Distribution of a trace element in a boundary layer with mass transfer

Abstract

A theory is presented for the downstream distribution of a tracer injected into a boundary layer of a flat plate or cone with self-similar mass transfer. The tracer, which is injected over a finite length of the surface, is assumed to be chemically inert. Its behavior is controlled by a frozen diffusion equation. The assumption of constant density-viscosity product, and unit Schmidt number results in a linear partial differential equation for the seed mass fraction. This is solved analytically in series form by separation of variables, and numerically by a finite difference technique. It is shown that far downstream of the tracer injection region the analytical results reduce to a simple form; namely, the mass concentration profile is proportional to the local shear even in the presence of mass transfer. However, near the tracer injection region, even the first ten terms are insufficient to yield profiles comparable to those produced by the numerical integration scheme. NomenclatureC = trace species D = diffusion coefficient / = Blasius function with mass transfer M = total mass rate of injection of trace species N = eigenfunction with mass transfer u,v = velocity components along and normal to the surface x,y = Cartesian coordinates along and normal to the surface z = transformed similarity variable £,17 = similarity variables X = eigenvalue 0 = mass fraction of tracer in ablating material p = density IJL = coefficient of viscosity (pv) w ~ mass transfer at solid surface Subscripts e = outer edge of boundary layer h = trailing edge of injection region Received May 12, 1969.