#### 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.

Tables 1 and 2 give values of the surface shear and the entrainment rate represented, respectively, by and , as well as the induced magnetic field components at the surface represented, respectively, by and . Refer to [13] for values of , , , and at different values of and , when .

Presented in Figures 4–9 are the results for the temperature constituents , , and obtained, in case of prescribed surface temperature , and in case of prescribed surface heat flux ; , 1 or 2. Table 3 summarizes the given surface values and the constituents involved in Figures 4–9, noting that (13)–(15) indicate that and are independent of the other constituents, while depends on .

**(a)**Temperature constituent at different values of

**(b)**Temperature constituent at different values of

**(a)**Temperature constituent at different values of

**(b)**Temperature constituent at different values ofThe following is noticed.(i)*Constant surface temperature*: at , . For , and decreases monotonically with , while for , and has a peak that gets farther from the surface as decreases.(ii)*Constant heat flux*: when , . As more heat is added to the fluid, that is, for increasing , rises and decreases monotonically with . Removing more heat from the fluid, that is, for decreasing , decreases and has a peak that gets farther from the surface.(iii)*Linear surface temperature and heat flux*: noting that and , the presented results for nonnegative and indicate that decreases monotonically with . Higher results in smaller , while higher results in higher .(iv)*Surface temperature *: at , . For , and decreases monotonically with . For , and has a peak that gets farther from the surface as decreases. For , and rises monotonically with . At , and drops from its zero surface value to a local minimum and then rises to its zero far field value. Similar behavior of is observed for decreasing , but with decreasing minimum and . For increasing , and rises to a higher peak. For , and rises to a peak, falls to zero and then to a bottom, and rises again to zero.(v)*Heat flux *: when , . For increasing , rises and decreases monotonically with . For decreasing , decreases from positive to negative values with having a positive peak. For , rises monotonically from a negative surface value to zero. For , is monotonically decreasing, while for , is monotonically increasing from its surface value to zero. For , decreases to a negative local minimum and then rises to zero.

To complement Figures 4–9, we give in Table 4 the numerical values of at different values of , at different values of , and and at different values of , and in Table 5 the numerical values of at different values of , at different values of , and and at different values of .

Table 6 shows the effect of the thermal slip coefficient . As the first legs of conditions (19)–(21) indicate, the sign of the surface derivative of the temperature constituent determines whether the surface value of the constituent increases or decreases with . Thus, for example, (0) increases when , for which , and decreases when , for which .

The shear work is represented by the term involving the velocity slip coefficient in the second leg of conditions (19) for . Table 7 demonstrates its importance. Neglecting the shear work reduces the predicted surface temperature.

On the right-hand side of (13) and (15), the first terms represent Joule heating and streamwise heat diffusion, respectively, while the second terms represent heat dissipation. Table 8 demonstrates the effect of neglecting these three processes. The predicted heat flux to the surface, represented by and , is reduced considerably by neglecting viscous dissipation, streamwise diffusion, and/or Joule heating. Neglecting one or more may even predict heat flux in the wrong direction.

#### 5. Conclusion

The problem of the flow due to a linearly stretching sheet in the presence of a transverse magnetic field has been formulated to include surface feed, velocity slip, and thermal slip. The problem has been shown to admit self-similarity of the full MHD fluid flow equations. Included in the thermal equation and conditions are physical processes such as viscous dissipation, Joule heating, streamwise heat diffusion, and shear work which were traditionally ignored or approximated.

The presented numerical results are sample results. They show that the self-similar model (with the several physical processes involved) and the method of solution are capable of producing meaningful and useful results. No attempt is made to present a detailed study of the individual or collective effect of the 12 parameters at hand. It is up to the interested reader to choose his/her set of processes and associated parameters to concentrate on. Moreover, other physical processes can be added to the model such as thermal radiation, heat generation, and/or flow through a porous medium, as long as self-similarity is preserved.

Nonetheless, the presented results pinpoint some interesting observations described below.

The velocity slip coefficient and the suction rate have opposite effects on the curvature of the streamwise velocity profile . While increasing the first flattens the profile causing the surface shear to decrease, increasing the second curls the profile causing to increase. The more the curling (flattening) of the profile, the higher (the lower) the far field entrainment rate , to compensate for the faster (the slower) streamwise flow close to the surface.

The induced magnetic field is primarily affected by the streamwise velocity component . As decreases due to higher surface slip or suction rate, both and decrease.

The effect of the velocity slip coefficient and the suction rate on the temperature is through their effect on the velocity and magnetic field components, with their involvement in viscous dissipation and Joule heating. Moreover, the velocity slip coefficient has the added effect of shear work, neglect of which results in considerable reduction in the predicted surface temperature.

Even when the surface is maintained at the ambient temperature , the fluid adjacent to the surface acquires a higher temperature that increases as the thermal slip coefficient increases. This is due to viscous dissipation and Joule heating, which are traditionally ignored. The same applies to the equivalent situation of convective heating when the temperature on the other side of the sheet is maintained at , with being the heat convection coefficient.

The streamwise heat diffusion, which is neglected in boundary-layer models, is as important as viscous dissipation and Joule heating. Neglecting one or more of these thermal processes may predict lower heat flux or even heat flux in the wrong direction.

Depending on the surface thermal conditions, the profiles of the temperature constituents my decay monotonically toward the far field or may have an extremum at a finite distance from the surface. The critical condition corresponding to the extremum being at the surface separates cases of heat transfer to and from the surface.

#### Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.