Heat transfer in MHD flow due to a linearly stretching sheet with induced magnetic field

The full MHD problem of the flow and heat transfer due to a linearly stretching sheet in the presence of a transverse magnetic field is put in a self-similar form. Traditionally ignored physical processes such as induced magnetic field, viscous dissipation, Joule heating, and work shear are included and their importance is established. Cases of prescribed surface temperature, prescribed heat flux, surface feed (injection or suction), velocity slip, and thermal slip are also considered. The problem is shown to admit self similarity. Sample numerical solutions are obtained for chosen combinations of the flow parameters.


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. Gupta and Gupta [3] added surface feed (suction or injection). These problems were recognized as being exact solutions of corresponding Navier-Stokes problems by Crane [1], Andersson [4] and Wang [5], respectively. To the Navier-Stokes problem, Wang [6] and Andersson [7], independently, added velocity slip. Fang et al. [8] 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 [3,9] or temperature or heat flux proportional to a power of the stretch-wise coordinate x [10,11,12]. Prasad et al. [13] retained viscous dissipation and Joule heating, in case of the surface temperature being proportional to 2 x . The abovementioned MHD problems adopted the small magnetic Reynolds number assumption, thus neglecting the induced magnetic field. In [14], it was shown that the full MHD problem, i.e., Navier-Stokes and Maxwell's equations with adherence conditions and appropriate magnetic conditions, allowed for self similarity. 2 In this article, we extend the work of [14] to the heat transfer problem including viscous dissipation and Joule heating, in cases of quadratic surface temperature or heat flux. Surface feed, velocity and thermal slip, and shear work are also included.

Mathematical model
An electrically conducting, incompressible, Newtonian fluid is driven by a non-conducting porous sheet, which is stretching linearly in the x -direction. At the surface, we consider cases of prescribed temperature or heat flux, and allow for velocity and thermal slip. In the farfield, the fluid is essentially quiescent under pressure  p and temperature  T , and is permeated by a stationary magnetic field of uniform strength B in the transverse y -direction.
The equations governing this steady two-dimensional MHD flow are with the surface conditions and the farfield conditions ) , ( v u are the velocity components in the ) , ( y x directions, respectively, and ) , ( s r are the corresponding induced magnetic field components. p is the pressure and T is the temperature.
Constants are the fluid density  , kinematic viscosity  , electric conductivity  , magnetic permeability  , specific heat c , and thermal conductivity k . The stretching rate  and the velocity and thermal slip coefficients w  and w  are also constant. In the condition for w q , the 3 last term represents the shear work [15]. In the farfield, the condition for r translates the physical requirement of the absence of any current density, while that on s indicates that B stands for the farfield total magnetic field imposed and induced [14].
The problem admits the similarity transformations where primes denote differentiation with respect to  . Note that the temperature is quadratic in is the magnetic Prandtl number, Consistent with the similarity transformations we take the surface values to be , we get the following conditions on the flow variables

Numerical method
We start by solving for ) ( f and ) ( g , since their problem is uncoupled from the problems for ) (        The following is noticed.
the presented results for non-negative 1  and 1 Q indicate that 1  decreases monotonically with  .

Higher 1
 results in smaller 1  , 0  rises to a peak, falls to zero then to a bottom, and rises again to zero.
 decreases to a negative local minimum then rises to zero. To complement Figs. 4-9, we give in Table 3 The shear work is represented by the term involving the velocity slip coefficient  in condition (20) for 2  (0). Table 6 demonstrates its importance. Neglecting the shear work reduces the predicted surface temperature.   14 On the right-hand-sides of Eqs. (12) and (14), the first terms represent Joule heating and streamwise heat diffusion, respectively, while the second terms represent heat dissipation. Table  7 demonstrates the effect of neglecting these three processes. The predicted heat flux to the surface, represented by 0  (0) and 2  (0), is reduced considerably by neglecting viscous dissipation, less by neglecting streamwise diffusion and lesser by neglecting Joule heating. Neglecting one or more may even predict heat flux in the wrong direction.

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 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 importance of these processes as well as the significance of the formulation has been established through samples of numerical results.