Stability analysis of gravity currents of a power-law fluid in a porous medium (Q1665240)

Summary: 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.