ASYMPTOTIC PROFILES WITH FINITE MASS IN ONE-DIMENSIONAL CONTAMINANT TRANSPORT THROUGH POROUS-MEDIA - THE FAST REACTION CASE

The paper considers the large-time behaviour of Positive solutions of the equation partial derivative(u + u(p))/partial derivative t = parti al derivative 2u/partial derivative x2 - partial derivative u/partial derivative x, p > 0 with - infinity < x < infinity and t greater-than- or-equal-to 0, for pulse-type initial data. In suitably scaled variabl es this equation models the one-dimensional flow of a solute through a porous medium with the solute undergoing absorption by the solid matr ix of the medium. With the total mass both absorbed and in solution in variant, it is shown that the asymptotic solution depends crucially on the value of p. For p > 2 the solution approaches the symmetric solut ion of the linear heat equation centred on x = t, while for p = 2 this becomes asymmetric due to the effect of nonlinearity. For 1 < p < 2 c onvection dominates at large time and the solution approaches the form of an asymmetric pulse moving at unit speed along the positive x-axis . Diffusion effects are confined to regions near the leading and trail ing edges of the pulse. For 0 < p < 1 the pulse is still convection-do minated but it no longer moves under a simple translation. Again diffu sion only becomes important near the leading and trailing edges. It is shown that for p greater-than-or-equal-to 2 the asymptotic solutions are uniformly valid in x but for 0 < p < 2 convection-dominated outer solutions have to be supplemented by diffusion boundary layers. In thi s latter case uniformly-valid composite solutions can be constructed. Finally our asymptotic analyses for the various values of p are compar ed with numerical solutions of the initial-value problem.