ANZIAM  J.  44 (2003), 365-380
n-Dimensional first integral and similarity solutions for two-phase flow

S. W. Weeks
  School of Mathematical Sciences
  Queensland University of Technology
  GPO Box 2434
  Brisbane QLD 4001
G. C. Sander
  Department of Civil and Building Engineering
  Loughborough University
  LE11 3TU
J.-Y. Parlange
  Department of Biological and Environmental Engineering
  Riley-Robb Hall
  Cornell University
  Ithaca NY

This paper considers similarity solutions of the multi-dimensional transport equation for the unsteady flow of two viscous incompressible fluids. We show that in plane, cylindrical and spherical geometries, the flow equation can be reduced to a weakly-coupled system of two first-order nonlinear ordinary differential equations. This occurs when the two phase diffusivity $D(\theta)$ satisfies $(D/D')'=1/\alpha$ and the fractional flow function $f(\theta)$ satisfies $df/d\theta = \kappa D^{n/2}$, where n is a geometry index (1, 2 or 3), $\alpha$ and $\kappa$ are constants and primes denote differentiation with respect to the water content $\theta$. Solutions are obtained for time dependent flux boundary conditions. Unlike single-phase flow, for two-phase flow with n = 2 or 3, a saturated zone around the injection point will only occur provided the two conditions $\int_0^1D/(1-f) \, d\theta <\infty$ and $f'(1) \ne 0$ are satisfied. The latter condition is important due to the prevalence of functional forms of $f(\theta)$ in oil/water flow literature having the property $f'(1) = 0$.
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