On a new numerical analysis of the Hall effect on MHD flow and heat transfer over an unsteady stretching permeable surface in the presence of thermal radiation and heat source/sink

This paper employs a computational iterative approach known as the spectral local linearization method (SLLM) to analyze the Hall effect on MHD flow and heat transfer over an unsteady stretching permeable surface in the presence of thermal radiation and heat source/sink. To demonstrate the reliability of our proposed method, we made comparison with the Matlab bvp4c routine technique, and an excellent agreement was observed. The governing partial differential equations are transformed into a system of ordinary differential equations by using suitable similarity transformations. The results are obtained for velocity, temperature, skin friction and the Nusselt number. MSC: 65PXX; 76-XX


Introduction
Theoretical studies of magnetohydrodynamic flow and heat transfer over stretching surfaces have received great attention by virtue of their numerous applications in the fields of metallurgy and chemical engineering. Such applications include geothermal reservoirs, wire and fiber coating, food stuff processing, reactor fluidization, enhanced oil recovery, packed bed catalytic reactors and cooling of nuclear reactors. The primary aim in extrusion is to maintain the quality of the surface of the extricate. When the magnetic strength is strong, Hall currents cannot be neglected. In an ionized gas where the density is low and/or the magnetic field is very strong, the conductivity normal to the magnetic field is reduced due to the free spiraling of electrons and ions about the magnetic lines of force before suffering collisions, and a current is induced in a direction normal to both electric and magnetic fields. Due to Hall currents, the electrical conductivity of the fluid becomes anisotropic and this causes the secondary flow. The Hall effect is important when the Hall parameter, which is the ratio between the electron-cyclotron frequency and the electron-atom-collision frequency, is high. Hall currents are of great importance in many http://www.boundaryvalueproblems.com/content/2014/1/170 astrophysical problems, Hall accelerators, Hall sensors and flows of plasma in MHD power generators.
Examples of such studies include Sakiadis [, ] who did pioneering work on boundary layer flow on a continuously moving surface. Shateyi and Motsa [] carried out a numerical analysis of the problem of magnetohydrodynamic boundary layer flow, heat and mass transfer rates on steady two-dimensional flow of an electrically conducting fluid over a stretching sheet embedded in a non-Darcy porous medium in the presence of thermal radiation and viscous dissipation. Shateyi  Pal and Mondal [] presented a model to study the effects of temperature-dependent viscosity and variable conductivity on mixed convective diffusion of species over a stretching sheet. When the conducting fluid is an ionized gas, and the strength of the applied magnetic field is large, the conductivity normal to the magnetic field is reduced due to the free spiraling of electrons and ions about the magnetic lines of force before collisions occur and a current is induced in a direction normal to both electric and magnetic fields. This phenomenon is known as the Hall effect or Hall current. The conductivity of the fluid is anisotropic and the effect of Hall current cannot be neglected when the medium is rarefied or if a strong magnetic field is present. The study of MHD viscous flows with Hall current has important applications in problems of Hall accelerators and sensors as well as flight magnetohydrodynamics. Pop  Governing equations modeling MHD flow and heat transfer over stretching surfaces are highly nonlinear, thereby exact solutions are impossible to obtain. Therefore, numerical solutions have always been developed and modified as a bid of getting more accurate and stable solutions. The current study seeks to investigate the Hall effect on MHD flow and heat transfer over an unsteady stretching permeable surface in the presence of thermal ra-http://www.boundaryvalueproblems.com/content/2014/1/170 diation and heat source/sink. We propose to numerically solve the present problem using a recently developed iterative method known as the spectral local linearization method (SLLM), Motsa []. The SLLM approach is based on transforming a nonlinear ordinary differential equation into an iterative scheme. The iterative scheme is then blended with the Chebyshev spectral method []. A similar approach to our current proposed method can be found in Motsa

