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

Authors: Atifa Latif; Muhammad Afzal Rana; Babar Ahmad; Muhammad Hussan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

The study of porous medium in context of free convective flow frequently has fascinated the researches over the last decades. This 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. Thus, it may not be necessary to consider the permeability or suction of the porous medium as constant. Several efforts [3–6] 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. They are routinely used in industries to heat, pump, stir, and levitate liquid metals. There 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. The 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 [11–14] analyzed viscoelastic fluids in various geometries under distinct physical states. Further, researchers [15–17] 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. [18–20] studied various non-Newtonian models with slip/hydromagnetic mechanism in free convective flow past a nonlinear stretching surface/through a porous medium. The workers [21–27] reported important results for free convective flows in three dimensions with periodic permeability of non-Newtonian fluids. Arpino with his co-authors [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. This constitutes the novelty of the present analysis. The 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.Section2 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.


## 2. Description and Modeling of Problem

The 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 thexz-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(1)Kz=K01+εcosπz/l,is the porous medium periodic permeability, where K0 presents the medium mean permeability, ε≪1 is the permeability variation’s amplitude, and l is the length of wave for the permeability distribution. The sinusoidal variation in permeability (1) causes the flow to be 3-dimensional. The following assumptions are taken into account:(i)
The 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 fieldB⟶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)
The electric field is assumed to be zero(vi)
Suction velocityU and free stream velocity V0 are constantFigure 1
Physical model of the problem.The form of velocity field here is taken:(2)V⟶=uy,zi+vy,zj+wy,zk,with velocity components u, v, w, respectively, in the x-, y-, z-directions. The 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.(3)∇⋅V⟶=0,ρdV⟶dt=∇⋅τ˜+ρb⟶+J⟶×B⟶−μV⟶K,ρdϱdt=−∇⋅q⟶,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=cpT,q⟶=−kt∇T and T is the temperature, kt is thermal conductivity, and cp is specific heat at constant pressure.(4)τ˜=−pI˜+μA˜1+α1A˜2+α2A˜12,where A˜1 and A˜2 are Rivlin–Ericksen tensors, defined in equation (5), p is pressure, I˜ is the identity tensor, and α1 and α2 are material constants.