Numerical Solution of MHD Convection and Mass Transfer Flow of Viscous Incompressible Fluid about an Inclined Plate with Hall Current and Constant Heat Flux

The present numerically study investigates the influence of the Hall current and constant heat flux on the Magneto hydrodynamic (MHD) natural convection boundary layer viscous incompressible fluid flow in the manifestation of transverse magnetic field near an inclined vertical permeable flat plate. It is assumed that the induced magnetic field is negligible compared with the imposed magnetic field. The governing boundary layer equations have been transferred into non-similar model by implementing similarity approaches. The physical dimensionless parameter has been set up into the model as Prandtl number, Eckert number, Magnetic parameter, Schmidt number, local Grashof number and local modified Grashof number. The numerical method of NactsheimSwigert shooting iteration technique together with Runge-Kutta six order iteration scheme has been used to solve the system of governing non-similar equations. The physical effects of the various parameters on dimensionless primary velocity profile, secondary velocity profile, and temperature and concentration profile are discussed graphically. Moreover, the local skin friction coefficient, the local Nusselt number and Sherwood number are shown in tabular form for various values of the parameters.


Introduction
Hall current has important contribution in the study of MHD viscous flows.It has many applications in problems of the Hall accelerators as well as in the flight MHD.The current trend is on the application of MHD towards a strong magnetic field and a low density of gas.For this reason, the Hall current and ion slip become important.Hydrodynamic flow of a viscous liquid through a straight channel in presence of Hall Effect is examined by Sato [1], Yamanishi [2], and Sherman and Sutton [3].The Hall current effects on the boundary layer flow past a semi-infinite plate are studied by Katagiri [4].Free convection flow of a conducting fluid permeated by a transverse magnetic field in the presence of the Hall effects and uniform magnetic field is analyzed by Pop and Watanabe [5].Aboeldahab and Elbarbary [6] studied the effect of the Hall current on the MHD free convection flow in the presence of foreign species over a vertical surface, where the flow is subjected to a strong external magnetic field.Eichhorn [7] investigated the similarity solution by considering the power-law variations in the plate temperature and transpiration velocity.Vedhanayagam et al. [8] worked on the free convection flow along a vertical plate with the arbitrary blowing and wall temperature.Lin and Yu [9] investigated the free convection flow over a horizontal plate.Recently, Hossain et al. [10] investigated the natural convection flow from a vertical permeable flat plate with the variable surface temperature, considering the temperature and transpiration rates to follow the power-law variation.Saha et al. [11] studied the effect of Hall current on the steady laminar natural convection boundary layer flow of MHD viscous and incompressible fluids.Lately, Saha et al. [12] examined the effect of Hall current on MHD natural convection flow from vertical permeable flat plate with uniform surface heat flux.In recent years a number of studies of MHD convective heat and mass transfer boundary layer flow of viscous incompressible fluid were reported in the literature [13]- [25].However, the effect of hall current and constant heat flux is still not getting promising attraction to the researchers.In this study MHD Free Convection and Mass Transfer Flow of Viscous Incompressible Fluid about an inclined Plate with Hall Current and Constant Heat Flux is investigated.

