Flow of Brinkman Fluid with Heat Generation and Chemical Reaction

Unsteady magnetohydrodynamics (MHD) flow of fractionalized Brinkman-type fluid over a vertical plate is discussed. In the model of problem, additional effects such as heat generation/absorption and chemical reaction are also considered. *e model is solved by using the Caputo fractional derivative. *e governing dimensionless equations for velocity, concentration, and temperature profiles are solved using the Laplace transformmethod and compared graphically.*e effects of different parameters like fractional parameter, heat generation/absorption Q, chemical reaction R, and magnetic parameter M are discussed through numerous graphs. Furthermore, comparison among ordinary and fractionalized velocity fields are also drawn. From the figures, it is observed that chemical reaction and magnetic field have decreasing effect on velocity profile, whereas thermal radiation and mass Grashof numbers have increasing effect on the velocity of the fluid.


Introduction
e important significance of non-Newtonian fluids can be seen in applied mathematics, engineering, and physics. It has various significances in many areas, such as uses of lubricants, biological fluid food processing, or plastic manufacturing. Some commonly examples of non-Newtonian fluids are custard, colloids, melted butter, paint, ketchup, starch suspensions, blood, toothpaste, gels, shampoo, and corn starch.
Mass transfer and heat transfer occurs mostly in nature due to temperature and concentration differences, respectively. Today, research work in magnetohydrodynamics (MHD) has substantial significance as these flows are absolutely prevailing in nature.
Convection flow with porous media has numerous applications such as flows in soils, solar power collectors, heat transfer correlated with geothermal systems, heat source in the field of agricultural storage system, heat transfer in nuclear reactors, heat transfer in aerobic and anaerobic reactions, heat evacuation from nuclear fuel detritus, and heat exchangers for porous material.
MHD fluid has many implementations in meteorology, distillation of gasoline, energy generators, geophysics, accelerators, petroleum industry, astrophysics, polymer technology, aerodynamics, and boundary layer control and in material processes such as glass fiber drawing, extrusion, and casting wire. e flow of viscous fluid through a perpendicular plate is analyzed by Swamy et al. [1]. e effect of mass diffusion on MHD fluid with porosity has been observed by Chaudhary et al. [2]. Exact solution for magnetohydrodynamics flow through a perpendicular plate in the existence of porosity is obtained by Sivaiah et al. [3]. e solution for unsteady flow of viscous fluid with porosity is obtained by Das and Jana [4]. radiation. Chamkha [7] discussed the effect of heat source on MHD fluid through a moving plate. e authors of [8] studied the flow of polar fluid through a plate. Rahman and Sattar [9] studied the flow of fluid with a heat source. Rajesh and Varma [10] studied the influence of mass diffusion on magnetohydrodynamic fluid flow. e authors of [11] analyzed the solution of convection flow through a vertical plate. ey also discussed the solution for time-dependent concentration and temperature. Convection flow immersed in a porous media through a surface is discussed in [12][13][14]. e impact of conjugate flow of MHD fluid is discussed by Khan et al. [15]. Rajesh et al. [16] discussed the MHD flow through a moving plate. MHD flow through an accelerated surface in the existence of porous media is discussed by Chaudhary et al. [17]. e authors also analyzed the solution of velocity field graphically. Das [18] analyzed the solution of magnetohydrodynamics of convection flow through a plate. Pal et al. [19] examined the solution of viscous fluid with thermal radiation on magnetohydrodynamics flow, whereas the solution for convection flow with nonuniform temperature through a moving plate is obtained by Seth et al. [20]. e solution of nanofluid with ramped temperature is studied by Khalid et al. [21]. e discussion of mass diffusion has empirical use in numerous areas of engineering and applied sciences. ese phenomena play a vital role in cooling of a nuclear reactor and tabular reactor, chemical industry, mixture of terracotta material, petroleum industry, and decomposition of rigid materials. Seddeek et al. [22] examined the MHD fluid flow with thermal radiation. An intensive study of chemical reaction with heat source/sink is studied by Shah et al. [23]. Seth et al. [24] obtained the solution of unsteady magnetohydrodynamic flow of the fluid over a plate with ramped condition. e solution of convection flow of MHD fluid over a plate with heat generation/absorption is obtained by Shateyi and Motsa [25]. MHD fluid flow with Ohmic heating and heat generation is analyzed by Kasim et al. [26]. e exact solution of MHD fluid with mass transfer immersed in a porous media is discussed by Ali et al. [27]. e exact solution of magnetohydrodynamic flow of a Brinkman fluid perpendicular to the plate is analyzed by Khan et al. [28]. e analytical investigation of Brinkman fluid flow with variable concentration, temperature, and velocity is obtained by Ali et al. [29]. e flow of nanofluid with thermal radiation is studied in [30][31][32][33][34][35]. Patel et al. [36] studied the effect of Joule's heating on ferrofluid. e influence of Brownian motion and thermophoresis is studied by Mittal and Kataria [37]. Kataria [38] studied the effect of radiation and magnetic field on Casson fluid. Hashemi et al. [39] analyzed the solution of a circular rod. Some flows of fluids with numerical and computational methods are discussed in [40][41][42].
In this problem, the model of unsteady magnetohydrodynamic free convection flow of Brinkman fluid through a plate is considered. e impact of chemical reaction and heat absorption/generation is added into account. Firstly, the governing equations have been made nondimensional and then solved semianalytically. e results for velocity profile, temperature profile, and concentration profile are obtained and then analyzed graphically. Various graphs are plotted and discussed for different parameters, which are used in the flow model. e comparison between ordinary and fractionalized fluid is drawn graphically and shows that Caputo fractional derivative is the best choice for controlled fluid velocity.

