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

Authors: Nadeem Ahmad Sheikh; Dennis Ling Chuan Ching; Ilyas Khan; Hamzah bin Sakidin
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

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. The 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. The 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. The regression analysis shows that the variation in the velocity for time parameter is statistically significant.


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## Body

## 1. Introduction

Because of its flexible and special properties, fractional calculus has evolved tremendously nowadays. The 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 [3–5]. 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. The solutions were presented in the form special functions. The work was then extended by Ali et al. [7] for the flow of blood. They 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. This 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. The flow of second-grade fluid with heat transfer and MHD effect was analysed by Sheikh et al. [16] using the Caputo–Fabrizio fractional derivatives. The modern concept of the fractional derivatives using the generalized exponential functions (Mittag-Leffler function) has been created in 2016 by Atangana and Baleanu [17]. The 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. The 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 [21–25]. 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, 26–30]. 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]. The 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. The 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. The 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. The authors find the exact solutions by using Fourier’s mathematical method. They 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 [37–42] 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. The 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.


## 2. Mathematical Modelling

We took Brinkman-type fluid motion into consideration in a vertical channel. It is assumed that the flow is in thex-axis direction, while the y-axis is taken perpendicular to the plates. With ambient temperature T1 and ambient concentration C1, 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 Uht in its own plane. At y=d, the plate temperature and the rate of concentration rose over time t to Θ1+Θ2−Θ1ft and C1+C2−C1gt, respectively.Figure 1
Geometry of the flow.We suppose the velocity profile as [22, 43](1)V⟶=u,0,0.With the aid of the well-known Boussinesq approximation, the free convection flow of fluid of the Brinkman-type, along with the heat and mass transfer, is regulated by the following partial differential equations [39, 40]:(2)ρ∂uy,t∂t+βruy,t=μ∂2uy,t∂y2−σB02uy,t+ρβΘgΘ−Θ1+ρβCgC−C1,(3)ρcp∂Θy,t∂t=−∂qy,t∂y,(4)qy,t=−k∂Θy,t∂y,(5)∂Cy,t∂t=−∂jy,t∂y,(6)jy,t=−D∂Cy,t∂y,with the initial and boundary conditions(7)uy,0=0,Θy,0=Θ1,Cy,0=C1,u0,t=0,Θ0,t=Θ1,C0,t=C1,ud,t=Uht,Θd,t=Θ1+Θ2−Θ1ft,Cd,t=C1+C2−C1gt,,where μ, ρ, u, βr, Θ, βΘ, βC, g, cp, 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(8)v=uU,ξ=yd,τ=νd2t,θ=Θ−Θ1Θ2−Θ1,Φ=C−C1C2−C1,δ=qdkΘ2−Θ1,λ=jdDC2−C1,fτ=fd2νt,gτ=gd2νt,hτ=hd2νt.into equations (2)–(6), we get(9)∂vξ,τ∂τ=∂2vξ,τ∂ξ2−Αvξ,τ+Grθξ,τ+GmΦξ,τ,(10)∂θξ,τ∂τ=−1Pr∂δξ,τ∂ξ,(11)δξ,τ=−∂θξ,τ∂ξ,(12)∂Φξ,τ∂τ=−1Sc∂λξ,τ∂ξ,(13)λξ,τ=−∂Φξ,τ∂ξ,(14)vξ,0=0v0,τ=0v1,τ=hτθξ,0=0θ0,τ=0θ1,τ=fτΦξ,0=0Φ0,τ=0Φ1,τ=gτ,where Gr=gd2βΘ/νUΘ2−Θ1 is the thermal Grashof number, Gm=gd2βC/νUC2−C1 is the mass Grashof number, γB=βrd2/ν is the Brinkman-type fluid parameter, M=σB02d2/μ is Hartman number, Pr=μcp/k is the Prandtl number, Sc=ν/D is the Schmidt number, and Α=M+γB.


