Peristaltic Pumping of Solid Particles Immersed in a Viscoelastic Fluid

Mathematical Modelling of Natural Phenomena, Jan 2011

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.

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Peristaltic Pumping of Solid Particles Immersed in a Viscoelastic Fluid

Peristaltic Pumping of Solid Particles Immersed in a Viscoelastic Fluid J. Chrispell 0 1 L. Fauci 0 1 0 Center for Computational Science, Tulane University , New Orleans, Louisiana 70118 , USA 1 Department of Mathematics, Tulane University , New Orleans, Louisiana 70118 , USA 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. viscoelastic fluid; peristaltic pumping; Oldroyd-B AMS subject classification; 76Z05; 92C35 Introduction Peristaltic pumping, the transport of a fluid in a tube due to waves of contraction, is fundamental to many physiological flows. Peristaltic contractions in the oviduct and uterus contribute to ovum transport and embryo implantation in the uterus [ 4, 11, 10, 13, 23 ], and peristaltic contractions are responsible for the passage of urine from the kidneys to the bladder [ 5 ]. In some instances, Article published by EDP Sciences and available at http://www.mmnp-journal.org or http://dx.doi.org/10.1051/mmnp/20116504 particles of appreciable size compared to the tube diameter are transported along with the fluid. For example, the embryo diameter at implantation is about 150 microns, whereas the uterine channel diameter is on the order of 1000 microns [ 10 ]. Similarly, peristalsis of urine through the ureter can sometimes be accompanied by particles such as kidney stones [ 21 ]. In this manuscript, we present a model of the transport of a single, macroscopic, solid particle in a 2D peristaltic channel. Since many biological fluids have suspended microstructures, they may exhibit complicated, non-Newtonian responses. For this reason, we consider a simple model of a viscoelastic fluid within the channel. We describe a mathematical model and numerical method that couples the motion of moving boundaries to a Navier-Stokes/Oldroyd-B description of an elastic Boger fluid using an immersed boundary framework [ 27 ]. In particular, we extend the immersed boundary model of peristaltic pumping of a Stokes viscoelastic fluid [ 31 ] to include inertial effects, and the immersed boundary model of pumping of a solid particle within a Newtonian fluid [ 12 ] to include viscoelastic effects. Many fluid dynamical studies of peristalsis, both analytical and numerical, have been examined in the last decades, e.g. [ 10, 25, 20, 19, 28, 30 ]. A recent analytical study of particle motion in a peristaltic fluid flow was presented in [21]. A long wavelength perturbation method is used for the fluid, along with the Basset-Boussinesq-Oseen equation for the small spherical particles. Particular application to the transport of calcium renal stones from the kidney to the ureter is studied. In this analysis, the fluid motion did influence the motion of the particles, but the particles did not affect the motion of the fluid. An experimental study of macroscopic particle transport in a peristaltic channel was presented in [ 18 ]. Immersed boundary simulations of the transport of a macroscopic particle in a Newtonian fluid were performed in [ 12 ], and a lattice Boltzmann model of this same system was recently presented [ 8 ]. Here, we study the transport of a single solid particle in a peristaltic channel with two-way coupling between the viscoelastic fluid and the particle. We find that the viscoelasticity of the fluid does hamper the overall transport of the particle in the direction of the wave. We examine the trajectories of the particle at different Weissenberg numbers and the temporal and spatial evolution of the viscoelastic stress field. Mathematical Model We choose an Oldroyd-B model that is derived from a microscopic description of viscoelasticity where the fluid is treated as a dilute suspension of polymers in a Newtonian solvent [ 3, 24 ]. The transport and distension of this immersed polymer field generates extra stress on the Newtonian solvent. The Navier-Stokes/Oldroyd-B equations that model the conservation of mass and momentum of this incompressible, viscoelastic fluid are: ¾ + rt Here u is the fluid velocity, p is pressure, ¾ is the viscoelastic contribution to (...truncated)


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J. Chrispell, L. Fauci. Peristaltic Pumping of Solid Particles Immersed in a Viscoelastic Fluid, Mathematical Modelling of Natural Phenomena, 2011, pp. 67-83, Volume 6, Issue 5, DOI: 10.1051/mmnp/20116504