Stationary Diffusion

Abstract

At steady state, pure diffusion of energy and momentum are described, respectively, by the Laplace equation and the Stokes equation. The former describes the stationary temperature (or concentration) profile, in the absence of any heat convection. The Stokes equation describes the steady state velocity and pressure profiles at low Reynolds number conditions (that is, again, in the absence of any convective effects) of an incompressible fluid. In this Chapter, we study some important properties of these equations, all centered upon the behavior of harmonic functions, described in Sect. 22.1. Then, in Sect. 22.2, these properties are applied to determine the general solution of the Stokes equation. In particular, the uniform flow past a sphere is studied, finding the celebrated Stokes law expressing the drag force in terms of the fluid velocity.