The Complex‐Variable Method

The velocity potential and the stream function of a two‐dimensional irrotational flow of an inviscid fluid satisfy relations identical to the Cauchy‐Riemann conditions for an analytic function of a complex variable. This allows the application of the full machinery of complex variable theory to formulate and analyze two dimensional irrotational flows of an inviscid fluid. More specifically, the velocity potential and the stream function can be combined to form an analytic function of a complex variable z = x + iy , in the x , y ‐plane occupied by the flow. Furthermore, one may use the idea of conformal mapping in complex variable theory to relate the solution to a two‐dimensional potential flow for one geometry with that for a simpler geometry. In cases where the flow separates from a rigid boundary and free streamlines appear, one may use the Schwarz‐Christoffel transformation to formulate and analyze the flow.