## 3. Fractional Model

The generalized laws of Fourier and Fick are utilized as follows to establish a fractional model for the convective part of the referred flow problem:(15)δξ,τ=−℘Cτ1−α∂θξ,τ∂ξ;0<α≤1,(16)λξ,τ=−℘Cτ1−α∂Φξ,τ∂ξ;0<α≤1,where ℘Cτα⋅ is the time-fractional operator developed by Caputo [44] and is described as [7](17)℘Cταry,t=1Γ1−α∫0tr˙y,st−s−αds=ηαt∗r˙y,t;0<α≤1.Here,ηαt=t−α/Γ1−α is the singular power-law kernel.Furthermore,(18)Lηαt=1s1−α,η1−α∗ηαt=1,η0t=L−11s=1,η1t=L−11=ζt,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(19)℘Cτ0ry,t=ry,t−ry,0,℘Cτ1ry,t=∂ry,t∂t.Utilizing the definition of Caputo time-fractional operator form equation (23) and using equations (10, 12, 15, and 16), we arrived at(20)∂θξ,τ∂t=1Pr℘Cτ1−α∂2θξ,τ∂2ξ,(21)∂Φξ,τ∂t=1Sc℘Cτ1−α∂2Φξ,τ∂2ξ.We recalled the time-fractional integral operator to get the finest form for the last two equations:(22)ℑtαry,t=η1−α∗rt=1Γα∫0try,st−sα−1ds.This is the inverse operator of the derivative operator℘Cτα⋅. Using the properties from equation (18), we have(23)ℑtα∘℘Cταry,t=ℑtα℘Cταry,t=η1−α∗ηα∗r˙t=η1−α∗ηα∗r•r˙t=1∗r˙t=ry,t−ry,0,(24)⇒ℑtα∘℘Cταry,t=ry,t,ifry,0=0.Using the propertyℑt1−αr˙y,t=ηα∗r˙t=℘Cταry,t, equations (20) and (21) can be written as(25)℘Cταθξ,τ=1Pr∂2θξ,τ∂2ξ,(26)℘CταΦξ,τ=1Sc∂2Φξ,τ∂2ξ,


## 4. Solution of the Problem

### 4.1. Energy Field

The following transformation is used:(27)χξ,τ=θξ,τ−ξfτ.Equation (25) takes the following form:(28)℘Cταχξ,τ−ξ℘Cταfτ=1Pr∂2χξ,τ∂2ξ.With the corresponding initial and boundary conditions,(29)χξ,0=0,χ0,τ=0,χ1,τ=0.Applying the Laplace and Fourier sine transform, we get(30)χ¯Fn,s=sf¯s−1nnπsα−1sα+nπ2/Pr.Inverting the integral transformations of equation (30), we have(31)χξ,τ=2∑n=1∞−1nsinξnπnπ∫0τf˙τ−tEα,α−1−nπ2Prtαdt.Therefore, the solution of the energy equation is(32)θξ,τ=χξ,τ+ξfτ.


### 4.2. Concentration Field

The following transformation is used:(33)Ψξ,τ=Φξ,τ−ξgτ.Equation (26) takes the following form:(34)℘CταΨξ,τ−ξ℘Cταgτ=1Sc∂2Ψξ,τ∂2ξ.With the corresponding initial and boundary conditions,(35)Ψξ,0=0,Ψ0,τ=0,Ψ1,τ=0.Applying the Laplace and Fourier sine transform, we get(36)Ψ¯Fn,s=sg¯s−1nnπsα−1sα+nπ2/Sc.Inverting the integral transformations of equation (36), we have(37)Ψξ,τ=2∑n=1∞−1nsinξnπnπ∫0τg˙τ−tEα,α−1−nπ2Sctαdt.The final solution for the concentration equation is(38)Φξ,τ=Ψξ,τ+ξgτ.


