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

Abstract

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 tα. 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(α+n)/(3+n). The current shape factor is derived in closed-form for an instantaneous release (α=0), and numerically for time-dependent injection (α≠0). For the general case α≠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 α≠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 α. The shape of the current is mostly sensitive to α and moderately to n; the case α=3 acts as a transition between decelerating and accelerating currents.