Advances in Mathematical Physics

Volume 2018, Article ID 5686089, 8 pages

https://doi.org/10.1155/2018/5686089

## Heat Transfer in MHD Flow due to a Linearly Stretching Sheet with Induced Magnetic Field

Correspondence should be addressed to Tarek M. A. El-Mistikawy; ge.ude.uc.gne@ywakitsimle

Received 4 November 2017; Accepted 28 January 2018; Published 25 February 2018

Academic Editor: Alkesh Punjabi

Copyright © 2018 Tarek M. A. El-Mistikawy. 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 traditionally ignored physical processes of viscous dissipation, Joule heating, streamwise heat diffusion, and work shear are assessed and their importance is established. The study is performed for the MHD flow due to a linearly stretching sheet with induced magnetic field. Cases of prescribed surface temperature, heat flux, surface feed (injection or suction), velocity slip, and thermal slip are considered. Sample numerical solutions are obtained for the chosen combinations of the flow parameters.

#### 1. Introduction

The problem of the two-dimensional flow due to a linearly stretching sheet, first formulated by Crane [1], has a simple exact similarity solution. This invited several researchers to add to it new features allowing for self-similarity. As a boundary-layer problem, Pavlov [2] added uniform transverse magnetic field. P. S. Gupta and A. S. Gupta [3] added surface feed (suction or injection). These problems were recognized as being exact solutions of the corresponding Navier-Stokes problems by Crane [1], Andersson [4], and Wang [5], respectively. To the Navier-Stokes problem, Andersson [6] added velocity slip. Fang et al. [7] combined the effects of transverse magnetic field, surface feed, and velocity slip.

Heat transfer was treated in several publications, mostly neglecting viscous dissipation and Joule heating (in MHD problems). This allowed self-similar formulation in cases of the surface having constant temperature [3, 8] or temperature or heat flux proportional to a power of the stretch-wise coordinate [9–11]. Prasad and Vajravelu [12] treated the boundary-layer flow of a power law fluid retaining viscous dissipation and Joule heating, in case of the surface temperature being proportional to .

The abovementioned MHD problems adopted the small magnetic Reynolds number assumption, thus neglecting the induced magnetic field. In [13], it was shown that the full MHD problem, that is, Navier-Stokes and Maxwell’s equations with adherence conditions and appropriate magnetic conditions, allowed for self-similarity.

In this article, the work of [13] is extended to the heat transfer problem including viscous dissipation and Joule heating, in cases of prescribed surface temperature or heat flux. Surface feed, velocity and thermal slip, and shear work are also included.

The problem is of both theoretical and practical value. Theoretically, it indicates the importance of the traditionally ignored physical processes of induced magnetic field, viscous dissipation, Joule heating, and shear work. Practically, the problem is encountered in several situations. For example, extrusion processes in polymer and glass industries involve stretching sheets extruded in an otherwise quiescent fluid. The quality of the product depends on the controlled heat transfer between the sheet and the fluid. Four control agents are in mind, in this study, the MHD effect of a magnetic field permeating a conducting fluid, surface feed (fluid injection or suction), fluid additives (possibly, nanoparticles) associated with velocity and thermal slip [14], and convective heating or cooling [15] which has the same effect as thermal slip.

#### 2. Mathematical Model

An electrically conducting, incompressible, and Newtonian fluid is driven by a nonconducting porous sheet, which is stretching linearly in the -direction. At the surface, we consider cases of prescribed temperature or heat flux and allow for velocity and thermal slip. In the far field, the fluid is essentially quiescent under pressure and temperature and is permeated by a stationary magnetic field of uniform strength in the transverse -direction.

The equations governing this steady MHD incompressible flow are the continuity and Navier-Stokes equations with Lorentz force [16]the energy equation with viscous dissipation and Joule heatingand Maxwell’s equations and Ohm’s law, in the absence of surplus charge and electric field, is the velocity vector, is the pressure, is the temperature, is the current density, is the magnetic field, and is the dissipation function. Constants are the fluid density , kinematic viscosity , specific heat , thermal conductivity , the electric conductivity , and magnetic permeability .

Use of Ohm’s law (6) to eliminate from (2) to (4) and casting in Cartesian components lead to the following equations for two-dimensional flow.They are complemented with the surface conditionsand the far field conditions are the velocity components in the directions, respectively, and are the corresponding induced magnetic field components. The stretching rate and the velocity and thermal slip coefficients and are assumed constant. In the condition for , the last term represents the shear work [17]. In the far field, the condition for translates the physical requirement of the absence of any current density, while that on indicates that stands for the far field total magnetic field imposed and induced [13].

The problem admits the similarity transformations where primes denote differentiation with respect to . The fact that the temperature is quadratic in allows its constituents , , and to be dependent on only.

The problem becomeswhere is the magnetic Prandtl number, is the magnetic interaction number, and is the Prandtl number.

Consistent with the similarity transformations, we take the surface values to bewhere , , , , , , and are prescribed values.

With and , we get the following conditions on the flow variables:

#### 3. Numerical Method

We start by solving for and , since their nonlinear problem is uncoupled from the problems for , , and . A closed form solution is not possible, so we seek an iterative numerical solution. In the th iteration, we solve, for , (11) with its right hand side evaluated using the previous iteration solutions and , together with conditions (17). Then we solve, for , (12) with the known , together with conditions (18). The iterations continue until the maximum error in , , , and becomes less than 10^{−10}. For the first iteration, we zero the right hand side of (11) which corresponds to .

The numerical solution of the problem for and utilizes Keller’s two-point, second-order accurate, finite-difference scheme [18]. A uniform step size is used on a finite domain . The value of is chosen sufficiently large in order to insure the asymptotic satisfaction of the far field conditions. The nonlinear terms in the problem for are quasi-linearized, and an iterative procedure is implemented, terminating when the maximum error in and becomes less than 10^{−10}.

Having determined and , we solve the linear problems: (13) with conditions (19) for , (14) with conditions (20) for , and then (15) with conditions (21) for , using Keller’s scheme on the same grid.

#### 4. Sample Results and Discussion

The problem for and involves four parameters: , , , and . For , , Figure 1 depicts at different values of , when , and at different values of , when . The corresponding results for together with are depicted in Figures 2 and 3, respectively. The induced magnetic field is primarily affected by the streamwise velocity component represented by . As decreases due to higher surface slip or suction rate, both and decrease.