Generalization of the Convective Flow of Brinkman-Type Fluid Using Fourier’s and Fick’s Laws: Exact Solutions and Entropy Generation

A new scheme to formulating the Caputo time-fractional model for the flow of Brinkman-type fluid between the plates was introduced by using the generalized laws of Fourier and Fick. Within a channel, free convection flow of the electrically conducted Brinkman-type fluid was considered. A newly generated transformation was applied to the heat and mass concentration equations. ,e governing equations were solved by the techniques of Fourier sine and the Laplace transforms. In terms of the special function, namely, the Mittag-Leffler function, final solutions were obtained.,e entropy generation and Bejan number are also calculated for the given flow. To explain the conceptual arguments of the embedded parameters, separate plots are represented in figures and are often quantitatively computed and presented in tables. It is worth noting that for increasing the values of the Brinkman-type fluid parameter, the velocity profile decreases. ,e regression analysis shows that the variation in the velocity for time parameter is statistically significant.


Introduction
Because of its flexible and special properties, fractional calculus has evolved tremendously nowadays. e noninteger derivatives of all the orders are solved utilizing fractional calculus techniques. Fractional calculus is the extension of the classical calculus which has a history of around three centuries. Fractional calculus is a versatile and important method to explain many processes including memories [1,2]. In recent years, fractional calculus has been used for many applications in different areas, such as electrochemistry, ground-level water distribution, electromagnetism, elasticity, diffusion, and heat stream conduction [3][4][5]. For the flow problems, the fractional derivatives approach is used by many researchers. Shah et al. [6] study the viscous fluid in a cylindrical geometry using the Caputo time-fractional derivatives. e solutions were presented in the form special functions. e work was then extended by Ali et al. [7] for the flow of blood. ey have considered magnetic particles suspended in the blood and obtained the exact solutions for the flow problem. Caputo fractional derivatives have been used by Vieru et al. [8], Shakeel et al. [9], and Ali et al. [10] for the flow problems, and some interesting and useful results are obtained. Another approach of fractional derivatives is Caputo-Fabrizio fractional derivatives [11] which are also termed as a derivative with nonsingular kernel in the literature. is approach is used by the numerical and analytical solvers for different phenomena in real life. For different situations and analysis, CF derivatives are widely used by Hristov [12], Ali et al. [13], and Fetecau et al. [14]. Abro et al. [15] studied the electrically conducted flow of nanofluid using the concept of CF derivatives and obtained the solutions using special functions. e flow of second-grade fluid with heat transfer and MHD effect was analysed by Sheikh et al. [16] using the Caputo-Fabrizio fractional derivatives. e modern concept of the fractional derivatives using the generalized exponential functions (Mittag-Leffler function) has been created in 2016 by Atangana and Baleanu [17]. e kernel of integral associated with that derivative is nonlocal and nonsingular. Atangana and Koca [18] used the modern concept of a fractional derivative to a simple nonlinear system to demonstrate the presence and uniqueness of a solution for the problem. Afterwards, some authors used fractional derivative of Atangana-Baleanu in their study. e theory of frictional derivatives has been used by Sheikh et al. [19,20] for the flow problem of non-Newtonian fluids, which give exact solutions. In these articles, they discussed the difference between the two new fractional operators: Atangana-Baleanu and Caputo-Fabrizio. In both papers, they suggest that the velocity profile obtained for Atangana-Baleanu fractional models falls into a steady-state faster than the velocity profile for CF derivatives.
