Effect of Relaxation and Retardation Time on Peristaltic Transport of the Oldroydian Viscoelastic Fluid

Abstract

The influence of relaxation and retardation time on peristaltic transport of an incompressible Oldroydian viscoelastic fluid by means of an infinite train of sinusoidal waves traveling along the walls of a two-dimensional flexible channel is investigated. A perturbation solution is obtained for the case in which the amplitude ratio (wave amplitude to channel half-width) is small. The results show that the values of the mean axial velocity of an Oldroydian viscoelastic fluid is smaller than that for a Newtonian fluid. The reflux phenomena are discussed. It is found that the critical reflux pressure gradient decreases with increasing retardation time and increases with increasing relaxation time. Numerical results are reported for different values of the physical parameters of interest.Introduction. The word "peristalsis" stems from the Greek word "peristalikos," which means clasping and compressing. Peristaltic transport of fluids occurs in the esophagus, the ureter, and the lower intestine. In addition, peristaltic pumping occurs in many practical applications involving biomechanical systems, such as roller and finger pumps. A mathematical analysis of peristaltic pumping in a two-dimensional formulation was presented by Latham [1]. Fung and Yih [2] investigated a perturbation solution of a two-dimensional case in which the amplitude ratio (wave amplitude to channel half-width) was small. Srivastava and Srivastava [3] studied the blood flow. El-Shehawey and Mekheimer [4] examined the effects of couple-stresses in peristaltic transport of fluid. Peristaltic transport of a particle-fluid suspension was considered in [5,6]. Antanovskii and Ramkissoon [7] studied peristaltic transport of a compressible viscous fluid in a finite pipe. Carew and Pedley [8] investigated periodic activation waves in an infinite tube.Most theoretical investigations were performed for Newtonian fluids, although it is known that most physiological fluids behave like non-Newtonian fluids. In this aspect, there is only limited information on transport of non-Newtonian fluids. The main reason is that additional nonlinear terms appear in equations of motion, rendering the problem more difficult to solve. Another reason is that a universal non-Newtonian constitutive relation that can be used for all fluids and flows is not available. The earliest studies date back to Raju and Devanathan [9, 10] who considered the motion of an inelastic power-law fluid and of a special differential-type viscoelastic fluid of grade two through a tube with sinusoidal small-amplitude corrugation in the axial direction. Bohme and Friedrich [11] investigated peristaltic flows of viscoelastic fluids under the assumptions that the relevant Reynolds number is small enough to neglect inertia forces and that the ratio of the wave length and the channel height is large, which implies that the pressure is constant over the cross section. Hayat et al. [12] investigated periodic unsteady flows of a non-Newtonian fluid. Misra and Pandey [13] studied a peristaltic flow of blood i...

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