Inertial effects on viscous fingering in the complex plane

We present a nonlinear unsteady Darcy's equation which includes inertial effects for flows in a porous medium or Hele-Shaw cell and discuss the conditions under which it reduces to the classical Darcy's law. In the absence of surface tension we derive a generalized Polubarinova-Galin equation in a circular geometry, using the method of conformal mapping. The linear stability of the base-flow state is examined by perturbing the corresponding conformal map. We show that inertia always has a tendency to stabilize the interface, regardless of whether a less viscous fluid is displacing a more viscous fluid or vice versa.