Analysis of newly developed fractal-fractional derivative with power law kernel for MHD couple stress fluid in channel embedded in a porous medium

Fractal-fractional derivative is a new class of fractional derivative with power Law kernel which has many applications in real world problems. This operator is used for the first time in such kind of fluid flow. The big advantage of this operator is that one can formulate models describing much better the systems with memory effects. Furthermore, in real world there are many problems where it is necessary to know that how much information the system carries. To explain the memory in a system fractal-fractional derivatives with power law kernel is analyzed in the present work. Keeping these motivation in mind in the present paper new concept of fractal-fractional derivative for the modeling of couple stress fluid (CSF) with the combined effect of heat and mass transfer have been used. The magnetohydrodynamics (MHD) flow of CSF is taken in channel with porous media in the presence of external pressure. The constant motion of the left plate generates the CSF motion while the right plate is kept stationary. The non-dimensional fractal-fractional model of couple stress fluid in Riemann–Liouville sense with power law is solved numerically by using the implicit finite difference method. The obtained solutions for the present problem have been shown through graphs. The effects of various parameters are shown through graphs on velocity, temperature and concentration fields. The velocity, temperature and concentration profiles of the MHD CSF in channel with porous media decreases for the greater values of both fractional parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α and fractal parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}β respectively. From the graphical results it can be noticed that the fractal-fractional solutions are more general as compared to classical and fractional solutions of CSF motion in channel. Furthermore, the fractal-fractional model of CSF explains good memory effect on the dynamics of couple stress fluid in channel as compared to fractional model of CSF. Finally, the skin friction, Nusselt number and Sherwood number are evaluated and presented in tabular form.


Formulation of the problem
In the present study we have considered the incompressible unsteady MHD flow of CSF in channel. The couple stress fluid is taken in channel in the presence of external pressure with body couples. The fluid is taken between two plates and porous medium is considered. Initially, both plates and fluid were at rest after some time the left plate start moving with constant velocity due to which the fluid flow in channel and the right plate is fixed. The governing equations for the present flow regime are given below: The continuity equation is given by 25,29,40 : The momentum equation of the given problem is given by: The energy equation can be written as: The concentration equation can be written as: where in the momentum equation the term ρ − → b shows the body forces which can be expressed as: Here ⇀ V, ⇀ T and ⇀ C are the velocity, temperature and concentration vectors. p, ρ, µ, ρ b, k, D, g and r is the pressure, density, dynamic viscosity, body forces, thermal conductivity, thermal diffusivity, gravitational acceleration and Darcy resistance respectively. ⇀ J and ⇀ B is the current density and total magnetic field. As we have considered unidirectional flow, therefore, velocity, temperature and concentration fields of the given flow are as: In the given study we have consider Darcy resistance in CSF, therefor, Darcy's law can be written in the following form: where φ represent porous media and k represent permeability of the porous medium. Maxwell equation can be defined as: From Maxwell equation: here E is the total electric field. By Ohm's Law (generalized form): Cross product with the magnetic field is: here B 0 is applied magnetic field and b is induced magnetic field by polarization. Now applying the vector scalar triple product, Eq. (11) becomes: (1) ∇ · ⇀ V= 0, www.nature.com/scientificreports/ In this article we have considered the incompressible unsteady MHD flow of CSF in channel. The MHD CSF laminar fluid is considered to flow through an open channel of two parallel plates separated by a distance d. The medium is considered as a porous medium with porosity K in the presence of constant external pressure gradient G and the induced magnetic field B 0 which is taken normal to the fluid flow. The motion of the fluid is considered in x-direction. Initially, for t ≤ 0 the fluid and both the plates are stationary with surrounding temperature T ∞ and constant concentration C ∞ . At t = 0 + , the temperature and concentration of the left plate raised to T and C respectively. As a result the left plate moving with constant velocity and the right plate is stationary. The geometrical sketch of the considered model is given in Fig. 1.
Using the assumptions which are considered in the problem the governing equations for the flow, energy and concentration equations are given by 2,25 : with the physical initial and boundary conditions: From the above initial and boundary conditions it can be observed that the fluid and plates were at rest initially. After some time the left plate u(0, t) = H(t)U 0 moving with constant velocity U 0 , where H(t) shows the Heaviside step function and the right plate is fixed. At the left plate the wall temperature and concentration and at the right plate there is ambient temeprature and concentration. Furthermore, ∂ 2 u(0,t) ∂y 2 = 0 , shows the couple shear stresses at the left and right plates.
For dimensional analysis, the following non-dimensional variables are introduced: www.nature.com/scientificreports/ After dimensionalization process we get the following dimensionless system of equations along with initial and boundary conditions.
where Gr represents Grashof number, Gm mass Grashof number, Pr represents Prandtl number, Re represents Reynolds number, Sc represents Schmidth number, M represnets magnetic parameter,K porosity parameter and H Hartmann number.

