Atangana-Baleanu and Caputo-Fabrizio Analysis of Fractional Derivatives on MHD Flow past a Moving Vertical Plate with Variable Viscosity and Thermal Conductivity in a Porous Medium

In this paper, a numerical investigation is presented for non-integer order derivatives with Atangana-Baleanu (AB) and Caputo-Fabrizio (CF) fractional derivatives for the variable viscosity and thermal conductivity over a moving vertical plate in a porous medium two dimensional free convection unsteady MHD flow. The effects of radiation have also been considered. The governing partial differential equations along with the boundary conditions are changed to ordinary form by similarity transformations. Hence physical parameters show up in the equations and interpretations on these parameters can be achieved suitably.By using ordinary finite difference scheme the equations are discritized and developed in fractional form. These discritized equations are numerically solved by the approach based on Gauss-seidel iteration scheme. Some numerical strategies are used to find the values of AB and CF approaches on time by developing programming code in MATLAB. The effects of all the physical parameters involved in the problem on velocity, temperature and concentration distribution are compared graphically as well as in tabular form. The effects of each parameter are found to be prominent. We have observed a significant variation of values under different parameters using AB and CF approaches on velocity, temperature and concentration distribution with respect to time. KeywordsAB and CF derivatives, Viscosity, Thermal conductivity, Porous medium, Radiation.


Introduction
The boundary layer and heat transfer flow of a viscous fluid over a moving vertical plate in a porous medium have been investigated in a number of technological approaches such as warm rolling, metallic extrusion, petroleum industries, polymer extrusion, wires drawing and metallic spinning. Natural convection flows driven by temperature differences are of great interest in a number of industrial applications. Bejan and Khair (1985) studied free convection heat and mass transfer in a porous medium. In recent there has been a growing interest in studing the combined application of MHD flow and porous media. Aldoss et al. (1995) investigated combined free and forced convection flow from vertical plate in aporous medium in the presence of magnetic field. Hossain and Munir (2000) analysed a two dimensional mixed convection flow of a viscous incompressible fluid of temperature dependent viscosity past a vertical plate. Javaherdeh et al. (2015) studied the natural convection heat and mass transfer in MHD fluid flow past a moving vertical plate with variable surface temperature and concentration in a porous medium. However, the impact of variable viscosity and thermal conductivity of a MHD free convection flow past a vertical plate embedded in porous medium has received a little attention. Mukhopadhyay and Layek (2008) presented the effects of variable viscosity through a porous medium over a stretching sheet in presence of thermal radiation.
Recently, fractional calculus has gained tremendous popularity among the researchers because of singular kernel with locality and non-singular kernel with non-locality problem. Caputo and Fabrizio used an exponential function in fractional derivative to avoid the singular kernel problem. Mirza and Vieru (2017) analyzed that the use of the time-fractional derivative without singular kernel is more advantageous than Caputo time-fractional derivative. Nehad et al. (2016) applied CF fractional derivatives to analyze the solutions for heat transfer of second grade fluids over vertical oscillating plates. "A comparative study has been obtained by using the AB and CF fractional derivatives for casson fluid model with chemical reaction and heat generation" by Sheikh et al. (2017). Few works using fractional derivatives have been studied in Histrov (2017), Atangana and Baleanu (2016). A comparative study of Atangana-Baleanu and Caputo-Fabrizio evaluation of fractional derivatives for heat and mass transfer of fluid over a vertical plate has been done by Khan et al. (2017).
The main objective of this paper is to investigate the effects of variable viscosity and thermal conductivity over a moving vertical plate in porous medium and comparing the results with the approaches AB and CF fractional derivatives. The non-dimensional governing equations with the non-dimensional boundary conditions are discretized with ordinary finite-difference kernel solved numerically with the help of AB and CF fractional derivative methods by developing suitable programming code in MATLAB. A comparative analysis under different parameters is represented graphically as well as in tabular form.

Mathematical Formulation
Consider a two-dimensional free convection steady heat and mass transfer flow of viscous incompressible electrically conducting fluid past a moving vertical plate in a porous medium (Figure 1). Here a uniform magnetic field is applied in a direction perpendicular to the fluid flow. The x -axis is taken along the vertical plate in the direction of the flow and y axis is normal to it. At   0 t , the fluid have gained the temperature w T , and concentration level near the plate is w C . The fluid properties are assumed to be constant except for the fluid viscosity and thermal conductivity which are assumed to vary as an inverse linear function of temperature. A uniform magnetic field of strength 0 B is applied normal to the plate. There is no chemical reaction between the fluid and diffusing species. Plate temperature w T is variable and  T is the free stream temperature assumed constant.
The variable viscosity and thermal conductivity are governed by the following equations under the boundary layer approximation: Equation of conservation of energy: The boundary conditions are:  where, u and v are the fluid velocities in the direction of x and y respectively,  is the kinematic viscosity,   is the kinematic viscosity of the fluid in the free stream,  is the fluid density,  is the viscosity of the fluid,  is the electrical conductivity, g is the acceleration due to gravity, t  is the volume expansion coefficient for heat transfer, c  is the volume expansion coefficient for mass transfer, T is the fluid temperature within the boundary layer,  T is the temperature at free stream of the fluid, C is the species concentration of the fluid,  C is the concentration at free stream of the fluid,  is the thermal conductivity of the fluid, where, By neglecting the higher order terms beyond 1 st degree in Using equations (6) and (7) Appling the following non dimensional quantities: According to Lai and Kulacki (1990) the viscosity and thermal conductivity of the fluid are assumed to be inverse linear function of temperature as follows: where,  and  are constants which depend on the thermal property of the fluid.
We define two parameters as, Using these two parameters in (10) and (11), we have the viscosity and thermal conductivity respectively as It is also important to note that r  is negative for liquid and positive for gases.

