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Title:  Peristaltic Flows With Some Applications 
Authors:  Mishra, Manoranjan 
Advisors:  Rao, A Ramachandra 
Keywords:  Fluid Dynamics Peristaltic Flow Peristaltic Transport Peristaltic Pumping Viscous Newtonian Fluid Porous Peripheral Layer Poroflexible Boundary Peristaltic Slip Flow Viscous Fluid Porous Tube 
Submitted Date:  Apr2004 
Abstract:  Peristalsis is a mechanism of pumping fluids in ducts when a progressive wave of area contraction or expansion propagates along the length of a distensible tube containing fluid. It induces in general propulsive and mixing movements and pumps the fluids against pressure rise. Physiologically, peristaltic action is an inherent property of smooth muscle contraction. It is an automatic and vital process that drives the urine from the kidney to the bladder, food through the digestive tract, bile from the gallbladder into the duodenum, movement of ovum in the fallopian tube and many other situations. A major industrial application of this principle is in the design of roller pumps, which are used in pumping fluids without being contaminated due to the contact with the pumping machinery.
Even though peristalsis is a wellknown mechanism in biological system, the first theoretical and experimental analysis of its fluid dynamics aspects were given four decades ago. In reality, the peristaltic flow problems are unsteady moving free boundary value problems where the shape of the wave on flexible tube wall is not known apriori. But the mathematical models on peristaltic transport considered in the literature deal with a prescribed train of waves moving with constant speed on the flexible boundaries and they are studied in either a fixed frame or a wave frame moving with constant velocity of the wave. In a wave frame the moving walls become stationary wavy walls. Further the motion could be treated steady under the assumptions that the peristaltic wave train is periodic, the pressure difference across the length of the tube is constant and the tube length is an integral multiple of the wavelength.
Some mathematical models of peristaltic flows representing some physiological situations are studied using a wave frame of reference in this thesis. The important characteristics of these flows namely pumping (variation of time averaged flux with difference in pressures across one wavelength), trapping (splitting of streamlines enclosing a bolus which moves as a whole along with the wave), reflux phenomena (the presence of some fluid particles whose mean motion over one cycle is against the net pumping direction) are discussed in detail. A brief general introduction to the peristaltic transport and their application in physiological fluid dynamics is presented in chapter one.
In the second chapter, the peristaltic transport of an incompressible viscous Newtonian fluid in an asymmetric channel is studied under long wavelength and lowReynolds number assumptions. Choosing the peristaltic wave train on the walls to have different amplitudes and phase produces the channel asymmetry. This study is motivated by the intrauterine fluid flow induced by uterine wall contractions which represent a peristaltic flow in an asymmetric channel and this flow is responsible for embryo transport to a successful implantation site. The solution for the stream function is obtained by neglecting inertia and curvature effects. The streamlines are plotted in both fixed and wave frames. The effects of different geometric parameters causing asymmetry like phase difference; varying channel width and wave amplitudes are investigated on the pumping characteristics, streamline pattern, trapping and reflux phenomena. It is observed that the pumping against pressure rise, trapping and reflux layer exists only when crosssection of the channel varies along the axis. The limits on the time averaged flux for trapping and reflux are obtained. The peristaltic waves on the walls with same amplitudes propagating in phase produce zero flux rate as the channel crosssection remains the same through out. The trapping and reflux regions reduce for asymmetric channels compared to symmetric channels.
The flow of an incompressible viscous fluid driven by the traveling waves along the boundaries of an asymmetric channel is studied in the chapter three, when inertia and streamline curvature effects are not negligible. It was well documented that the inertial forces cannot be ignored in the pharyngeal phase of bolus transport. Choosing the wave train on the walls to have different amplitudes and phases produces the channel asymmetry here. An asymptotic solution is obtained in powers of a geometric parameter £, the ratio of the channel width to the wavelength, giving curvature and inertia effects. A domain transformation is used to transform the channel of variable cross section to a uniform cross section, and this facilitates in easy way of finding closed form solutions at higher orders. The solutions are presented upto second order in 6. It has been found that, the relation connecting the pressure gradient and time average flux rate is a cubic leading to a nonunique of flux for a given pressure gradient. A uniqueness criterion is derived which restricts the parameters to get a unique flux for a prescribed pressure difference. The effects of inertia and curvature on peristaltic pumping, trapping and shear stress are discussed for various parameters governing the flow for symmetric and asymmetric channels and compared with the existing results in the literature. Even under a favourable pressure gradient the possibility of fluid flow in a direction opposite to the direction of the waves propagating on the walls is detected as in the case of some nonNewtonian fluids. It is noticed that the Reynolds number and asymmetry of the wall geometry may play an important role in producing mixing. The appearance of a second trapped bolus near the down streamside of the channel for some Reynolds number is a new feature. Further, the nonzero curvature produces three trapped boluses for high Reynolds number in symmetric channel as well as for inertia free flow in an asymmetric channel. Another interesting phenomena is that the shear stress distribution on the walls vanishes at some points but it does not indicate any flow separation as the MRS criteria is not satisfied.