Definition of Fractal-Fractional Derivative with Power Law Kernal
Let assume that f (t) is continuous in the interval (a, b) and let the function is fractal differentiable on (a, b) having order β , then the fractal-fractional derivative of f having order α in Riemann-Liouville RL sense with power law kernel is given by 10 : where

Solutions of CSF with fractal-fractional derivative
In order to transform the classical CSF model into fractal-fractional derivative Eqs. (19)(20)(21) can be written in the following form: where FFP D α,β t (., .) shows the fractal-fractional derivative, 0 < α ≤ 1 is the fractional order and 0 < β ≤ 1 is the fractal dimension.

Solution of Energy Equation.
From Eq. (26), the following result is obtained: www.nature.com/scientificreports/ Equation (28) can be written as: the above result can be written in the following form: using the given initial condition from Eq. (22), Eq. (30) reduces to the following form: by discretizing the above equation at (ξ i , τ = τ n+1 ), the following form is obtained: From the above equation the following results are obtained: (27), the following result is obtained:

Solution of concentration equation. From equation
Equation (35) can be written as: here using the given initial condition from Eq. (22), Eq. (37) reduces to the following form: by discretize the above equation at (ξ i , τ = τ n+1 ), one can get the following result: the above result can be written as:

Solution of momentum equation
From Eq. (25), one can get the following fractal-fractional form of momentum equation: Equation (42) can be written as: here using the given initial condition from Eq. (22), Eq. (44) reduces to the following form: by discretizing the above equation at (ξ i , τ = τ n+1 ), the discretized form is given as under: from the above step the following result is obtained: Re .

Limiting case
In this section the present obtained solutions are reduced to already published work in order to verify the obtained solutions. Therefore, by putting (Gr = 0),(Gm = 0),P = 0 and 1 K → 0 present solutions reduced to the solutions recently obtained by Akgül and Siddique 12 which verify the present results.
Using the above assumptions Eq. (25) reduces to the following form: Equation (49) can be written as: here using the given initial condition from Eq. (22), Eq. (51) gives to the following form: by discretizing the above equation at (ξ i , τ = τ n+1 ), one can get the following form: From the above equation the following result is obtaiend: Re .

Skin friction. Skin friction for CSF is as under:
As the given flow model is between two parallel plates. Therefore, the skin friction at the left and right plates is as under: where Sf lp (.) and Sf rp (.) denotes the skin friction at left and right plates respectively.

