Stability Analysis of Gravity Currents of a Power-Law Fluid in a Porous Medium

Abstract

We analyse the linear stability of self-similar shallow, two-dimensional and axisymmetric gravity currents of a viscous power-law non-Newtonian fluid in a porous medium. The flow domain is initially saturated by a fluid lighter than the intruding fluid, whose volume varies with time astα. The transition between decelerated and accelerated currents occurs atα= 2 for two-dimensional and atα= 3 for axisymmetric geometry. Stability is investigated analytically for special values ofαand numerically in the remaining cases; axisymmetric currents are analysed only for radially varying perturbations. The two-dimensional currents are linearly stable forα< 2 (decelerated currents) with a continuum spectrum of eigenvalues and unstable forα= 2, with a growth rate proportional to the square of the fluid behavior index. The axisymmetric currents are linearly stable for anyα< 3 (decelerated currents) with a continuum spectrum of eigenvalues, while forα= 3 no firm conclusion can be drawn. Forα> 2 (two-dimensional accelerated currents) andα> 3 (axisymmetric accelerated currents) the linear stability analysis is of limited value since the hypotheses of the model will be violated.