(5)A˜1=gradV⟶T+gradV⟶,A˜2=dA˜1dt+gradV⟶TA⟶1+A⟶1gradV⟶.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:(6)α1+α2=0,α1≥0,μ≥0.In the absence of displacement currents, Maxwell’s equations modified Ohm’s law [31, 32] can be written as(7)∇⋅B⟶=0,∇×B⟶=μmJ⟶,∇×E⟶=−∂B⟶∂t,J⟶=σE⟶+V⟶×B⟶,where μm 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(8)ρb⟶=−ρ01−β0T−T∞g,where g denotes the gravity, β0 the coefficient of thermal expansion, T∞ reference temperature, and ρ0 a constant density which is going to symbolize as ρ throughout the article for convenience. The problem defined in equation (3) can be restraint in the following mathematical model with the support of equations (1)–(8).(9)∂v∂y+∂w∂z=0,(10)ρv∂u∂y+w∂u∂z=gρβ0T−T∞+μ∂2u∂y2+∂2u∂z2−σB02u+α1v∂3u∂y3+w∂3u∂y2∂z+w∂3u∂z3+v∂3u∂y∂z2−μKu−U,(11)ρv∂v∂y+w∂v∂z=−∂p∂y+μ∂2v∂y2+∂2v∂z2+α1v∂3v∂y3+w∂3v∂y2∂z+v∂3v∂y∂z2+w∂3v∂z3+∂v∂z∂2v∂y∂z+∂u∂z∂2u∂y∂z+5∂v∂y∂2v∂y2+∂v∂z∂2w∂y2+2∂u∂y∂2u∂y2+2∂w∂y∂2w∂y2+∂u∂y∂2u∂z2+∂v∂y∂2v∂z2−μKv,(12)ρv∂w∂y+w∂w∂z=−∂p∂z+μ∂2w∂y2+∂2w∂z2−σB02w+α1w∂3w∂y2∂z+v∂3w∂y3+v∂3w∂y∂z2+w∂3w∂z3+∂w∂y∂2w∂y∂z+∂u∂y∂2u∂y∂z+5∂w∂z∂2w∂z2+∂w∂y∂2v∂z2+2∂u∂z∂2u∂z2+2∂v∂z∂2v∂z2+∂u∂z∂2u∂y2+∂w∂z∂2w∂y2−μKw,(13)ρv∂T∂y+w∂T∂z=ktcp∂2T∂y2+∂2T∂z2,with the boundary conditions [6].(14)Wheny=0,v=−V0,T=Tw,u=0,w=0,Aty⟶∞,p=p∞,T=T∞,u=U,w=0,where V0 is a positive constant suction velocity and -ve sign arises because of the suction towards plate, and the constant pressure in free stream is denoted as p∞ while plate temperature and the fluid temperature (far away from the plate) both are, respectively, Tw and T∞. Now, the variables are assigned the following dimensionless values: (15)Uu⊳=u,ly⊳=y,lz⊳=z,V0v⊳=v,V0w⊳=w,ρV02p⊳=p,l2K0⊳=K0,θ=T−T∞Tw−T∞,where the nondimensional variables are u⊳,v⊳,w⊳,y⊳,z⊳,p⊳, and K0⊳. Then, after the omission of symbol “⊳”,equations (9)-(14) have the following form for ease:(16)∂v∂y+∂w∂z=0,(17)v∂u∂y+w∂u∂z=GReθ−1ReK01+εcosπzu−1+1Re∂2u∂y2+∂2u∂z2−Mu+Lv∂3u∂y3+w∂3u∂y2∂z+w∂3u∂z3+v∂3u∂y∂z2,(18)v∂v∂y+w∂v∂z=−∂p∂y−1ReK01+εcosπzv+1Re∂2v∂y2+∂2v∂z2+Lv∂3v∂y3+w∂3v∂y2∂z+v∂3v∂y∂z2+w∂3v∂z3+∂v∂z∂2v∂y∂z+α2∂u∂z∂2u∂y∂z+5∂v∂y∂2v∂y2+∂v∂z∂2w∂y2+2α2∂u∂y∂2u∂y2+2∂w∂y∂2w∂y2+α2∂u∂y∂2u∂z2+∂v∂y∂2v∂z2,(19)v∂w∂y+w∂w∂z=−∂p∂z−1ReK01+εcosπzw+1Re∂2w∂y2+∂2w∂z2−Mw+Lw∂3w∂y2∂z+v∂3w∂y3+v∂3w∂y∂z2+w∂3w∂z3+∂w∂y∂2w∂y∂z+α2∂u∂y∂2u∂y∂z+5∂w∂z∂2w∂z2+∂w∂y∂2v∂z2+2α2∂u∂z∂2u∂z2+2∂v∂z∂2v∂z2+α2∂u∂z∂2u∂y2+∂w∂z∂2w∂y2,(20)v∂θ∂y+w∂θ∂z=1RePr∂2θ∂y2+∂2θ∂z2,and the associated boundary conditions are(21)Asy⟶∞:θ=0,u=1,w=0,p=p∞,Fory=0:θ=1,u=0,w=0,v=−1,where Reynolds number, Grashof number, elastic parameter, Prandtl number, magnetic parameter, and suction parameter, respectively, are given below: (22)Re=V0lν,G=νgβ0Tw−T∞UV02,L=α1ρl2,Pr=μcpkt,M=σB02lρV0,α=UV0.


## 3. 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:(23)g=g0+εg1+ε2g2+⋯,where g takes the position for all of θ, p,u,v, and w and ε is a very small parameter.