Results and discussion
This section provide fractal-fractional derivative model of unsteady MHD generalized Couette flow of CSF in channel with embedded in porous media with power law kernel. Numerical solutions for the proposed problem are obtained using the implicit finite difference method. In this study we have found the influence of fractal dimension and fractional operator on the fluid motion, fluid temeprature and concentration graphically. Furthermore, for clear understanding the influence of all parameters is shown through graphs which effect the fluid motion, temperature and concentration. Figure 1 shows the physical sketch of the proposed problem. The effect of fractal dimension β on velocity profile is highlighted in Fig. 2. From the graph it is quite clear that for greater values of fractal parameter result a decay in the fluid velocity it is due to the power law kernel. The effect of fractional parameter α is shown in Fig. 3. From the figure a decreasing in the CSF velocity is noticed. The influence of α on CSF velocity is similar to the effect of fractal dimension β on the velocity field. The comparison between fractal-fractional CSF velocity and fractional velocity is plotted in Fig. 4. From the figure it can be noticed that the magnitude of fractional velocity is greater than fractal-fractional velocity. In this paper we have added a parameter β known as fractal-fractional dimension. This parameter β shows the combined effect of fractal-fractional derivative with fractional derivative.  Fig. 7. From the figure it is clear that increasing M result a decrease in the fractal-fractional CSF velocity. This is due to the fact that for greater values of M Lorentz forces increases in the CSF which control the boundary layer thickness as a result velocity of the fractal-fractional CSF decreases. The influence of porosity K on fractalfractional CSF velocity is highlighted in Fig. 8. From this figure it seems that the CSF velocity increases with the greater values of K it is due to the fact that increasing K increases the pores in the channel as a result the fluid velocity accelerates. The effect of couple stress parameter η A is depicted in Fig. 9. This figure shows the influence of η A on the fractal-fractional velocity in channel. Increasing the values of couple stress parameter η A increases the viscosity of the fluid as a result the CSF velocity retards in channel. From this figure it can also be noticed that for η A = 0 shows the comparison of simple Newtonian viscous fluid with fractal-fractional velocity in channel. The influence of external pressure P is highlighted in Fig. 10. From the figure it can be noticed that increasing the values of P result an increase in the CSF velocity in channel.
The comparison of fractal-fractional temperature with fractal temperature β and fractional temperature α is highlighted in Fig. 11. From the comparison we can see that the classical temperature is higher than fractalfractional, fractal and fractional temperature. The influence of fractional parameter α and fractal parameter β on temperature is depicted in Figs. 12 and 13 respectively. From both the figures it can be noticed that for the greater values of fractional parameter α and fractal parameter β the temeprature of the CSF in channel reduces.  www.nature.com/scientificreports/ The effect of Pr on temperature profile is highlighted in Fig. 14. From the figure we can observe that for larger values of Pr the CSF temperature decreases it is due to the fact that increasing Pr results a decrease in thermal conductivity of the fluid as a result the temperature of the fluid decreases. The influence of Reynolds umber on temeprature profile is shown in Fig. 15. From the figure it can be observed that greater values of Reynolds number decrease the temperature of CSF in channel. The comparison of fractal-fractional, fractal and fractional concentration is highlighted in Fig. 16. From the figure one can noticed that the magnitude of classical concentration is higher than the concentration for fractal and fractional concentration. The effect of fractional parameter α and fractal parameter β on concentration profile is highlighted in Figs. 17 and 18 respectively. From both the figures it can be noticed that for the greater values of fractional parameter α and fractal parameter β in both the cases the concentration profile of the CSF in channel reduces. The influence of Reynolds number Re on concentration profile is highlighted in Fig. 19.   www.nature.com/scientificreports/ the change in skin friction in that specific parameter. Similarly, Table 3 shows the Nusselt number variation for different parameters. Table 4 shows the variation in Sherwood number for different parameters. In the Tables 1, 2, 3 and 4 bold values represents the changes in the specific parameter and its effect on the ski friction, Nusselt number and Sherwood number.

Conclusion
The present paper is focused to study the applications of fractal and fractional derivative on the unsteady MHD CSF in channel with power law kernel. The fractal-fractional CSF is assumed to flow in channel embedded in porous medium. The unsteady CSF with heat and mass transfer passes through the channel in the presence of external pressure. The implicit finite difference method is applied to obtain the numerical solutions of the