∂x = ∂^{2 φ}/∂x ∂y
Therefore with respect to φ in x and y directions
∂u/∂y =∂ v/∂x

∂y/ ∂x =∂ u/∂v

This gives the slope of the velocity potential as dy/dx = u/v

Multiple of slopes of stream function and velocity potential  gives –1

Thus stream function and velocity potential are at right angles i.e. are ORTHOGONAL TO EACH OTHER.

PROPERTIES OF THE VELOCITY POTENTIAL

If the velocity potential (φ) exists, the flow should be ir-rotational. If velocity potential
function satisfies Laplace Equation, It represents the possible case of steady,
in-compressible, ir-rotational flow. Assumption of a velocity potential is equivalent to
the assumption of ir-rotational flow (non viscous flow).

CAUCHY-RIEMANN EQUATIONS

 These equations connect the stream function and the velocity potential. It has seen that the velocity components of the flow are given in terms of velocity potential and stream function by the equations given below.
u = ∂φ/ ∂x = ∂ ψ/ ∂y
v = ∂φ/ ∂y =  — ∂ ψ/ ∂x

These are the Cauchy-Riemann equations and that  φ and ψ are orthogonal and that both  φ and  ψ obey Laplace Equation. This helps to calculate the stream function when the velocity potential is given and vice versa.

STREAM FUNCTION VS VELOCITY POTENTIAL

It is easily noticeable that velocity potential φ and stream function Ψ are connected with velocity components. It is necessary to understand the similarities and differences between them.
Similarities and Differences Between Stream Function Ψ and Velocity Potential φ
We notice that velocity potential φ and stream function Ψ are connected with velocity components. It is necessary to bring out the similarities and differences between them.

TABLE: Properties of stream function and velocity potential

Property	stream function, Ψ	velocity potential, φ
Continuity Equation	Automatically satisfied means flow is there	Will be satisfied if Laplace equation ∇2 φ=0, Only then the flow will exist
Condition for ir-rotational Flow	Will be satisfied if Laplace equation  ∇2 Ψ=0 and the flow will be an ir-rotational flo	Automatically satisfied

EXIST FOR	Only 2-D	1D, 2D, 3D, all types of flow
EXIST FOR	Viscous flow (rotational flow)  or non- viscous flow (ir-rotational flow)	Only for non –viscous flow or Irrotational flow
EXIST FOR	Incompressible (steady or unsteady flow)	Incompressible (steady or unsteady flow)
EXIST FOR	Compressible flow (only Steady flow)	Compressible flow (Steady or unsteady flow)

Thus both functions exit for a 2-D flow
(i) in-viscid, in-compressible and steady state flow
(ii) in-viscid, compressible and steady state flow

There are lines of constant velocity potential, φ.

There are lines of constant stream function,Ψ.

Constant velocity potential lines and constant stream function lines are at right angles to each other. For example φ lines are circles and Ψ lines are radial lines.

∂φ/∂x = u = ∂Ψ/∂x

∂φ/∂y = v = – ∂Ψ/∂x

 

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