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

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 as t^alpha.The transition between decelerated and accelerated currents occurs at alpha = 2 for two-dimensional and at alpha = 3 for axisymmetric geometry. Stability is investigated analytically for special values of alpha and numerically in the remaining cases; axisymmetric currents are analysed only for radially varying perturbations.The two-dimensional currents are linearly stable for alpha < 2 (decelerated currents) with a continuum spectrum of eigenvalues and unstable for alpha = 2, with a growth rate proportional to the square of the fluid behavior index. The axisymmetric currents are linearly stable for any alpha < 3 (decelerated currents) with a continuum spectrum of eigenvalues, while for alpha = 3 no firm conclusion can be drawn. For alpha > 2 (two-dimensional accelerated currents) and alpha > 3 (axisymmetric accelerated currents) the linear stability analysis is of limited value since the hypotheses of the model will be violated.