Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 286487, 11 pages

http://dx.doi.org/10.1155/2015/286487

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

^{1}Dipartimento di Ingegneria Civile, Ambiente Territorio e Architettura (DICATeA), Università di Parma, 43124 Parma, Italy^{2}Dipartimento 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.

#### 1. Introduction

A thorough comprehension of numerous class of flows in porous media is firmly based on approximations and simplifying assumptions leading to differential problems amenable to analytical or quasi-analytical solutions. A notable example is the shallow water assumption, with negligible vertical velocity and a horizontal length scale much larger than the vertical one [1–4]; the approximation leads in turn to modelling a wide class of propagation problems as one-dimensional ones. An important class of solutions arising in several one-dimensional transient flows driven by gravity is constituted by self-similar solutions, representable in terms of functions of one variable and describing the “intermediate asymptotic” behavior of the system in a region where the solution is no longer dependent on the specific initial and/or boundary conditions adopted [3]. Whenever self-similar solutions can be obtained in analytical or in approximate analytical form, they represent an excellent test to verify model correctness by means of comparison with experiments; they also constitute a reliable benchmark for numerical integration.

Noteworthy examples of self-similar solutions to single-phase gravity driven flows in porous media include flow over a horizontal surface in plane [2] and radial geometry, Lyle et al. [5]; these solutions were extended to incorporate two-layer flow [6], the effect of a sloping bottom [7], the action of impermeable confining boundaries [8], drainage effects [9], and non-Newtonian power-law fluids [10], also in a vertically graded porous medium [11]. The dipole solution, originally derived by Barenblatt [3], was extended by King and Woods [12] to include drainage, and by Mathunjwa and Hogg [13] to incorporate a vertical variation in permeability.

An important step following the derivation of a self-similar solution is the check of its stability. In fluid mechanics, such an analysis can essentially be reduced to the study of the spatiotemporal evolution of disturbances applied to the system. In a broad sense, the evolution of instabilities leads to new solutions of the differential problem describing the flow and provides hints on possible stable solutions. Stability analyses were performed in the context of several filtration problems amenable to self-similar solutions. The linear stability of the Barenblatt-Pattle (B-P) self-similar solutions of the porous medium equation, describing a number of flows including viscous and porous media gravity currents, was investigated by Grundy and McLaughlin [14] for plane and axisymmetric geometry. Their analysis was later extended to nonradially symmetric perturbations by Mathunjwa and Hogg [15]. The linear stability of a class of self-similar solutions to the filtration-absorption equation was demonstrated by Barenblatt et al. [16] and Chertok [17]. Mathunjwa and Hogg [13] showed the linear stability of the dipole self-similar solution of the first kind [3]. With respect to all these analyses the present study is new since it considers non-Newtonian fluids.

The study of gravity currents in porous media was recently extended to non-Newtonian fluids by Di Federico et al. [18, 19], respectively, for two-dimensional and axisymmetric geometry; self-similar solutions in analytical and numerical form were obtained for a power-law fluid spreading in a uniform medium subject to the injection of a volume of intruding fluid increasing with time with the exponent . The motivation for their study was the nonlinear rheological nature displayed by fluids flowing in porous media in a variety of industrial and environmental applications ([20] and references therein).

In this paper, we analyse the stability of the aforementioned solutions. Section 2 presents a linear stability analysis for two-dimensional geometry. First, the special cases (correspondent to an instantaneous release of a finite volume of fluid) and (correspondent to a linear increase in time of injected volume) are discussed, deriving a closed-form expression for the eigenvalues of the associated Sturm-Liouville (SL) problem. Second, the stability of the general case with is demonstrated numerically. In Section 3, a similar analysis is conducted for axisymmetric geometry, considering radially varying perturbations. Stability is investigated analytically for and numerically for . Concluding remarks are outlined in Section 4. Further mathematical details are included in the Appendices.

#### 2. Stability Analysis in Two-Dimensional Geometry

Di Federico et al. [18] (Figure 1) consider the motion of shallow two-dimensional gravity currents of a purely viscous and relatively heavy non-Newtonian fluid of uniform density in a homogeneous porous layer saturated with a lighter fluid of uniform density . The flow is driven by the release from a source at the boundary of a volume , above a horizontal impermeable bottom.