Stream Function, Constancy of ψ on a Streamline and Stream function for an irrotational flow

Stream Function

Let us consider a two-dimensional incompressible flow parallel to the x - y plane in a rectangular cartesian coordinate system. The flow field in this case is defined by

u = u(x, y, t) v = v(x, y, t) w = 0

The equation of continuity is

(10.1)

If a function ψ(x, y, t) is defined in the manner

	(10.2a)
	(10.2b)

so that it automatically satisfies the equation of continuity (Eq. (10.1)), then the function is known as stream function. 
Note that for a steady flow, ψ is a function of two variables x and y only.

Constancy of ψ on a Streamline

Since ψ is a point function, it has a value at every point in the flow field. Thus a change in the stream function ψ can be written as

The equation of a streamline is given by

It follows that dψ = 0 on a streamline.This implies the value of ψ is constant along a streamline. Therefore, the equation of a streamline can be expressed in terms of stream function as

ψ(x, y) = constant

(10.3)

Once the function ψ is known, streamline can be drawn by joining the same values of ψ in the flow field.

 Stream function for an irrotational flow
In case of a two-dimensional irrotational flow

Conclusion drawn:For an irrotational flow, stream function satisfies the Laplace’s equation