MSI Banner

[Back][Index][Help][MSI][ANU Online]

Research Report MRR97-056

Analytical solution of the spatially variable coefficient advective-diffusion equation in one-, two- and three-dimensions

C. Zoppou and J.H. Knight

Abstract: Analytical solutions are provided for the one-, two- and three-dimensional advective-diffusion equation with spatially variable velocity and diffusion coefficients. By assuming that the velocity is proportional to distance and the diffusion coefficient is proportional to the square of the velocity, there is a simple transformation which will reduce the spatial variable equation into a constant coefficient problem. There are available a large number of known analytical solutions for general initial and boundary conditions to the constant coefficient problem. These solutions are also solutions to the spatially variable advective-diffusion equation. The special form of the spatial coefficients have practical relevance and for divergent free flow represent corner or straining flow.

Unlike many other analytical solutions, we use the transformation to obtain solutions of the spatially variable coefficient advective-diffusion equation in one-, two- and three-dimensions. The analytical solutions, which are simple to evaluate, can be used to validate numerical models for solving the advective-diffusion equation with spatially variable coefficients. For numerical schemes which cannot handle flow stagnation points, we provide analytical solution to the spatially variable coefficient advective-diffusion equation for two-dimensional corner flow which contains an impermeable flow boundary. The impermeable flow boundary coincides with a streamline along which the fluid velocity is finite but the concentration vanishes. This example is useful for validating numerical schemes designed to predict transport around an irregular boundary.

This service is maintained by the Mathematical Sciences Institute (MSI)
Comments to webmaster@maths.anu.edu.au URL: http://wwwmaths.anu.edu.au/