### 4.3. Velocity Profile

Applying the Laplace and Fourier transform to equation (9) using equation (14), we arrived at(39)v¯Fn,s=−1n+1h¯snπ+ℜ5s+ℜ6s+ℜ1−1nsh¯snπ+Grθ¯Fn,ss+ℜ1+GmΦ¯Fn,ss+ℜ1,where(40)ℜ1=Α+nπ2,ℜ2=ℜ1nπ2,ℜ3=1−nπ2ℜ1,ℜ4=ℜ1−nπ2,ℜ5=ℜ4ℜ1,ℜ6=ℜ3ℜ1+ℜ2ℜ1+ℜ4ℜ1.Inverting the Laplace and Fourier sine transformations, the final solution is(41)vξ,τ=hτξ+2∑n=1∞−1nnπh˙τ∗ℜ5Hτ+ℜ6exp−ℜ1τsinξnπ+2Grℜ∑n=1∞−1nnπexp−ℜ1τ∗∫0τf˙τ−tEα,α−1−nπ2Prtαdt+fτsinξnπ+2Gmℜ∑n=1∞−1nnπexp−ℜ1τ∗∫0τg˙τ−tEα,α−1−nπ2Sctαdt+gτsinξnπ,where Hτ is the unit step function and Ea,b⋅ is the Mittag-Leffler function [45].


## 4.1. Energy Field

The following transformation is used:(27)χξ,τ=θξ,τ−ξfτ.Equation (25) takes the following form:(28)℘Cταχξ,τ−ξ℘Cταfτ=1Pr∂2χξ,τ∂2ξ.With the corresponding initial and boundary conditions,(29)χξ,0=0,χ0,τ=0,χ1,τ=0.Applying the Laplace and Fourier sine transform, we get(30)χ¯Fn,s=sf¯s−1nnπsα−1sα+nπ2/Pr.Inverting the integral transformations of equation (30), we have(31)χξ,τ=2∑n=1∞−1nsinξnπnπ∫0τf˙τ−tEα,α−1−nπ2Prtαdt.Therefore, the solution of the energy equation is(32)θξ,τ=χξ,τ+ξfτ.


## 4.2. Concentration Field

The following transformation is used:(33)Ψξ,τ=Φξ,τ−ξgτ.Equation (26) takes the following form:(34)℘CταΨξ,τ−ξ℘Cταgτ=1Sc∂2Ψξ,τ∂2ξ.With the corresponding initial and boundary conditions,(35)Ψξ,0=0,Ψ0,τ=0,Ψ1,τ=0.Applying the Laplace and Fourier sine transform, we get(36)Ψ¯Fn,s=sg¯s−1nnπsα−1sα+nπ2/Sc.Inverting the integral transformations of equation (36), we have(37)Ψξ,τ=2∑n=1∞−1nsinξnπnπ∫0τg˙τ−tEα,α−1−nπ2Sctαdt.The final solution for the concentration equation is(38)Φξ,τ=Ψξ,τ+ξgτ.


## 4.3. Velocity Profile

Applying the Laplace and Fourier transform to equation (9) using equation (14), we arrived at(39)v¯Fn,s=−1n+1h¯snπ+ℜ5s+ℜ6s+ℜ1−1nsh¯snπ+Grθ¯Fn,ss+ℜ1+GmΦ¯Fn,ss+ℜ1,where(40)ℜ1=Α+nπ2,ℜ2=ℜ1nπ2,ℜ3=1−nπ2ℜ1,ℜ4=ℜ1−nπ2,ℜ5=ℜ4ℜ1,ℜ6=ℜ3ℜ1+ℜ2ℜ1+ℜ4ℜ1.Inverting the Laplace and Fourier sine transformations, the final solution is(41)vξ,τ=hτξ+2∑n=1∞−1nnπh˙τ∗ℜ5Hτ+ℜ6exp−ℜ1τsinξnπ+2Grℜ∑n=1∞−1nnπexp−ℜ1τ∗∫0τf˙τ−tEα,α−1−nπ2Prtαdt+fτsinξnπ+2Gmℜ∑n=1∞−1nnπexp−ℜ1τ∗∫0τg˙τ−tEα,α−1−nπ2Sctαdt+gτsinξnπ,where Hτ is the unit step function and Ea,b⋅ is the Mittag-Leffler function [45].


## 5. Special Cases

### 5.1. Flow in the Absence of Mass Transfer

In the absence of mass concentration (Gm=0), equation (41) takes the following form:(42)vξ,τ=hτξ+2∑n=1∞−1nnπh˙τ∗ℜ5Hτ+ℜ6exp−ℜ1τsinξnπ+2Grℜ∑n=1∞−1nnπexp−ℜ1τ∗∫0τf˙τ−tEα,α−1−nπ2Prtαdt+fτsinξnπ.