Mathematical formulation
We consider an unsteady two-dimensional laminar MHD boundary layer flow and heat transfer of an incompressible, viscous and electrically conducting fluid over a continuously moving stretching permeable surface. We take the x-axis along the stretching surface in the direction of the motion, and the y-axis is perpendicular to the sheet in the outward direction and the z-axis is transverse to the xy-plane. The flow is subjected to a transverse magnetic field of strength B  and the Hall current is taken into account in this study. The effects of the Hall current give rise to the Lorentz force in the z-direction, which induces a cross flow in this direction. We assume that there is no variation of flow and heat transfer quantities in the z-direction. The temperature at the sheet is maintained at a prescribed constant T w . The flow is generated due to a stretching sheet which is caused by simultaneously applying two equal but opposite forces. The velocity is assumed to be proportional to the distance from the origin. Under these assumptions, together with the usual boundary layer approximations, the dimensional governing equations of continuity, momentum and energy in the presence of a strong magnetic field, heat source and thermal radiation become as follows: The associated boundary conditions to the current problem are the following: where u, v and w are the velocity components along x, y and z directions, respectively, and t is the time. T is the temperature within the fluid, c p is the specific heat at constant pressure, α is the thermal diffusivity, ν is the kinematic viscosity of the fluid density, T w (x, t) is the temperature on the stretching surface, T ∞ is the ambient temperature with T w > T ∞ . We have V w = -(νU w /x) / f () representing the mass transfer at the surface with V w >  for injection and V w <  for suction. We also have U w (x, t) = ax/(ct), where a (stretching rate) and c are positive constants, with ct < . It is noted that the stretching rate a/( -http://www.boundaryvalueproblems.com/content/2014/1/170 ct) increases with time since a > . The surface temperature of the sheet varies with the distance x from the origin and time t and takes the form We must remark that the particular forms of U w and T w (x, t) have been specifically chosen in order to come up with similarity transformations which then transform the governing partial differential equations into a set of nonlinear ordinary differential equations.
Following Ishak et al.
[], we introduce the following dimensionless functions f and θ , and the similarity variable η, By using the Rosseland approximation, the radiative heat flux is given by where σ * and K * are, respectively, the Stephan-Boltzman constant and the mean absorption coefficient. Assume that the temperature differences within the flow are small such that T  can be expressed as a linear function. Expanding T  in a Taylor series about T ∞ and neglecting higher order terms, we get By using the above transformations, the governing partial differential equations are transformed into a system of non-dimensional nonlinear and coupled ordinary differential equations as follows: Here M  = σ B   /ρa is the magnetic parameter, A = c/a is a parameter that measures the unsteadiness, Pr = ν/α is the Prandtl number, R = σ * T  ∞ /kK s is the thermal radiation parameter. The boundary conditions are as follows: where f () = f w with f w <  or f w >  corresponding to injection or suction, respectively. The physical engineering quantities of interest in this problem are the skin friction coefficients in the x-and z-directions and the local Nusselt number, Nu x which are defined as follows: where τ w = μ( ∂u ∂y ) y= is the wall shear stress, and q w = -κ( ∂T ∂y ) y= is the surface heat flux, where μ and κ are the dynamic viscosity and thermal conductivity, respectively.

Basic idea of the spectral local linearization method (SLLM)
Consider the problem of finding Z = [Z  (η), Z  (η), . . . , Z m (η)] that satisfies the system Also, L i and N i are linear and nonlinear components, respectively. Basically, the SLLM is an iterative method for solving differential equations such as () which begins with an initial approximation Z  of Z. Successive application of the SLLM generates approximations Z  , Z  , . . . , where Z r = [Z ,r , . . . , Z m,r ] for each r = , , , . . . . Upon linearizing nonlinear component N i , differential equation () shall be solved numerically using the Chebyshev spectral collocation method [].
First, let denote the ith differential equation () after the first r +  iterations of the SLLM. Let w r be an n-tuple of Z i,r and its derivatives. If we assume that N i is a function of w r only, then linearization of N i at w r is which, upon inserting into equation () and re-arranging the result, gives Equation () shall be solved using the Chebyshev spectral collocation method. For the sake of brevity, we leave out the details of this method. We urge the interested reader to see, for example, []. http://www.boundaryvalueproblems.com/content/2014/1/170

