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Journal of Applied Mathematics
Volume 2019, Article ID 3679373, 19 pages
https://doi.org/10.1155/2019/3679373
Research Article

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

1Department of Mathematics, College of Education for Pure Science, University of Basrah, Basrah, Iraq
2Department 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 [814]. 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.

Figure 1: Schematic diagram for the flow problem.

Rheological equation for Casson fluid is defined [22] aswhere and is the component of the deformation rate, is the product of the component of deformation rate with itself, is the critical value of the said product, is the plastic dynamic viscosity of the non-Newtonian fluid, and is the yield stress of slurry fluid. We are also applying the following assumptions on the flow model: (i)The effects of induced magnetic and electric field produced due to the flow of electrically conducting fluid are negligible.(ii)No external electric field is present.

Under aforementioned constraints the conservation equations for the flow arewhere and are the velocity components in and directions, respectively, is the pressure, is the dynamic viscosity of the fluid (ratio of Kinematic viscosity and density), is the Casson fluid parameter, and is the magnitude of imposed magnetic field. Supporting conditions for the problem are as follows:and we can simplify the above system of equations by eliminating the pressure term equations (3)-(4) and using (2). After cross differentiation and introducing vorticity we get Transform introduced by Wang [7] for a two dimensional flow is stated as whereSubstituting (8)-(10) in (7) and using (6) yield a nonlinear ordinary differential equation describing the Casson fluid flow aswhere denotes the nondimensional Squeeze number and is magnetic number. Boundary conditions for the problem by using (8)-(10) reduce toand squeezing number describes the movement of the plates ( corresponds to the plates moving apart, while corresponds to collapsing movement of the plates). It is pertinent to mention here that for and our study reduces to the one obtained by Wang [7]. Skin friction coefficient is defined as where

3. The Basic Steps of the Novel Algorithm

This section describes how to obtain a novel algorithm to calculate the coefficients of the power series solution resulting from solving nonlinear ordinary differential equations resulting from using transforms ((8)-(10)) to find analytical-approximate solution. These coefficients are important bases to construct the solution formula; therefore they can be computed recursively by differentiation ways. To illustrate the computation of these coefficients and derivation of the novel algorithm, we summarized the detailed new outlook in the following steps.

Step 1. Consider the nonlinear differential equation as follows:and integrating (16) with respect to on yieldswhere

Step 2. Assume thatrewriting (19)and substituting (20) in (17), we obtainwhere

Step 3. We focus on computing the derivatives of with respect to which is the crucial part of the proposed method. Let us start calculating , We see that the calculations become more complicated in the second and third derivatives because of the numerous calculations. Consequently, the systematic structure on calculation is extremely important. Fortunately, due to the assumption that the operator and the solution are analytic functions, the mixed derivatives are equivalent.

We note that the derivatives function to is unknown, so we suggest the following hypothesis Therefore (23)-(26) are evaluated by

Step 4. Substituting (28)-(31) in (21) we will get the required analytical-approximate solution for (16).

4. Application

The novel algorithm described in the previous section can be used as a powerful solver to the nonlinear differential equations of squeezing flow between two parallel plates (11) - (12) in order to find a new analytical-approximate solution. From Step 1 we have and we rewrite (32) as followswhere From the boundary conditions, (33) becomesand from Step 2 we have Step 3 yields We note that the derivatives of with respect to that are given in (27) can be computed by (37)-(40) as Now, we need to extract the first derivatives of G as follows: and from (36) by using (41)-(45), we obtain