On Two-Dimensional Convection-Diffusion Past a Circle
Charles Knessl · SIAM Journal on Applied Mathematics · 2001
We consider the concentration p(x,y) of some substance at the point $(x,y)\in \mbox{\boldmath R}^2$ exterior to the unit circle. The substance diffuses in the plane and is convected to the right by a uniform field. We assume that the substance cannot penetrate the circular obstacle, so its flux must vanish on the obstacle's boundary. We also assume that the concentration is uniform far away from the obstacle, so we take $p\lraw 1$ as $x^2+y^2\lraw\infty$. The concentration satisfies a linear elliptic PDE. We explicitly solve this problem and evaluate the solution in the asymptotic limit where convection dominates diffusion (i.e., the Peclet number is large). We also develop singular perturbation methods to treat problems of this type, and these should apply to situations where the obstacle has a more complex shape. Numerical studies are used to back up the asymptotic analysis.