Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 6904161, 8 pages

https://doi.org/10.1155/2017/6904161

## Swirling Flow in a Permeable Tube at Slowly Expanding and Contracting Wall

Department of Quantitative Methods, College Business Administration, University of Dammam, Dammam 31441, Saudi Arabia

Correspondence should be addressed to Sufian Munawar

Received 5 September 2016; Accepted 27 December 2016; Published 19 January 2017

Academic Editor: Sébastien Poncet

Copyright © 2017 Sufian Munawar. 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

The swirling flow inside a circular elastic tube with expanding and contracting permeable wall in the presence of a uniform magnetic field is studied analytically. The tube is also assumed to be rotating around its axis with an angular velocity. The governing equations for this multidimensional flow are reduced to nonlinear differential equations with similarity transformations. An analytic series solution is obtained by homotopy analysis method (HAM). The effects of physical parameters on various flow characteristics, such as the velocity, skin friction, and pressure variation, have been analysed briefly. The impact of surface expansion/contraction and rotation has been investigated on the internal boundary-layer flow inside the tube of uniform cross-section.

#### 1. Introduction

The laminar boundary-layer flows inside elastic pipes or tubes have received great attention by numerous researchers since they are widely used in various engineering and biological applications. The examples include the binary gas diffusion, transport of biological liquids through contracting or expanding vessels, pulsating porous diaphragms, urine flow in urethras, natural transpiration and cooling, and the regression of the burning surface in solid rocket motors. Uchida and Aoki [1] initiated the study on the flow inside a tube with deformable wall and one closed end. Ohki [2] discussed the laminar incompressible flow in a semi-infinite porous pipe with expanding and contracting radius in axial direction. A theoretical investigation was performed by Goto and Uchida [3] to study the suction/injection effects in an expanding/contracting pipe. Bujurke et al. [4] investigated the same flow problem by calculating a series solution. Recently, Si et al. [5] calculated the multiple solutions for the contracting or expanding porous pipe at large suction Reynolds number using singular perturbation method. Srinivas et al. [6] obtained analytic series solution for the thermal-diffusion and diffusion-thermo effects in a channel with moveable permeable walls. Currently, the effect of mass transfer and chemical reaction inside a porous tube with expanding or contracting wall was addressed by Srinivas et al. [7]. The effect of MHD on the same type of flow phenomenon was studied by Srinivas et al. [8]. The flow inside a twisted pipe was investigated by O’Dea and Waters [9] and calculated the solute uptake inside a twisting pipe. Makinde [10] presented a mathematical model describing the flow inside a collapsible tube and obtained an analytic series solution. The effect of variable viscosity and viscous dissipation on the flow in a moving pipe was discussed by Makinde [11]. Xinhui et al. [12] investigated the slip effects in the micropolar fluid inside a porous channel with expanding/contracting walls. Another type of three-dimensional flows in polar coordinates is the boundary-layer flow over cylinder. In this regard, Sprague and Weidman [13] investigated the boundary-layer flow over a twisted cylinder and an asymptotic solution for large range of the Reynolds number was obtained. Keeping this fact in mind the boundary-layer flow over cylinder can also be considered three-dimensional if the surface of pipe moves in two lateral directions. Recently, Munawar et al. [14] made a complete thermodynamical analysis of a three-dimensional viscous flow over a stretching and rotating cylinder. Fang and Yao [15] and Fang et al. [16] discussed other sorts of the three-dimensional boundary-layer flows over cylinder which was rotating, expanding, and stretching with respect to its length.

This analysis extends the idea by embarking on the impact of swirling on the flow inside a contracting/expanding semi-infinite porous tube. Since most of the biological fluids are electrically conducting, therefore a uniform magnetic field is taken along the radial direction. An approximate series solution is obtained by using homotopy analysis method (HAM). The convergence and accuracy of the solution are illustrated by plotting convergence tables, plotting the -curves, and calculating the squared residual errors. To validate our HAM solution a comparison has also been made with the results reported by Li et al. [17] using numerical and analytic techniques.

#### 2. Mathematical Formulation

Consider an axisymmetric flow of an incompressible viscous fluid inside a semi-infinite horizontally placed expanding and contracting tube. The tube is rotating with a time-dependent angular velocity . The radius of tube is a continuous function of time and the surface of tube is assumed to be uniformly permeable. One end of the tube is kept closed and the wall of the tube expands and contracts with a time-dependent rate . A uniform magnetic field with intensity is applied along the radial direction. Assuming small magnetic Reynolds number, the induced magnetic field is assumed to be negligible. The -axis is measured along the axis of tube and the -axis is perpendicular to the surface (see Figure 1). Under these assumptions the governing equations of motion are given by