A class of exactly solvable free-boundary inhomogeneous porous medium flows

We describe a class of inhomogeneous two-dimensional porous medium flows, driven by a finite number of multipole sources; the free boundary dynamics can be parametrized by polynomial conformal maps. 1 Inhomogeneous Porous-Medium Flows A class of two-phase porous medium flows in two dimensions involves the dynamics of the boundary ∂Ω(t) in the (x, y) plane separating two disjoint, open regions, the liquid (saturated) region Ω = Ω(t) and the unsaturated region C\Ω̄. The velocity of the liquid is proportional to the gradient of the pressure v = −κ∇P, (1) where the permeability κ = κ(z, z̄) is a real function, sufficiently regular in Ω, and z = x + iy, z̄ = x− iy are complex coordinates on the plane. The flow is incompressible and the velocity satisfies the continuity equation ∇ · v = 0. (2) The free boundary conditions are P (∂Ω) = 0 (3) and the normal velocity of the boundary is vn = n · v, for z ∈ ∂Ω, (4) where n denotes the outward normal to the boundary. The flow is driven by a point multipole source of order k+1 located at z = z1, so that the pressure is singular at this point. From (1), (2), (7) it follows that, away from singularities, the pressure satisfies ∇ · (κ∇P ) = 0, z 6= z1. (5)