Mathematical Analysis
Steady natural convection boundary layer flow of an electrically conducting and viscous incompressible fluid from a semi-infinite heated permeable inclined flat plate with a uniform surface heat flux and transverse magnetic field with the effect of the Hall current is considered.Here x axis is taken along the vertically upward direction and y axis is normal to it.The leading edge of the permeable surface is taken along z axis.The uniform heat is supplied from the surface of the plate to the fluid, which is maintained uniformly throughout the fluid flow.The temperature and concentration at the wall are w T and w C respectively.The temperature and con- centration outside the boundary layer are T ∞ and C ∞ respectively.Uniform magnetic field of magnitude 0 B is imposed to perpendicular to the flow along the y axis.Let the angle of inclination of the plate is γ and the plate is semi finite.The x component momentum equation reduces to the boundary layer equation if and only if body force is made by gravity, then the body force per unit mass is 0 cos , We have the generalized ohm's law in the absence of electric field to the case of short circuit problem is of the form ( ) where, e µ is the magnetic permeability, e τ is the electron collision time, σ is the electrical conductivity, e ω is the cyclotron frequency, 0 B is the applied magnetic field.Since no applied or polarized voltage exist, so the effect of polarization of fluid is negligible, i.e.
( ) e e e J J B q B B ω τ σµ If is assumed that induced magnetic field generated by fluid motion is negligible in comparison to the applied one i.e.
( ) . This assumption is valid because magnetic Reynolds number is very small for liquid metals and partially ionized fluids.
Since the Hall coefficient is e e m ω τ = , so the Equation ( 2) we can write ( ) ( ) where 0 y J = .The fundamental equations for the steady incompressible MHD flow with the generalized Ohm's law and Maxwell's equations, under the assumptions that the fluid is quasi-neutral, and the ion slip and thermoelectric effects can be neglected.Since the plate is semi-infinite and motion is steady, all physical equations will be the functions of x and y.Thus mathematically the problem reduces to a two dimensional problem given as follows: Subjected to the boundary conditions 0, 0, 0, , at 0 0, 0, , as where , , u v w are the velocity components in the , , x y z direction respectively, υ is the kinematics viscosity, ρ is the density.T, w T and T ∞ are the temperature of the fluid inside the thermal boundary layer, the plate temperature and the fluid temperature in the free stream, respectively, while C, w C , C ∞ are the corresponding concentrations.Also, σ is the electric conductivity of the medium, k is the thermal conductivity of the medium, m D is the coefficient of mass diffusivity, p c is the specific heat at constant pressure, Q is the constant heat flux per unit area and other symbols have their usual meaning.
In order to solve the above system (Figure 1) of Equations ( 6)-( 9) with the boundary conditions (10), we adopt the well-defined similarity analysis to attain similarity solutions.For this purpose, the following similarity transformations are now introduced; Thus, Equations ( 6)-( 10) becomes; ( ) ( ) The corresponding boundary conditions are where is the local modified Grashof number.Similarity transformations expressions also satisfy the continuity Equation (5).

Skin-Friction Coefficients, Nusselt Number and Sherwood Number
The quantities of chief physical interest are the skin friction coefficients, the Nusselt number and the Sherwood number.The equation defining the wall skin frictions are , hence we have ( ) . The numerical values of the skin-friction coefficients, the Nusselt number and the Sherwood number are sorted in Tables 1-8.

