Mathematical modeling and analysis of ciliary induced flow of Newtonian and non-Newtonian fluids

Abstract

The present thesis deals with a mathematical study of ciliary induced flows of various non- Newtonian fluids through a planar channel and in an axially symmetric tube. The main motivation of the present research work is concerned with an investigation of the propulsion mechanics of ciliary induced flows of some biological fluids through certain physiological systems of the human body. In particular, we want to study the role.of ciliary movement in the transport of spermatic fluids through the ductuli efferentes of male reproduction system in the human beings through mathematical modeling. The spermatic fluids or the efferent duct materials are assumed as Casson, Carreau, micropolar, hydromagnetic conducting Newtonian fluids and the geometry of the ductus efferentes of the human male reproduction system is approximated with a planar channel of uniform dimensions and an axially symmetric uniform cylinderical tube. The mathematical equations governing the flow of the present problem are formulated in Cartesian and axisymmetric cylindrical coordinate systems. These are highly nonlinear and coupled partial differential equations. However, implication of the well known creeping flow approximation along with the long wavelength assumption permits us to obtain closed form exact solutions for the resulting simplifying system of equations governing the flow problems. This is a valid approach for the low Reynolds number flows and is widely used in the literature of physiological dynamics. The flow is produced under the action of ciliary beating that generates a metachronal wave and the analysis is made in the wave frame travelling at the speed of metachronal wave in the direction of flow. Exact solutions for velocity components, axial. pressure.gradient and the stream. function are obtained. For Carreau fluid model the governing system of equations is.reduced to a.system of nonlinear but ordinary differential equations by employing the creeping flow i.e., the low Reynolds number assumption along with the long wavelength approximation. In this case, we utilize the well known regular perturbation method to tackle the nonlinear.terms of the governing system of equations. Consequently, series form solutions for the stream.function, the velocity.distributions and the pressure.gradient are computed. In last two chapters, we have investigated the magnetohydrodynamics (MHD) effects on the ciliary induced flows by assuming that the efferent duct material is a conducting fluid. The applications of magnetohydrodynamics principles in physiological-type flows have been the subject of intensive research and study during the last few years. The study of MHD effects on the flow of spermatic fluids through the ductus efferents is a relatively new aspect of the problem. We have studied the problem theoretically through a mathematical model. The quantities of physical interest like the pumping characteristics, the ciliary trapping phenomenon, the axial pressure gradient, the velocity distribution and the volume flow rates are discussed in this study. Extensive analytical and numerical computations are carried out to obtain the results of various flow characteristics of physiological interest. The influence of pertinent parameters on the analytical results obtained by these models are analyzed and discussed through graphs. The numerical values of the volumetric flow rate obtained by the proposed models of our study are ix also compared with the experimentally estimated value of the flow rate of the human semen, 6103 mlh . This estimated value was suggested by Lardner and Shack [1] in human based on the flow rates in the ductus efferentes in the other animals, e.g., rat, ram and bull. These values are found to be in excellent agreement with the estimated value as compared to the value obtained by the Newtonian model of Lardner and Shack which was0.12103 mlh1 .