Laplace's equation for convective scalar transport in potential flow

Laplace's equation for the stream function and the velocity potential has been used for a long time for the solution of irrotational flow problems in hydrodynamics. Here, a similar formulation is presented for the convective transport of a scalar such as heat or mass. It is discovered that, in suitable coordinates, the so-called heat function also satisfies Laplace's equation. This function, which was introduced earlier for the visualization of heat flow in two-dimensional convection, can be used to build up solutions for scalar transport, both analytical and numerical, by superposition, in much the same way that the stream function or velocity potential can. A vast array of solutions and techniques, developed over centuries, therefore becomes available for the scalar transport problem.