Journal of Fluids

Volume 2016, Article ID 7506953, 8 pages

http://dx.doi.org/10.1155/2016/7506953

## Mathematical Analysis on Heat Transfer during Peristaltic Pumping of Fractional Second-Grade Fluid through a Nonuniform Permeable Tube

^{1}Department of Applied Sciences and Humanities, Kamla Nehru Institute of Technology, Sultanpur, Uttar Pradesh 228118, India^{2}Department of Mechanical Engineering, Kamla Nehru Institute of Technology, Sultanpur, India

Received 8 March 2016; Accepted 19 May 2016

Academic Editor: Jose M. Montanero

Copyright © 2016 Siddharth Shankar Bhatt et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This mathematical study is related to heat transfer under peristaltic flow of fractional second-grade fluid through nonuniform cylindrical tube with permeable walls. The analysis is performed under low Reynolds number and long wavelength approximation. The analytical solution for pressure gradient, friction force, and temperature field is obtained. The effects of appropriate parameters such as Grashof number, nonuniformity of tube, permeability of tube wall, heat source/sink parameter, material constant, fractional time derivative parameter and amplitude ratio on pressure rise, friction force, and temperature distribution are discussed. It is found that an increase in amplitude ratio and material constant causes increase in pressure but increase in nonuniformity of the tube causes decrease in pressure. It is also observed that variation of friction force against flow rate shows opposite behavior to that of pressure. Increase in temperature is also observed due to increase in heat source/sink parameter at inlet as well as downstream.

#### 1. Introduction

The word “peristalsis” originated from the Greek word “peristaltikos,” which means clasping and compressing. Peristalsis is a mechanism of fluid transport through deformable vessels with the aid of a progressive contraction/expansion wave along the vessel. It is an important mechanism of fluid transport in different parts of the entire physiological system. Peristaltic flow appears in urine transport from kidney to bladder, the movement of spermatozoa in the ductus efferentes of the male reproductive tract and in the vasomotion of small blood vessels such as arterioles, venules, and capillaries. Peristaltic motion finds application in industry such as heart lung machine and roller pump. Peristaltic motion was first studied clinically in an article given by Bayliss and Starling [1] and much later Latham [2] theoretically investigated peristalsis using fluid mechanics principles. The work of Jaffrin and Shapiro [3] throws light on various parameters involved in the analysis of peristaltic pumping. Kumar et al. [4] studied unsteady peristaltic pumping in finite length tube with permeable wall.

Interaction of peristalsis with heat transfer plays an important role in biomedical science. Models of microvascular heat transfer are being increasingly used for optimizing thermal therapies such as hyperthermia treatment. Thermodynamic aspects of blood become significant in processes like oxygenation and hemodialysis. Victor and Shah [5] studied heat transfer to blood flowing in a tube. Srinivas and Kothandapani [6] investigated peristaltic transport in an asymmetric channel with heat transfer. Muthuraj and Srinivas [7] studied mixed convective heat and mass transfer in a vertical wavy channel with traveling thermal waves and porous medium. Lots of investigations have been done for uniform channel or tube; however, most physiological vessels, for example, ureters, esophagus, intestine, and ductus efferentes of the reproductive tract, possess nonuniform geometries. Some researcher have used nonuniform geometry for analysis such as Radhakrishnamacharya and Radhakrishna [8] who discussed heat transfer to peristaltic transport in a nonuniform channel. Ellahi et al. [9] discussed effect of heat and mass transfer on peristaltic flow in a nonuniform rectangular duct. However, the study of heat flow during peristalsis has not been given much consideration by investigators.

Most physiological fluids exhibit both viscous and elastic properties. Viscoelastic models are derived by using classical mechanics laws, that is, Newton’s law for viscous liquids and Hooke’s law for elastic solids. In modern days, fractional calculus is a rapidly growing field of research in physics, biology, and medical engineering. By using fractional calculus, the viscoelastic behavior of fluid can be successfully explained. Fractional second-grade calculus operator is actually a generalization to deal with integrals and derivatives of noninteger order. Fractional second-grade model can be obtained by replacing ordinary time derivative to fractional time derivative. This model is applied to the study of movement of chyme through small intestine, esophagus, and so forth. Using the concept of fractional calculus some visoelastic models have been developed such as fractional Maxwell, fractional Zener, fractional anti-Zener or fractional Jeffrey, and fractional Burgers’ models. Number of researchers has studied unsteady flows of viscoelastic fluids using different models like fractional Maxwell model, fractional second-grade fluid model, fractional Burgers’ model and fractional generalized Burgers’ model, and fractional Oldroyed-B model in channels, annulus, or tubes. Qi and Xu [10] studied unsteady flow of viscoelastic fluid with fractional Maxwell model in a channel. Tripathi [11] explored the transportation of a viscoelastic fluid with fractional second-grade model by peristalsis through cylindrical tube under the assumptions of long wavelength and low Reynolds number. Tripathi and Bég [12] studied peristaltic propulsion of generalized Burgers’ fluid through a nonuniform porous medium with chyme dynamics through diseased intestine. Hameed et al. [13] analysed the heat transfer on peristaltic flow of the fractional second-grade fluid confined in a uniform cylindrical tube in presence of magnetic field. Rathod and Tuljappa [14] have studied peristaltic flow of fractional second-grade fluid through a cylindrical tube with heat transfer under the assumption of long wavelength and low Reynolds number assumption.

The objective of this study is to investigate the effect of various concerned parameters on fractional second-grade fluid with heat transfer and peristaltic flow through nonuniform tube with permeable walls. The effects of these parameters have been studied under long wave length and the low Reynolds number approximation. The problem is solved analytically by the use of fractional calculus. The obtained expressions are utilized to discuss the influences of various physical parameters.

#### 2. Basic Definitions

*Definition 1. *The Riemann-Liouville fractional integral operator of order of a function : is given by [15]

*Definition 2. *The fractional derivative of order of a continuous function is given by [15] where , provided that right-hand side is point-wise defined on .

*Remark 3. *For example, ; we quote for ; in (2) one can getgiving in particular , , where is the smallest integer greater than or equal to .

#### 3. Mathematical Modelling

Consider the flow of an incompressible fractional second-grade fluid (as shown in Figure 1) due to peristaltic transport induced by sinusoidal wave trains propagating with constant speed . The temperature of walls of tube is . The constitutive equation for viscoelastic fluid with fractional second-grade model is given bywhere , , , and is time, shear stress, rate of shear strain, and material constant, respectively, is viscosity, and is fractional time derivative parameters such that . This model reduces to second-grade models when and classical Navier-Stokes model is obtained by substituting .