On the Concentration Gradient across a Spherical Source Washed by Slow Flow

Abstract

A model has been numerically analyzed to help interpret the orienting effects of flow upon cells. The model is a sphere steadily and uniformly emitting a diffusible stuff into a medium otherwise free of it and moving past with Stokes flow. Its properties depend primarily upon the Peclet number, Pe, equal to a . v(infinity)/D, i.e., the sphere's radius, a, times the free stream speed, v(infinity), over the stuff's diffusion constant, D. As Pe rises, and washing becomes more effective, the average surface concentration, C(s a) falls (Figs. 2 and 5) and the residual material becomes relatively concentrated on the sphere's lee pole (Figs. 2 and 4). Specifically, as Pe rises from 0.1 to 1, the relative concentration gradient, G, rises from 0.7 to 5.0 per cent and to the point where it is rising at about 8 per cent per decade; by Pe 1000, G = 22.1 per cent. From Pe 1 through 1000, G/(1 - C(s a)), or the gradient per concentration deficiency remains at about 26 per cent suggesting that G approaches a ceiling of about 26 per cent. Also from Pe 1 through 1000, the average mass transfer co-efficient nearly equals that previously calculated for spheres maintaining constant surface concentration instead of flux. The complete differential equation without approximations, the Gauss-Seidel method, and an approximation for the outer boundary condition were used.