MHD Three-Dimensional Free Convective Flow with Periodic Permeability and Heat Transfer of a Second-Grade Fluid

The present study delivers the mathematical model and theoretical analysis of a three-dimensional ﬂow in a free convection for an electrically conducting incompressible second-grade ﬂuid through a very high porous medium circumscribed by an inﬁnite vertical porous plate subject to a constant suction. A uniform magnetic ﬁeld along the normal to the surface of plate is applied. Periodic permeability for the medium is assumed, while velocity of free stream is taken to be uniform. Analytic expressions are presented for velocity and temperature ﬁelds, pressure, and skin friction components by perturbation technique. The impacts on these physical quantities by the physical parameters existing in the model are discussed and envisioned graphically. It is interesting to note that elastic and permeability parameters are able to control the skin friction along the main ﬂow direction, magnetic ﬁeld to reduce the pressure, and Reynolds number to control the thermal boundary layer thickness. It is also noted that temperature distribution does not depend upon permeability parameter.


Introduction
e study of porous medium in context of free convective flow frequently has fascinated the researches over the last decades.
is area has appealing quality due to its wide spread applications in the field of science, technology, and engineering. For example, the processes of purification and filtration in the arena of chemical engineering, the study of seeping water in the river basin, underground water resources in agriculture engineering, and evaporative cooling air conditioners in context of technology are few practical models of porosity in daily routine. Raptis [1] elaborated a free convective time free Newtonian fluid flow past porous medium, and Raptis and Perdikis [2] investigated oscillatory free convective flow of a Newtonian fluid past porous medium.
In the above cited work, both permeability and suction of the porous medium have been supposed to be constant or transient. Since a porous medium, in general, is not a homogeneous channel, there can exist several inhomogeneities in such mediums. us, it may not be necessary to consider the permeability or suction of the porous medium as constant. Several efforts [3][4][5][6] by Singh et al. have been done in this regard on the motion of Newtonian fluids in three dimensions with periodic variation of permeability or suction velocity passing through an extremely high porous medium. Further, Vafai and Hadim [7] took an overview of the studies of heat transfer in porous beds with natural convection and mixed convection applications. Jain et al. [8] delivered the impact of free convective temperature and sinusoidal permeability of 3-dimensional Newtonian fluid flow past a porous medium with the existence of slip on flow parameters.
Moreover, in the above studies the fluid flows were supposed electrically nonconducting. However, magnetic fields influence many natural and man-made flows. ey are routinely used in industries to heat, pump, stir, and levitate liquid metals. ere is the terrestrial magnetic field which is maintained by fluid motion in the earth's core, the solar magnetic field which generates sunspots and solar flares, and the galactic field which influences the formation of stars. e flow problems of an electrically conducting fluid under the influence of magnetic field have attracted the interest of many authors in view of their applications to geophysics, astrophysics, and engineering, and to the boundary layer control in the field of aerodynamics. Ahmed [9] put forward getting the effects of electrically conducted fluid for mixed convective flows with periodic suction velocity and magnetic field through porous vertical plate. Reddy et al. [10] observed that the velocity of 3-dimensional fluid past a porous medium with sinusoidal permeability reduces due to magnetohydrodynamic flow, and the parameter of heat absorption causes enhancing the heat transfer coefficient. Various workers [11][12][13][14] analyzed viscoelastic fluids in various geometries under distinct physical states. Further, researchers [15][16][17] investigated electrically conducting viscoelastic fluids over a stretching sheet/in highly porous mediums with the MHD effects and presented very interesting results.
In many practical applications, a situation may arise when slip of particles at the boundary may occur. For example, the surfaces of air-craft and rockets move at a very high altitude, where particles adjacent to the surface possess a finite tangential velocity which slips along the surface. Seth et al. [18][19][20] studied various non-Newtonian models with slip/hydromagnetic mechanism in free convective flow past a nonlinear stretching surface/through a porous medium. e workers [21][22][23][24][25][26][27] reported important results for free convective flows in three dimensions with periodic permeability of non-Newtonian fluids. Arpino with his coauthors [28,29] analyzed transient thermal natural convection in porous and partially porous channels. Khanafer and Vafai [30] recently investigated porous medium with the applications of nanofluids.
In the present study, a second-grade free convective fluid flow in three dimensions through a very highly porous medium with periodic permeability in the presence of magnetic field is explored. To the author's knowledge, such a study for the second-grade fluid model has not been addressed.
is constitutes the novelty of the present analysis. e nondimensional highly nonlinear partial differential equations subject to appropriate boundary conditions are solved analytically using regular perturbation technique. A detailed parametric study of the Hartmann number, permeability parameter, Grashof number, Prandtl number, Reynolds number, and non-Newtonian parameter on velocity components, skin friction components, and the coefficient of heat transfer is visualized graphically. Elaborate interpretation of the physics of the flow is also conducted. In view of the above considerations, the setup of the article is as given below.
Section 2 of the article narrates the description and modeling of the problem, while Section 3 gives solutions of the model in different dimensions for main flow, secondary flow, and energy equation by the regular perturbation technique, friction coefficients along z-direction and x-direction, and the coefficient of heat transfer rate is also demonstrated at the end of this section. Results and discussions are interpreted in Section 4 and conclusions are described in Section 5.

