# A Novel Design of Three-Dimensional MHD Flow of Second-Grade Fluid past a Porous Plate

Authors: Muhammad Shoaib; Rizwan Akhtar; Muhammad Abdul Rehman Khan; Muhammad Afzal Rana; Abdul Majeed Siddiqui; Zhu Zhiyu; Muhammad Asif Zahoor Raja
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this study, a novel theoretical model for three-dimensional (3D) laminar magnetohydrodynamic (MHD) flow of a non-Newtonian second-grade fluid along a plate of semi-infinite length is developed based on slightly sinusoidal transverse suction velocity. The suction velocity involves a steady distribution with a low superimposed perpendicularly varying dispersion. The strength of the uniform magnetic field is incorporated in the normal direction to the wall. The variable suction transforms the fluidic problem into a 3D flow problem because of variable suction velocity in the normal direction to the plane wall. The proposed mathematical modeling and its dynamical analysis are prescribed for the boundary layer flow keeping the magnetic effects without taking into consideration the induced magnetic field. An analytical perturbation technique is employed for the series solutions of the system of ordinary differential equations of velocity profile and pressure. Graphical illustrations are used to analyze the behavior of different proficient parameters of interest. The magnetic parameter is responsible for accelerating the main flow velocity, while controlling the cross flow velocities.


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## Body

## 1. Introduction

The electrically conducting fluid flows due to the effect of the transverse magnetic field have received attention of a large section of research community because of their use in engineering, astrophysics, and geophysics and in the field of aerodynamics to control the boundary layer. In industries, the induced magnetic field-based procedures are used for excursion, heat exchanger, pump, and levitate liquid metals. Gersten and Gross [1] investigated the heat transfer effects on the three-dimensional fluid flows based on main and cross flow velocity components along a porous plane wall based on the transverse sinusoidal suction velocity distribution. Singh et al. [2] analyzed the problem of the 3D porous medium based on convective flow. The authors also incorporated heat transfer effects in the fluidic problem. Furthermore, Singh [3, 4] analyzed the hydromagnetic and magnetohydrodynamic impacts on 3D free convective fluid flow. The authors considered a vertical porous wall for the impact of porosity on the flow problem. Furthermore, the effect of the magnetic field with unvarying strength is incorporated perpendicular to the free convective fluid flow which is electrically conducting through a semi-infinite plate [5]. Later on, Helmy [6] discussed the heat transfer impact on the flow of an electrically conducting fluid across an infinite plane wall with variable suction.Researchers have extensively exploited the field of magnetohydrodynamics with reference to geometrical configuration, various types of formulations, and different dimensions using analytical and numerical methods [7–19]. The authors investigated the influence of the magnetic field on different structures of flow dimensions for distributions of concentrations and temperatures. Moreover, the magnetic parameter is proved to be a controlling parameter to restrain the fluid and heat flows under the closed spaces. Different types of non-Newtonian fluids bargain applications in many magnetohydrodynamic devices and in power generation as well. Magnetohydrodynamics with different types of convective flows, heat generation/absorption, and transfer analysis [20–25] have been investigated in detail.Gupta and Johari [26] presented the analysis of 3D magnetohydrodynamic flow across a porous plane wall, and the fluid taken into account is laminar having the power of conducting heat. The authors considered a magnetic flux in the normal direction to the plate. Moreover, Guria and Jana [27] considered a vertical porous wall for the three-dimensional hydrodynamic fluid flow problem. Furthermore, Greenspan and Carrier [28], Rossow [29], and Singh [30] presented their studies extensively on the magnetohydrodynamic impacts on the flow across a plane wall based on injection or suction. There are some non-Newtonian models presenting the three-dimensional fluidic problems though a porous wall with variable suction [31–35]. Periodic suction has received very much attention in the field of aerodynamics [36–38]. The porous medium has very much importance in the fluid dynamics [39–45]. Abbas et al. [46] studied 3D peristaltic flow fluid with hyperbolic tangent in the nonuniform channel along flexible walls. Bhatti and Lu [47] studied analytical analysis of the head-on collision mechanism among hydroelastic solitary waves with uniform current. Jhorar et al. [48] analyzed the microfluid in the channel for the electroosmosis-modulated biomechanical transport.The motivation of this study is to analyze the 3D MHD flow of the simplest non-Newtonian second-grade fluidic model through a semi-infinite wall which is based on succession of waves or curves with fluctuating velocity distribution. A uniform suction velocity along the surface of the plane wall transforms the problem into a 2D asymptotic suction velocity solution [49]; however, because of the variable suction velocity distribution in the normal direction, the fluidic system turns out to be 3D. The analytical perturbation technique is incorporated for finding the series solution of this problem. The proposed outcomes are estimated for different parameters of interest such as the suction parameter α, second-grade parameter K, Reynolds number Re, and Hartmann number M. This article is organized as follows: Section 2 consists of the statement of the problem, Section 3 presents the perturbation method, Section 4 specifies the design of the problem, Sections 5 and 6 describe analytical solutions, Section 7 integrates results and discussion, and Section 8 contains conclusions. In addition to these, appendices and nomenclature are given thereafter.


