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Mathematical Problems in Engineering
Volume 2015, Article ID 286487, 11 pages
http://dx.doi.org/10.1155/2015/286487
Research Article

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

1Dipartimento di Ingegneria Civile, Ambiente Territorio e Architettura (DICATeA), Università di Parma, 43124 Parma, Italy
2Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali (DICAM), Università di Bologna, 40136 Bologna, Italy

Received 7 April 2015; Accepted 21 May 2015

Academic Editor: F. M. Mahomed

Copyright © 2015 Sandro Longo and Vittorio Di Federico. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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