Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-17T07:30:04.671Z Has data issue: false hasContentIssue false
  • English
  • Français

Peristaltic Pumping of Solid Particles Immersed in aViscoelastic Fluid

Published online by Cambridge University Press:  10 August 2011

J. Chrispell  and
L. Fauci
J. Chrispell*
Affiliation:
Department of Mathematics, Tulane University, New Orleans, Louisiana 70118, USA Center for Computational Science, Tulane University, New Orleans, Louisiana 70118, USA
L. Fauci
Affiliation:
Department of Mathematics, Tulane University, New Orleans, Louisiana 70118, USA Center for Computational Science, Tulane University, New Orleans, Louisiana 70118, USA
*
⋆ Corresponding author. E-mail: [email protected]
Get access

Abstract

Peristaltic pumping of fluid is a fundamental method of transport in many biologicalprocesses. In some instances, particles of appreciable size are transported along with thefluid, such as ovum transport in the oviduct or kidney stones in the ureter. In some ofthese biological settings, the fluid may be viscoelastic. In such a case, a nonlinearconstitutive equation to describe the evolution of the viscoelastic contribution to thestress tensor must be included in the governing equations. Here we use an immersedboundary framework to study peristaltic transport of a macroscopic solid particle in aviscoelastic fluid governed by a Navier-Stokes/Oldroyd-B model. Numerical simulations ofperistaltic pumping as a function of Weissenberg number are presented. We examine thespatial and temporal evolution of the polymer stress field, and also find that theviscoelasticity of the fluid does hamper the overall transport of the particle in thedirection of the wave.