### 3.1. Two-Dimensional Solution

Forε=0, the problem becomes two-dimensional and consequently, we have(24)dv0dy=0,(25)Ld3u0dy3−1Red2u0dy2−du0dy+1ReK0+Mu0=1ReK0+GReθ0,(26)2Lα2du0dyd2u0dy2+2Ldw0dyd2w0dy2+1ReK0=dp0dy,(27)LRed3w0dy3+d2w0dy2+Redw0dy−1K0+MRew0=0,(28)d2θ0dy2+RePrdθ0dy=0,subject to boundary conditions(29)θ0=1,u0=0,v0=−1,w0=0,aty=0,θ0=0,u0=1,w0=0,p0=p∞asy⟶∞.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:(30)u00+Lu01+OL2=u0,taking the parameter L very small. Solving the equations (24) and (26)–(28), we get the following solutions:(31)v0=−1,w0=0,p0=p∞,θ0=e−RePry.Solving equation (25) with the help of equation (30) and comparing coefficients of order of L0 and L, we obtain the following boundary value problems: (32)d2u00dy2+Redu00dy−1K0+MReu00=−1K0−GRe2e−RePry,u000=0,u00∞=1,(33)Red3u00dy3−d2u01dy2−Redu01dy+1K0+MReu01=0,u010=0,u01∞=0.Solving the boundary value problems (32) and (33), then the zeroth-order solution yields(34)u0y=Gλ0−11+K0MRee−λy+11+K0MRe−Gλ0e−RePry+LGλ02Re2Pr3e−RePry−e−λy.Results of [6, 21] are retrieved for M=0 and for both M=0=L, respectively.


### 3.2. Three-Dimensional Solution

Forε≠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. Then by comparing first-order terms of ε, we acquired the following partial differential equations from equations (16)–(20):(35)∂v1∂y+∂w1∂z=0,(36)−∂u1∂y+v1∂u0∂y=GReθ1−1ReK0+Mu1+1Re∂2u1∂y2+∂2u1∂z2+L−∂3u1∂y3−∂3u1∂y∂z2+v1∂3u0∂y3−1ReK0u0−1cosπz,(37)−∂v1∂y=−∂p1∂y+1Re∂2v1∂y2+∂2v1∂z2−L∂3v1∂y3+∂3v1∂y∂z2−1ReK0v1−cosπz,(38)−∂w1∂y=−∂p1∂z+1Re∂2w1∂y2+∂2w1∂z2−L∂3w1∂y3+∂3w1∂y∂z2−1ReK0+Mw1,(39)−∂θ1∂y+v1∂θ0∂y=1RePr∂2θ1∂y2+∂2θ1∂z2.Similarly, the boundary conditions (21) yield (40)Aty=0,u1=0=v1=w1=θ1,Asy⟶∞,u1=0=w1=θ1,p1=0.The set of coupled PDEs (partial differential equations) subject to the boundary conditions from (35)–(40) describes the 3-dimensional free convective fluid flow.


### 3.3. 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 forp1, v1, and w1 as(41)p1y,z=p11ycosπz,v1y,z=−v11ycosπz,w1y,z=1πv11′ysinπz,where v11′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(42)LRev11″′−v11″−Lπ2+1Rev11′+π2+1K0v11=Rep11′−1K0,(43)LRev11iv−v11″′−Lπ2+1Rev11″+π2+1K0+MRev11′=π2Rep11,with the boundary conditions(44)v11′∞=0,v110=0,v11′0=0.Simultaneously solving both equations (42) and (43) and eliminating p11, the pressure term, we obtain the following differential equation:(45)LRev11v−v11iv−2Lπ2+1Rev11″′+2π2+1K0+MRev11″+π2Lπ2+1Rev11′−π2π2+1K0v11=π2K0,and equation (45) can be solved by the perturbation technique, assuming the following solution for equation (45) with a very small parameter L:(46)v11=v00+Lv01+OL2.Putting equations (46) in equations (45) and (44), comparing like powers of L, and then solving the resulting boundary value problems, we obtain(47)v11=−d2ed1y+d1ed2y−d1+d2d1−d2π2K0+11+L,where d1 and d2 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(48)vy,z=−1+ε−d2ed1y+d1ed2y−d1+d2d1−d2π2K0+11+Lcosπz,(49)wy,z=ε1+Ld1d2πd1−d2π2K0+1ed2y−ed1ysinπz.


### 3.4. Temperature Field and Pressure

The value of pressure can be obtained by using equations (23), (31), (41), (43), and (48), which is given by (50)py,z=p∞+ε1+Ld1d2cosπzπ2Red1−d2π2K0+1×−LRed13+d12+Lπ2+1Red1−π2−1K0−MReed1y+LRed23−d22−Lπ2+1Red2+π2+1K0+MReed2y.Now, assume the following solution for getting the result of temperature distribution:(51)θ1y,z=θ11ycosπz.Then, PDE (39) with the boundary conditions(52)θ11∞=0,θ110=0,yields(53)θy,z=e−RePry+εRe2Pr21+Lcosπzd1−d2π2K0+1A−B+d1−d2π2e−βy+Aed1−RePry+Bed2−RePry+d1−d2π2e−RePry,where A=−d2/d12−d1RePr−π2, B=d1/d22−d2RePr−π2, λ0=Re2/Re2PrPr−1−1/K0+MRe, λ=Re/2+Re/22+1/K0+MRe, λ1=Re/2+Re/22+π2+1/K0+MRe, β=RePr/2+RePr/22+π2.


