Some exact solutions of convection‐diffusion and diffusion equations

Abstract

We examine solutions of the convection‐diffusion equation during steady radial flow in two and three dimensions with diffusivity proportional to Pn, with P the Péclet number. In two dimensions, closed‐form solutions of the Laplace‐transformed equation occur only for n = 1 and n = 2; and in three dimensions only for n = 5/4 and n = 2. The inversions for n = 1 (two dimensions) and 5/4 (three dimensions) cannot be expressed in closed form. On the other hand, the value n = 2, proposed by de Gennes (1983), leads to simple exact solutions, both in two dimensions and in three. In both cases a change of variable gives a one‐dimensional equation with constant coefficients. Exact instantaneous and continuous point source solutions for radial flow in two and three dimensions are thus established. In general, finite difference methods are needed for arbitrary n. Relevant instantaneous and continuous point source diffusion solutions are developed for radial two‐ and three‐dimensional systems. These represent small‐time solutions useful for initiating the finite difference solutions for convection‐diffusion with arbitrary n.