Heat Transfer in Hydromagnetic Flow over an Unsteady Stretching Permeable Sheet

Unsteady laminar boundary layer flow of viscous incompressible electrically conducting fluid along a continuous stretched permeable surface with the magnetic field effect is investigated. The defining characteristics of unsteady laminar boundary layer flow are governed a more than one independent variables, stretching velocity and surface temperature of the field. Governing equations are obtained for influencing parameters and transformed into ordinary differential equations by taking convenient similarity variables. Runge-Kutta fourth order method in corporation by the shooting technique is introduced to carry out numerical computations of the investigation. Velocity and temperature profiles are computed and represented graphically for the influences of suction/injection parameter, unsteadiness parameter, magnetic parameter and Prandtl number, while numerical solutions of local skin friction coefficient and local Nusselt number are discussed through tables. For non-magnetic condition, results are found in concordance with earlier research work. KeywordsHeat transfer, Hydromagnetic, Unsteady flow, Stretching permeable sheet.


Introduction
Studies of heat transfer and hydrodynamic flow towards a stretchable sheet have been receiving substantial importance over to its roles in upgrading technological applications and growing industrial processes. These types of problems have numerous characterizations with the small velocities in areas of geothermal where the layers of the shallow sheet are being stretched. The concept of stretching has useful applications in the hydrothermal system where fluid properties at depth influence the geothermal energy work. This study has value in important engineering applications such as drawing, tinning, and annealing of copper wire, crystal growing, etc. in the area of metallurgy and chemical engineering procedures. The above mentioned applications affect the continuous filaments cooling or strips via drawing them by a quiescent liquid. Besides, it has widespread applications in geophysics and thermal science, particularly in the packed bed reactors, petroleum resources recovery, geothermal energy technology, building thermal insulation, and underground disposal of chemical and nuclear waste. The final product quality is influenced by the heat transfer rate at the stretching wall. Sakiadis (1961) initiated the boundary layer flow over a moving continuous solid surface with constant velocity. Further, Crane (1970) extended it and obtained an effective analytical solution for the steady two-dimensional boundary layer flow problem in a quiescent incompressible fluid caused through the elastic flat surface stretching which acts with it plane over linearly varying surface speed. Many researchers such as Andersson (2002), Hayat and Sajid (2007), and Mukhopadhyay and Layek (2012) have instigated the problems of steady flow due to a stretching sheet in different conditions. Recently, , and Ram et al. (2018) have conducted numerous analysis on the stretching surface problems in steady boundary layer flow.
The findings of the above studies that deal with stretching sheet are more significant with considering that the flow conditions are to be steady. In the unsteady flow condition where flow vary with respect to both time and space, the stretchable plate can vary along time and thus the appearance of unsteady stretching surface grow into practical problems. To enhance the physically realistic applications of the flow situation, that is worthwhile to carry unsteadiness in the governing equations of fluid flow analysis. A considerable number of engineering problems such as startup process and periodic fluid motion are defined by unsteady boundary layer conditions. Wang (1990) was the first one who studies the hydrodynamics of an unsteady boundary layer fluid flow of a liquid film past a stretched surface. Many researchers such as Elbashbeshy and Bazid (2004), Ishak et al. (2008), Elbashbeshy and Aldawody (2010), , and Ram et al. (2017) are determined the problems of unsteady flow over a stretching plate in different situations.
Furthermore, the studies of heat transfer and flow along a permeable area are important in predicting the behavior of practical applications particularly in geophysical fluid dynamics like as small blood vessels, the human lung, wood, sandstone, limestone and beach sand. In polymer processing industries, the flow along a permeable stretching surface has extensive characterizations. The advancement in engineering and industrial applications of heat transfer and flow due to the permeable surface have a lot of attention by researchers. Beavers and Joseph (1967) were pioneered the exploration of the boundary condition at a naturally permeable wall. Many researchers such as Magyari and Keller (2000), Liao (2007), and Khader and Megahed (2014) are considered the permeable stretching surface flow problem under different situations. Recently, Chaudhary and Kumar (2015), Maity et al. (2016), and Ghadikolaei et al. (2018) are considered the heat transfer and flow over a stretched sheet in a porous medium and found the numerical solution.
Unsteadiness in hydromagnetic flow has received attention to some considerable extent in recent years due to the facts that numerical modeling of problems has developed a better understanding of practical and physical processes involved in engineering and technological applications. The field of solar energy collection, the design of cooling systems for electronic devices, an infinite metallic plate cooling in a cooling bath, a polymer plate or filament extruded continuously through a dye, geothermal reservoirs, heat exchangers, MHD marine propulsion, the boundary layer over a liquid film in condensation procedures, casting of magnetic-levitation and molten metal MHD stirring etc. are continuously driven by the influences of hydromagnetic unsteadiness of flow in variable situations. The problem of variable surface heat and mass flux of an unsteady natural convection MHD fluid flow over an inclined plate was presented by Jat and Chaudhary (2009), and Rashad (2014). Recently, hydromagnetic flow through unsteady stretching sheet was investigated by Mabood and Khan (2016), and Tian et al. (2017). Until, very recently, some scientists and With reference to above discussed applications and explorations, the primary objective of the present investigation is to expand the analysis of Ishak et al. (2009) for an electrically conducting fluid with the impact of the magnetic field associated to specified boundary conditions.