Description and Modeling of Problem
e present study is the investigation of a three-dimensional second-grade fluid through an extremely high porous medium circumscribed by an infinite vertical porous plate placed on the xz-plane with x-axis pointing upward along the plate and y-axis pointing along the normal to the plane of the plate ( Figure 1) and is the porous medium periodic permeability, where K 0 presents the medium mean permeability, ε( ≪ 1) is the permeability variation's amplitude, and l is the length of wave for the permeability distribution. e sinusoidal variation in permeability (1) causes the flow to be 3-dimensional. e following assumptions are taken into account: (i) e fluid is incompressible and the fluid flow is laminar (ii) All fluid's properties are considered to be constant; however, fluid density variation effect with temperature is contemplated in the term of body force (iii) A uniform magnetic field B → 0 is applied along the y-axis (iv) Magnetic Reynolds number is assumed to be very small so that the induced magnetic field is negligible [31,32] (v) e electric field is assumed to be zero (vi) Suction velocity U and free stream velocity V 0 are constant e form of velocity field here is taken: with velocity components u, v, w, respectively, in the x-, y-, z-directions. e physical quantities will not be x dependent as the plate length is infinite in x-direction and definitely the flow remains in 3 dimensions because of the variation of sinusoidal permeability. Now, consider the equations of motion [33,34] governing the given fluid flow.
where τ, a Cauchy stress tensor for the second-grade fluid [35], is defined below in (4), V → is velocity field defined in (2), ∇ is the vector operator, fluid density is ρ, b → is the generated body force per unit mass, B → is the total magnetic field, J → is the electric current density, μ is the dynamic viscosity, ϱ � c p T, q → � − k t ∇T and T is the temperature, k t is thermal conductivity, and c p is specific heat at constant pressure.

Mathematical Problems in Engineering
where A 1 and A 2 are Rivlin-Ericksen tensors, defined in equation (5), p is pressure, I is the identity tensor, and α 1 and α 2 are material constants.
Now, for model (4) to be compatible with the thermodynamics in the sense that all motions meet the Clausius-Duhem inequality and with the supposition that the specific Helmholtz free energy is a minimum in equilibrium [35], then the following conditions must satisfy the material parameters: In the absence of displacement currents, Maxwell's equations modified Ohm's law [31,32] can be written as where μ m is the magnetic permeability, σ is the electrical conductivity of the fluid, and E → is the electric field. Using the usual Boussinesq assumption for the body force [33], we have where g denotes the gravity, β 0 the coefficient of thermal expansion, T ∞ reference temperature, and ρ 0 a constant density which is going to symbolize as ρ throughout the article for convenience. e problem defined in equation (3) can be restraint in the following mathematical model with the support of equations (1)- (8). zv zy ρ v zu zy zv zz Mathematical Problems in Engineering 5 and the associated boundary conditions are where Reynolds number, Grashof number, elastic parameter, Prandtl number, magnetic parameter, and suction parameter, respectively, are given below:

Analysis of the Mathematical Model
In this section, we discuss the solutions of equations (16)- (20) in two and three dimensions so we assume the following type of a solution in the neighbourhood of the channel: where g takes the position for all of θ, p, u, v, and w and ε is a very small parameter.

Two-Dimensional Solution.
For ε � 0, the problem becomes two-dimensional and consequently, we have LRe subject to boundary conditions Clearly, due to the presence of elasticity parameter in equations (25)- (27), the order of these differential equations has increased from 2 to 3. For unique solution of equations (25)- (27), three boundary conditions are required here. To take off this difficulty, consider the solution of the following type: taking the parameter L very small. Solving the equations (24) and (26)- (28), we get the following solutions: Solving equation (25) with the help of equation (30) and comparing coefficients of order of L 0 and L, we obtain the following boundary value problems: Re Solving the boundary value problems (32) and (33), then the zeroth-order solution yields 6 Mathematical Problems in Engineering Results of [6,21] are retrieved for M � 0 and for both M � 0 � L, respectively.

ree-Dimensional Solution.
For ε ≠ 0, the flow turns into three-dimensional and solution of the type described in equation (23) can be taken as the assumed solution of the obtaining expression. en by comparing first-order terms of ε, we acquired the following partial differential equations from equations (16)-(20): Similarly, the boundary conditions (21) yield e set of coupled PDEs (partial differential equations) subject to the boundary conditions from (35)-(40) describes the 3-dimensional free convective fluid flow.

