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\¯Ω. The velocity of the liquid is proportional to the gradient of the pressure v = −κ P, (1) where the permeability κ = κ(z, ¯z) is a real function, sufficiently regular in Ω, and z = x + iy, ¯z = 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 bo...

• Type: Journal
• Year: 2007
• Where: APPML
• Authors: Sam D. Howison, Igor Loutsenko, John R. Ockendon
• Keywords: APPML 2007, Free Boundary Dynamics, Multipole Source, Porous Medium