Mathematical Description of the Model
e magnetohydrodynamic flow of Brinkman fluid through a plate with mass and heat transfer is considered. e fluid is flowing along the x · axis. e motion of fluid depends on y · -axis and time t · 1 . e plate and fluid have concentration C · ∞ and temperature T · ∞ at constant t · 1 � 0 with zero velocity. But, for t · 1 > 0, the plate starts to move in the plane with uniform velocity U 1 e at · 1 . e concentration and temperature of the plate increased linearly to C · w and T · w with time t · . A constant strength β 0 of magnetic field is applied normally. In view of the above assumption and using Boussinesq's approximation, the convection flow of Brinkman fluid with chemical reaction, and magnetic field through a plate, the linear momentum equation is Shear stress τ is According to Fourier's Law, q 1 (y · , t · 1 ) is given by Diffusion equation is According to Fick's Law, J 1 (y · , t · 1 ) is given by e boundary conditions for the flow model are 2 Complexity To write the flow model in dimensionless form, we used the following dimensionless variables: Using nondimensional variables from equations in (10) into the equations (1)-(9), we have zT(y, t) zt zC(y, t) zt with boundary conditions as where where Gr, B, Sc, Q, M, Pr, Gm, and v represent the Grashof number for heat transfer, Brinkman parameter, Schmidt number, nondimensional heat source, magnetic field, Prandtl number, mass Grashof number, and velocity of the fluid, respectively.

Solution of the Problem
Equations (25)- (27) with initial and boundary conditions are solved semianalytically. (27) is

Calculation of Concentration. Solution of equation
Boundary conditions satisfying equation (31) are Equation (31) is solved by using conditions given in equation (32), and we have which is complicated and cannot be solved analytically. e numerical result of equation (33) is obtained by using the algorithm in [46,47]. (26) is

Complexity
Boundary conditions satisfying equation (34) are Equation (34) is solved by using conditions given in equation (35), which results in which is complicated one and cannot be solved analytically. e numerical result of equation (36) is obtained by using the algorithm in [46,47]. (25) is

Calculation of Velocity. Solution of equation
Boundary conditions satisfying equation (37) are Equation (37) is solved by using conditions given in equation (38), and we obtain which is much complicated, so it cannot be solved analytically. Numerical result of equation (39) will be obtained by using the algorithm in [46,47].