The gastrointestinal tract is surrounded by a number of muscle layers having smooth muscle. The most important smooth muscle layers in gastrointestinal tract are submucosa and a layer of epithelial cells and these are responsible for the absorption of nutrients and water in the intestine. These layers consist of many folds and there are pores through out the tight junctions of them. Thus a study of peristaltic transport with porous peripheral layer and porous boundaries of a duct are important. Motivated by this the flow in gastrointestinal tract is mathematically modeled by a peristaltic flow of two fluid system in a twodimensional channel with a porous peripheral layer and a Newtonian fluid core layer, in chapter four. The fluid flow is investigated under the assumptions of long wavelength and low Reynolds number in a wave frame of reference. Brinkman extended Darcy equation is utilized to model
the flow in the porous peripheral layer. A shear stress jump boundary condition of OchoaTapia and Whitaker is used at the interface between porous and fluid regions together with continuity of velocity and normal stress conditions. Here one needs an extra assumption that the fluid interface and the peristaltic wave on the boundary have the same period in addition to the constant pressure difference at the ends of channel and the length of the channel to be an integral multiple of the wavelength, to consider the flow to be steady. The interface is determined as a part of the solution using the conservation of mass in both the porous and fluid regions independently. Matlab packages are used to solve the transcendental equation governing it. An interval of critical time averaged flux Q is obtained for the existence of a unique solution for the interphase. The physical quantities of importance in peristaltic transport namely, pumping, trapping and reflux are discussed for various parameters of interest governing the flow like Darcy number Da, porosity 6, shearstress jump constant /3, viscosity ratio /i. It is observed that the peristalsis works as a pump against greater pressure rise with a porous medium in the peripheral layer than a viscous fluid. The limits on the time averaged flux Q for trapping in the core layer are obtained. The existence of reflux near the axis is observed for small values of Darcy number and large values of /?.
Chapter five deals with the peristaltic transport in a tube with a poroflexible wall and having a porous material layer in the peripheral region and a Newtonian fluid in the core region. Flow in tube may be more realistic to model a flow in gastrointestinal system. At the poroflexible wall, a slip boundary condition of Saffman.type is used. The fluid flow is studied in a wave frame of reference under lubrication approach. Brinkman extended Darcy equation in cylindrical polar coordinates is considered for the porous medium with a shearstress jump boundary condition of OchoaTapia and Whitaker at the interface of porous and fluid regions together with the continuity of velocity and normal stress. The interface is found as a part of the solutions using the conservation of mass in both the regions of deformable porous medium and fluid medium independently. The interface equation turns out to be a transcendental equation involving modified Bessel functions and it is solved by using Matlab packages. The uniqueness criterion of the solutions for the interface equation in the flow region is determined for certain values of time averaged flux Q. Pumping characteristics, trapping and reflux phenomena are discussed for various parameters of interest governing the flow like, wall slip constant fc, Darcy number Da, viscosity ratio /x. shear stress jump constant f) and peripheral layer thickness 7. The slip condition at the boundary, arising due to the poroflexible nature of the wall, enhances pumping. The trapped bolus volume in the core layer is observed to decrease with a decrease in Da, /i and k and an increase in /?. The reflux phenomena is discussed in detail. The trapping limits on the flux rate Q in the core region are obtained.
As the behaviour of most of the physiological fluids is known to be nonNewtonian, the peristaltic flows of powerlaw and micro polar fluids are investigated in the next two chapters.
In chapter six, the peristaltic transport of a powerlaw fluid in an axisymmetric tube having poroflexible wall is studied. The powerlaw model of Ostwaldde Waele type is considered, which accommodates the study of both shear thinning and shear thickening fluids. The flow characteristics are studied in wave frame analysis under lubrication approach. The slip boundary conditions of BeaversJoseph and Saffman type are considered at the wall in obtaining solutions for the flow and resulting pumping characteristics are investigated with a straight section dominated (SSD) wave form other than the sinusoidal one. It is observed that the time mean flow becomes negative in free pumping for a shear thickening fluid with a SSD expansion wave and the same is observed for a SSD contraction wave in the case of shear thinning fluid. The pressure rise increases with increasing of Darcy number Da against which the peristalsis works as a pump and decreases for an increase in BeaverJoseph constant a. Peristalsis works as a pump against a greater pressure rise for a shear thickening fluid and the opposite happens for a shear thinning fluid compared with Newtonian fluid. Trapping and reflux phenomena are discussed for various parameters of interest governing the flow like Da, a and the fluid behaviour index n. The trapping limits on Q are derived. The trapped bolus volume for sinusoidal wave is observed to decrease as the fluid behaviour index decreases, i.e as the fluid behaviour changes from shear thickening to shear thinning, where as it increases with an increase in Darcy number. The rheological properties of fluid, wave shape and porous nature of the wall play an important role in the peristaltic transport and may be useful in understanding the transport of chyme in small intestine.
The chapter seven contains the study of peristaltic transport of a micropolar fluid in an axisymmetric tube. Micropolar fluids exhibit some microscopic effects arising from the local structure and micro motion of the fluid elements. Further, they can sustain couple stresses. It is speculated that, in microcirculation, peristalsis may be involved as well in the vasomotion of small blood vessels which change their diameters periodically. Therefore, modelling blood by a micropolar fluid may be more appropriate. The closed form solutions are obtained for velocity, microrotation components, as well as the stream function under the assumption of long wavelength and low Reynolds number. The solution contains new additional parameters namely, N the coupling number and m the microploar parameter. In the case of free pumping (pressure difference Ap = 0) the difference in pumping flux is observed to be very small for Newtonian and micropolar fluids but in the case of pumping (Ap > 0) the characteristics are significantly altered for different N and m. It is observed that the peristalsis in micropolar fluids works as a pump against a greater pressure rise compared with a Newtonian fluid. Streamline patterns which depict trapping phenomena aie presented for different parameter ranges. The limit on the trapping of the center streamline is obtained. The effects of N and m on friction force for different Ap are discussed.
The nomenclature of symbols in each chapter is independent of the other. Each of the chapter has its own appendix and they are numbered with the corresponding roman number of the chapters. The purpose of the study here is not to represent exactly the functioning of various physiological applications, but rather to understand the fluidmechanical aspects inherent in the problems of peristaltic transport. 
URI:  http://hdl.handle.net/2005/339 
Appears in Collections:  Mathematics (math)

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