Mathematical Model
, and both a and c have dimension time ̶ 1 . All properties of the fluid are summarized to be constant all over the flow. Under these considerations along with the boundary layer approximations and neglecting the viscous dissipation, the basic equations that describe the case are as follows where u and v are the velocity components in the x and y directions respectively,     is the kinematic viscosity,  is the coefficient of fluid viscosity,  is the fluid density, e  is the electrical conductivity, e  is the magnetic permeability, T is the temperature of the fluid,

Similarity Variables
The continuity equation (1) (2) and (3) are transformed to the ordinary differential equations by defined the following similarity variables (Ishak et al., 2009) is the non-dimensional stream function,  is the similarity variable and    is the non-dimensional temperature. The converted equations are 0 2 with the associated boundary conditions where primes ) ' ( denote differentiation with respect to  , a c A  is the unsteadiness parameter,

Method of Solution
Apply a perturbation technique for the computational solution of the equations (6) and (7) along to the corresponding boundary conditions equations (8), by surmising the power series in terms of the small magnetic parameter M as Substituting equations (9) and (10) and its derivatives in equations (6) and (7) and comparing the coefficients of like powers of M , the following set of equations are found 0 2 with the relevant boundary conditions The equation (11) was obtained by Ishak et al. (2009) in the absence of magnetic impact and the resting equations are solved numerically via Runge-Kutta fourth order method in association with shooting technique (Na, 1979;Hoffman, 2001) with step size 0.001. The above process is imitated until finding the converged results up to the aimed level of accuracy 5 10  . Consequently, the numerical values of velocity and temperature fields are sketched in Figures 2 to 4 and 5 to 8 respectively for illustrations of the result.

Local Skin Friction Coefficient and Local Nusselt Number
In addition to velocity and temperature, the physical quantities of engineering interest are the local skin friction coefficient     Figure 6. Also, Figure 5 represents that the surface heat flux is higher for suction 0  S as compared by injection 0  S . Therefore, suction can be handling for the surface cooling enough faster than injection. Furthermore, the greater Prandtl number fluid has relatively a higher viscosity or a lower thermal conductivity which concludes in the thinner thermal boundary layer, decrease in the fluid temperature and hence, a higher rate of heat transfer at surface. Also, for greater Prandtl number, the thermal boundary layer falls down at a faster rate. This leads that the cooling rate is faster for greater Prandtl number. Therefore, the Prandtl number can be handling to evolve the cooling rate in resulting in fluid flows.

Concluding Remarks
A model for the numerical study is developed to investigate the analysis of the heat transfer on unsteady boundary layer flow of an incompressible viscous and electrically conducting fluid past a stretchable permeable surface. Employ the similarity transformations to convert the governing partial differential equations to ordinary differential equations. Finally, the numerical solutions of the set of resulting equations are found by implementing the Runge-Kutta fourth order method in association with the shooting technique. The numerical results for the surface heat flux are in congruous with the earlier results in the absence of the magnetic parameter. In present work, it is observed that the fluid flow, as well as the surface gradient, depreciates along with the raising values of the suction/injection parameter, the unsteadiness parameter and the magnetic parameter. Subsequently, the fluid velocity enlarges for a booming value of the unsteadiness parameter for 5 . 2   . Also, the thickness of thermal boundary layer as well as the temperature gradient stepsdown over the developing nature of the suction/injection parameter, the unsteadiness parameter, and the Prandtl number, whereas these increase along to the increasing values of the magnetic parameter. Furthermore, the fluid temperature boosts with an evolvement in the value of the unsteadiness parameter for 5 . 2   .

Conflict of Interest
Authors declare that they have no conflict of interest.