Solution of Cross Flow.
To get the solutions of PDEs (35)-(39), we explore the equations (35), (37), and (38) firstly as these three equations are independent of temperature field and main flow.
Let us suppose the solutions for p 1 , v 1 , and w 1 as where v 11 ′ (y) is the derivative. All the equations in (41) satisfy the continuity equation (35). Putting values from equations (41) into equations (37) and (38), we have with the boundary conditions Simultaneously solving both equations (42) and (43) and eliminating p 11 , the pressure term, we obtain the following differential equation: and equation (45) can be solved by the perturbation technique, assuming the following solution for equation (45) with a very small parameter L: Putting equations (46) in equations (45) and (44), comparing like powers of L, and then solving the resulting boundary value problems, we obtain where d 1 and d 2 are the real roots of equation (45) having long expressions that are not shown here for the sake of brevity. In view of (23), (31), and (41), finally we get

Temperature Field and Pressure.
e value of pressure can be obtained by using equations (23), (31), (41), (43), and (48), which is given by Now, assume the following solution for getting the result of temperature distribution: en, PDE (39) with the boundary conditions 3.5. Solution of Main Flow. Now, finally the solution of main flow can be acquired from the partial differential equation (36). Similar to the previous solutions, we assume as the solution for (36), and for perturbation on small parameter L, we take Subsequently computing, we have the result and equations (56) and (57)  Mathematical Problems in Engineering B 3 � Re 2 A 2 Pr Re 2 Pr 2 − π 2 + Gλ 2 0 /K 0 Pr 3 Re 2 Pr(Pr − 1) − π 2 + 1/K 0 + MRe ,

Skin Friction Coefficients.
e important physical quantity, skin friction components, can be achieved after obtaining the velocity field. In x-direction, the nondimensional skin friction component is given by We obtain the following result after omitting the symbol of "⊳" to make it easy: where Similarly, in the z-direction the nondimensional skin friction component is where (63)

Heat Flux.
After getting the temperature distribution, we may get e Nusselt number, Nu, from the temperature field.
Getting nondimensional form and after simplifying the resulting equation, we have where Nu 0 � − 1,