Keywords

viscoelastic fluidperistaltic pumpingOldroyd-B
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 5: Complex Fluids , 2011 , pp. 67 - 83
Copyright
© EDP Sciences, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baranger, J., Machmoum, A.. Existence of approximate solutions and error bounds for viscoelastic fluid flow: characteristics method. Comput. Methods Appl. Mech. Engrg. , 148 (1997), No. 1-2, 39–52. CrossRefGoogle Scholar
Baranger, J., Sandri, D.. Finite element approximation of viscoelastic fluid flow: existence of approximate solutions and error bounds. I. Discontinuous constraints. Numer. Math., 63 (1992), No. 1, 13–27. CrossRefGoogle Scholar
R.B. Bird, R.C. Armstrong, O. Hassager. Dynamics of Polymeric Liquids. Wiley-Interscience, 1987.
Blake, J.R., Vann, P.G., Winet, H. A model of ovum transport. J. Theor. Biol., 102 (1983), No. 1, 145–166. CrossRefGoogle ScholarPubMed
S. Boyarski, C. Gottschalk, E. Tanagho, P. Zimskind. Urodynamics: Hydrodynamics of the Ureter and the Renal Pelvis. Academic Press, New York, 1971.
Brooks, A., Hughes, T.. Streamline Upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 32 (1982), No. (1-3), 199–259. CrossRefGoogle Scholar
Chrispell, J.C., Ervin, V.J., Jenkins, E.W.. A fractional step [theta]-method approximation of time-dependent viscoelastic fluid flow. Journal of Computational and Applied Mathematics, 232 (2009), No. 2, 159–175. CrossRefGoogle Scholar
Connington, K., Kang, Q., Viswanathan, H., Abdel-Fattah, A., Chen, S.. Peristaltic particle transport using the lattice boltzmann method. Phys. of Fluids, 21 (2009), No. 5, 053301. CrossRefGoogle Scholar
El-Kareh, A.W., Leal, L.G.. Existence of solutions for all deborah numbers for a non-Newtonian model modified to include diffusion. Journal of Non-Newtonian Fluid Mechanics, 33 (1989), No. 3, 257–287. CrossRefGoogle Scholar
Eytan, O., Elad, D.. Analysis of intra-uterine fluid motion induced by uterine contractions. Bull. Math. Biol., 61 (1999), No. 2, 221–238. CrossRefGoogle ScholarPubMed
Eytan, O., Jaffa, A.J., Har-Toov, J., Dalach, E., Elad, D.. Dynamics of the intrauterine fluid-wall interface. Ann. Biomed. Engr., 27 (1999) No. 3, 372-9. CrossRefGoogle ScholarPubMed
Fauci, L.. Peristaltic pumping of solid particles. Comp. & Fluids, 21 (1992), No. 4, 583–598. CrossRefGoogle Scholar
Fauci, L., Dillon, R.. Biofluidmechanics of reproduction. Annu. Rev. Fluid. Mech., 38 (2006), No. 1, 371–394. CrossRefGoogle Scholar
Griffith, B.E., Peskin, C.S.. On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems. Journal of Computational Physics, 208 (2005), No. 1, 75–105. CrossRefGoogle Scholar
Guy, R., Fogelson, A.. A wave propagation algorithm for viscoelastic fluids with spatially and temporally varying properties. Comput. Methods Appl. Mech. Engr., 197 (2008), No. 1, 2250–2264. CrossRefGoogle Scholar
Harlow, F. H., Welch, J. E.. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. of Fluids, 8 (1965), No. 12, 2182–2189. CrossRefGoogle Scholar
Hinch, E. J.. Uncoiling a polymer molecule in a strong extensional flow. Journal of Non-Newtonian Fluid Mechanics, 54 (1994), No. C, 209–230. CrossRefGoogle Scholar
Hung, T.K., Brown, T.D.. Solid-particle motion in two-dimensional peristaltic flows. J. Fluid Mech, 73 (1976), No. 1,77–96. CrossRefGoogle Scholar
Jaffrin, M. Y. and Shapiro, A. H.. Peristaltic pumping. Annu. Rev. Fluid Mech., 3 (1971), No. 1, 13–37. CrossRefGoogle Scholar
Jaffrin, M. Y., Shapiro, A. H., Weinberg, S. L.. Peristaltic pumping with long wavelengths at low reynolds number. J. Fluid Mech., 37 (1969), No. 4, 799–825. Google Scholar
Jimenez-Lozano, J., Sen, M., Dunn, P.. Particle motion in unsteady two-dimensional peristaltic flow with application to the ureter. Phys. Rev. E, 79 (2009), No. 4, 041901. CrossRefGoogle ScholarPubMed
Kim, J., Moin, P.. Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comp. Physics, 59 (1985), No. 2, 308–323. CrossRefGoogle Scholar
Kunz, G., Beil, D., Deiniger, H., Einspanier, A., Mall, G., Leyendecker, G.. The uterine peristaltic pump. normal and impeded sperm transport within the female genital tract. Adv. Exp. Med. Biol., 424 (1997), No. 1, 267–277. CrossRefGoogle ScholarPubMed
R.G. Larson. The Structure and Rheology of Complex Fluids. Oxford University Press, 1998.
Li, M., Brasseur, J.. Non-steady peristaltic transport in finite-length tubes. J. Fluid Mech., 248 (1993), No. 1, 129–151. CrossRefGoogle Scholar
Lu, C.Y., Olmsted, P.D., Ball, R.C.. Effects of nonlocal stress on the determination of shear banding flow. Phys. Rev. Lett., 84 (2000), No. 4, 642–645. CrossRefGoogle ScholarPubMed
Peskin, C.S.. The immersed boundary method. Acta Numerica, 11 (2002), 479–517. CrossRefGoogle Scholar
Pozrikidis, C.. A study of peristaltic flow. J. Fluid Mech. 180 (1987), 180:515. CrossRefGoogle Scholar
Rallison, J.M.. Dissipative stresses in dilute polymer solutions. Journal of Non-Newtonian Fluid Mechanics, 68 (1997), No. 1, 61–83. CrossRefGoogle Scholar
Takabatake, S., Ayukawa, K., Mori, A.. Peristaltic pumping in circular cylindrical tubes: a numerical study of fluid transport and its efficiency. J. Fluid Mech., 194 (1988), 193:267. Google Scholar
Teran, J., Fauci, L., Shelley, M.. Peristaltic pumping and irreversibility of a Stokesian viscoelastic fluid. Phys. of Fluids, 20 (2008), No. 7, 073101. CrossRefGoogle Scholar
Teran, J., Fauci, L., Shelley, M.. Viscoelastic fluid response can increase the speed and efficiency of a free swimmer. Phys. Rev. Letters, 104 (2010), No. 3, 038101. CrossRefGoogle Scholar
Thomases, B., Shelley, M.. Transition to mixing and oscillations in a Stokesian viscoelastic flow. Phys. Rev. Lett., 103 (2009), No. 9, 094501. CrossRefGoogle Scholar