Diffusion-Controlled Reaction in a Vortex Field

Abstract

A two-dimensional model of a constant-density diffusion-controlled reaction between unmixed species initially occupying adjacent half-spaces is formulated and analyzed. An axisymmetric viscous vortex field satisfying the Navies-Stokes equations winds up thc interface between the species as they diffuse together and react. A flame-sheet approximation of the rapid reaction is made using a mixture fraction dependent variable. The problem was originally proposed by F. Marble, who performed a local analysis and determined the total consumption rate along thc flame sheet. The present paper describes a global similarity solution to the problem which is Fourier analyzed in a Lagrangian coordinate system. The Fourier amplitudes are determined both by an asymptotic analysis, valid for large Schmidt numbers,and by numerical solution of the two-point boundary-value ordinary differential equations. The solution is evaluated in both Lagrangian and Eulerian coordinate systems. Comparisons are made between the asymptotic and the numerical solutions for a variety of values of the governing parameters, the Reynolds and Schmidt numbers.