Self-similar solutions for spreading of non-Newtonian gravity currents in a porous layer

We consider the motion of a shallow two-dimensional current of a purely viscous power-law fluid in a uniform porous layer above a horizontal impermeable boundary, driven by the instantaneous injection of a fixed volume of fluid. The equation of motion for power-law fluids in porous media is a modified Darcy’s law taking into account the nonlinearity of the rheological equation. Coupling the flow law with the mass balance equation yields a nonlinear partial differential equation governing the unknown free-surface height h(x, t) , with x and t being spatial coordinate and time. In dimensionless form, the governing equation admits an asymptotic solution in the self-similar form h(x,t) = eta_N^(n+1)*t ^(n/(2+n))*Phi(eta/eta_N), where eta=x*t^(-n/(2+n) is the self-similar variable, eta_N its value at the current tip, and Phi(eta/eta_N) the shape of the interface. Asymptotically a fixed volume of fluid spreads as t^(n/(2+n)), generalizing earlier results for Newtonian fluids. When an axisymmetric geometry is considered, the asymptotic spread rate becomes proportional to t^(n/(3+n)).