Influence of Hall Current on MHD Flow and Heat Transfer over a slender stretching sheet in the presence of variable fluid properties

MHD flow and heat transfer of an electrically conducting fluid over a stretching sheet with variable thickness in the presence of variable fluid properties is analyzed. Wall temperature and the velocity of the stretching sheet are assumed to vary. Also the external magnetic field perpendicular to the sheet and the effects of Hall current are taken into account. The governing nonlinear differential equations are solved numerically by an implicit finite difference scheme. To validate the numerical method, comparisons are made with the available results in the literature for some special cases and the results are found to be in excellent agreement. The effects of physical parameters on the flow and temperature fields are analyzed graphically. The Hall current gives rise to a cross flow and the variable fluid properties have strong effects on the shear stress and the Nusselt number.


Introduction
During the last several decades, the study of magnetohydrodynamic (MHD) flow has attracted several scientists and researchers due to its importance and applications to various technological and industrial problems such as petroleum industries, purification of crude oil, plasma studies, magneto-hydrodynamic electrical power generation, glass manufacturing, paper production, geothermal energy extractions, and boundary layer control in the field of aerodynamics.In view of these applications, Chakrabarti and Gupta [1] obtained similarity solution for electrically conducting fluid analyzed the effect of hydromagnetic flow and heat transfer over a stretching sheet.Andersson et al. [2] extended the work of Chakrabarti and Gupta [1] to power law fluid and Andersson [3] presented two-dimensional Navier-Stokes equations by considering uniform magnetic field for viscous http://www.ispacs.com/journals/cna/2016/cna-00251/International Scientific Publications and Consulting Services flow past a stretching sheet.Vajravelu and Nayfeh [4] analyzed hydromagnetic flow of a dusty fluid over a stretching sheet.Ishak et al. [5] studied the effect of a uniform transverse magnetic field on the stagnation point flow toward a vertical stretching sheet.Cortell [6] examined the flow and heat transfer of an electrically conducting second grade fluid in the presence of transverse magnetic field past a semi-infinite stretching sheet.Sweet et al. [7] obtained the analytical solution of the MHD flow of a viscous fluid between two moving parallel plates via the homotopy analysis method.Further, Robert and Vajravelu [8] obtained explicit exact solutions for fourth-order nonlinear differential equations arising in the hydromagnetic flow of a second grade fluid over a stretching or shrinking sheet.Recently, Abbasbandy et al. [9] obtained both numerical and analytical solutions for Falkner-Skan flow of MHD Maxwell fluid and emphasized the variations of viscoelastic and magnetic parameters.Many researchers have made the valuable contribution to the literature of hydromagnetic flow and heat transfer over a stretching sheet by considering different geometry and obtained exact solutions [10][11][12][13][14][15].However, not much attention has been paid to the effects of Hall currents.When the magnetic field is strong and the density of electrons is small, the Hall effect cannot be ignored as it has a significant effect on the flow pattern of an ionized gas.However, for ionized gases, the conventional MHD is not valid.In an ionized conducting fluid where the density is low and/or magnetic field is very strong, the conductivity normal to the magnetic field is reduced due to the free spiraling of electrons and a current is induced in a direction normal to both electric and magnetic fields.This phenomenon is called Hall effect.The effect of Hall current cannot be neglected when the medium is rarefied or in a strong magnetic field.The study of MHD flows with Hall current has important industrial applications in many geophysical and astrophysical situations and in several engineering problems such as Hall accelerators, Hall effect sensors, constrictions of turbines, centrifugal machines, and flight magneto-hydrodynamics.In view of these applications, it is essential to analyze the influence of Hall current on the fluid flow.Gupta [16] considered a strong magnetic field which permeated the fluid and investigated the effect of Hall current on the fluid past an infinite porous flat plate and Jana et al. [17] continued the work of Ref. [16] for different physical situations.Hossain and Rashid [18] extended the work of Gupta [16] to unsteady free convection flow.Further, Rana et al. [19] analyzed the effect of Hall current on Hartmann flow between two parallel electrically insulated infinite planes.Chaudhary and Jha [20] studied the effect of heat and mass transfer on the flow of elastico-viscous fluid past an impulsively started infinite vertical plate with mass transfer and Hall effect taken into account.Recently, Hayat et al. [21] obtained an analytical solution for unsteady three-dimensional MHD flow due to a stretching surface in a porous medium via the homotopy method.In general, studies are centered on a linear or nonlinear stretching sheet.However, not much work has been carried out for a special type of non-linear stretching, namely, for different values of n (that is, a stretching sheet with variable thickness) in a thermally stratified environment.Such a study would have practical application since deforming substances like needles and nozzles have variable sheet thickness.Also variable thickness (Lee [22]) is one of the significant properties in the analysis of vibration of orthotropic plates.Special form of non linear stretching sheet with variable thickness is often used in machine design, architecture, nuclear reactor technology, naval structures, and acoustical components.Very recently, Fang et al. [23] observed the behavior of boundary layer flow over a stretching sheet with variable thickness and explained significant effects of the non-flatness of the sheet on the velocity and shear stress.
All the above mentioned researchers restricted their analyses to the hydromagnetic flow over a horizontal or a vertical plate and assumed thermo-physical properties of the fluid as constant.However, these physical properties may change with temperature; especially the viscosity and the thermal conductivity (see for details Chiam [24], Hassanien [25], Subhas Abel et al. [26], and Prasad et al. [27]).To the best knowledge of the authors, a study on the combined effects of variable fluid properties and the Hall effect for flow over a slender stretching sheet with variable thickness has not been carried out.In view of this, the  [27] to the special form of stretching sheet with variable thickness in the presence of strong magnetic field.The coupled non-linear partial differential equations governing the problem have been transformed to a system of coupled non-linear ordinary differential equations.The transformed equations are solved numerically by a second order finite difference scheme known as the Keller-box method.Computed numerical results for the flow and heat transfer characteristics are analyzed.The analysis of the results shows that the fluid flow is appreciably influenced by the sundry parameters.It is expected that the results obtained will not only provide useful information for industrial application but also complement the previous works.

