Analysis of Hartmann boundary layer peristaltic flow of Jeffrey fluid: Quantitative and qualitative approaches

Abstract

Peristalsis of Jeffrey fluid in axisymmetric tube is studied when the fluid is subject to strong magnetic field. Mathematical analysis is made under the assumption of small Reynolds number and long wave length approximations. An alternate assumption of lubrication theory can also be applied to the peristalsis problem. Both the approaches lead to same mathematical expressions. The effects of magnetic field are investigated in the boundary layer (Hartman boundary layer) and on the boundary layer thickness. The effects of strong magnetic field, in the boundary layer, are explored using asymptotic analysis to find analytical solution. We notice that this approach facilitates to unveil the effects of strong magnetic field explicitly and determines the boundary layer thickness mathematically. The objective behind this study is: how to control the boundary layer thickness through the application of strong magnetic field. This phenomenon is central from theoretical and applied points of view. While going for the asymptotic analysis of large magnetic field; the mathematical model leads to singular perturbation problem which is an apparent diversion to most of the peristalsis problems– attempted by regular perturbation method.The boundary value problem is solved analytically using singular perturbation approach together with higher order matching technique. The important features of peristalsis like stream function, velocity, and pressure rise are calculated. In addition to the analytical solution, we explore the qualitative behavior of the flow using the theory of dynamical systems. The velocity field is determined by the phase plane analysis and the stability of the solution is found through bifurcation diagrams. Qualitative analysis of the solution has been carried out for magnetic parameter, amplitude ratio, flow rate and the Jeffrey fluid parameter. The concomitant change of stability is given through topological flow patterns. Equilibrium plots (bifurcation diagrams) give a complete description of the various flow patterns developed for the complete range of a flow parameters in contrast to most of the studies that describe the flow patterns at some particular value of a parameter. The study helps to analyze the behavior of the fluid at the critical points that represent steady solution. The mix of the analytical and qualitative approach will help to broaden the scope and understanding of peristaltic transport of fluid in channels, tubes and curved tubes (this study).