A two-dimensional, potential flow can be represented by a single scalar, may it be the potential function or the stream function . The governing equation is nothing but the Laplace equation of or , and .
Either equation, with appropriate boundary conditions, could be used to solve a given potential flow problem. If one ) is solved for, the other could be easily obtained as they are harmonic-conjugates of each other, or in other words, the complex potential W defined as is an analytic function as it satisfies the Cauchy-Riemann equations.
Given that one solves the Laplace’s equation, and along with Cauchy-Riemann conditions, one can get as well as for the whole flow field. Next, the task is to plot iso-lines of these quantities, and , i.e. lines which represent and . This isoline, when overlayed on top of each other, makes a net-like structure, which is called the flow-net.
It can be shown that is orthogonal to at every point in the flow-field, except for isolated points which are singular. Therefore, if the gradients of the and are mutually orthogonal to each other, the iso-lines corresponding to and would also be perpendicular to each other.
Furthermore, from complex analysis, harmonic-conjugates of an analytic complex function are known to be mutually orthogonal trajectories (from the notion of conformal transformation). The iso-lines of is called the equipotential lines, whereas the same for is called the streamlines. In conclusion, the streamlines and the equipotential lines are drawn together is the flow-net.
However, application-oriented, real problems are never potential or irrotational, though it could give a first estimate of the solution to the problem.