Mathematical Problems in Engineering

Volume 2016, Article ID 2568785, 10 pages

http://dx.doi.org/10.1155/2016/2568785

## On Combined Effects of Heat Transfer and Chemical Reaction for the Flow through an Asymmetric Channel with Orthogonally Deformable Porous Walls

^{1}Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan^{2}Department of Mathematics, COMSATS Institute of Information Technology, Attock, Pakistan^{3}Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan^{4}School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 19 November 2015; Revised 26 January 2016; Accepted 21 February 2016

Academic Editor: Francesco Franco

Copyright © 2016 Syed Tauseef Mohyud-Din 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

The combined effects of heat transfer and chemical reaction are studied for the flow through a semi-infinite asymmetric channel with orthogonally deformable porous walls. The similarity transforms have been used to reduce the conservation laws to a corresponding system of nonlinear ordinary differential equations. The resulting equations are solved, both analytically and numerically, by using Homotopy Analysis Method (HAM) and the fourth-order Runge-Kutta (RK-4) method, respectively. The convergence of the analytical solution is assured through the so-called total squared residual error analysis. The optimal values of auxiliary parameters are obtained by minimizing the total squared residual error.

#### 1. Introduction

The flow between porous channels can rightly be considered as one of the most important problems in fluid mechanics. Its practical applications in diversified fields of science and technology have been the basic reason why scientists from all over the world have tried to understand these flows in a better way. These applications include uniformly distributed irrigation, interbody biological transport, ablation processes, sublimation mechanisms, propellant combustion, and industrial cleansing systems.

Formative work regarding these kinds of flows can be traced back to Berman [1], who initiated a mathematical model to describe the flows between parallel plates. He established that the normal component of the velocity must be independent of streamwise coordinate. This in fact enabled him to reduce Navier-Stokes equations to a single, fourth-order, nonlinear ordinary differential equation. Following his footsteps, numerous studies have been carried out later on. The detailed history about these investigations can be found in [2].

Heat transfer coupled with chemical reaction plays an important role in many fields of science and engineering. Its practical applications may include combustion systems, atomic reactor safety, dying of cloths, metallurgy, and chemical engineering. A chemical reaction is said to be of first order if the rate of reaction varies directly with respect to the concentration. Almost in all chemical industries, a relatively inexpensive raw material is put through some specifically designed chemical processes to obtain high value products. In most of the situations, these chemical processes are based on chemical reactions in the presence of heat transfer. Bridges accompanied by Rajagopal [3] investigated the pulsatile flow of a chemically reacting fluid, whose viscosity was dependent on the concentration of the constituents. The flow was governed by a convection-reaction-diffusion equation and the velocity gradient, which could thicken or thin the fluid. The effects of chemical reaction and the space porosity on mixed convective MHD flow in an asymmetric channel were investigated by Srinivas and Muthuraj [4]. The details of heat transfer analysis can be found in [5–11].

In a very recent article, Reddy et al. [12] discussed the combined effects of heat and chemical reaction on an asymmetric laminar flow between slowly expanding and contracting walls. To make their perturbation solution valid, they had to impose physical constraints such as lower wall deformation rate and weak permeability. Besides, the presence of other nonperturbed physical parameters makes their solution more vulnerable as the perturbation solution strictly relies on the existence of very small parameters in equation.

In this study, we have tried to remove those physical constraints, which have been imposed in [13], only to insure the convergence of their perturbation solution. A probable invalidity of their solution for concentration profile has also been pointed out even for smaller values of wall deformation rate and the permeation Reynolds number.

We have used a well-known analytical method called Homotopy Analysis Method (HAM) [14–18] to guarantee the convergence of the analytical solution. To further ensure the validity of the solution, a numerical solution (fourth-order Runge-Kutta coupled with shooting method) has also been sought. The aforesaid numerical method has effectively been used in several studies [19–21]. We have compared the analytical and numerical results and have found an excellent agreement between them. Finally, the effects of physical parameters on temperature and concentration profiles are displayed and analyzed with the help of graphs accompanied by comprehensive discussions.

#### 2. Mathematical Analysis

The laminar flow of a viscous incompressible fluid is taken into account, in a rectangular domain of infinite length, and having two deformable porous walls. The fluid may enter or exit, during the successive contractions/expansions, through the permeable walls. The leading end of the container is closed with a solid but flexible membrane, which deforms incorporation with the movement of the walls. A narrow gap assumption has been imposed; that is, the height () of the channel is taken to be very small as compared to its width. It enables us to confine the whole problem in to a half domain and a planer cross section of the simulating domain is presented in Figure 1.