Self-similar solutions of the second kind of nonlinear diffusion-type equations

Abstract

We study the self-similar solutions of the problem of one-dimensional nonlinear diffusion of a passive scalar u u (diffusivity D ∞ u m , m ≥ 1 D \infty {u^m}, m \ge 1 ) towards the centre of a cylindrical or spherical symmetry. It is shown that this problem has a self-similar solution of the second kind. The self-similarity exponent δ \delta is found by solving a nonlinear eigenvalue problem arising from the requirement that the integral curve that represents the solution must join the appropriate singular points in the phase plane of the diffusion equation. In this way the integral curves that describe the solution before and after the diffusive current arrives at the centre of symmetry can be determined. The eigenvalues for different values of the nonlinearity index m m and for cylindrical and spherical geometry are computed. Numerical integration of the equations allows us to determine the shape of the solution in terms of the physical variables. The application to the case m = 3 m = 3 , corresponding (for cylindrical symmetry) to the creeping gravity currents of a very viscous liquid, is worked out in detail.