### 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(54)u1y,z=u11ycosπz,as the solution for (36), and for perturbation on small parameter L, we take(55)u11=u100+Lu110+OL2.Subsequently computing, we have the result(56)u100y=A1+A2+A11e−λ1y−A1e−λy−A2e−RePry−A11−Red1−d2π2K0+1A3e−βy+A4ed1−RePry+A5ed2−RePry+A6e−RePry+A7ed1−λy−A8ed2−λy−A9e−λy−A10e−λ1y,(57)u110y=B3−B2e−λ1y−B2e−λy+B3e−RePry+Red1−d2π2K0+1B4e−βy+B5ed1−RePry+B6ed2−RePry+B7e−RePry+B8ed1−λy+B9ed2−λy−B10e−λy−B11e−λ1y,and equations (56) and (57) in view of equations (55), (54), and (23) attain the final solution for main flow velocity. It is good to reveal that the results of both [6, 21] are successfully retrieved for L=0=M and M=0, respectively. The constants of integration that involved the solutions of (56) and (57) are as follows: (58)A1=Gλ0−1K0π2,A2=Gλ0K0Re2PrPr−1−π2+1/K0+MRe,A3=GRe3Pr2A+B+d1−d2/π2β2−Reβ−π2+1/K0+MRe,A4=GRePrR2ePrA−λ0d2d1−RePr2+Red1−RePr−π2+1/K0+MRe,A5=GRePrRe2PrB−λ0d1d2−RePr2+Red2−RePr−π2+1/K0+MRe,A6=GRePrd1−d2Re2Pr/π2−λ0Re2PrPr−1−π2+1/K0+MRe,A7=λd2Gλ0−1/1+K0MRed1−λ2+Red1−λ−π2+1/K0+MRe,A8=λd1Gλ0−1/1+K0MRed2−λ2+Red2−λ−π2+1/K0+MRe,A9=λd1−d2Gλ0−1/1+K0MReπ2,A10=A7+A6+A5+A4+A3−A8−A9,A11=−MRe1+MReK0π2+1/K0+MRe,B2=A1Reλλ2−π2−Gλ02/K0Re2Pr3π2,B3=Re2A2PrRe2Pr2−π2+Gλ02/K0Pr3Re2PrPr−1−π2+1/K0+MRe,B4=ReA3β−β2+π2+GRe3Pr2A+B+d1−d2/π2β2−Reβ−π2+1/K0+MRe,B5=A4Red1−RePrd1−RePr2−π2+AGRe3Pr2+Gλ0d2RePr−1+Re2Pr21+λ0Prd1−RePr2+Red1−RePr−π2+1/K0+MRe,B6=A5Red2−RePrd2−RePr2−π2+BGRe3Pr2+Gλ0d1RePr1−Re2Pr21+λ0Prd2−RePr2+Red2−RePr−π2+1/K0+MRe,B7=A6Re2Prπ2−Re2Pr2+GRe3Pr2d1−d2/π2+Gλ0d1−d2RePr−1+Re2Pr21+λ0PrRe2PrPr−1−π2+1/K0+MRe,B8=A7Red1−λd1−λ2−π2+λd2Gλ0−1/1+K0MRe1−λ2−Gλ02Re2Pr3d1−λ2+Red1−λ−π2+1/K0+MRe,B9=A8Red2−λ−d2−λ2+π2+λd1Gλ0−1/1+K0MReλ2−1+Gλ02Re2Pr3d2−λ2+Red2−λ−π2+1/K0+MRe,B10=A9Reλλ2−π2+λd1−d2Gλ0−1/1+K0MRe−λ2+1−Gλ02Re2Pr3π2,B11=B9+B8+B7+B6+B5+B4−B10.


