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Lagrangian Mesh Model with Regridding for Planar Poiseuille Flow

Part of: Physiological, cellular and medical topics Partial differential equations, initial value and time-dependent initial-boundary value problems Special subfields of solid mechanics Numerical methods Biological fluid mechanics Coupling of solid mechanics with other effects

Published online by Cambridge University Press:  03 May 2017

Jingxuan Zhuo ,
Ricardo Cortez  and
Robert Dillon
Jingxuan Zhuo*
Affiliation:
Department of Mathematics and Statistics, Washington State University, USA
Ricardo Cortez*
Affiliation:
Department of Mathematics, Tulane University, USA
Robert Dillon*
Affiliation:
Department of Mathematics and Statistics, Washington State University, USA
*
*Corresponding author. Email addresses:[email protected] (J. Zhuo), [email protected] (R. Cortez), [email protected] (R. Dillon)
*Corresponding author. Email addresses:[email protected] (J. Zhuo), [email protected] (R. Cortez), [email protected] (R. Dillon)
*Corresponding author. Email addresses:[email protected] (J. Zhuo), [email protected] (R. Cortez), [email protected] (R. Dillon)
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Abstract

Many biological settings involve complex fluids that have non-Newtonian mechanical responses that arise from suspended microstructures. In contrast, Newtonian fluids are liquids or mixtures of a simple molecular structure that exhibit a linear relationship between the shear stress and the rate of deformation. In modeling complex fluids, the extra stress from the non-Newtonian contribution must be included in the governing equations.

In this study we compare Lagrangian mesh and Oldroyd-B formulations of fluid-structure interaction in an immersed boundary framework. The start-up phase of planar Poiseuille flow between two parallel plates is used as a test case for the fluid models. For Newtonian and Oldroyd-B fluids there exist analytical solutions which are used in the comparison of simulation and theoretical results. The Lagrangian mesh results are compared with Oldroyd-B using comparable parameters. A regridding algorithm is introduced for the Lagrangian mesh model. We show that the Lagrangian mesh model simulations with regridding produce results in close agreement with the Oldfoyd-B model.

Keywords

Lagrangian mesh modelOldroyd-Bimmersed boundary methodviscoelastic fluidregridding methods

MSC classification

Secondary: 74F10: Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 76Z05: Physiological flows 65M06: Finite difference methods 92C10: Biomechanics 74S20: Finite difference methods 74L15: Biomechanical solid mechanics
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 1 , July 2017 , pp. 112 - 132
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Alpkvist, E., and Klapper, I. Description of mechanical response including detachment using a novel particle method of biofilm/flow interaction. Wat. Sci. Tech. 55 (2007), 265–273.CrossRefGoogle ScholarPubMed
[2] Bottino, D. C. Modeling viscoelastic networks and cell deformation in the context of the immersed boundary method. J. Comput. Phys. 147 (1998), 86–113.CrossRefGoogle Scholar
[3] Chrispell, J., and Fauci, L. Peristaltic pumping of solid particles immersed in a viscoelastic fluid. Mathematical Modelling of Natural Phenomena 6 (2011), 67–83.CrossRefGoogle Scholar
[4] Chrispell, J. C., Cortez, R., Khismatullin, D. B., and Fauci, L. J. Shape oscillations of a droplet in an oldroyd-b fluid. Physica D: Nonlinear Phenomena 240, 20 (2011), 1593–1601. Special Issue: Fluid Dynamics: From Theory to Experiment.CrossRefGoogle Scholar
[5] Dasgupta, M., Liu, B., Fu, H. C., Berhanu, M., Breuer, K. S., Powers, T. R., and Kudrolli, A. Speed of a swimming sheet in newtonian and viscoelastic fluids. Phys. Rev. E 87 (Jan 2013), 013015.CrossRefGoogle ScholarPubMed
[6] Dillon, R., and Othmer, H. G. A mathematical model for outgrowth and spatial patterning of the vertebrate limb bud. J. Theor. Biol. 197, 3 (1999), 295–330.CrossRefGoogle Scholar
[7] Dillon, R., and Zhuo, J. Using the immersed boundary method to model complex fluidsstructure interaction in sperm motility. DCDS-Series B 15, 2 (2011), 343–355.CrossRefGoogle Scholar
[8] Du, J., Guy, R. D., and Fogelson, A. L. An immersed boundary method for two-fluid mixtures. J. Comput. Phys. 262 (2014), 231–243.CrossRefGoogle ScholarPubMed
[9] Duarte, A. S. R., Miranda, A. I. P., and Oliveira, P. J. Numerical and analytical modeling of unsteady viscoelastic flows: The start-up and pulsating test case problems. J. Non-Newtonian Fluid Mech. 154 (2008), 153–169.CrossRefGoogle Scholar
[10] Eytan, O., and Elad, D. Analysis of intra-uterine fluid motion induced by uterine contractions. Bull. Math. Biol. 61 (1999), 221.CrossRefGoogle ScholarPubMed
[11] Fauci, L. Peristaltic pumping of solid particles. Comput. Fluids 21 (1992), 583.CrossRefGoogle Scholar
[12] Grove, R. R., and Chrispell, J. Immersed Boundary Modeling of Journal Bearings in a Viscoelastic Fluid. PhD thesis, Indiana University OF Pennsylvania, 2013.Google Scholar
[13] Joseph, D. D. Fluid Dynamics of Viscoelastic Liquids. Springer-Verlag, New York, 1990.CrossRefGoogle Scholar
[14] Larson, R. G. The Structure and Rheology of Complex Fluids. Oxford University Press, Oxford, 1998.Google Scholar
[15] Li, M., and Brasseur, J. Non-steady peristaltic transport in finite-length tubes. J. Fluid Mech. 248 (1993), 129151.CrossRefGoogle Scholar
[16] Oldroyd, J. On the formulation of rheological equations of state. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 200, 1063 (1950), 523–541.Google Scholar
[17] Peskin, C. S. The immersed boundary method. Acta Numer. 11 (2002), 479–517.CrossRefGoogle Scholar
[18] Teran, J., Fauci, L., and Shelley, M. Peristaltic pumping and irreversibility of a stokesian viscoelastic fluid. Phys. Fluids 20 (2008), 73–101.CrossRefGoogle Scholar
[19] Teran, J., Fauci, L., and Shelley, M. Viscoelastic fluid response can increase the speed and efficiency of a free swimmer. Phys. Rev. Lett. 104, 3 (2010), 038101.CrossRefGoogle Scholar
[20] Villone, M., Davino, G., Hulsen, M., Greco, F., and Maffettone, P. Particle motion in square channel flow of a viscoelastic liquid: Migration vs. secondary flows. J. Non-Newtonian Fluid Mech. 195, (2013), 1–8.CrossRefGoogle Scholar
[21] Waters, N. D., and King, M. J. Unsteady flow of an elastico-viscous liquid. Rheol. Acta 9 (1970), 345–355.CrossRefGoogle Scholar
[22] White, F. M. Fluid Mechanics. McGraw-Hill, 1999.Google Scholar
[23] Wróbel, J. K., Cortez, R., and Fauci, L. Modeling viscoelastic networks in stokes flow. Phys. Fluids 26, 11 (2014).Google Scholar
[24] Wróbel, J. K., Lynch, S., Barrett, A., Fauci, L., and Cortez, R. Enhanced flagellar swimming through a compliant viscoelastic network in stokes flow. J. Fluid Mech. 792, 4 (2016), 775–797.CrossRefGoogle Scholar