In the manufacturing and scientific fields, the physical properties of non-Newtonian fluids strongly affect [21][22][23][24][25]. Moreover, in magnetohydrodynamic (MHD) flows, the effects of heat transfer include a wide variety of applications from geosciences to engineering and the chemical sciences field [23,24,[26][27][28][29][30]. Non-Newtonian fluids demonstrate a complex process which ultimately needs to be described and represented by mathematical modelling. In a porous medium or in clay water contact, this phenomenon of non-Newtonian flow becomes much more complex [23,24]. In case of flow of non-Newtonian fluid in a porous medium, the effective viscosity becomes nonlinear function of pore velocity and shear flow [30]. e flow velocity behaves inversely under the influence of porosity and magnetohydrodynamics [22]. Due to its huge applications in textile factories, nuclear waste reservoirs, heat pipes, grain storage, and enhanced oil reservoirs storage, the heat and mass transfer through a porous media saturated with fluid has gained great attention. Darcy [31] identified the theoretical analysis and the respective mathematical model of the viscous fluid flow through a medium containing pore. In general, Darcy's implemented law can describe the flows passing over a low permeable region. However, Darcy's law is not effective and applicable to such fluids that move through a medium with high porosity, so Brinkman's model [32] is valid and suitable for such fluids. Extensive research has been done using the Brinkman model on the issue of fluid flow through a porous medium. e properties of the viscous fluid flowing through a porous channel were discussed in [33], using the Brinkman-type fluid model. In two cases, (1) where both walls are porous and (2) where the upper wall is rigid and the lower wall is porous, this problem has been solved. e flow was through the channel, which was highly permeable, and so Brinkman's model was considered. Lin and Payne [34] were analyzing the structural stability of the Brinkman model. Asif et al. [35] addressed the Brinkman-type insight view of fluid flow between two parallel walls. In their study, the oscillatory stress that is applied to the lower plate causes fluid motion. e authors find the exact solutions by using Fourier's mathematical method. ey reduced their solutions to already published work by neglecting some of the incorporated parameters to confirm the authenticity of the calculi. Kumar et al. [36] examined Dufour's effect on the thermal-radiating flow of viscous fluid that underlies the highly permeable region. Also, in their work, the authors mention the effects of MHD and chemical reaction. In the light of relative constitutive equations, the governing equations are derived. Some other interesting and relevant studies can be found in [37][38][39][40][41][42] and the references therein.
In this article, the unsteady MHD flow of Brinkman-type fluid in a vertical channel is considered with heat and mass transfer, keeping in mind the above discussion. Using the Caputo time-fractional derivative principle via generalized Fick's and Fourier's laws, the governing equations are translated into fractional PDEs. e latest transformation is used to transform the energy and concentration equation with initial and boundary conditions. With the joint applications of Laplace and Fourier sine transformation technique, the equations are solved. All the physical conditions are met and can be shown in graphs and tables.

