Reciprocal identities and integral formulations for diffusive scalar transport and Stokes flow with position-dependent diffusivity or viscosity

Abstract

Reciprocal relations between any two solutions of the equations of diffusive scalar transport with position-dependent diffusivity are derived, and integral representations involving the Green’s function of Laplace’s equation and accompanying integral equations are developed. When the diffusivity exhibits mild variations over the solution domain, the rate of transport across an isoscalar surface can be predicted from the solution for uniform diffusivity. A corresponding analysis is presented for the equations of Stokes flow with position-dependent fluid viscosity. When the viscosity exhibits mild variations, the force and torque exerted on a surface moving as a rigid body can be predicted from the solution for constant viscosity. Numerical methods for solving the integral equations based on boundary-element and domain discretizations into triangular elements are developed in two dimensions.