## 2. Statement of the Problem

In this problem, the 3D steady, laminar magnetohydrodynamic flow of an incompressible non-Newtonian second-grade fluid passing through an infinite plate is considered. Thex∗z∗ plane is considered where y∗-axis is normal to the plane (see Figure 1). A time-independent distribution is a basic steady distribution [1], in which l and ε indicate the wavelength and amplitude of the variable suction velocity, respectively. Thus, variable suction velocity has the following form:(1)vz∗=−v01+εcosπz∗l.Figure 1
Geometrical representation of the problem.The fluidic system becomes two-dimensional because of constant suction velocity, whereas it is three-dimensional in case of variable suction velocity [49]. A constant magnetic field B0 in the normal direction to the x∗z∗ wall is applied. Also, the following are considered: (i) the fluid has electric conduction; (ii) the fluid has steady and laminar flow; (iii) the fluid has uniform free stream velocity; (iv) the magnetic Reynolds number is at small scale, and also the induced magnetic field is inconsiderable; (v) Hall and polarization effects are neglected; and (vi) all physical properties of the parameters are independent of x∗ because of the infinite extended length of the plate in the x∗ direction, but the flow remains 3D because of the variable suction velocity (1).


## 3. Perturbation Method

Perturbation methods [50–52] are strong mathematical tools to find the series/approximate solutions of those problems whose analytic or exact solutions are not possible or hard to find. These techniques have frequently been used for the problems arising in the fields of engineering and science. The function uy,z,ε is convoluted in physical problems, and then it can be shown mathematically by the differential equation(2)Lu,y,z,ε=0,subject to the boundary condition(3)Bu,ε=0,where y is a vector or scalar independent variable and ε is a parameter. One cannot solve this problem exactly, in general. However, if there exists an ε=ε0 (ε can be scaled so that ε=0) for which the above problem can be solved exactly, then one explores to obtain the solution for small ε in the form(4)uy,z,ε=u0y+εu1y,z+ε2u2y,z+⋯,where un n=0,1,2,… does not depend on ε and u0y is the solution of the problem for ε=0. One then substitutes this expansion into equations (2) and (3), which expands for small ε and collects coefficients of each power of ε. Since these equations must hold for all values of ε, each coefficient of ϵ must vanish independently because sequences of ϵ are linearly independent. These usually are simpler equations governing un n=0,1,2,… which can be solved successively.