### 3.6. Skin Friction Coefficients

The important physical quantity, skin friction components, can be achieved after obtaining the velocity field. Inx-direction, the nondimensional skin friction component is given by(59)τx⊳=τyxρUV0=νV0l∂u∂yy=0.We obtain the following result after omitting the symbol of “⊳” to make it easy:(60)τx=S0+εS1Re,Pr,G,K0,L,Mcosπz,where(61)S0=1Redu0dyy=0−Ld2u0dy2y=0,S1Re,Pr,G,K0,L,M=1Redu11dyy=0−Ld2u11dy2y=0.Similarly, in thez-direction the nondimensional skin friction component is(62)τz=τyzμV/l=∂w∂yy=0+LRev∂2w∂y2y=0=−εS2Re,L,K0,Msinπz,where(63)S2Re,L,K0,M=1+Ld1d2ππ2K0+1−1+LRed1+d2.


### 3.7. Heat Flux

After getting the temperature distribution, we may get The Nusselt number,Nu, from the temperature field.(64)Nu=−qρV0cpTw−T∞.Getting nondimensional form and after simplifying the resulting equation, we have(65)Nu=ktρV0cpl∂θ∂yy=0=1RePrdθ0dy+ε∂θ1∂yy=0=Nu0+εf3Re,Pr,K0,Mcosπz,where(66)Nu0=−1,S3Re,Pr,K0,M=Repr1+Ld1−d2π2K0+1−βA+B+d1−d2π2+Ad1−RePr+Bd2−RePr−d1−d2π2RePr.


## 3.1. Two-Dimensional Solution

Forε=0, the problem becomes two-dimensional and consequently, we have(24)dv0dy=0,(25)Ld3u0dy3−1Red2u0dy2−du0dy+1ReK0+Mu0=1ReK0+GReθ0,(26)2Lα2du0dyd2u0dy2+2Ldw0dyd2w0dy2+1ReK0=dp0dy,(27)LRed3w0dy3+d2w0dy2+Redw0dy−1K0+MRew0=0,(28)d2θ0dy2+RePrdθ0dy=0,subject to boundary conditions(29)θ0=1,u0=0,v0=−1,w0=0,aty=0,θ0=0,u0=1,w0=0,p0=p∞asy⟶∞.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:(30)u00+Lu01+OL2=u0,taking the parameter L very small. Solving the equations (24) and (26)–(28), we get the following solutions:(31)v0=−1,w0=0,p0=p∞,θ0=e−RePry.Solving equation (25) with the help of equation (30) and comparing coefficients of order of L0 and L, we obtain the following boundary value problems: (32)d2u00dy2+Redu00dy−1K0+MReu00=−1K0−GRe2e−RePry,u000=0,u00∞=1,(33)Red3u00dy3−d2u01dy2−Redu01dy+1K0+MReu01=0,u010=0,u01∞=0.Solving the boundary value problems (32) and (33), then the zeroth-order solution yields(34)u0y=Gλ0−11+K0MRee−λy+11+K0MRe−Gλ0e−RePry+LGλ02Re2Pr3e−RePry−e−λy.Results of [6, 21] are retrieved for M=0 and for both M=0=L, respectively.


## 3.2. Three-Dimensional Solution

Forε≠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. Then by comparing first-order terms of ε, we acquired the following partial differential equations from equations (16)–(20):(35)∂v1∂y+∂w1∂z=0,(36)−∂u1∂y+v1∂u0∂y=GReθ1−1ReK0+Mu1+1Re∂2u1∂y2+∂2u1∂z2+L−∂3u1∂y3−∂3u1∂y∂z2+v1∂3u0∂y3−1ReK0u0−1cosπz,(37)−∂v1∂y=−∂p1∂y+1Re∂2v1∂y2+∂2v1∂z2−L∂3v1∂y3+∂3v1∂y∂z2−1ReK0v1−cosπz,(38)−∂w1∂y=−∂p1∂z+1Re∂2w1∂y2+∂2w1∂z2−L∂3w1∂y3+∂3w1∂y∂z2−1ReK0+Mw1,(39)−∂θ1∂y+v1∂θ0∂y=1RePr∂2θ1∂y2+∂2θ1∂z2.Similarly, the boundary conditions (21) yield (40)Aty=0,u1=0=v1=w1=θ1,Asy⟶∞,u1=0=w1=θ1,p1=0.The set of coupled PDEs (partial differential equations) subject to the boundary conditions from (35)–(40) describes the 3-dimensional free convective fluid flow.