Solving current problem using the SLLM
If we let f = p, then equations ()-() become Equations ()-() together with the change of variable f = p may be written as where As an illustration we derive equation () when i = . In this case, equation () is which, upon substituting () and simplifying, becomes A similar approach for each i = , ,  yields Equations ()-() are subject to the boundary conditions respectively. Chebyshev differentiation [] replaces equations () through () with the discrete form or equivalently subject to the boundary conditions and O,  are zero matrices of order (N + ) × (N + ), (N + ) × , respectively. Before we solve linear system (), we need to apply boundary conditions. This is easier to illustrate using the equivalent form, equations ()-(), as follows: We choose initial approximations g  (η) = ηe -η that satisfy boundary conditions ()-() so that the SLLM generates subsequent approximations f r , p r , g r , θ r for each r = , , . . . .

Results and discussion
The numerical results iteratively generated by the SLLM for the main parameters that have significant effects on the flow properties are presented in this section. All the SLLM results presented in this work were obtained using N =  collocation points, and we are glad to highlight that convergence was achieved in as few as six iterations. We take the infinity value η ∞ to be . The magnetic field is taken quite strong by assigning large values of M to ensure the generation of Hall currents. In order to validate our numerical method, it was compared to MATLAB routine bvpc which is an adaptive Lobatto quadrature iterative scheme.
In Table  we present a comparison between the SLLM approximate results and the bvpc results for selected default values of the stretching parameter A. It can be clearly seen from this table that there is an excellent agreement between the results from the two methods. Also Table  shows that an increase in the unsteadiness parameter leads to an increase in the skin-friction coefficients in both directions. Also the heat transfer gradient increases as the values of the unsteadiness parameter increase. The negative values of f () mean that the solid surface exerts a drag force on the fluid. This is due to the development http://www.boundaryvalueproblems.com/content/2014/1/170   of the velocity boundary layer which in the current study is caused solely by the stretching sheet.

