Development of Mathematical Models for Peristalsis with Rheological Characteristics

Abstract

In recent years the study of peristalsis of non-Newtonian fluids has attracted the attention of many investigators due to its extensive applications in engineering and physiology. The non-Newtonian fluids do not obey the Newton's law of viscosity and cannot be described by using the Navier-Stokes equations. Therefore, various constitutive equations have been proposed due to the diversity in the physical structure of non-Newtonian fluids. Most of them are empirical or semi-empirical and give rise to equations which are much more non-linear, higher order and more complicated than the Navier-Stokes equations. Peristaltic flows are of fundamental importance in processes such as swallowing food tJu'ough the esophagus, vasomotion of small blood vessels such as arterioles, venues and capillaries, bile flow from the gall bladder into the duodenum. urine transport from kidney to the bladder through the ureter etc. Devices like roller and finger pumps are also operated on the principle of peristaltic pumping that are used to pump corrosive fluids, slurries, blood and foods in order to avoid their direct contact with the machinery. The design of many modern medical devices are also based upon the principle of peristaltic pumping for instance one may consider the blood in the heart-lung machine. Peristaltic flows with heat transfer analysis have vital role especially in chemical engineering processes. In physiology, the heat transfer is used to analyze the properties of tissues. Radio frequency therapy is helpful in the treatment of diseases like tissue coagulation, the primary living cancer, the lung cancer and the reflux of stomach acid. Moreover heat transfer analysis is also significant in hemodialysis and oxygenation processes. Magnetohydrodynamic fluid flow in a channel/tube with elastic and rhythmically contracting walls is of great interest with certain problems of the movement of conductive physiological fluids and for operating peristaltic MI-ID compressor. Flows in presence of magnetic field are significant in MHD power generators, MHD pumps and accelerators etc. Specific examples in this direction may include flow of nuclear fuel slulTies, flow of liquid metals and alloys, flow of plasma, flow of mercury amalgams, lubrications of hcavy oils and greases. In medical sciences the effect of magnetic field is used in the development of magnetic devices for cell separation, magnetic wounds and cancer tumor treatment, reduction of bleeding during surgeries. However, very little has been reported yet on the pelistaitic flows in the presence of magnetic field with Hall effect. Such effect cannot be over looked when flow subject to high magnetic field is considered. Hall effect can be taken into account when the Hall parameter (the ratio between the electron-cyclotron frequency and the electron-atom-collision frequency) is high. This happens, when the magnetic field is high or when the collision frequency is low. Also the current trend in the application of magnetohydrodynamics is towards a strong magnetic field, therefore one has to consider the influence of Hall current as it has great effect on the magnitude and direction of the CUITent density and consequently on the magnetic-force term. However it is ilOted that scarce literature is available for MHD peristaltic flow of non-Newtonian fluids with Hall effect. Motivated by all such facts we structure the present dissertation as follows: In chapter one we presented some definitions, fundamental equations and existing literature review regarding peristaltic flow in various flow configurations. Chapter two discusses the peristaltic flow of Johnson Segalman fluid with nanoparticles. Johnson Segal man fluid is useful for explaining 'spurt' phenomenon. Experimentalists associate 'spurt' with slip at wall. The problem is first non-dimensionalized and then solved by homotopy perturbation method (HPM) under long wavelength and low Reynolds number approximation. A net flow due to the travelling wave is obtained up to second order approximation. The results of this chapter are published in Journal of Aerospace Engineering 27 (2013) 404-413. The influence of heat and mass transfer on peristaltic flow of Jeffrey six-constant fluid with nanoparticles in an asymmetric channel has been discussed in chapter three. Asymmetry in the flow is induced by sinusoidal wave with different amplitudes and phases. Flow is induced by peristaltic wave along the length of channel walls. The analysis for axial velocity, pressure gradient and stream functions are carried out under lubrication approximation. The resulting non-linear equations are then solved for the series solutions. Graphical results are obtained to see the effect of various parameters of interest. The contents ofthis chapter have been submitted in "Meccanica". Chapter four investigates the effect of velocity slip on the peristaltic transpOli of PowellEyring fluid. Powell-Eyring fluid model deserves attention due to the fact that its stress constitutive relation is deduced from kinetic theory of liquids rather than