Atangana-Baleanu (AB) Fractional Derivatives
To express AB fractional approach, "The governing partial differential equations can be written with respect to time by the AB fractional operator of the order is the Mittag-Leffler function.

Caputo-Fabrizio (CF) Fractional Derivatives
To express Caputo-Fabrizio fractional derivatives approach, "The governing partial differential equations can be written with respect to time by the CF fractional operator of the

Solution of the Problem
Solutions of equations (17) -(20) or (21) -(24) are obtained by using ordinary finite difference scheme. Discritization is performed using the following formulae: , , The fractional derivatives given by (20) or (24) are calculated using numerical integration. Finally the set of equations (17) -(19) or (21) -(23) together with boundary condition (16) completely discritized and the discritized equations are solved by using an iterative method based on Gaussseidel scheme.
The boundary conditions given in equation (16), now reduces to the form:

Important Physical Parameters
The physical parameter Skin-friction coefficient indicates physical wall shear stress, Nusselt number indicates physical rate of heat transfer and Sherwood number indicates mass transfer.

(a) Coefficient of Skin Friction
By the Newton's law of viscosity: The non-dimensional skin-friction is given by,

(b) Nusselt Number
By the Fourier's law of heat conduction in the form: The rate of heat transfer coefficient is given by Nusselt Number

(c) Sherwood Number
The mass flux m q from the plate to the fluid is given by the Fick's law, The rate of mass transfer is given by Sherwood number

Results and Discussion
By applying non dimensional quantities, the non dimensional discretized governing equations along with the non-dimensional boundary conditions are solved with the help of AB and CF fractional derivative methods by developing suitable programming code in MATLAB using finite difference scheme. This analysis has been done to study the effects of various parameters such

Graphical Representation
In Figure 2, it is seen that with the increasing value of the Hartmann number, velocity decreases. The presence of magnetic field in the normal direction of the flow in an electrically conducting fluid produces Lorentz force which opposes the flow. To overcome this opposing force, some extra work should be done which is transformed to heat energy. Hence temperature increases (Figure 3). This is because that the applied magnetic field opposes the fluid motion and therefore enhancing the temperature for this response. With the increase of M species concentration also increases ( Figure 4). increases the transportation of heat from hot region to colder region increases. Since temperature within boundary layer is more than the outside so temperature is decreased. Again species concentration increases with the increasing value of c  (Figure 7).
The effects of viscosity parameter r  on velocity, temperature and species concentration distribution are plotted in Figure 8 to Figure 10. Figure 8 displays that dimensionless velocity decreases with the increase of r  . This is due to the fact that with the increase of the viscosity parameter the thickness of the velocity boundary layer decreases. Physically, this is because of that a larger r  implies higher temperature difference between the fluid and the surface. Figure 9 shows that temperature increases with the increasing value of r  . The viscosity causes a rise in the friction, when friction increases the area of the stretching surface in contact with the flow increases therefore generated heat from the friction on the surface is transferred to the flow which leads to a rise in the surface temperature and the flow is heated. The species concentration decreases for increasing value of r  ( Figure 10).
The Eckert number Ec signifies the viscous dissipation of the fluid, on temperature it is plotted in Figure 11. It is seen that an increase in viscous dissipation of the fluid tends to increase in fluid temperature with increase of Ec. In Figure 12, it is noticed that with the increase of radiation parameter Kr temperature increases. This is due to the fact that the thermal boundary layer thickness increases with the increase of Kr and hence temperature. Velocity decreases with the increasing value of Prandtl number Pr (Figure 13). This is due to the fact that with the increase of Pr, viscosity increases, so velocity decreases. In Figure 14, it is noticed that with the increasing value of Pr temperature of the fluid decreases. For higher Prandtl number the fluid has a relatively high thermal conductivity which decreases the temperature.           In Figure 15, Sc is increased the concentration boundary layer becomes thinner than the viscous boundary layer, as a result of which velocity reduces. With thinner concentration boundary layer the concentration gradients are enhanced causing a decrease in concentration of species in the boundary layer.

Comparision of AB and CF Fractional Derivatives for Various Values of the Parameters in Tabular Form
Here we compare between AB and CF fractional derivative for various values of the parameters M , c  , r  , Sc , Kr , Pr , Ec taking 4 . 0  y and 4 . 0  t . From the following tables it is found that the values of the velocity, temperature and concentration profiles for various parameters are almost the same for both the methods-AB and CF fractional derivative.   and CF fractional derivatives. It is observed that, if time is less than 1, the velocity obtained using AB approach is greater than the velocity computed with CF approach. Otherwise if the time is just greater than 1, the velocity obtained by CF is greater than the velocity obtained by AB approach. Again for temperature profile under the parameters the value obtained by CF approach just greater than AB approach but for concentration profile the value obtained by AB approach is just greater than CF approach.

Comparison of AB and CF method for Coefficient of Skin Friction, Nusselt number and Sherwood Number
We are considering the values of the AB operator ( ) and the CF operator (  Table 6 to Table 8.