## 5.1. Flow in the Absence of Mass Transfer

In the absence of mass concentration (Gm=0), equation (41) takes the following form:(42)vξ,τ=hτξ+2∑n=1∞−1nnπh˙τ∗ℜ5Hτ+ℜ6exp−ℜ1τsinξnπ+2Grℜ∑n=1∞−1nnπexp−ℜ1τ∗∫0τf˙τ−tEα,α−1−nπ2Prtαdt+fτsinξnπ.


## 6. 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](43)SG=kΘ12∂Θy,t∂y2︸entropy generationdue to heat transfer+μΘ1∂uy,t∂y2+βrΘ1u2y,t︸entropy generationdue to fluid friction+σB02Θ1u2y,t︸entropy generationdue to magnetic field.The dimensionless form of equations (43) is given as(44)Nsξ,τ=SGS0=∂θξ,τ∂ξ2+BrΩ∂vξ,τ∂ξ2+BrγBΩv2ξ,τ+BrMΩv2ξ,τ,with S0=kUΘ2−Θ1/ν2Θ12,Br=U2μ/kΘ2−Θ1,Ω=Θ2−Θ1/Θ1 which are the characteristics of entropy generation, Brinkman number, and dimensionless temperature difference, respectively.


### 6.1. The Bejan Number

The Bejan number is defined as the irreversibility distribution parameter mathematically:(45)Be=entropy generation due to heat transfertotal entropy generation.


## 6.1. The Bejan Number

The Bejan number is defined as the irreversibility distribution parameter mathematically:(45)Be=entropy generation due to heat transfertotal entropy generation.


## 7. Nusselt Number

Nusselt number is an important physical quantity specially for engineers and industrialists. In nondimensional form, Nusselt number is given by(46)Nu=∂θξ,τ∂ξ|ξ=1.


## 8. Sherwood Number

The gradient of mass concentration is termed as Sherwood number. In nondimensional form, Sherwood number is given by(47)Sh=∂Φξ,τ∂ξ|ξ=1.