the empirical relation as in the case of power-law model. It also correctly reduces to Newtonian model for high shear rate. The channel is assumed symmetric. The governing equations are prescnted in a wave frame. Solutions for strcam functioll and pressure gradient are derived by employing long wavelength and low Reynolds number assumptions. It is found that the magnitude of shear stress decreases for velocity slip parameter. The contents of this chapter are published in Applied Bionics and Biomechanics 11(2014) 69-79. Chapter five discloses the effect of Hall CUlTent on peristaltic flow of an electrically conducting Powell-Eyring fluid. The motion is induced by a sinusoidal wave traveling along the flexible walls of the channel. The flow is analyzed in a wave frame of reference moving with the velocity of wave. The equations governing the flow are solved by adopting lubrication approach. Series solution for stream function and axial pressure gradient are obtained. Impact of rheological parameters, Hartman number and Hall parameter on the flow quantities of interest are analyzed. It is noted that Hall parameter assists the flow. Moreover effect of Hall parameter is quite opposite to that of Hartman number. Main observations of this chapter are submitted in J. Magnetism and Magnetic Materials. Chapter six looks at the peristaltic motion of an incompressible third order fluid in a symmetric channel. This study is performed in the presence of applied magnetic field and the effect of Hall currents is also considered. The third order fluid described shear thinning/shear thickening effect but it lacks the stress relaxation and retardation effects. Mathematical formulation is given in a wave frame of reference. Series solutions up to first order for small Deborah number are obtained for the stream function, longitudinal velocity and pressure gradient. Numerical integration is carried out for the pressure rise and frictional forces. Influence of emerging parameters on the pressure rise, frictional forces, axial pressure gradient, velocity profile and h-apping are discussed. It is observed that pressure rise is a decreasing function of Hall parameter. The outcome of this chapter is submitted for publication in Int. J. Numerical Methods for Heat and Fluid flow. Hall and ion-slip effects on the peristaltic flow of hyperbolic fluid are reported in chapter seven. Long wavelength and low Reynolds number assumptions are employed in the problem fonnulation . The govcrning nonlinear problem is solved using pelturbation approach. Graphical results are reported and discussed for various parameters of interest. It is found that effect of Hall and ion-slip parameters on velocity is quite similar. Findings of this chapter have been submitted for publication in Journal of Aerospace Engineering. Heat transfer analysis for the peristaltic transport of an incompressible Williamson fluid with Hall and ion-slip effects is carried out in chapter eight. Joule heating effects is also taken in to account. The flow analysis is modeled in a frame moving with the velocity of the wave. Lubrication approach is adopted in the mathematical formulation. Series solutions for stream function, pressure gradient and temperature profile are constructed for small values of Weissenberg number. Variations of emerging physical parameters on the axial velocity, shear stress, pressure gradient and temperature are analyzed graphi cally. Increase in Weissenberg number leads to an enhancement in the pressure rise. It is also noted that pressure rise is a decreasing function of Hall and ion-slip parameters. The results of this chapter have been submitted for publication in "Applied Bionics and Biomechanics". Chapter nine contains the study of peristaltic flow of Phan-Thein-Tannar (PTT) fluid with Joule heating. The fluid is electrically conducting in the presence of uniform applied magnetic field. Hall and ion-slip effects are considered. PTT model is derived from the Lodge-Yamamoto network theory and is lmown as the simple best differential model which exhibits viscoelastic and shear thinning properties of polymer solution. The problem formulation is developed in a wave frame ofreference. The resulting problem is solved for the slream function, longitudinal pressure gradient and temperature. The phenomena of pumping and trapping are discussed. The contents of this chapter have been submitted for publication in "Applied Bionics and Biomechanics". Chapter ten is devoted to study the peristaltic flow of a MHD Prandtl fluid with Hall and ion-slip effects. Flow configuration is taken asymmetric. Asymmetry in the flow is induced by sinusoidal wave with different amplitudes and phases. Flow is induced by peristaltic wave along the length of c11annel walls. Both the magnetic field and channel are inclined. Mathematical modeling of the governing equations is developed. Series solutions for stream function and pressure gradient are obtained under the assumption of small wave number. Results of pressure rise are analyzed through numerical integration. Main results of this chapter are submitted for publication in "Applied Mathematics and Mechanics"