## 3.3. 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 forp1, v1, and w1 as(41)p1y,z=p11ycosπz,v1y,z=−v11ycosπz,w1y,z=1πv11′ysinπz,where v11′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(42)LRev11″′−v11″−Lπ2+1Rev11′+π2+1K0v11=Rep11′−1K0,(43)LRev11iv−v11″′−Lπ2+1Rev11″+π2+1K0+MRev11′=π2Rep11,with the boundary conditions(44)v11′∞=0,v110=0,v11′0=0.Simultaneously solving both equations (42) and (43) and eliminating p11, the pressure term, we obtain the following differential equation:(45)LRev11v−v11iv−2Lπ2+1Rev11″′+2π2+1K0+MRev11″+π2Lπ2+1Rev11′−π2π2+1K0v11=π2K0,and equation (45) can be solved by the perturbation technique, assuming the following solution for equation (45) with a very small parameter L:(46)v11=v00+Lv01+OL2.Putting equations (46) in equations (45) and (44), comparing like powers of L, and then solving the resulting boundary value problems, we obtain(47)v11=−d2ed1y+d1ed2y−d1+d2d1−d2π2K0+11+L,where d1 and d2 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(48)vy,z=−1+ε−d2ed1y+d1ed2y−d1+d2d1−d2π2K0+11+Lcosπz,(49)wy,z=ε1+Ld1d2πd1−d2π2K0+1ed2y−ed1ysinπz.


## 3.4. Temperature Field and Pressure

The value of pressure can be obtained by using equations (23), (31), (41), (43), and (48), which is given by (50)py,z=p∞+ε1+Ld1d2cosπzπ2Red1−d2π2K0+1×−LRed13+d12+Lπ2+1Red1−π2−1K0−MReed1y+LRed23−d22−Lπ2+1Red2+π2+1K0+MReed2y.Now, assume the following solution for getting the result of temperature distribution:(51)θ1y,z=θ11ycosπz.Then, PDE (39) with the boundary conditions(52)θ11∞=0,θ110=0,yields(53)θy,z=e−RePry+εRe2Pr21+Lcosπzd1−d2π2K0+1A−B+d1−d2π2e−βy+Aed1−RePry+Bed2−RePry+d1−d2π2e−RePry,where A=−d2/d12−d1RePr−π2, B=d1/d22−d2RePr−π2, λ0=Re2/Re2PrPr−1−1/K0+MRe, λ=Re/2+Re/22+1/K0+MRe, λ1=Re/2+Re/22+π2+1/K0+MRe, β=RePr/2+RePr/22+π2.


