A Fractal-Fractional Model for the MHD Flow of Casson Fluid in a Channel

An emerging definition of the fractal-fractional operator has been used in this study for the modeling of Casson fluid flow. The magnetohydrodynamics flow of Casson fluid has cogent in a channel where the motion of the upper plate generates the flow while the lower plate is at a static position. The proposed model is non-dimensionalized using the Pi-Buckingham theorem to reduce the complexity in solving the model and computation time. The non-dimensional fractal-fractional model with the power-law kernel has been solved through the Laplace transform technique. The Mathcad software has been used for illustration of the influence of various parameters, i.e., Hartman number, fractal, fractional, and Casson fluid parameters on the velocity of fluid flow. Through graphs and tables, the results have been implemented and it is shown that the boundary conditions are fully satisfied. The results reveal that the flow velocity is decreasing with the increasing values of the Hartman number and is increasing with the increasing values of the Casson fluid parameter. The findings of the fractal-fractional model have elucidated that the memory effect of the flow model has higher quality than the simple fractional and classical models. Furthermore, to show the validity of the obtained closed-form solutions, special cases have been obtained which are in agreement with the already published solutions.


Introduction
The applications of non-Newtonian fluids are quite remarkable and play an important role in the flow modeling in industry and engineering sectors [1][2][3][4]. The magnetohydrodynamic (MHD) flows of these fluids are widely used in the field of MHD generators, magnetic drug targeting (MDT), MRI, and controlled flows, etc. with increased interest by many researchers [3,5,6]. Non-Newtonian fluids exhibit a complex structure that essentially requires some robust mathematical modeling for explanation and depiction. The non-Newtonian fluids flow behaves inversely with the influence of porosity and magnetohydrodynamics [2]. To solve the model for the MHD flow of micropolar fluid, the Laplace transformation was applied by Ali et al. [7]. MHD flow has been studied for the oscillating plates which include the effects of porous media under heat transfer and radiation [2,5]. The convective flow of non-Newtonian fluid over an oscillating plate in porous media was studied by Krishna et al. [8]. They have shown the relationship between the pressure gradient and flow velocity. This pressure gradient study has also discussed the relationship with the magnetohydrodynamics and the effects of different materials on the velocity using the Lattice Boltzmann method. The unsteady flow of Casson fluid for the oscillating plate to show the velocity relation in the porous media was studied by Khan et al. [9], who also discussed the velocity and shear stress by obtaining closed-form solutions. Ullah et al. [10] studied the flow of Casson fluid with slip on the stretching sheet.
Fractional derivatives are quite applicable for efficiently discussing the complex real-world problem for the flow of various types of fluids [11]. Several numerical simulations have studied water pollution with different applications through Caputo Fabrizio (CF) fractional derivatives [12]. These CF fractional derivatives have also been used in heat transfer analysis to get closed-form solutions for unsteady fluid flow over an oscillating plate where the fractional parameter is directly proportional to temperature. At the same time, it is inversely related to the velocity of the fluid. Fractional fluids have higher velocity values, which positively influence fluid flow [13]. These tenuous effects of the fluid flow could be efficiently resolved with the help of fractional derivatives [11]. The CF fractional derivatives and Atangana-Baleanu (AB) are very useful to identify and calculate the velocity values [14]. Fractional partial differential equations had also solved for closed-form solutions by using finite Hankel and Laplace transformation techniques. The numerical analysis of the fractional derivatives highlights a more significant effect on the velocity as compared to ordinary derivatives [15]. The fractional model could also apply to Casson fluid along with the energy equations. The Caputo fractional model and the Laplace transformation have also been used to get the solutions with Wright function by Ali et al. [16]. AB fractional derivatives and CF fractional derivatives could be used for the demonstration of the heat and mass transfer analysis of free convection flow of Casson fluid. For these types of flows, solutions and results were obtained through the Laplace transformation technique by Sheikh et al. [17]. Free convection generalized (Caputo-Fabrizio time-fractional derivatives) flow of Jeffrey fluid was analyzed via the Laplace transformation technique by Saqib et al. [18]. They had obtained the exact solutions for velocity and temperature. The results emphasize that fractional flows are relatively swifter than classical flows. Saqib et al. [11] discussed the Atangana-Baleanu (AB) derivatives model for Casson fluid in a microchannel. They have concluded that the fractional parameter considerably influences the viscosity and buoyancy forces. The numerical analysis of nanofluids through a porous media in a vertical channel by applying AB fractional derivative for convection free flow was discussed by Saqib et al. [19]. The flow of Casson fluid was considered by Sheikh et al. [20]. They studied the Caputo fractional derivative and the flow had modeled using the generalized Fourier and Fick laws.
Keeping in mind the above literature survey, in this paper we have considered electrically conducted flow of Casson fluid in a channel. The fractal-fractional model has been created for the subject flow using the concept of fractional calculus. The exact solutions have been obtained using the Laplace transform technique.

Mathematical Modelling
We have considered the motion of Casson fluid in a vertical channel. The flow is assumed in the direction of the x-axis while the y-axis is taken perpendicular to the plates. The fluid and the plates are at rest when t 0. At t ¼ 0 þ , the plate at y ¼ d begins to move in its plane with velocity U as shown in Fig. 1.

Fractal-Fractional Model
The fractal-fractional model for the mentioned flow problem, in the generalized form is: If we assume that vðjÞ is continuous in open interval, if vðjÞ is fractal differentiable on (a, b) with order d, then the fractal fractional derivative of vðjÞ with order a in Riemann-Liouville sense with power law is presented as [43] FFP 0 } a;d j vðjÞ ¼ where

Limiting Cases
Validating results, the present solutions have reduced to the already published results in the literature, and the solutions for some other well-known flows have been obtained.

Case 1 (Newtonian Fluid)
For b ! 1 the obtained solution is reduced to the following form: v n; s ð Þ ¼ ; (14) which is identical to the solution calculated by [23].

Case 2 (Solution for fractional model)
For d ¼ 1 the obtained solution is reduced to the corresponding fractional model:   The effect of the fractional parameter is shown in Fig. 3 and presented in Tab. 2. Variations in velocity are shown for different values of a, and despite the increasing or decreasing behavior of the velocity, this figure shows that we can draw many graphs for the velocity while keeping the physical parameters constant. This effect is known as a memory effect, which can be described by a.
The graph is plotted for d in Fig. 4 and presented in Tab. 3. Despite the increasing or decreasing behavior of the velocity, this figure shows that we can draw many graphs for the velocity while keeping the physical parameters constant. The important fact is that both the fractional and fractal parameters are describing the memory, and together these can describe the memory better than the fractional and classical models.
To show the effect of the Casson fluid parameter b, Fig. 5 has been plotted. It has been depicted from this figure that velocity increases with the increasing values of b. Physically, the increasing values of b reduces the thickness of the boundary layer. We can say that for a huge value of b the fluid will behave like a Newtonian fluid.        Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.