Skip to main content

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

  • Jian Du (a1), Robert D. Guy (a2), Aaron L. Fogelson (a3) (a4), Grady B. Wright (a5) and James P. Keener (a3) (a4)...

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.

Corresponding author
Hide All
[1]Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., and Walter, P., Molecular Biology of the Cell, Garland Science, New York, 2007.
[2]Fahy, J.V. and Dickey, B. R., Airway Mucus Function and Dysfunction, N. Engl. J. Med., 363(2010), 223347.
[3]Schreiber, Sören and Scheid, Peter, Gastric mucus of the guinea pig: Proton carrier and diffusion barrier, Am. J. Physiol. Gastrointest, Liver Physiol., 272(1997), G63G70.
[4]Verdugo, P., Deyrup-Olsen, I., Martin, A.W. and Luchtel, D. L., Polymer gel phase transition: The molecular mechanism of product release in mucin secretion, Mechanics of Swelling, Springer-Verlag, NATO ASI Series (1992), H64 671681.
[5]Weisel, J.W., Fibrinogen and Fibrin, Adv. Protein Chem., 70 (2005), 247299.
[6]Gersh, K.C., Edmondson, K.E. and Weisel, J. W., Flow rate and fibrin fiber alignment, J. Thromb. Haemost., 8(2010), 28268.
[7]Campbell, R.A., Aleman, M.M., Gray, L.D., Falvo, M.R. and Wolberg, A.S., Flow Profoundly Influences Fibrin Network Structure: Implications for Fibrin Formation and Clot Stability in Hemostasis, Thromb. Haemost., 104(2010), 12814.
[8]Neeves, K.B., Illing, D.A. and Diamond, S.L., Thrombin flux and wall shear rate regulate fibrin fiber deposition state during polymerization under flow, Biophys. J., 98(2010), 134452.
[9]Barocas, V. H. and Tranquillo, R. T., An anisotropic biphasic theory of tissue-equivalent mechanics: The interplay among cell traction, fibrillar network deformation, fiber alignment, and cell contact guidance, J. Biomech. Eng., 119 (1997), 137145.
[10]Byrne, H. and Preziosi, L., Modelling solid tumour growth using the theory of mixtures, Math. Med. Biol., 20 (2003), 341366.
[11]Cogan, N.G. and Guy, R.D., Multiphase flow models of biogels from crawling cells to bacterial biofilms, HFSP J., 4(2010), 1125.
[12]He, X. and Dembo, M., On the mechanics of the first cleavage division of the sea urchin egg, Exp. Cell. Res., 233 (1997), 252273.
[13]Keener, J. P. and Sircar, S. and Fogelson, A. L., Kinetics of swelling gels, SIAM J. Appl. Math., 71 (2011), 854875.
[14]Levine, A. J. and MacKintosh, F. C., The mechanics and fluctuation spectrum of active gels, J. Phys. Chem. B, 113 (2009), 38203830.
[15]Mow, V. C. and Holmes, M. H. and Lai, W. M., Fluid transport and mechanical properties of articular cartilage: A review, J. Biomech., 17 (1984), 377394.
[16]Wolgemuth, C. W. and Mogilner, A. and Oster, G., The hyrdration dynamics of polyelectrolyte gels with applications to cell motility and drug delivery, Eur. Biophys. J., 33 (2004), 146158.
[17]Doi, M. and See, H., Introductin to Polymer Physics, Oxford University Press, Oxford, England, 1996.
[18]Tanaka, T. and Fillmore, D. J., Kinetics of swelling in gels, J. Chem. Phys., 70 (1979), 12141218.
[19]Alt, W. and Dembo, M., Cytoplasm dynamics and cell motion: two-phase flow models. Math. Biosci., 156 (1999), 20728.
[20]Wright, G.B., Guy, R.D. and Fogelson, A.L., An efficient and robust method for simulating two-phase gel dynamics, SIAM J. Sci. Comput., 30(2008), 25352565.
[21]Wright, G.B., Guy, R.D., Du, J. and Fogelson, A.L., A high-resolution finite-difference method for simulating two-fluid, viscoelastic gel dynamics, J. Non-Newton. Fluid Mech., 166(2011), 11371157.
[22]Guy, R.D., Fogelson, A.L. and Keener, J. P., Modeling Fibrin Gel Formation in a Shear Flow, Mathematical Medicine and Biology, 24(2007), 111130.
[23]Glimm, J., Grove, J.W., Li, X., Shyue, K.-M., Zeng, Y. and Zhang, Q., Three dimensional front tracking, SIAM J. Sci. Comp., 19(1998), 703727.
[24]Boettinger, W. J., Warren, J. A., Beckermann, C. and Karma, A., Phase-field simulation of solidification, Annu. Rev. Mater. Res., 32 (2002), 16394.
[25]Osher, S. and Sethian, J., Front propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79(1988), 1249.
[26]Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys., 87(1990), 171200.
[27]Leer, B. van, Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov’s method, J. Comput. Phys., 32(1979), 101136.
[28]Washio, T. and Oosterlee, C. W., Krylov subspace acceleration for nonlinear multigrid schemes, Electron. Trans. Numer. Anal., 6 (1997), 271290.
[29]Oosterlee, C. W. and Washio, T., An evaluation of parallel multigrid as a solver and a precon-ditioner for singularly perturbed problems, SIAM J. Sci. Comput., 19(1998), 87110.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 10 *
Loading metrics...

Abstract views

Total abstract views: 130 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th July 2018. This data will be updated every 24 hours.