Utilization of nonlinear diffusion solutions forn-diffusion and power-law materials

Abstract

The equation ofn-diffusion, as originally formulated by J. R. Philip for various problems involving unsteady turbulent flows, applies directly to the one-dimensional flow of a non-Newtonian power-law fluid as well as indirectly to the laminar boundary layer flow of such fluids over a flat plate. The purpose of this note is to utilize a simple relation between one-dimensionaln-diffusion and nonlinear diffusion with power-law diffusivity such that ifp(x, t) denotes ann-diffusion pressure thenc(x, t)=|ϖp/ϖx| satisfies the nonlinear diffusion equation with power law diffusivity. This means in particular that the large number of solutions presently known for nonlinear diffusion can be utilized in the context ofn-diffusion. Known solutions ofn-diffusion are obtained via this procedure as well as newn-diffusion solutions, including the source solution and a solution of the problem of fluid withdrawal at a constant flow rate for a non-Newtonian fluid in a porous medium of infinite extent. Solutions arising from other exact nonlinear diffusion profiles are also investigated as well as the limiting case ofn-diffusion forn tending to infinity and the results are displayed graphically.