## 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(54)u1y,z=u11ycosπz,as the solution for (36), and for perturbation on small parameter L, we take(55)u11=u100+Lu110+OL2.Subsequently computing, we have the result(56)u100y=A1+A2+A11e−λ1y−A1e−λy−A2e−RePry−A11−Red1−d2π2K0+1A3e−βy+A4ed1−RePry+A5ed2−RePry+A6e−RePry+A7ed1−λy−A8ed2−λy−A9e−λy−A10e−λ1y,(57)u110y=B3−B2e−λ1y−B2e−λy+B3e−RePry+Red1−d2π2K0+1B4e−βy+B5ed1−RePry+B6ed2−RePry+B7e−RePry+B8ed1−λy+B9ed2−λy−B10e−λy−B11e−λ1y,and equations (56) and (57) in view of equations (55), (54), and (23) attain the final solution for main flow velocity. It is good to reveal that the results of both [6, 21] are successfully retrieved for L=0=M and M=0, respectively. The constants of integration that involved the solutions of (56) and (57) are as follows: (58)A1=Gλ0−1K0π2,A2=Gλ0K0Re2PrPr−1−π2+1/K0+MRe,A3=GRe3Pr2A+B+d1−d2/π2β2−Reβ−π2+1/K0+MRe,A4=GRePrR2ePrA−λ0d2d1−RePr2+Red1−RePr−π2+1/K0+MRe,A5=GRePrRe2PrB−λ0d1d2−RePr2+Red2−RePr−π2+1/K0+MRe,A6=GRePrd1−d2Re2Pr/π2−λ0Re2PrPr−1−π2+1/K0+MRe,A7=λd2Gλ0−1/1+K0MRed1−λ2+Red1−λ−π2+1/K0+MRe,A8=λd1Gλ0−1/1+K0MRed2−λ2+Red2−λ−π2+1/K0+MRe,A9=λd1−d2Gλ0−1/1+K0MReπ2,A10=A7+A6+A5+A4+A3−A8−A9,A11=−MRe1+MReK0π2+1/K0+MRe,B2=A1Reλλ2−π2−Gλ02/K0Re2Pr3π2,B3=Re2A2PrRe2Pr2−π2+Gλ02/K0Pr3Re2PrPr−1−π2+1/K0+MRe,B4=ReA3β−β2+π2+GRe3Pr2A+B+d1−d2/π2β2−Reβ−π2+1/K0+MRe,B5=A4Red1−RePrd1−RePr2−π2+AGRe3Pr2+Gλ0d2RePr−1+Re2Pr21+λ0Prd1−RePr2+Red1−RePr−π2+1/K0+MRe,B6=A5Red2−RePrd2−RePr2−π2+BGRe3Pr2+Gλ0d1RePr1−Re2Pr21+λ0Prd2−RePr2+Red2−RePr−π2+1/K0+MRe,B7=A6Re2Prπ2−Re2Pr2+GRe3Pr2d1−d2/π2+Gλ0d1−d2RePr−1+Re2Pr21+λ0PrRe2PrPr−1−π2+1/K0+MRe,B8=A7Red1−λd1−λ2−π2+λd2Gλ0−1/1+K0MRe1−λ2−Gλ02Re2Pr3d1−λ2+Red1−λ−π2+1/K0+MRe,B9=A8Red2−λ−d2−λ2+π2+λd1Gλ0−1/1+K0MReλ2−1+Gλ02Re2Pr3d2−λ2+Red2−λ−π2+1/K0+MRe,B10=A9Reλλ2−π2+λd1−d2Gλ0−1/1+K0MRe−λ2+1−Gλ02Re2Pr3π2,B11=B9+B8+B7+B6+B5+B4−B10.


## 3.6. Skin Friction Coefficients

The important physical quantity, skin friction components, can be achieved after obtaining the velocity field. Inx-direction, the nondimensional skin friction component is given by(59)τx⊳=τyxρUV0=νV0l∂u∂yy=0.We obtain the following result after omitting the symbol of “⊳” to make it easy:(60)τx=S0+εS1Re,Pr,G,K0,L,Mcosπz,where(61)S0=1Redu0dyy=0−Ld2u0dy2y=0,S1Re,Pr,G,K0,L,M=1Redu11dyy=0−Ld2u11dy2y=0.Similarly, in thez-direction the nondimensional skin friction component is(62)τz=τyzμV/l=∂w∂yy=0+LRev∂2w∂y2y=0=−εS2Re,L,K0,Msinπz,where(63)S2Re,L,K0,M=1+Ld1d2ππ2K0+1−1+LRed1+d2.


## 3.7. Heat Flux

After getting the temperature distribution, we may get The Nusselt number,Nu, from the temperature field.(64)Nu=−qρV0cpTw−T∞.Getting nondimensional form and after simplifying the resulting equation, we have(65)Nu=ktρV0cpl∂θ∂yy=0=1RePrdθ0dy+ε∂θ1∂yy=0=Nu0+εf3Re,Pr,K0,Mcosπz,where(66)Nu0=−1,S3Re,Pr,K0,M=Repr1+Ld1−d2π2K0+1−βA+B+d1−d2π2+Ad1−RePr+Bd2−RePr−d1−d2π2RePr.


