Lie Group Solutions of Advection-Diffusion Equations

Transport phenomena in homogeneous and inhomogeneous media are commonly encountered in many practical and industrial applications, which are modeled by advection-diffusion equations (ADEs) with constant or variable diffusivities, respectively. This paper provides a new perspective on how to solve advection-diffusion equations that model different transport phenomena in low Reynolds number flows. A mathematical description of the Lie group method is conducted first and then its potential in solving advection-diffusion equations for passive scalars transport with no-slip and no-flux boundary conditions is explored. The key step is to recast advection-diffusion equations as homogeneous diffusion processes on unimodular matrix Lie groups. Consequently, an approximate solution can be obtained from mean and covariance propagation techniques developed for diffusion equations on these Lie groups. The motivation to transform the advection-diffusion equation from Euclidean space to Lie groups is to exploit the available solutions of diffusion equation on these Lie groups so that the original equation can be solved in a simple way. In this paper, methodological details have been illustrated in solving ADEs modeling three kinds of transport phenomena. Two of them govern homogeneous transport and the solutions from mean and covariance propagation on the Lie group agree well with available results in published papers. We also use this method to solve more complicated ADEs governing inhomogeneous transport in one-dimensional compressible flows with spatially varying diffusivity, which is beyond the capabilities of existing approaches. The three real problems solved by the Lie group method illustrate the potential of this method. Instead of numerical calculations, the proposed closed-form method provides a simple alternative to study mass transfer encountered in various complex physical and industrial processes.