Spreading of axisymmetric non-Newtonian power-law gravity currents in porous media

A relatively heavy, non-Newtonian power-law fluid of flow behavior index n is released from a point source into a saturated porous medium above an horizontal bed; the intruding volume increases with time as t(alpha). Spreading of the resulting axisymmetric gravity current is governed by a non-linear equation amenable to a similarity solution, yielding an asymptotic rate of spreading proportional to t((alpha+n)/(3+n)). The current shape factor is derived in closed-form for an instantaneous release (alpha = 0), and numerically for time-dependent injection (alpha not equal 0). For the general case alpha not equal 0, the differential problem shows a singularity near the tip of the current and in the origin; the shape factor has an asymptote in the origin for n >= 1 and alpha not equal 0. Different kinds of analytical approximations to the general problem are developed near the origin and for the entire domain (a Frobenius series and one based on a recursive integration procedure). The behavior of the solutions is discussed for different values of n and alpha. The shape of the current is mostly sensitive to alpha and moderately to n; the case alpha = 3 acts as a transition between decelerating and accelerating currents. (c) 2012 Elsevier B.V. All rights reserved.