Results and Discussions
In this study the MHD Free Convection and Mass Transfer Flow of Viscous Incompressible Fluid about an  inclined Plate with Hall Current and Constant Heat Flux have been investigated using the Nachtsheim-Swigert shooting iteration technique.To study the physical situation of this problem, we have computed the numerical values of the velocity, temperature, and concentration within the boundary layer and also find the skin friction coefficient, Nusseltnumber, Sherwood number at the plate.It can be seen that the solutions are affected by the parameters, namely suction parameter w f , Grashof number r G , modified Grashof number m G , magnetic parameter M , Prandtl number r P , Eckert number c E , Schimidt number.The values of M and r G are taken to be large for cooling Newtonian fluid keeping the plate at different angle.The values 0.2, 0.5, 0.73, 2, 3, 4, 5 are considered for r P .The values 0.1, 0.5, 0.6, 1.0, 2.0, 3.0, 4.0 also considered for c S .The values of other parameters are however chosen arbitrarily.
Figures 2-5, respectively, show the primary velocity, secondary velocity, temperature and concentration profiles for different values of suction parameter w f .Here 0 w f > corresponds to suction and 0 w f < cor- responds to injection at the plate or blowing.From Figure 2-5, it can be seen that the primary velocity, secondary velocity, temperature and concentration profiles decreases with the increase of suction parameter w f .Figures 6-9, respectively, show the primary velocity, secondary velocity profiles decreased and temperature and concentration profiles increases for different values of M .Figure 10 & Figure 11, respectively, show the cross-flow of primary velocity and secondary velocity, at first increases then decreases with the increase of c E .Figure 12 & Figure 13 shows that the temperature profile increase and concentration profile decreas- es with the increase of c E .Figures 14-17 show that the primary velocity, secondary velocity profile and con- centration profile decreases and temperature profile increases with the increase of c S .Figure 18 & Figure 19, respectively, shows the cross flow of the primary velocity and secondary velocity with the increase of r P both of the profile is decrease then increase.Figure 20 & Figure 21 shows that temperature decrease and concentration profile increase with the increase of c S .Figure 22 & Figure 23, show the cross flow of the primary velocity and secondary velocity with the increase of γ both of the profile is decrease then increase.Finally the effect of various parameters on the skin friction coefficients ( x τ , w τ ), Nusselt number ( u N ) and Sherwood ( h S ) are tabulated in Tables 1-8.Table 1 shows that the skin friction coefficient ( x τ , w τ ) decreases and Nusselt number ( u N ) and Sherwood number ( h S ) increase with the increase of w f .Table 2 shows that the skin friction coefficient x τ decreases and w τ increases and Nusselt number ( u N ) and Sherwood number ( h S ) decreases with the increase of M .Table 3 shows that the skin friction coefficient ( x τ , w τ ) and Sher- wood number ( h S ) increases and Nusselt number ( u N ) decreases with the increase of c E .Table 4 shows that the skin friction coefficient ( x τ , w τ ) and Sherwood number ( h S ) decreases and Nusselt number ( u N ) in- creases with the increase of r P .Table 5 shows that the skin friction coefficient ( x τ , w τ ) and Nusselt number ( u N ) decreases and Sherwood number ( h S ) increases with the increase of c S .Table 6 shows that the skin friction coefficients ( x τ , w τ ), Nusselt number ( u N ) and Sherwood number ( h S ) decreases with the increase of γ .Table 7 & Table 8 shows that the skin friction coefficients ( x τ , w τ ), Nusselt number ( u N ) and Sher- wood number ( h S ) increases with the increase of r G and m G .

Conclusions
The effect of viscous incompressible fluid flow about an inclined plate with hall current is analyzed in the present study for constant heat flux.A range of physical parameter values tested over the boundary layer flows.The variation of different physical parameters instigated different flow pattern as increasing, decreasing and cross flow in the dimensionless primary and secondary velocity, temperature and concentration distribution as well as in the profile of skin friction coefficients, Nusseltand Sherwood number.The findings of the present investigation are briefly: • As suction parameter increases, the primary velocity, secondary velocity, temperature and concentration profiles decrease gradually.However for the same parameter effects, the skin friction coefficient decreases and Nusselt and Sherwood numbers increase.• The primary velocity, secondary velocity profiles as well as the skin friction coefficient, Nusselt and Sherwood numbers decreased as magnetic parameter increased whereas the reverse effects found in the profile of temperature and concentration.• The cross-flow of primary and secondary velocity observed as Eckert number increases whereas temperature profile increases and concentration profile decreases for the same parameter effects.Also for the similar parameter effects skin friction coefficient and Sherwood number increase whereas Nusselt number decreases.• The increasing effect of Schmidt number causes primary and secondary velocity profile and concentration profile as well as the skin friction coefficient and Nusselt number decrease whereas the reverse situation observed in the temperature and Sherwood number profiles.• The cross flow of the primary and secondary velocity with the increase of Prandlt number has been observed, where both of the profiles first decrease and near the layer the profiles increase.However within the same parameter effects, the skin friction coefficient and Sherwood number decrease and Nusselt number increases.
• As the parameter γ rises the cross flow pattern observed in primary and secondary velocity profiles whe- reas temperature and concentration profile increase.However for the same parameter effects the skin friction coefficient, Nusselt number and Sherwood number profiles decrease.• The increasing effect of Grashof and modified Grashof number causes the cross flow of the primary and secondary velocity whereas temperature and concentration profile decrease for the similar parameter effects.However the skin friction coefficient, Nusselt number and Sherwood number increase.

