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An Interface-Capturing Regularization Method for Solving the Equations for Two-Fluid Mixtures

Published online by Cambridge University Press:  03 June 2015

Jian Du ,
Robert D. Guy ,
Aaron L. Fogelson ,
Grady B. Wright  and
James P. Keener
Jian Du*
Affiliation:
Department of Mathematical Science, Florida Institute of Technology, Melbourne, FL 32901, USA
Robert D. Guy*
Affiliation:
Department of Mathematics, University of California Davis, Davis, CA 95616, USA
Aaron L. Fogelson*
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA Department of Bioengineering, University of Utah, Salt Lake City, UT 84112, USA
Grady B. Wright*
Affiliation:
Department of Mathematics, Boise State University, Boise, ID 83725, USA
James P. Keener*
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA Department of Bioengineering, University of Utah, Salt Lake City, UT 84112, USA
*
Corresponding author.Email:jdu@fit.edu
Email:guy@math.ucdavis.edu
Email:fogelson@math.utah.edu
Email:wright@math.boisestate.edu
Email:wright@math.boisestate.edu
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Abstract

Many problems in biology involve gels which are mixtures composed of a polymer network permeated by a fluid solvent (water). The two-fluid model is a widely used approach to described gel mechanics, in which both network and solvent coexist at each point of space and their relative abundance is described by their volume fractions. Each phase is modeled as a continuum with its own velocity and constitutive law. In some biological applications, free boundaries separate regions of gel and regions of pure solvent, resulting in a degenerate network momentum equation where the network volume fraction vanishes. To overcome this difficulty, we develop a regularization method to solve the two-phase gel equations when the volume fraction of one phase goes to zero in part of the computational domain. A small and constant network volume fraction is temporarily added throughout the domain in setting up the discrete linear equations and the same set of equation is solved everywhere. These equations are very poorly conditioned for small values of the regularization parameter, but the multigrid-preconditioned GMRES method we use to solve them is efficient and produces an accurate solution of these equations for the full range of relevant regularization parameter values.

Keywords

76T9976D0776D2765F2276M2065M12Gel swellingpreconditionertwo-fluid model
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 5 , November 2013 , pp. 1322 - 1346
Copyright
Copyright © Global Science Press Limited 2013

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