Revised Formulation of Fick's, Fourier's, and Newton's Laws for Spatially Varying Linear Transport Coefficients

Abstract

We argue that for situations involving spatially varying linear transport coefficients (diffusivities, thermal conductivities, and viscosities), the original Fick's, Fourier's, and Newton's law equations should be modified to place the diffusivity, thermal conductivity, and viscosity inside the derivative operator, that is, in one-dimensional rectilinear situations, j=-∂(Dc)∂x, q=-∂(kT)∂x, and τxy=-∂(μvy)∂x. We present simple derivations of these formulas in which diffusive mass transfer, conductive heat transfer, and molecular momentum transfer processes are described using lattice random walk models. We also present simple examples demonstrating how these modifications affect calculations.