Hall and Ion-Slip Currents Effects on Unsteady MHD Fluid Flow over an Inclined Plate with Inclined Magnetic Field and Variable Temperature

A numerical study is performed to investigate the effect of Hall and Ion-slip current on unsteady MHD viscous electrically conducting fluid over an electrically non-conducting inclined plate, which is along the plate under the strong inclined magnetic field. The magnetic Reynolds number of flows is taken very small so that the induced magnetic field equation is ignored in our analysis. The governing equations are derived from the Navier- Stokes equation, Energy equation and Concentration equation and boundary layer approximation has been employed. The obtained non-linear, non-dimensional coupled governing equations are solved by using numerical methods of explicit finite difference for velocity, temperature and concentration. The flows are affected by the inclined angle. The effect of necessary important relevant parameters on the velocity, temperature and concentration distributions has been discussed in detail and also the shear stress, Nusselt number and Sherwood number are explained and compare it with its natural behavior, as well as graphically.


Introduction
The influence of the magnetic field of MHD fluid flows with hall and ion slip current has a great importance of experimental and theoretical applications in magnetic material processing, astrophysics, nuclear engineering, geophysical and industrial fields. Abel and Veena (1998) discussed the visco-elastic fluid flow in a saturated porous medium with internal heat generation and frictional heating over a stretching surface. A free-streamline theory is carried out by Abernathy (1962) for a flat plate of inclination towards the infinite flow field at an angle of attack. Measurement of the position of the free-rotation layers, the frequency of the rotation street and back-pressure plate of a sharp-edged plate have been reported to be effective both in attacking angle and lateral contraction of the flow. Ajay (2003) explained the combined effects of hall current, viscous dissipation, Joule heating and thermal diffusion on the magneto-hydrodynamic free convection flow of an electrically conducting fluid with mass transfer past an infinite vertical porous plate. Angirasa and Peterson (1997) presented a numerical study on heat transfer in natural transmission from an isothermal vertical surface which is a stable layered to a fluid-saturated thermally stratified porous medium. Basant and Malgwi (2019) have analyzed hall and ion-slip current effect on free convection of an incompressible viscous fluid flow in a vertical microchannel with an induced magnetic field. Beg et al. (2010) have developed a numerical solution for MHD viscous, incompressible flow in a rotating porous channel comprising two infinite parallel plates with hall currents and inclined magnetic field. Bhpendra et al. (2007) presented the hall current effect on the unsteady viscous incompressible mixed convective fluid flow past a vertical infinite porous plate with the heat source (or sink). Two immiscible convective flow and heat transfer with porous past an inclined channel with pressure gradient is discussed by Daniel and Daniel (2013). Eraslan (1969) expressed the fully developed distributions of temperature for the MHD channels with Hall Effect. They have found the presence of porous structure, material constants and different liquids can efficiently control fluid flow. Gupta (1997) studied the heat and mass transfer of an electromagnetic fluid flow on a stretching sheet with suction as well as with blowing. Javeri (1975) investigated the effect of combined hall, ion slip, Joule heating and viscous dissipation on the laminar MHD channel heat transfer. Jyotsna et al. (2017) have analyzed the MHD heat and mass transfer free convection flow past an exponentially accelerated inclined plate embedded in a saturated porous medium with radiation and mass transfer effect. Kumar and Singh (1999) analyzed the natural convection of heat transfer in a poachy porous medium from a vertically isothermal surface under the influence of thermal stratification. Navnet et al. (2016) examined the inclined magnetic field effect on unsteady viscous, incompressible fluid flow past on a vertical moving plate with variable temperature. Opanuga et al. (2018) have investigated the rate of entropy generation on a couple of stress fluids in the presence of hall and ion-slip current with velocity slip and jump of temperature. The influence of hall current of the generalized Couette flow of a viscous, conducting fluid under an inclined magnetic field is analyzed by Prasada et al.(1986). Rajput and Gupt (2016) have considered the unsteady free convection MHD fluid flow along an exponentially accelerated plate past in a porous medium with inclined magnetic field, variable temperature and constant mass diffusion. Ram (1995) considered the hall and ion-slip current effects of free convective heat-generating rarefied gas in a rotating frame with a strong magnetic field imposed perpendicular to the plate. Rishab et al. (2017) have presented the flow of visco-elastic electrically conducting fluid through two parallel plates filled with a porous medium placed in an inclined magnetic field with heat and mass transfer. Sato has discussed the electrical conductivity of an ionized gas with the Hall Effect, where the conductivity is presented by a tensor in a similar form for both fully and partially ionized gases. Seth et al. (1961Seth et al. ( , 2009Seth et al. ( , 2010Seth et al. ( , 2016 have studied the Hall Effect in the Viscous Flow of Ionized Gas Between two Parallel Plates under Transverse Magnetic Field; MHD Couette Flow in the presence of an inclined magnetic field with rotation; Hartmann's flow of a viscous incompressible electrically conducting fluid in the presence of the inclined magnetic field with hall effects and with rotation and also studied on the MHD free convection flow of a viscous, incompressible, electrically conducting fluid past an impulsively started vertical plate through a saturated porous medium with ramped heat flux in the presence of the inclined magnetic field. Suman and Nazibuddin (2018) presented the unsteady MHD flow with mass transfer along an infinite inclined plate where the velocity is strung in a poachy porous medium with a variant temperature.
Recently Krishna et al. (2020) have modified the work of Jyotsna et al. and investigated the hall and ion slip current effects on unsteady magneto-hydrodynamic free convective flow with rotation through an exponentially accelerated inclined plate entrenched in a saturated porous medium with the effect of inclined angle also with the change of reference frame. They have used Laplace transform to solve these problems analytically. In this paper, our main aim is to modify the work of Jyotsna et al. This study presented the result of numerical modeling of the unsteady MHD fluid flow over an inclined plate and inclined induced magnetic field without change of reference frame with time variation measurements by the effect of Dufour, hall and ion-slip currents.

