It is possible to demonstrate that the condition of irrotationality implies the existence of a velocity potential such that
On substituting the definition of potential into the continuity equation we obtain
The velocity potential must then satisfy the Laplace equation and it consequently is a harmonic function of space.
Solution of the Laplace equation, with an appropriate set of boundary conditions, leads then to the determination of the flow field.
Laplace equation has been widely studied in many fields, and shows some interesting properties. Among the latter, one of the most important is its linearity. Given two solutions of the Laplace equation, any linear combination of them (and in particular their sum and difference) is again a valid solution. The potential of a complex flow can then be obtained by superimposing potentials of simpler flows.