In the present analysis of an irrotational plane flow, the velocity field can be obtained in terms of a stream function instead of a potential function.
We can in fact define a (scalar) stream function
that satisfies identically the continuity equation for the Schwarz theorem on mixed derivatives. Such a function is called the stream function because its isolines are streamlines (that is lines such that at any given time they are tangent to the velocity vector). Note that, by definition, the component of the velocity normal to a streamline is always zero so that there is no mass flux across a streamline. Every solid body or boundary must then be represented by a streamline.
If we now make use of the irrotationality of the flow we obtain:
So the stream function satisfies the Laplace equation, hence being a harmonic function of space.
Stream function and velocity potential are both harmonic functions of space and are related by the following equations
Two bi-dimensional harmonic functions that satisfy the above conditions (the so-called Cauchy-Riemann conditions) are said to be conjugate.
It is not difficult to demonstrate that by using the Cauchy-Riemann conditions, lines along which the stream function is constant (streamlines) and lines along which the velocity potential is constant (isopotential lines) always intersect at right angles.