# Peristaltic Pumping of Solid Particles Immersed in a Viscoelastic Fluid

Mathematical Modelling of Natural Phenomena (2011)

- Volume: 6, Issue: 5, page 67-83
- ISSN: 0973-5348

## Access Full Article

top## Abstract

top## How to cite

topChrispell, J., and Fauci, L.. "Peristaltic Pumping of Solid Particles Immersed in a Viscoelastic Fluid." Mathematical Modelling of Natural Phenomena 6.5 (2011): 67-83. <http://eudml.org/doc/222311>.

@article{Chrispell2011,

abstract = {Peristaltic pumping of fluid is a fundamental method of transport in many biological
processes. In some instances, particles of appreciable size are transported along with the
fluid, such as ovum transport in the oviduct or kidney stones in the ureter. In some of
these biological settings, the fluid may be viscoelastic. In such a case, a nonlinear
constitutive equation to describe the evolution of the viscoelastic contribution to the
stress tensor must be included in the governing equations. Here we use an immersed
boundary framework to study peristaltic transport of a macroscopic solid particle in a
viscoelastic fluid governed by a Navier-Stokes/Oldroyd-B model. Numerical simulations of
peristaltic pumping as a function of Weissenberg number are presented. We examine the
spatial and temporal evolution of the polymer stress field, and also find that the
viscoelasticity of the fluid does hamper the overall transport of the particle in the
direction of the wave.},

author = {Chrispell, J., Fauci, L.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {viscoelastic fluid; peristaltic pumping; Oldroyd-B},

language = {eng},

month = {8},

number = {5},

pages = {67-83},

publisher = {EDP Sciences},

title = {Peristaltic Pumping of Solid Particles Immersed in a Viscoelastic Fluid},

url = {http://eudml.org/doc/222311},

volume = {6},

year = {2011},

}

TY - JOUR

AU - Chrispell, J.

AU - Fauci, L.

TI - Peristaltic Pumping of Solid Particles Immersed in a Viscoelastic Fluid

JO - Mathematical Modelling of Natural Phenomena

DA - 2011/8//

PB - EDP Sciences

VL - 6

IS - 5

SP - 67

EP - 83

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

LA - eng

KW - viscoelastic fluid; peristaltic pumping; Oldroyd-B

UR - http://eudml.org/doc/222311

ER -

## References

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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.