Spreading of axisymmetric non-Newtonian powerlaw 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 nonlinear 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 closedform for an instantaneous release (alpha= 0 ), and numerically for time-dependent injection ( alpha different 0 ). For the general case alpha different 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 different 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.