## 9. Results and Discussion

In this analysis, we address the unsteady flow of Brinkman-type fluid in a vertical channel. The cumulative effects of heat and mass transfer have been considered. A fractional model was developed using the generalized Fick’s and Fourier’s laws. In order to convert the equation into simpler form, a new transformation is applied, and then the Laplace and Fourier sine transformations are used to obtain solutions in closed form. Figures and tables demonstrated the effect of different embedded parameters on the velocity, temperature, and concentration profiles. Regression analysis is introduced to estimate the time-based velocity profile.The influence of the fractional parameterαis shown in Figure 2. From this figure, the fluid velocity is the increasing function of fractional parameter. Figures 3 and 4 are sketched to depict the influence of thermal and mass Grashof numbers. These figures show that the velocity is increasing with the increasing values of Gr and Gm. This is due to buoyancy forces, these forces become dominant with the increasing values of Grand Gm, and as a result, the boundary layer thickness decreases. Both the figures show that imposed boundary conditions are satisfied. Prandtl number is the ration of viscous forces to the thermal forces. The increasing values of Prandtl number mean that the viscous forces become dominant over the thermal forces, and as a result, the fluid velocity decreases as shown in Figure 5. The variations in velocity profile due to change in Brinkman-type fluid parameter are depicted in Figure 6. From this figure, interesting results are obtained. The velocity is decreasing with increasing values of the material parameter γB. Figure 7 is plotted in order to show the influence of Schmidt number on the fluid flow. It is noticed that the fluid velocity is decelerating with the increasing values of Sc. The influence of Hartman number is depicted in Figure 8. Velocity of the fluid is decreasing with the increasing values of M. As the increasing values of M strengthen the Lorentz forces which are the flow opposing forces. In order to show the validation of the obtained solution, Figure 9 is plotted. In this figure, hτ=τ(τ=0.5,1,1.5 and 2) is taken and the conditions are satisfied. This means that for all type of boundary conditions, the obtained solutions are valid.Figure 2
Variations in velocity profile againstξ if hτ=0 for different values of α.Figure 3
Variations in velocity profile againstξ if hτ=0 for different values of Gr.Figure 4
Variations in velocity profile againstξ if hτ=0 for different values of M.Figure 5
Variations in velocity profile againstξ if hτ=0 for different values of Pr.Figure 6
Variations in velocity profile againstξ if hτ=0 for different values of γB.Figure 7
Variations in velocity profile againstξ if hτ=0 for different values of Sc.Figure 8
Variations in velocity profile againstξ if hτ=0 for different values of M.Figure 9
Variations in velocity profile againstξ if hτ=τ for different values of τ.The influence of fractional parameter on the temperature profile is shown in Figure10. The figure shows the temperature-increasing function of the fractional parameter. The effect of Prandtl number on the temperature profile is shown in Figure 11. The 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. The same behavior is noticed as in the case of temperature in this figure. The concentration of mass is decreasing with the increasing values of Schmidt number which is shown in Figure 13. The influence of the Brinkman-type fluid parameter on the entropy generation is shown in Figure 14. It is revealed from this figure that near the plates, the entropy generation decreases with increasing vales of the Brinkman-type fluid parameter, while the entropy generation increases in the middle of the channel for increasing values of γB. The impact of γB on the Bejan number is shown in Figure 15. From this figure, it is clear that the Bejan number increases near the walls of the channel while decreases at the middle of the channel for increasing vales of γB. 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. The 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. This variation is statistically significant [47] as the P value is less than 0.05 as shown in Table 3.Figure 10
Variations in temperature profile againstξ for different values of α.Figure 11
Variations in temperature profile againstξ for different values of Pr.Figure 12
Variations in Concentration profile againstξ for different values of α.Figure 13
Variations in Concentration profile againstξ for different values of Sc.Figure 14
Variations Entropy Generation againstξ for different values of γB.Figure 15
Variations Bejan Number againstξ for different values of γB.Table 1
Variations in velocity profile againstξif hτ=0 for different values of τ.
ξvξ,τ at τ=0.5vξ,τ at τ=1vξ,τ at τ=1.5vξ,τ at τ=20.040.0890.0940.0960.0980.080.1780.1880.1920.1950.120.2660.280.2860.290.160.3520.370.3790.3840.20.4360.4590.4690.4760.240.5180.5440.5560.5640.280.5960.6250.6390.6490.320.670.7020.7180.7280.360.7390.7740.7910.8020.40.8020.8390.8580.870.440.8580.8980.9170.9290.480.9070.9480.9670.9810.520.9470.9881.0081.0220.560.9761.0171.0381.0510.60.9931.0341.0541.0680.640.9971.0361.0561.070.680.9851.0221.0421.0550.720.9550.991.0081.0210.760.9050.9370.9540.9650.80.8330.8610.8750.8850.840.7340.7580.770.7780.880.6050.6240.6330.640.920.4430.4560.4630.4670.960.2430.250.2530.25510000Figure 16
The prediction of velocity for increasing time atξ=0.4.Table 2
Variation in velocity againstτ.
τ00.20.40.60.811.21.41.61.82vξ,τ00.70.790.810.830.840.850.860.90.870.9Table 3
Regression table of velocity forτ.
CoefficientsP valueIntercept0.7048329.26E-16τ0.1035410.000225


## 10. Conclusion

The fractional Brinkman-type fluid model is developed using a new methodology in this research. The generalized Fick and Fourier laws are used to develop a fractional model. The Laplace and Fourier transformation methods are used to solve the problem. The results produced are drawn up and displayed in tables. The main findings of this analysis are as follows:(1)
The new transformation is more reliable for the solution of the fractional model. It is easier to solve the fractional model using this transformation.(2)
This transformation is reducing the computational time for finding the exact solutions of such problems.(3)
The velocity reduces with higher values of Hartman’s number and the Brinkman-type fluid parameter.(4)
For various values ofα, 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. This result demonstrates the fluid’s memory effect, which cannot be seen from the integer order model.(5)
The variation in the velocity profile for different values of the fractional parameter is statistically significant.(6)
The entropy generation and Bejan number show different behaviors for the Brinkman-type fluid parameter near the walls and in the middle of the channel.



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