Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On diffusive slowdown in three-layer Hele-Shaw flows


Authors: Prabir Daripa and Gelu Pasa
Journal: Quart. Appl. Math. 68 (2010), 591-606
MSC (2000): Primary 76E17, 76T30, 76R50, 65F99, 65Q05
DOI: https://doi.org/10.1090/S0033-569X-2010-01174-3
Published electronically: May 19, 2010
MathSciNet review: 2676978
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Abstract: In a recently published article of Daripa and Pasa [Transp. Porous Media (2007) 70:11-23], the stabilizing effect of diffusion in three-layer Hele-Shaw flows was proved using an exact analysis of normal modes. In particular, this was established from an upper bound on the growth rate of instabilities which was derived from analyzing stability equations. However, the method used there is not constructive in the sense that the upper bound derived from actual numerical discretization of the problem could be significantly different from the exact one reported depending on the scheme used. In this paper, a numerical approach to solve the stability equations using a finite difference scheme is presented and analyzed. An upper bound on the growth rate is derived from numerical analysis of the discrete system which also shows the diffusive slowdown of instabilities. Upper bounds obtained by this numerical approach and by the analytical approach are compared. The present approach is constructive and directly leads to the implementation of the numerical approach to obtain approximate solutions in the presence of diffusion. The contributions of the paper are the novelty of the approach and a bound on the growth rates that does not depend on the solution itself.


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Additional Information

Prabir Daripa
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: daripa@math.tamu.edu

Gelu Pasa
Affiliation: Institute of Mathematics “Simion Stoillow” of Romanian Academy, Bucharest, Romania 70700
Email: Gelu.Pasa@imar.ro

DOI: https://doi.org/10.1090/S0033-569X-2010-01174-3
Received by editor(s): February 1, 2009
Published electronically: May 19, 2010
Article copyright: © Copyright 2010 Brown University
The copyright for this article reverts to public domain 28 years after publication.


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