Mathematical formulation
Consider a steady, laminar boundary layer flow of a viscous, incompressible and electrically conducting fluid in the presence of a strong magnetic field B(x) past an impermeable stretching sheet with variable thickness.The positive x coordinate is measured along the stretching sheet in the direction of motion and the positive y coordinate is measured normal to the sheet in the upward direction (see Fig. 1 for details).An external magnetic field is applied in the positive y-direction with a constant flux density B 0 .In general, for an electrically conducting fluid, Hall current affects the flow in the presence of a strong magnetic field.The effect of Hall current gives rise to a cross flow and hence the flow becomes three-dimensional.We assume that there is no variation of flow quantities in the z-direction.This assumption is valid for a surface of infinite extent.The generalized Ohm's law including Hall currents in the usual form is given by 11 =.
e ee p en en Here   = , , x y z J J J J is the current density vector, E is the intensity vector of the electric field, V is the velocity vector,   0 = 0, ,0 B B is the magnetic induction vector, is the electrical conductivity, and e p is the electronic pressure.Since there is no applied or polarization voltage is imposed on the flow we have, http://www.ispacs.com/journals/cna/2016/cna-00251/International Scientific Publications and Consulting Services 0.

 E
For weakly ionized gases, the electron pressure gradient and the ion slip effects can be neglected.The generalized Ohm's law under the above conditions for electrically non-conducting sheet becomes 0. y J  Hence Eq. (2.1) reduces to and . 1 1 Here , v,and uw are the , ,and x y z components of the velocity vector V and m is the Hall parameter.
The following assumptions are made  Joule heating and viscous dissipation are neglected. The fluid is isotropic, homogeneous, and has constant viscosity and electric conductivity.
 The sheet is being stretched with a velocity   00 ( ) = where is constant, b is the physical parameter related to stretching sheet, and n is the velocity exponent parameter.
 The sheet is not flat and is defined as where the coefficient A is chosen as small so that the sheet is sufficiently thin, to avoid pressure gradient along the sheet ( / = 0) px  .
Under these assumptions, along with the boundary layer approximations, the governing equations can be written as (see Chaudhary and Jha [20] for details) v 0, Here, the subscript denotes partial differentiation with respect to the independent variable,   is fluid density, p C is the specific heat at constant pressure, and T is the temperature.Further,  is the coefficient of viscosity which varies as an inverse function of temperature (see for details Prasad et al. [27], and Lai and Kulacki [28]) and is as follows where, a =    and 1 .Further, the special form of magnetic field is the temperature-dependent thermal conductivity.We consider the temperature dependent thermal conductivity in the following form (see for details Chiam [24]): T  is the temperature of the fluid away from the sheet, C is a constant, l is the characteristic length, r is the wall temperature,  is a small parameter known as the thermal conductivity parameter, and  k is thermal conductivity of the fluid away from the sheet.Substituting (2.7) and (2.8), into (2.4) to (2.6), we obtain The appropriate boundary conditions for the problem are where C is a constant and r is the wall temperature.It should be noted that the positive and negative value of n indicate cases of surface stretching and surface shrinking respectively.Now we transform the system of equations (2.3) -(2.6) into a dimensionless form.To this end, let us define the dimensionless similarity variable Now in terms of  , we define the dimensionless stream function ( , )  xy  and the dimensionless where ( , ) xy  identically satisfies the continuity equation (2.3).With the help of equation (2.14), the velocity components can be written as Here prime denotes differentiation with respect to .With the use of equations (2.13)-(2.15),equations (2.9)-(2.11)and condition (2.12) can be reduced to The non-dimensional parameters and Pr Mn respectively denote the magnetic parameter and the Prandtl number and are defined as and the corresponding boundary conditions are   where the prime denotes differentiation with respect to .
