EXACT-SOLUTIONS FOR REDISTRIBUTION BY NONLINEAR CONVECTION-DIFFUSION

The nonlinear convection-diffusion equation has been studied for 40 years in the context of nonhysteretic water movement in unsaturated soil. We establish new similarity solutions for instantaneous sources of finite strength redistributed by nonlinear convection-diffusion obeying the dimensionless equation partial derivative theta/partial derivative t = partial derivative(theta(m) partial derivative theta/partial derivative z)/partial derivative z - theta(m+1) partial derivative theta/partial derivative z(m greater-than-or-equal-to 0). For m = 0 (Burger's equation) solutions involve the error function, and for m = 1 Airy functions. Problems 1, 2, and 3 relate, respectively, to the regions 0 less-than-or-equal-to z less-than-or-equal-to infinity, - infinity less-than-or-equal-to z less-than-or-equal-to infinity, and - infinity less-than-or-equal-to z less-than-or-equal-to 0. Solutions for m = 0 have infinite tails, but for m > 0 and finite t, theta > 0 inside, and zero = 0 outside, a finite interval in z. At the slug boundary, theta(z) is tangential to the z-axis for 0 < m < 1 ; and it meets the axis obliquely for m = 1 and normally for m > 1. Illustrative results are presented. For Problems l and 2 (but not 3) finiteness of source strength sets an upper bound on THETA-0, the similarity "concentration" at z = 0. The magnitude of convection relative to diffusion increases with THETA-0 ; and apparently the dynamic equilibrium between the two processes, implied by the similarity solutions, ceases to be possible when THETA-0 is large enough.