## 4. Design of the Problem

The equations of continuity and momentum are presented in the following way:(5)∂w∗∂z∗+∂v∗∂y∗=0,(6)ρw∗∂u∗∂z∗+v∗∂u∗∂y∗=μ∂2u∗∂y∗2+∂2u∗∂z∗2+α1v∗∂3u∗∂y∗3+w∗∂3u∗∂y∗2∂z∗+v∗∂3u∗∂y∗∂z∗2+w∗∂3u∗∂z∗3−ρB02u∗,(7)ρw∗∂v∗∂z∗+v∗∂v∗∂y∗=−∂p∗∂y+μ∂2v∗∂y∗2+∂2v∗∂z∗2+α1v∗∂3v∗∂y∗3+w∗∂3v∗∂y∗2∂z∗+v∗∂3v∗∂y∗∂z∗2+w∗∂3v∗∂z∗3+∂v∗∂z∗∂2v∗∂y∗∂z∗+∂u∗∂z∗∂2u∗∂y∗∂z∗+5∂v∗∂y∗∂2v∗∂y∗2+∂v∗∂z∗∂2w∗∂y∗2+2∂u∗∂y∗∂2u∗∂y∗2+2∂w∗∂y∗∂2w∗∂y∗2+∂u∗∂y∗∂2u∗∂z∗2+∂v∗∂y∗∂2v∗∂z∗2,(8)ρv∗∂w∗∂y∗+w∗∂w∗∂z∗=−∂p∗∂z∗+μ∂2w∗∂y∗2+∂2w∗∂z∗2+α1w∗∂3v∗∂y∗3+v∗∂3w∗∂y∗3+v∗∂3w∗∂y∗∂z∗2+w∗∂3w∗∂z∗3+∂w∗∂y∗∂2w∗∂y∗∂z∗+∂u∗∂y∗∂2u∗∂y∗∂z∗+5∂w∗∂z∗∂2w∗∂z∗2+∂w∗∂y∗∂2v∗∂z∗2+2∂u∗∂z∗∂2u∗∂z∗2+2∂v∗∂z∗∂2v∗∂z∗2+∂u∗∂z∗∂2u∗∂y∗2+∂w∗∂z∗∂2w∗∂y∗2−ρB02w∗,with the boundary conditions [7](9)u∗=0,v∗=−v01+εcosπz∗l,w∗=0aty∗=0,u∗=U,v∗=−v0,w∗=0,p∗=p∞∗asy∗⟶∞.Now the following dimensionless parameters are introduced [20]:(10)x=x∗l,y=y∗l,z=z∗l,u=u∗U,v=v∗U,w=w∗U,p=p∗ρU2.Then, equations (5)–(8) become(11)∂v∂y+∂w∂z=0,(12)w∂u∂z+v∂u∂y=1Re∂2u∂y2∂2u∂z2+Kw∂3u∂y2∂z+v∂3u∂y3+v∂3u∂y∂z2+w∂3u∂z3−Mu,(13)w∂v∂z+v∂v∂y=−∂p∂y+1Re∂2v∂y2+∂2v∂z2+Kv∂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,(14)w∂w∂z+v∂w∂y=−∂p∂z+1Re∂2w∂y2+∂2w∂z2+Kw∂3v∂y3+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−Mw,and the boundary conditions (9) take the forms(15)u=0,v=vz=−α1+εcosπz,w=0aty=0,u=1,v=−α,w=0,asy⟶∞,where(16)Re=Ulν,α=v0U,M=σB02lρU,K=α1ρl2.Sinceε is a small number, solutions are assumed as follows:(17)uy,z=u0+εu1+ε2u2+⋯,vy,z=v0+εv1+ε2v2+⋯,wy,z=w0+εw1+ε2w2+⋯,py,z=p0+εp1+ε2p2+⋯.Forε=0, the problem becomes two-dimensional because of constant suction velocity given in equation (1), which is resulted as follows:(18)KαRed3u0dy3−d2u0dy2−αRedu0dy+MReu0=0,subject to boundary conditions(19)u0=0,aty=0,u0=1,asy⟶∞.Consider the following form of the solution:(20)u0=u00+Ku01+OK2,where K is a small elastic parameter. Using equation (20) in equations (18) and (19) and correlating the coefficients of K0 and K, the following boundary value problems are obtained:(21)d2u00dy2+αRedu00dy−MReu00=0,u000=0,u00∞=1,(22)d2u01dy2−αRed3u00dy3+αRedu01dy−MReu01=0,u010=0,u01∞=0.Solving equations (21) and (22), we get(23)u00y=1−e−S1y,(24)u01y=αReS13αRe−2S1ye−S1y.Therefore, in the light of equations (23) and (24), equation (20) gives(25)u0y=1−e−S1y+KαReS13αRe−2S1ye−S1y.Whenε≠0, equation (17) is substituted into equations (11)–(14) to get the system of partial differential equations corresponding to terms of first order:(26)∂v1∂y+∂w1∂z=0,(27)−α∂u1∂y+v1∂u0∂y=1Re∂2u1∂y2+∂2u1∂z2+K−α∂3u1∂y3−α∂3u1∂y∂z2+v1∂3u0∂y3−Mu1,(28)−α∂v1∂y=−∂p1∂y+1Re∂2v1∂y2+∂2v1∂z2−Kα∂3v1∂y3+∂3v1∂y∂z2,(29)−α∂w1∂y=−∂p1∂z+1Re∂2w1∂y2+∂2w1∂z2−Kα∂3w1∂y3+∂3w1∂y∂z2−Mw1.The corresponding conditions on the boundary (15) take the form(30)u1=0,v1=vz=−αcosπz,w1=0aty=0,u1=0,v1=0,w1=0,asy⟶∞.