 For all practical purposes, the important physical quantities of interest are the horizontal skin friction

Exact solutions for some special cases
Here we present exact solutions for certain special cases and these solutions serve as a baseline for computing general solutions through numerical schemes.We notice that in the absence of variable fluid properties, magnetic field, and Hall current, heat transfer Eq. (2.20) reduces to those of Fang et al. [23].http://www.ispacs.com/journals/cna/2016/cna-00251/International Scientific Publications and Consulting Services Further, in the absence of variable fluid properties, Hall current, heat transfer and variable thickness of the wall, Eq. (2.20) reduces to those of Andersson [3], Van Gorder and Vajravelu [8], Vajravelu and Rollins [11], Pop and Na [12], Fang et al. [13], and Fang and Zhang [14]., where 0

  
To obtain finite solution it is essential to consider 2.

 
The solution is Appling for free boundary condition we obtain The final solution is  (2.23) has no exact analytical solutions, they are solved numerically via a second order finite difference scheme.These equations are highly non-linear and coupled ordinary differential equations with variable coefficients.Exact analytical solutions are not possible for the complete set of equations.Hence, we use an efficient numerical method with second order finite difference scheme known as the Keller-box method (For details see Cebeci and Bradshaw [29], Keller [30], and Vajravelu and Prasad [31]).For the sake of brevity, the details of the numerical solution procedure are not presented here.For numerical calculations, a uniform step size of 0.01   is found to be satisfactory and the shooting error was controlled with an error tolerance of 6 10  in all the cases.In order to validate the method used in this study and to judge the accuracy of the present analysis, the horizontal skin friction and the wall temperature gradient are compared with the previously published results of Andersson et al. [2], Grubka and Bobba [32], Chen [33], Ali [34], and Prasad et al. [27] for several special cases, and the results are found to be in good agreement:

Results and discussion
In  are tabulated in Table III.It can be seen that for increasing values of n both ( ) and ( ) f h    increase.This implies that the momentum boundary layer thickness increases but exactly the opposite is true in the case of the transverse velocity distribution.That is, there is a significant effect of (for 0, 0 and 0) n n n n    on the flow pattern: Here, it may be observed that for negative value of n, the sheet is shrinking along the axis, and is stretching for positive n. becomes very small and hence the resistive effect of the magnetic field is diminished.The effects of  and  on  can be seen from Figure 6.Temperature of the fluid is found to increase with increasing values of  .That is, the assumption of temperature dependent thermal conductivity suggests a reduction in the magnitude of the velocity by a quantity () k T y  which can be seen in Eq.
(2.6).Therefore, the rate of cooling is much faster for the coolant material having small thermal conductivity parameter.By a careful scrutiny of table III, one can observe that (0) f  decrease as  increases for < 1.