## 4. Results and Discussion

This 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. The 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 K0, respectively, on the obtained physical quantities are envisioned graphically.In this regard, Figure2 depicts the result of permeability parameter K0 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 K0, the permeability parameter. Figure 2(a) illustrates the influence of K0 on the temperature distribution. Here, we acquired that temperature distribution weakly depends on permeability parameter K0. On the other hand, Figure 2(b) depicts that, near the plate, increase in permeability parameter K0 causes increasing the pressure and at the free surface it reaches its maximum value. It is viewed from Figure 2(c) that the increase of permeability parameter K0 leads to decrease in the main flow velocity component u. Minimum value of the velocity component occurs at the lower boundary and the maximum value of the velocity component occurs at upper boundary. Approximately, the same consequences of permeability parameter K0 are observed in Figure 2(d) on the secondary flow velocity component v. It is detected from Figure 3(e) that velocity component w near the plate increases exponentially for a fixed value of K0 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 K0.Figure 2
Impact of permeability parameter on temperature distribution, pressure, and velocity field. (a) Temperature fieldθ. (b) Pressure p. (c) Velocity component u. (d) Velocity component v. (e) Velocity component w.
(a)(b)(c)(d)(e)Figure 3
Impact of Reynolds number on temperature distribution, pressure, and velocity field. (a) Temperature fieldθ. (b) Pressure p. (c) Velocity component u. (d) Velocity component v. (e) Velocity component w.
(a)(b)(c)(d)(e)Next, Figure3 depicts impact of Re on θ, temperature distribution, pressure, and the components of velocity field. The impact of Re on the temperature distribution is demonstrated in 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.Figure4 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 temperature distribution which illustrates that temperature distribution weakly depends on magnetic parameter M. Figure 4(b) displays that pressure decreases with an enhancement in M and reaches its maximum value at the free surface. It is investigated further by Figure 4(c) that main flow velocity component u decreases when parameter M is enhanced. Minimum value of the velocity component occurs at the lower boundary and the maximum value of the velocity component takes place at upper boundary. The same impact of magnetic parameter M on v is observed from Figure 4(d). It is viewed from Figure 4(e) that near the plate velocity component w 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.Figure 4
Impact of magnetic parameter on temperature field, pressure, and velocity field. (a) Temperature fieldθ. (b) Pressure p. (c) Velocity component u. (d) Velocity component v. (e) Velocity component w.
(a)(b)(c)(d)(e)The impact of Prandtl number on temperature field and main flow velocity componentu 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. Thus, 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 5
Impact of Prandtl number on temperature field and velocity component. (a) Temperature field. (b) Velocity componentu.
(a)(b)Figure6 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 K0. 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. The 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.Figure 6
Impact of non-Newtonian parameter on temperature field, pressure, and velocity field. (a) Temperature field. (b) Pressure. (c) Velocity componentu. (d) Velocity component v. (e) Velocity component w.
(a)(b)(c)(d)(e)As for the Figure7, it perceives the impact of Grashof number that is free convective parameter on u. This effect shows the cooling of the plate that happens due to the greater Grashof number. The 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.Figure 7
Main flow velocity with the impact of Grashof number.Further, in Figure8, the dimensionless skin friction component along the x−axis is depicted for distinct values of L, MK0, 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 K0 has an inverse effect shown in Figure 8(b).Figure 8
Impact of dimensionless parameters on skin friction component alongx-axis. (a) Effect of elastic parameter. (b) Effect of permeability parameter. (c) Effect of Grashof number. (d) Effect of Prandtl number. (e) Effect of magnetic parameter.
(a)(b)(c)(d)(e)Figure9 is framed for nondimensional component of skin friction along the secondary flow direction for distinct values of L, M, and K0 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 skin friction increases; however, permeability parameter K0 has an inverse effect (Figure 9(c)).Figure 9
Impact of dimensionless parameters on skin friction component alongz-axis. (a) Effect of elastic parameter. (b) Effect of magnetic parameter. (c) Effect of magnetic parameter.
(a)(b)(c)In Figure10, variation of dimensionless coefficient of heat transfer for the different values of L,Pr,M, and K0 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 K0.Figure 10
Impact of dimensionless parameters on Nusselt number. (a) Effect of elastic parameter. (b) Effect of permeability parameter. (c) Effect of magnetic parameter. (d) Effect of Prandtl number.
(a)(b)(c)(d)


## 5. Final Remarks

The 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)
The temperature distribution weakly depends upon the Hartmann number.(iv)
The fluid pressure rises due to increase in values of non-Newtonian parameter.(v)
The 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 forL=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)
The 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.



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*Source: 2501842-2021-02-17.xml*