Journal of Applied Mathematics

Volume 2019, Article ID 3679373, 19 pages

https://doi.org/10.1155/2019/3679373

## A Novel Algorithm for Studying the Effects of Squeezing Flow of a Casson Fluid between Parallel Plates on Magnetic Field

^{1}Department of Mathematics, College of Education for Pure Science, University of Basrah, Basrah, Iraq^{2}Department of Mathematics, College of Science, University of Basrah, Basrah, Iraq

Correspondence should be addressed to Abeer Majeed Jasim; moc.oohay@messaj.reeba

Received 4 December 2018; Revised 30 January 2019; Accepted 5 March 2019; Published 1 April 2019

Academic Editor: Oluwole D. Makinde

Copyright © 2019 Abdul-Sattar J. A. Al-Saif and Abeer Majeed Jasim. 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

In this paper, the magneto hydrodynamic (MHD) squeezing flow of a non-Newtonian, namely, Casson, fluid between parallel plates is studied. The suitable one of similarity transformation conversion laws is proposed to obtain the governing MHD flow nonlinear ordinary differential equation. The resulting equation has been solved by a novel algorithm. Comparisons between the results of the novel algorithm technique and other analytical techniques and one numerical Range-Kutta fourth-order algorithm are provided. The results are found to be in excellent agreement. Also, a novel convergence proof of the proposed algorithm based on properties of convergent series is introduced. Flow behavior under the changing involved physical parameters such as squeeze number, Casson fluid parameter, and magnetic number is discussed and explained in detail with help of tables and graphs.

#### 1. Introduction

Squeezing flow between parallel walls occurs in many industrial and biological systems. The unsteady squeezing flow of a viscous fluid between parallel plates in motion normal to their own surfaces is a great interest in hydrodynamical machines. The pioneer work and the basic formulation of squeezing flows under lubrication were assumed by Stefan [1]. In past researches over few decades, Reynolds [2] analyzed the squeezing flow between elliptic plates while Archibald [3] suggested the same problem for rectangular plates. The Reynolds equation was studied for squeezing flows which were not sufficient for some cases as was demonstrated by Jackson [4] and Usha and Sridharan [5]. The study on the motion of electrically conducting fluid in the presence of a magnetic field is called magneto hydrodynamics (MHD). In engineering, the application of MHD can be seen in the electromagnetic pump. The pumping motion of this device is caused by the Lorentz force. This force is produced when mutually perpendicular magnetic fields, and electric currents are applied in the direction perpendicular to the axis of a pipe containing conducting fluid [6]. The laws of conservations under the similarity transformation for the squeezing flow of an electrically conducting Casson fluid, which was suggested by Wang [7], have been used to extract a highly nonlinear ordinary differential equation governing the magneto hydrodynamic (MHD). Many researchers have conducted numerous research attempts for the purpose of understanding and analyzing squeezing flows [8–14]. Duwairi et al. [15] investigated the effects of squeezing on heat transfer of a viscous fluid between parallel plates. Mohyud-Din et al. [16] have studied heat and mass transfer analysis for the flow of a nanofluid between rotating parallel plates while Mohyud-Din and Khan [17] analyzed the nonlinear radiation effects on squeezing flow of a Casson fluid between parallel disks. Ahmed et al. [18] assumed MHD flow of an incompressible fluid through porous medium between dilating and squeezing permeable plates. Khan et al. [19] have explained MHD squeezing flow between two finite plates, and Hayat et al. [20] have demonstrated the effect of squeezing flow of second grade fluid between two parallel disks. Naveed et al. [21] studied effects on magnetic field in squeezing flow of a Casson fluid between parallel plates. The main scope of this paper is to implement novel algorithm to obtain analytical-approximate solution which depends mainly on the coefficients of powers series. This is close to the numerical solution for the squeezing flow of a Casson fluid between parallel plates; although the resulting series of the novel algorithm which contains initial conditions are known, the others are unknown and their values can be found through the boundary conditions that are defined. These series also contain physical parameters which can be compensated by constants, and the geometrical effects of these parameters would be investigated depending on the resulting solutions. In this work, it can be clearly seen that a novel algorithm is successfully applied to solve the equations of an unsteady squeeze of flow between parallel plates as well as to find an analytical-approximate solution. In contrast, it was found that our results from algorithm completely agree with the results obtained from the numerical solution through the application of Range-Kutta fourth-order method [RK-4], the numerical solutions of Wang [7], and variation of parameter method (VPM) [21]. The organization of this paper is as follows: The governing equations are derived in Section 2. Details of derivation of the novel algorithm have been written as steps in Section 3. The performance of the novel algorithm for the squeezing flow is applied in Section 4. In Section 5 the convergence analysis is presented. Results and discussions are given in Section 6. Finally, the conclusions are indicated in Section 7.

#### 2. Governing Equations

We consider an incompressible flow of a Casson fluid between two parallel plates separated by a distance , where is the initial gap between the plates (at a time ). Additionally corresponds to a squeezing motion of both plates until they touch each other at , for ; the plates bear a receding motion and dilate as described in Figure 1.