A two-dimensional, potential flow can be represented by a single scalar, may it be the potential function ϕϕ or the stream function ψψ. The governing equation is nothing but the Laplace equation of ϕϕ or ψψ,∇2ϕ=0∇2ϕ=0 and ∇2ψ=0∇2ψ=0.

Either equation, with appropriate boundary conditions, could be used to solve a given potential flow problem. If one (ϕorψ(ϕorψ) 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 ϕ+iψϕ+iψ 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 ϕϕ as well as ψψ for the whole flow field. Next, the task is to plot iso-lines of these quantities, ψψ and ϕϕ, i.e. lines which represent ϕ=constantϕ=constant and ψ=constantψ=constant . This isoline, when overlayed on top of each other, makes a net-like structure, which is called the *flow-net*.

Also Read: What Is Mohr’s Circle

It can be shown that ∇ϕ∇ϕ is orthogonal to ∇ψ∇ψ at every point in the flow-field, except for isolated points which are singular. Therefore, if the gradients of the ϕϕand ψψ are mutually orthogonal to each other, the iso-lines corresponding to ϕϕand ψψ 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 ϕϕ is called the equipotential lines, whereas the same for ψψ is called the streamlines. In conclusion, the streamlines and the equipotential lines are drawn together is the flow-net.

Also Read: What Is Mohr’s Circle

However, application-oriented, real problems are never potential or irrotational, though it could give a first estimate of the solution to the problem.