F
is the accele- ration due to gravity.Further no body force exists in the direction of y and z, i.e. = .The x component of pressure gradient at any point in the boundary layer must equal to the pressure gra- dient in the region outside the boundary layer, in this region 0 u = , 0 v = .Hence x component of pressure gra- is the density of the surrounding fluid at temperature T ∞ .The quantity ρ ρ ∞ − is related to the temperature difference T T ∞ − and concentration (or mass) differences C C ∞ − through the thermal volume expansion coefficient β and concentration volume expansion coefficient β * by the relation,

Figure 1 .
Figure 1.Physical configuration and co-ordinate system.

Figure 24 &
Figure 25, show that temperature and concentration profile increases with the increase of γ .

Figure 26 &
Figure 26 & Figure 27, show the cross flow of the primary velocity and secondary velocity with the increase of r G . Figure 28 & Figure 29, shows that the temperature and concentration profile decreases with the in- crease of r G . Figure 30 & Figure 31, show the cross flow of the primary velocity and secondary velocity with the increase of m G . Figure 32 & Figure 33, shows that the temperature and concentration profile decreases

Figure 2 .
Figure 2. Primary velocity profile for w f .

Figure 3 .
Figure 3. Secondary velocity profile for w f .

Figure 4 .
Figure 4. Temperature profile for w f .

Figure 5 .
Figure 5. Concentration profile for w f .

Figure 6 .
Figure 6.Primary velocity profile for M .

Figure 8 .
Figure 8. Temperature profile for M .

Figure 9 .
Figure 9. Concentration profile for M .

Figure 10 .
Figure 10.Primary velocity profile for c E .

Figure 11 .
Figure 11.Secondary velocity profile c E .

Figure 12 .
Figure 12.Temperature profile for c E .

Figure 13 .
Figure 13.Concentration profile for c E .

Figure 14 .
Figure 14.Primary velocity profile for c S .

Figure 15 .
Figure 15.Secondary velocity profile for c S .

Figure 16 .
Figure 16.Temperature profile for c S .

Figure 17 .
Figure 17.Concentration profile for c S .

Figure 18 .
Figure 18.Primary velocity profile r P .

Figure 19 .
Figure 19.Secondary velocity profile r P .

Figure 20 .
Figure 20.Temperature profile for r P .

Figure 21 .
Figure 21.Concentration profile for r P .

Figure 26 .
Figure 26.Primary velocity profile for r G .

Figure 27 .
Figure 27.Secondary velocity profile for r G .

Figure 28 .
Figure 28.Temperature profile for r G .

Figure 29 .
Figure 29.Concentration profile for r G .

Figure 30 .
Figure 30.Primary velocity profile for m G .

Figure 31 .
Figure 31.Secondary velocity profile for m G .

Figure 32 .
Figure 32.Temperature profile for m G .

Figure 33 .
Figure 33.Concentration profile for m G .

Figure 34 .
Figure 34.Primary velocity profile for m.

Figure 35 .
Figure 35.Secondary velocity profile for m.

Figure 36 .
Figure 36.Temperature profile for m.

Figure 37 .
Figure 37. Concentration profile for m.

Table 1 .
Numerical values of Skin friction coefficient x

Table 2 .
Numerical values of Skin friction coefficient x

Table 3 .
Numerical values of Skin friction coefficient x x τ

Table 4 .
Numerical values of Skin friction coefficient x

Table 5 .
Numerical values of Skin friction coefficient x

Table 6 .
Numerical values of Skin friction coefficient x

Table 7 .
Numerical values of Skin friction coefficient x

Table 8 .
Numerical values of Skin friction coefficient x