Mathematical Model
Let an electrically non-conducting plate be fixed at 0 = y in the fluid which makes an angle α with - x axis vertically upward and y-axis is normal to the -xz plane. Consider an unsteady MHD incompressible viscous electrically conducting fluid flow in the direction of parallel to the plate. Initially, we assumed that fluid velocity, temperature and concentration of the fluid are respectively Due to the effect of hall and Ion-slip current, the generalized Ohm's law may state as follows: Where e e e β τ ω = may treat as Hall parameter. We assumed that the magnetic field B is imposed for the electrically non-conducting plate at a right angle to the plate. The magnetic Reynolds number of flows is taken very small, so that induced magnetic field equation is not considerable in our analysis. And therefore the induced magnetic field vector becomes Physical model of the system is shown in the following Fig.1.

Fig.1 Physical configuration and coordinate system
It is assumed that the plate is of semi-infinite extended length, then all the physical variables in the problem are functions of x , y and t . Therefore the problem is two-dimensional. According to the above-mentioned assumptions along with the Boussinesq and boundary layer approximations, the governing equations of fluid and generalized Ohm's law with Hall current and ion-slip influence are given as follows: 3 with corresponding boundary conditions:

Similarity Analysis
The following non-dimensional variables are introduced to make the dimensionless form of the above equations.
With the corresponding boundary conditions:

Method of Solutions
The dimensionless, nonlinear, coupled partial differential equations (6)-(10) together with associated boundary conditions are solved by using the explicit finite difference method. The maximum length of the plate is taken Here, the subscript 'i' refers to x; 'j' to y and superscript 'k' refers to time t. Similarly, we have used finite difference schemes for the other variables. We have used Matlab code to get the solution of finite difference schemes and its graphical presentation.

Shear stresses, Nusselt number and Sherwood number:
The effects of various parameters on local and average shear stress have been investigated from the velocity profile. The velocity gradient at the plate is defined as the shear stress; the non-dimensional form of the local shear stress and average shear stress in x -

Results and Discussions
The results carried for the influence of the relevant non-dimensional parameters namely Hartmann number

Primary Velocity Distributions
At first, it is mentioned from our physical configuration of the model that "the plate is inclined means induced magnetic field is also inclined because induced magnetic field is applied to the right angle to the plate". Fig.4 indicates that the primary velocity u decreases with the increase of inclined angle α . It is observed that the velocity in x-direction is highly increasing when the plate is We observed that the velocities are very affected by the inclined angle α .       Please see the Manuscript PDF le for the complete gure caption  Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption   Please see the Manuscript PDF le for the complete gure caption  Please see the Manuscript PDF le for the complete gure caption Figure 12 Please see the Manuscript PDF le for the complete gure caption Figure 13 Please see the Manuscript PDF le for the complete gure caption Figure 14 Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption Figure 16 Please see the Manuscript PDF le for the complete gure caption Figure 17 Please see the Manuscript PDF le for the complete gure caption Figure 18 Please see the Manuscript PDF le for the complete gure caption Figure 19 Please see the Manuscript PDF le for the complete gure caption  Please see the Manuscript PDF le for the complete gure caption  Please see the Manuscript PDF le for the complete gure caption    Please see the Manuscript PDF le for the complete gure caption  Please see the Manuscript PDF le for the complete gure caption