Mathematical models for peristaltic motion in a channel

Abstract

Many nonlinear problems in theory of viscous fluids are governed by the Navier-Stokes eq uations. Such equations are inadequate for the flow description of non-Newtonian fluids. In literature, many theoretical investigations have been carried out by taking the physiological lluids to behave like a Newtonian fluid which is not true in reality. In such situations, the analysis of rheological characteristics associated with non-Newtonian fluids cannot be ignored. III patticular, peristalsis appears extensively in physiological and industrial applications. Especially, the peristaltic motion of magnetohydrodynamic (MHO) flows of electrically conducting fluids has become the subject of growing interest for the researchers in recent times. This is due to the fact that such studies are useful particularly for having a proper understanding of the functioning of different machines used by clinicians for pumping blood and magnetic resonance imaging (MRI). The effect of a magnetic field on the flow of blood in atherosclerotic vessels also finds application in a blood pump used by cardiac surgeons during the surgical procedure. On the other hand, the theoretical study of MHD channel flows have many practical app lications in designing cooling systems with liquid metals in many devices such as accelerators, MHO pumps, MHD power generators, electrostatic precipitation, petroleum industry, electrostatic precipitation, purification of crude oil, aerodynamics heating and fluid dropl ets sprays. The fluid flow in a porous space is significant specifically in geophysical fluid dynamics. The distribution of fatty cholesterol and artery clogging blood clots in the lumen of coronary artery also behave like a porous medium in the pathological situations. Moreover, the process of heat transfer is useful for the analysis of tissues. Thus, the application of heat (hyperthermia), radiation (laser therapy) and coldness (cryosurgery) has attracted the attention of the investigators in them1~1 modeling for the destruction of undesirable tissues such as cancer. Keeping in view all the above mentioned facts, the present thesis is arranged as follows: Chapter one provides the survey regarding the existing relevant literature for peristalsis m viscous/non-Newtonian fluids under various aspects. The slip effects on the peristaltic transport of viscous fluid are analyzed in chapter two. The flow in an asymmetric channel is considered. Closed form solutions have been established under the assumption of long wavelength and low Reynolds number. The discussion for pressure rise and Crictional forces is provided through numerical integration. The results of this chapter have been published in Numerical Methods for Partial Differential Equations 5 (2011) 1003-1015. The goal of chapter three is to develop a mathematical model in order to examine the slip and heat transfer effects on the MHO peristaltic flow in a channel with compliant walls. The velocity slip condition is imposed in terms of shear stress. Solutions of the axial velocity, stream function, temperature and heat transfer coefficient are derived. Further, some flow quantities of interest are analyzed through graphical results. The contents of this chapter have been published in AsiaPacitic Journal of Chemical Engineering DOl: 10.1002/apj.470. The compliant wall effects on the peristaltic flow of viscolls fluid in a curved channel are analyzed in chapter fOllt'. In addition, the heat transfer is considered. The series solution have been first computed and then examined by graphical illustrations. This research is published in Int. J. Heat and Mass Transfer 54 (2011) 1615-1621. Chapter five presents the analysis for peristaltic flow of non-Newtonian fluid in a channel with compliant walls. Constitutive equations of a subclass of rate type fluids namely an Oldroyd-B fluid have been used. The flow is induced by the sinusoidal waves on the channel walls. Results are given and discussed for the free pumping case. This research has been published in Int. J. Numerical Methods in Fluids DOl: 10.1002/fld.2439. Chapter six studies the peristaltic transport of Johnson-Segalman fluid in a compliant wall channel. This fluid model is developed to allow non-affine deformations and has been used by many investigators to explain the "spurt" phenomenon. The fluid is electrically conducting in the presence of a constant applied magnetic field. Expressions for mean velocity at the boundaries of the channel, the mean-velocity perturbation function and the time-averaged mean axial velocity distribution are derived. The effects of various emerging flow parameters are shown and discussed thro ugh graphs. This work has been published in Phys. Lett. A 372 (2008) 5026-5036. effects of wall properties on the peristaltic f10w of power-law fluid in an asymmetric channel have been investigated in chapter seven. Long wavelength and low Reynolds number approximations have been adopted in the presentation of mathematical developments. Closed form solutions are constructed for the stream function and velocity. The streamlines pattern and trapping are also discussed. The observations of this chapter are published in AppJ. Math. Mech. (English edition) 31 (2010) 1231-1240. The analysis of an electrically conducting Jetfrey Huid with peristalsis is presented in chapter eight. This nuid model is simplest and can describe the rheological characteristics in terms of relaxation and retardation time parameters. The nonlinear differential equations subject to appropriate boundary conditions are solved for the ti'ee pumping case. The effects of pertinent parameters on the fl ow quantities of interest are discussed. The research in this chapter has been published in Zeitschrift fur Naturforschung A 66a (2011) 106-116. Chapter nine discusses the MHO peristaltic transport of Jeffrey fluid in a compliant wall channel with porous space. Heat transfer analysis is also considered. A regular perturbation technique is employed to solve the resulting problem. Solutions are presented in a power of small wave number. Expressions for the stream functi on, temperature distribution, velocity and heat transfer coefficient are computed. The influence of emerging parameters is shown on velocity, temperature distribution, heat transfer coefficient and trapping. These contents are submitted for publication in Nonlinear Analysis: Modeling and Control. Chapter ten rep0l1s the peristaltic transport of compressible Je1frey fluid in a compliant wall channel. Pel1urbation approach has been employed when the ratio of the wave amplitude to the radius of the pore is small. Expressions of mean axial velocity distribution, mean velocity at the boundaries and critical values are derived. The effects of various embedded parameters are discussed. This research has been accepted for publication in Journal of Mechanics in Medicine and Biology.