Similarity Solutions of Radially Symmetric Two-Phase Flow

Abstract

When both the diffusivityD and fractional flow functionf have a power law dependence on the water content θ, i.e.D=Doθα andf=θα+1, the nonlinear transport equation for radially symmetric two phase flow can, in certain circumstances, be reduced to a weakly coupled system of two first order nonlinear ordinary differential equations. Numerical solutions of these equations for a constant flux boundary conditionVwo and comparison with experimental data are given. In particular, when the fluxVwo and a are related byVwo(α + 1)/Do=2, a new fully explicit analytical solution is found as θ(r, t)=(1 − αr2/4Dot)1/α forr2 < 4Dot/α and θ(r, t)=0 forr2 ≥ 4Dot/α We show that the existence of this exact soution is due to the presence of a Lagrangian symmetry.