Conclusion
Some of the interesting findings of the study are summarized below. An increase in the wall thickness parameter leads to reduction in the skin-friction coefficient in the xdirection for < 1 n while quite the opposite is true for 1 n  .
 Hall current has a strong effect on the shear stress and the Nusselt number. For large values of the Hall parameter m, the resistive effect of the magnetic field is reduced. The effect of the variable thermal conductivity parameter is to enhance the temperature field. The variable thermophysical property parameters have strong effects on the skin friction and the heat transfer characteristics.

Acknowledgement
The authors appreciate the constructive comments of the reviewer which led to definite improvement in the paper.

Figure 1 :
Figure 1: Schematic diagram of the stretching sheet with variable thickness model.The continuous sheet moves in its own plane with a velocity

T
are constants, and their values depend on the reference state and the thermal property of the fluid.In general, gases.T is the temperature; T ∞ and µ ∞ are the constant values of the temperature and coefficient of viscosity respectively away from the sheet. 17 ) http://www.ispacs.com/journals/cna/2016/cna-00251/International Scientific Publications and Consulting Services


Hall current for the case of a flat plate  since the boundary layer flow and heat transfer problem degenerates.The solution for the velocity in the presence of a magnetic field turns out to be The solution for the temperature field can be written as a two parameter solution in terms of the confluent hypergeometric function, namely, the Kummer's function,  , order to get a clear insight into the physical problem, numerical computations have been carried out for different values of the non-dimensional parameters such as the Hall parameter m , the fluid viscosity parameter r  , the power index parameter n, the variable thickness .Throughout the computation, the values of the wall temperature parameter r, the Prandtl number Pr, and the magnetic parameter Mn are fixed.Analytical solutions are obtained for the special cases of r   , n = 1, 0 m  , and Mn = 0. We presented the numerical results graphically for the horizontal velocity profile ( ) , f   the transverse velocity profile ( ), h  and the temperature field ()  for different values of pertinent parameters in Figs.2-6.It can be perceived from these graphs that all three profiles ( ), ( ) f     and () h  decrease monotonically and tend asymptotically to zero as the distance increases from the boundary.The computed numerical values for the horizontal skin friction (0) f  , the transverse skin friction (0), h and the rate of heat transfer (0)

Figures 2 (
Figures 2(a) through 2(c) depict the effect of increasing values of

Figures 4 (
a) to 4(c) are the graphical representations of ( ) , ( ), and ( ) of .It is noticed that all the profiles increase as  increases.Figures 5(a) to 5(c) demonstrate the effect of increasing values of m on ( ) , ( ), and ( ) and hence increases propelling effect on ( ). f   Further, the transverse flow in the z-direction initially increases with m, reaching a maximum for m = 1.5 and 1.0 respectively when n=1.0 and n = 5.0 and then decreases.For large values of m, the term

Nomenclatureb 0 BC
constant in equation (2.8) known as stretching rate b>0 uniform magnetic field C constant defined in (2.8) skin friction coefficient in the x-and z-directions respectively e electric charge f dimensionless stream function J current density vector K thermal conductivity K  thermal conductivity of the fluid far away from the sheet l characteristic length m Hall Effect parameter Mn magnetic parameter n velocity power index parameter n e electron number density

No variable fluid properties, no magnetic field, no Hall currents, no heat transfer but in the presence of variable boundary thickness
  The results are shown in Tables I and II.http://www.ispacs.com/journals/cna/2016/cna-00251/ International Scientific Publications and Consulting Services

Table I :
Comparison of skin friction

Table III :
values of Skin friction, transverse skin friction and wall temperature gradient for different values of the pertinent parameters with pr = 2.0, r = 1.0,Mn =1.0 http://www.ispacs.com/journals/cna/2016/cna-00251/ International Scientific Publications and Consulting Services