Results and Discussion
is effort reveals the mathematical modeling and theoretical analysis of a steady flow of second-grade fluid in three dimensions through a medium (porous) with periodic permeability and heat transfer with the existence of magnetic field applied normal to plate. With the help of regular perturbation method, analytical solutions for velocity field, pressure, heat flux, and skin friction are attained. e consequences of nondimensional parameters such as Prandtl number, elastic parameter, Reynolds number, Grashof number, magnetic parameter, and permeability parameter, Pr, L, Re, G, M, and K 0 , respectively, on the obtained physical quantities are envisioned graphically.
In this regard, Figure 2 depicts the result of permeability parameter K 0 on temperature distribution, velocity components, and pressure when all other nondimensional physical parameters are static (L � 0.01, Pr � 7, Re � 1, G � 1, M � 1, ε � 0.01, z � 0) except K 0 , the permeability parameter. Figure 2(a) illustrates the influence of K 0 on the temperature distribution. Here, we acquired that temperature distribution weakly depends on permeability parameter K 0 . On the other hand, Figure 2(b) depicts that, near the plate, increase in permeability parameter K 0 causes increasing the pressure and at the free surface it reaches its maximum value. It is viewed from Figure 2 Figure 3(e) that velocity component w near the plate increases exponentially for a fixed value of K 0 and attains its maximum height by obtaining parabolic profile here, then decreases sharply, and alternately approaches to zero as y ⟶ ∞. It can also be observed easily that w decreases with the increase of permeability K 0 . Next, Figure 3 depicts impact of Re on θ, temperature distribution, pressure, and the components of velocity field. e impact of Re on the temperature distribution is demonstrated in Mathematical Problems in Engineering Figure 3(a). It is noted that as we increase Reynolds number Re, the thermal boundary layer starts to decline. Figure 3(b) shows that pressure increases near the plate due to the increment of Re. Physically, it can be said that inertial forces are dominant near the plate over viscous forces. At the free surface, pressure has its maximum value. It is analyzed (Figure 3(c)) that the increment in Re causes increasing u, the main flow velocity component, and also for each value of Re the main flow velocity component u reaches its maximum value at the boundary level. Moreover, the thickness of boundary layer decreases as Reynolds number  increases, and the increase of Re also causes decreasing the magnitude of velocity component v (Figure 3(d)) which is naturally true with the existence of magnetic field B → 0 . It is detected from Figure 3(e) that velocity component w near the plate increases exponentially for a fixed value of Re and attains its maximum height by making parabolic profile here and then speedily decreases, and at last w ⟶ 0 as y ⟶ ∞. It is clearly noted that w increases due to an increment of Re. Figure 4 exhibits the impact of M on temperature distribution, pressure, and velocity components when all nondimensional parameters are fixed except magnetic parameter. Figure 4(a) demonstrates the impact of M on increases exponentially for a fixed value of magnetic parameter M and w approaches its maximum value by making parabolic profile here; afterwards, it decreases quickly and w ⟶ 0 for y ⟶ ∞. Clearly, it can be checked that w starts to decrease by increasing the values of M. e impact of Prandtl number on temperature field and main flow velocity component u is elaborated in Figure 5. It is obvious in Figure 5(a) that the fluid temperature reduces by the enhancement of Pr, Prandtl number, and this behavior of Pr reduces the thickness of thermal boundary layer. Actually, when we increase the Prandtl number, there exists a low thermal conductivity in fluid which causes reducing thermal layer thickness. us, the graphical observation of problem from this figure absolutely agrees with the physical principle that the increase of Pr causes decrease in boundary layer thickness. Figure 5(b) shows that with an increment of Prandtl number the velocity component u starts to decrease. Figure 6 reveals the impact of elastic parameter L on temperature field, pressure, and velocity components when all other nondimensional parameters are fixed except elastic parameter. Here, it is viewed from Figure 6(a) that the influence of L on the temperature distribution is the same as K 0 . Here, again we acquired that temperature distribution weakly depends on elastic parameter L. It is also exhibited (Figure 6(b)) here clearly that the pressure starts to increase by an increase in L, which is evident of above theory, because of the fluid thickness. Figure 6(c) shows that main flow velocity component u starts to decrease with the increase of non-Newtonian parameter L which is quite obvious physically as increase in non-Newtonian parameter L causes greater thickening of fluid that produces reduction in the velocity. e impact of L from Figure 6(d) on v is observed that minimum value of the velocity component occurs at the lower boundary and the maximum value of the velocity component takes place at upper boundary. It is viewed from Figure 6(e) that near the plate velocity component w increases exponentially for a fixed value of L and it decreases quickly after making parabolic profile; then w ⟶ 0 for y ⟶ ∞. It can be seen that w starts to increase by increasing the values of L.
As for the Figure 7, it perceives the impact of Grashof number that is free convective parameter on u. is effect shows the cooling of the plate that happens due to the greater Grashof number. e figure elaborates that the velocity component of main flow u starts to increase with an increase of Grashof number G which leads to the high cooling of the medium.
Further, in Figure 8, the dimensionless skin friction component along the x− axis is depicted for distinct values of L, M K 0 , G, and Pr through the direction of main flow and against Reynolds number. It is observed in all cases (Figures 8(a), 8(c), 8(d), and 8(e)) that with an increase in Reynolds number Re the skin friction is imposed by the plate on the fluid, which increases with the increase of each of these dimensionless parameters. It is worth mentioning that the skin friction is zero for L � 0.1. However, permeability parameter K 0 has an inverse effect shown in Figure 8(b). Figure 9 is framed for nondimensional component of skin friction along the secondary flow direction for distinct values of L, M, and K 0 against Reynolds number. In both cases (Figures 9(a) and 9(b)), it is perceived that the increase in Re originates the increase in component of skin friction. With the increase in either of the parameters L and M, the  Mathematical Problems in Engineering skin friction increases; however, permeability parameter K 0 has an inverse effect (Figure 9(c)).
In Figure 10, variation of dimensionless coefficient of heat transfer for the different values of L, Pr, M, and K 0 is demonstrated against Reynolds number. It is evident that coefficient of heat transfer is enhanced with an increase in either of the parameter L, Pr, and M. In contrast, the heat transfer coefficient decreases by increasing K 0 .

Final Remarks
e main outcomes observed in the article are described in the following for the focus of reader: (i) Magnetic field reduces the velocity field.
(ii) High cooling of the medium causes an increase in main flow velocity component.
(iii) e temperature distribution weakly depends upon the Hartmann number.
(iv) e fluid pressure rises due to increase in values of non-Newtonian parameter. (v) e skin friction components increase as the elastic parameter increases. It is also interesting to note that the skin friction component along the main flow direction reduces to zero for L � 0.1. (vi) Reynolds number causes controlling boundary layer thickness. It also provides a tool to control thermal boundary layer. (vii) Permeability plays a role of root to minimize the components of skin friction.
(viii) e plate friction decreases due to increase in permeability parameter. (ix) It is observed that permeability parameter weakly depends on temperature field. (x) It is a noteworthy evaluation that the solutions of [21] are retrieved in the absence of M as well as results of [6] recaptured while eliminating both M and L. Constants involved in solution c p :

Data Availability
All the supporting data for this research to obtain the findings are included within this article.

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