Results and Discussion
Semianalytical solution for MHD flow of Brinkman fluid with a combined concentration and temperature gradient over a plate is obtained. e generalized model is solved with a Caputo fractional derivative. e graph of concentration profile, temperature profile, and velocity profile are plotted for different parameters. Figure 1 represents the effect of B on v(y, t). It is noted that the v(y, t) decreases with increasing values of the Brinkman parameter. Physically, Brinkman is the relation between drag force and density; therefore, drag force increases with increasing values of the Brinkman parameter which decays down the fluid motion. e behavior of Gm is reported in Figure 2. From this graph, it is concluded that the magnitude of fluid velocity rises by raising the values of Gm. Gm is the relative strength of viscous force and concentration buoyancy force. As Gm increases, the motion of fluid is accelerated due to an increment of buoyancy force.  v(y, t). From this graph, it is noted that velocity distribution is directly proportional with Gr. Physically, Gr is a relation between viscous force and buoyancy force. erefore, with an increment in the values of Gr, buoyancy force is increased which raises the magnitude of v(y, t).  Figure 5 represents the behavior of R and Pr on the v(y, t). e graph shows that v(y, t) falls down for larger values of R. e impact of various values of Pr on v(y, t) is displayed in Figure 5. Pr represents the ratio of momentum (product of mass and velocity) diffusion to thermal diffusion. In the problems of heat transfer, Pr manages the thickness of boundary layer and momentum (velocity). For larger value of Pr, diffusion of heat becomes slow as compared to the fluid momentum (velocity) which decreases the thermal conductivity (thickness) and raises the boundary layer momentum. e influence of R on fluid motion is shown in Figure 5. From the figure, it is concluded that fluid motion decays by increasing the values of R. Physically, boundary layer thickness is increased by increasing values of Complexity 5 R which slows down the velocity distribution. Figure 6 shows the influence of Sc and α, β, and c on v(y, t). e graph shows that for increasing values of Sc, the diffusion of the molecule increases which reduces the fluid level. However, fluid velocity rises with increasing values of fractional parameters. e behavior of heat generation Q and Pr on T(y, t) is displayed in Figure 7. is figure shows that temperature increases with increment in the values of Q. Figure 7 indicates the influence of Pr on temperature T(y, t). Temperature distribution is accelerated with decreasing values of Pr as shown in the graph. Physically, the increase in Pr minimizes viscosity which reduces the thermal boundary layer. e behavior of chemical reaction R and Sc on C(y, t) is shown in Figure 8. e concentration level is accelerated with decreasing R as depicted in the graph. Physically, boundary layer thickness is increased by increasing values of R which slows down the concentration distribution. Figure 8 shows the influence of Sc on C(y, t). e concentration level increases with reducing values of Sc as highlighted in the figure. e graph shows that for increasing values of Sc, the diffusion of molecule increases which reduces the fluid level. Figure 9 shows the comparison of Brinkman-type fractional fluid with Olisa [48]. From the figure, it is concluded that fractional derivative is the best choice to enhance the fluid motion. Figure 9(b) represents that if we take fractional parameters β � c � α ⟶ 1, Gm � F(s) � 0, and B � 0, the Sc=0.4 Sc=0.5    Ordinary fluid [48] Brinkman fractional fluid Ordinary fluid [48] Brinkman fractional fluid fluid profiles are identical which shows the authenticity of the present work. Figure 10 represents the validity of inversion algorithms for concentration and temperature profiles. e overlapping velocity profiles show the validity of inversion algorithms as shown in Figure 11.

Conclusion
Solution of free convection magnetohydrodynamic flow of Brinkman-type fluid has been obtained via Laplace transform. Different parameters used in the model are plotted and discussed. e model is solved with a fractional derivative known as Caputo fractional derivative.
Here are the main points which have been summarized for this model: (i) Velocity distribution retards with decreasing values of fractional parameter (ii) ermal buoyancy forces lead to accelerate the v(y, t) (iii) e v(y, t) decreases as magnetic parameter, chemical reaction parameter, Prandtl number, and Sc increases (iv) e Brinkman parameter is a decreasing function of velocity field (v) e larger values of Q increased the T(y, t) (vi) e larger values of Pr reduced the T(y, t) (vii) e concentration level is a decreasing function of Sc (viii) e smaller values of R reduce the concentration profile (ix) Caputo fractional derivative is the best choice to enhance the fluid motion as compared to ordinary fluid

Data Availability
All used data are included within the manuscript.

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