Mathematical Modelling
We took Brinkman-type fluid motion into consideration in a vertical channel. It is assumed that the flow is in the x-axis direction, while the y-axis is taken perpendicular to the plates. With ambient temperature T 1 and ambient concentration C 1 , both the fluid and plates are at rest when t ≤ 0. At t � 0 + , as seen in Figure 1, the plate at y � d starts to move with velocity Uh(t) in its own plane. At y � d, the plate temperature and the rate of concentration rose over time t to Θ 1 + (Θ 2 − Θ 1 )f(t) and C 1 + (C 2 − C 1 )g(t), respectively.
We suppose the velocity profile as [22,43] With the aid of the well-known Boussinesq approximation, the free convection flow of fluid of the Brinkmantype, along with the heat and mass transfer, is regulated by the following partial differential equations [39,40]: with the initial and boundary conditions where μ, ρ, u, β r , Θ, β Θ , β C , g, c p , k, and D are dynamic viscosity, density, fluid velocity, material parameter, temperature, thermal expansion coefficient, coefficient of concentration, acceleration due to gravity, specific heat capacity, thermal conductivity, and mass diffusivity, respectively. Introducing the following dimensionless variables into equations (2)-(6), we get where Gr � ( Mathematical Problems in Engineering 3

Fractional Model
e generalized laws of Fourier and Fick are utilized as follows to establish a fractional model for the convective part of the referred flow problem: where C ℘ α τ (·) is the time-fractional operator developed by Caputo [44] and is described as [ Here, η α (t) � (t − α /Γ(1 − α)) is the singular power-law kernel. Furthermore, where L · { } is the Laplace transform, ζ(·) is Dirac's delta function, and s is the Laplace transform parameter.
Using the above properties and second form equation (17), it is convenient to show that Utilizing the definition of Caputo time-fractional operator form equation (23) and using equations (10, 12, 15, and 16), we arrived at We recalled the time-fractional integral operator to get the finest form for the last two equations: is is the inverse operator of the derivative operator C ℘ α τ (·). Using the properties from equation (18), we have Using the property (20) and (21) can be written as

Energy
Field. e following transformation is used: Equation (25) takes the following form: With the corresponding initial and boundary conditions, Applying the Laplace and Fourier sine transform, we get Inverting the integral transformations of equation (30), we have Figure 1: Geometry of the flow.
erefore, the solution of the energy equation is

Concentration
Field. e following transformation is used: Equation (26) takes the following form: With the corresponding initial and boundary conditions, Applying the Laplace and Fourier sine transform, we get Inverting the integral transformations of equation (36), we have e final solution for the concentration equation is

Velocity Profile.
Applying the Laplace and Fourier transform to equation (9) using equation (14), we arrived at where

Mathematical Problems in Engineering 5
where H(τ) is the unit step function and E a,b (·) is the Mittag-Leffler function [45].

Flow in the Absence of Mass Transfer.
In the absence of mass concentration (Gm � 0), equation (41) takes the following form:

Entropy Generation
For the flow of Brinkman-type fluid with the magnetic field and in the absence of mass concentration, taking into account the velocity from equation (42), the entropy generation is defined by [46] S G � , Ω � (Θ 2 − Θ 1 /Θ 1 ) which are the characteristics of entropy generation, Brinkman number, and dimensionless temperature difference, respectively.

e Bejan Number.
e Bejan number is defined as the irreversibility distribution parameter mathematically: Be � entropy generation due to heat transfer total entropy generation .

Nusselt Number
Nusselt number is an important physical quantity specially for engineers and industrialists. In nondimensional form, Nusselt number is given by

Sherwood Number
e gradient of mass concentration is termed as Sherwood number. In nondimensional form, Sherwood number is given by

Results and Discussion
In e influence of fractional parameter on the temperature profile is shown in Figure 10. e figure shows the temperature-increasing function of the fractional parameter. e effect of Prandtl number on the temperature profile is shown in Figure 11. e temperature is decreasing with the increasing values of Prandtl number, because the thermal forces weaken with the increasing values of Prandtl number. Concentration profile is plotted for various values of the fractional parameter in Figure 12.
e same behavior is noticed as in the case of temperature in this figure. e concentration of mass is decreasing with the increasing values of Schmidt number which is shown in Figure 13 Table 1 is presented to show the influence of the dimensionless time parameter on the fluid velocity, the velocity is showing an increasing trend for higher values of τ. e variation in the velocity and the predicted velocity against τ is presented in Figure 16 and Table 2 for fixed value of ξ, which shows that velocity is directly related to τ. is variation is statistically significant [47] as the P value is less than 0.05 as shown in Table 3.

Conclusion
e fractional Brinkman-type fluid model is developed using a new methodology in this research. e generalized Fick and Fourier laws are used to develop a fractional model. e Laplace and Fourier transformation methods are used to solve the problem. e results produced are drawn up and displayed in tables. e main findings of this analysis are as follows: (1) e new transformation is more reliable for the solution of the fractional model. It is easier to solve the fractional model using this transformation.
(2) is transformation is reducing the computational time for finding the exact solutions of such problems.
(3) e velocity reduces with higher values of Hartman's number and the Brinkman-type fluid parameter.
(4) For various values of α, variations in all the three profiles are shown. It is important to note here that for a fixed value of time, we have different lines in the graph. is result demonstrates the fluid's memory effect, which cannot be seen from the integer order model.

Data Availability
e data used to support the findings of this study are included in this paper and available without any restriction.

Conflicts of Interest
e authors declare that they have no conflicts of interest.