Interaction of peristaltic motion with rheological properties of fluids in the symmetric and asymmetric channels

Abstract

The range of peristaltic fl ows that occurs in physiology and engineering is very large. In patti cular such [1 ows are encountered in the esophagus, bile ducts, the ureter, the gastrointestinal tract, small blood vessels and many other glandular ducts tlu-oughout the body. The principle of peristalsis is also quite common in the industrial applications for instance the transpOlt of sanitary and corrosive fluids and blood pumps in the heart lung machine. It has been an established fact that most of the fluids in physiology are nonNewtonian. The motion of non-Newtonian fluids is an impOltant topic in the field of chemical, biomedical and environmental engineering and science. The governing equations in the non-Newtonian fluids are of higher order than the Navier-Stokes equations. The mathematical models of peristaltic flows involving non-Newtoni an fluids are of more intricate in nature. Such flows in the context of magnetohydrodynamics are of great interest for the movement of physiological fluids, for exampl e, the blood and in view of analysis of peristaltic MHD compressor. However, very little has been reported yet on the peristalti c flows in the presence of an induced magnetic field. On the other hand, majority of available literature on the peristaltic flows analyzed the situation when no-slip condition has been considered. Such condition is not reliable especially in polymeric liquids with high molecular weight. No-slip condition is also not appropriate in physiological flows, thin film problems, rarefied fluid problems and flow on multiple interfaces. Another aspect whi ch is not yet given due attention in the literature is the heat and mass transfer effects on the peristaltic flows of non-Newtonian fl uids. No doubt, heat transfer in tissues is subject to heat conduction in tissues, heat convection because of blood fl ow through the pores of tissues and heat radiati on between surface and its environment. The heat transfer consideration U1 blood is very important in the oxygenation and hemodialysis processes. The simultaneous influence of heat and mass transfer in the peristalsis is also significant when one desires to analyze the Soret and Dufour effects. Motivated by the above discussion, the main objective here is to examine the effects of rheological properties, an induced magnetic field, pattial slip features and heat and mass transfer on the peristaltic flows. This thesis is structured as follows. Chapter one is prepared for the brief review on the peristaltic flows and some relevant definitions. Analysis for the peristaltic flow of a Carreau fl uid in a planar channel has been carried out in chapter two. Symmetric nature of flow is considered when the Reynolds number is low and wavelength is long. The results for different wave forms are established and compared. The pumping and trapping phenomena are given proper attenti on. It is noticed that the velocity at the center of chaImel and bolus size decrease when there is an increase in the Weissenberg number. The findings of this chapter have been published in Numerical Methods for Partial Differential Equations 26, 519 (2010). Chapter three extends the research work of chapter two in the presence of an applied magnetic field. The governing equations are developed and analysis has been performed when magnetic Reynolds number is small. It is concluded that longitudinal velocity reduces in a magnetohydrodynamic (MHO) fluid. FUl1her, the bolus size in MHO case also decreases when compared with the hydrodynamic fluid. These observations have been accepted for publications U1 Zeitschrift Naturforschung A 66a, 215 (2011). The influence of an induced magnetic field on the flow ana lysis discussed in chapter three is seen in chapter four. Mathematical modelling is presented in detail. Besides the flow quantities constructed in the previous chapters, the expressions of magnetic force function and axial induced magnetic fie ld have been developed additionally. It is found that an axial induced magnetic field exhibits symmetric nature about the origin. Moreover, axial induced magnetic fie ld is decreased in MHD fluid. The behaviour of current density near the channel walls in MHD case is quite opposite to that of an induced magnetic field. Such conclusions have been published in Comm. Nonlinear Sci. Numer. Simulation 15, 2407 (2010). Chapter five describes the influence of an induced magnetic field on the peristaltic flow of a Can'eau fluid in an asymmetric channel. The heat transfer is also taken into account. The walls of channel have different temperatures. The relevant equations are first modeled and then solved. It is shown that pumping rate in Jo.1l-ID fluid decreases. The axial induced magnetic field about the origin is not symmetric. This is because of the phase difference in the considered shapes of the channel walls. The temperature is an increasing function of Brinkman number. The contents of this chapter have been submitted for publication in Comm. Nonlinear Sci. Numer. Simulation 16, 3559 (2011). Chapter six is devoted to the peristalti c flow of hydrodynamic Carreal! fluid in an asymmetric channel in the presence of pattial slip and heat transfer. The associated equations and boundary conditions are developed. The partial slip condition in terms of shear stress is accounted. It is seen that there is a critical value of mean flow rate for whi ch the frictional forces resists the flow along the chatmel walls. Below this critical value, the frictional force is an increasing function of slip parameter. This research material has been accepted for publi cation In Zeitschrift Naturforschung A 65a, J12J (20JO). Chapter seven provides the analysis for peristaltic fl ow of a second order fluid in a planar channel. The flow is symmetric and fluid is electrically conducting. Induced magnetic field effect is included. The relevant equations are modeled and solved for small wave number. Trapping and pumping are also examined. Better pumping performance is achieved when there is an increase in the material parameters of second order fluid. The magnetic force function is an increasing function of viscoelastic parameter. The role of viscoelastic parameter on the current density distribution is qualitatively simi lar to that of the magnetic fo rce function. The size of the trapped bolus decreases when the viscoelastic parameter increases. The contents of this chapter have been accepted for publication in Int. J. Numerical Methods Fluids. The considered flow problem in chapter seven in the presence of heat and mass transfer has been studied in chapter eight. The energy and concentration laws are examined additionally. Main conclusions of this chapter are accepted now in Zeitschrift Naturforschung A. The peristaltic flow of micropolar fluid in an asymmetric channel is discussed in chapter nine. The simultaneous effects of p3l1ial slip and heat and mass transfer are seen. Results for flow quantities like stream function, axial and microrotation velocities, temperature and concentration etc are presented and analyzed 111 detail. It is found that behaviour of slip and microrotation parameters on the longitudinal pressure gradient IS qualitatively similar. The pressure gradient is an increasing function of coupling parameter. The pressure rise increases when slip parameter increases. The slip parameter on the temperature has opposite effect when compared with the pressure rise. However, the effect of slip parameter on r the concentration field is similar to that of pressure rise in a qualitative sense. Such observations have been accepted for publication in Chiense Phys. Letters.