Table 1 Comparison of the SLLM results of -f (0), g (0), -θ (0) with those obtained by bvp4c for different values of the unsteadiness parameter
In Table  we display the effect of the magnetic parameter on the skin friction coefficients and the Nusselt number. The magnetic parameter M represents the significance of the magnetic field on the flow properties. As the magnetic strength increases, the dragging effect is clearly seen by the significant increments in the skin friction. We also observe that increasing the values of the Hartman number leads to the lowering of the values of the Nusselt number. Application of a strong magnetic field reduces the velocity, which in turn increases heat diffusion within the fluid flow. This physically explains why heat transfer at the wall is reduced as M is increased.
Table  displays the influence of the Hall current on the skin friction coefficients as well as the Nusselt number. The skin friction coefficient is reduced as the values of the Hall current parameter increase. This explains why the skin friction coefficient in the axial direction decreases. However, in the transverse direction the skin friction increases as the Hall current increases. There is a small effect of the Hall current on heat transfer rate on the stretching surface.
The effect of the unsteadiness parameter A on the axial velocity f (η) is presented in Figure . It can be observed from this figure that the axial velocity profiles decrease as A increases. Increasing the unsteadiness parameter A causes the velocity boundary layer thickness to decrease. This in turn may cause delays in transition from laminar to turbulent flows.
The influence of the suction/injection parameter f w on the axial velocity is depicted in Figure . The axial velocity is significantly influenced by this parameter. The velocity boundary layer is greatly enhanced when fluid is injected (f w < ) into the flow system thereby increasing the velocity profiles. However, removing fluid from the flow system http://www.boundaryvalueproblems.com/content/2014/1/170  through suction, as expected, drastically reduces the velocity profiles as can be clearly seen in Figure .
The application of a magnetic field perpendicular to the flow produces a drag force known as the Lorentz force. This force reduces the axial velocity as can be observed in  in our current study. Any further increase of the Hall current would make the magnetic effect insignificant. Figure  displays the effect of the unsteadiness parameter on the transverse velocity profiles. This parameter has very significant influence on the transverse velocity. It can clearly be seen from this figure that the effect of increasing the unsteadiness parameter A is to greatly reduce the transverse velocity g(η) near the plate. In Figure , we display the influence of the injection/suction parameter on the transverse velocity. Injecting/blowing fluid into the flow system causes the transverse velocity profiles to significantly increase, whereas suction, as expected, reduces the transverse velocity distribution. http://www.boundaryvalueproblems.com/content/2014/1/170    the values of A reduces the thermal boundary layer thickness thus reducing the fluid temperature distribution.
In Figure  we have the effect of the heat source/sink parameter δ on the temperature profiles. As expected, it is observed in this figure that the temperature in the boundary layer increases with increasing values of δ. The heat absorption due to a uniform sink (δ < ) leads to the reduction of the thermal boundary layer thickness, whereas this layer increases significantly with increases in δ > . Figure  shows the effect of injection/suction on the temperature profiles. Injecting fluid into the flow system causes enhancement of the thermal boundary layer thickness. http://www.boundaryvalueproblems.com/content/2014/1/170  This in turn causes the temperature distribution in the flow system to greatly increase. On the other hand, suction reduces the thermal boundary layer thickness. Figure  shows that the thermal boundary layer becomes thicker when the values of the magnetic parameter are increased. Application of the magnetic field gives rise to a resistive-type force which slows down the motion of the fluid and in turn increases the thermal boundary layer, hence increasing the temperature of the fluid. Figure  is plotted to depict the influence of the thermal radiation parameter R on the temperature profiles. We clearly observe that the temperature in the boundary layer increases with increasing values of the thermal radiation parameter. This is due to the fact http://www.boundaryvalueproblems.com/content/2014/1/170  that the divergence of the radiative heat flux increases as the Rosseland radiative absorption K * decreases, which in turn increases the rate of radiative heat transfer to the fluid. Thus the presence of thermal radiation enhances thermal state of the fluid causing its temperature to significantly increase.

Conclusion
The present work analyzed MHD unsteady flow and heat transfer of an electrically conducting fluid over a stretching sheet in the presence of thermal radiation and the Hall effect. Much attention was given to trying to investigate how the velocity field, skin friction, http://www.boundaryvalueproblems.com/content/2014/1/170 temperature distribution and heat transfer are influenced by the parameters of importance in this study. The governing partial differential equations are transformed into a system of nonlinear ordinary differential equations by using suitable similarity variables. The resultant system of nonlinear ordinary differential equations is solved numerically by a recently developed technique known as the spectral local linearization method. The accuracy of the SLLM is validated against the MATLAB in-built bvpc routine for solving boundary value problems. The following conclusions were drawn in our investigation.
• An excellent agreement was observed between our results and those obtained using the bvpc routine technique giving confidence to our present results. • The unsteadiness parameter A has significant effects on the velocity components and temperature profiles. The maximum axial velocity, transverse velocity and temperature profiles are attained when the flow is steady (A = ). • Increasing the values of the magnetic field strength decreases the momentum boundary layer thickness while increasing the thermal boundary layer thickness. • The velocity components are enhanced as the Hall parameter increases. • The fluid temperature increases with increasing values of thermal radiation as well as a heat source. • The heat transfer rate and the skin friction coefficient in the x-direction are increased while the skin friction in the z-direction decreases as the unsteadiness parameter increases. • The skin friction coefficients are enhanced while the heat transfer rate is depressed by increasing the values of magnetic strengths.