## 5. Cross Flow Solution

The cross flow velocity componentsv1 and w1 along with pressure p1 are considered and presented in the following way:(31)v1y,z=v11ycosπz,(32)w1y,z=−1πv11′ysinπz,(33)p1y,z=p11ycosπz.Substituting equations (31) and (32) in equations (28) and (29), we obtain(34)KαRev11″′−π2v11′−v11″+π2v11−αRev11′=−Rep11′,(35)KαRe−v11″″+π2v11″+v11″′−π2v11′+αRev11″−MRev11′=π2Rep11.Eliminating the termsp11 and p11′ from equations (34) and (35), we get(36)KαRe−v11″″′+2π2v11″′−π4v11′+v11″″+αRev11″′−2π2v11″−MRev11″−π2αRev11′+π4v11=0.The conditions on the boundary of the plate become(37)v110=−α,v11′0=0.We assume that(38)v11=v110+Kv111+OK2,p11=p110+Kp111+OK2.Then, the corresponding conditions on the boundary take the form(39)v1100=−α,v111′0=v1110=v110′0=0.From equations (36) and (38) with the boundary conditions (39), we obtain(40)v11=αλ1−λ2λ2e−λ1y−λ1e−λ2y−Kα2ReA1−A2λ1−λ2e−λ1y−A1−A2λ1−λ2e−λ2y+yA1e−λ1y−A2e−λ2y.The expression ofp11 is not presented here for the purpose of saving space. Substituting equation (40) in equations (31) and (32), we get(41)v1y,z=αλ1−λ2λ2e−λ1y−λ1e−λ2y−Kα2ReA1−A2λ1−λ2e−λ1y−A1−A2λ1−λ2e−λ2y+yA1e−λ1y−A2e−λ2ycosπz,(42)w1y,z=αλ1λ2πλ1−λ2−e−λ2y+e−λ1y+Kα2Reπ−λ1A1−A2λ1−λ2e−λ1y+λ2A1−A2λ1−λ2e−λ2y+A1e−λ1y−A2e−λ2y+y−λ1A1e−λ1y+A2λ2e−λ2ysinπz.


## 6. Main Flow Solution

The solution of equation (27) with conditions on the boundary (30) is considered in this section. The main flow velocity component u1 is assumed as(43)u1y,z=u11ycosπz.Then, the conditions on the boundary of the plate are reduced to(44)u11=0aty=0,u11=0asy⟶∞.Furthermore, it is assumed that(45)u11=u110+Ku111+OK2.Then, the analogous boundary conditions (30) are(46)u111=u110=0aty=0,u111=u110=0asy⟶∞.In view of equations (25), (41), and (43)–(46), equation (27) yields(47)uy,z=1−e−S1y+KαReS13αRe−2S1ye−S1y+εαReS1λ1−λ2A3e−λ1+my−A4e−λ2+my+A4−A3e−λycosπz+εKαReS1λ2−λ1F2C16+F3C15+F4C13+F5C11e−λycosπz+εKαReS1λ2−λ1−F1αRe−2λye−λy−F2C16e−λ1+my−F3C15e−λ2+my−F4C13e−λ1+my+C14ye−λ1+my−F5C11e−λ2+my+C12ye−λ2+mycosπz.It should be noted that the limiting velocityu1 as M=0 and K⟶0 differs from that computed by Gersten and Gross [1]. This is because of some calculation mistakes in their work.


## 7. Results and Discussion

The 3D steady, laminar MHD flow of incompressible second-grade fluid across a horizontal plate with infinite length subjected to variable suction is analyzed. A well-known perturbation technique is employed to solve the governing equations for the velocity profile and pressure. Graphical and tabular illustrations are used to analyze the behavior of different proficient parameters of interest.


### 7.1. Main Flow Velocity Field

The velocity profiles are presented for dimensionless parameters for the dynamics of the present flow problem such as the suction parameterα, second-grade parameter K, Reynolds number Re, and magnetic parameter M. The ranges of the parameters of interest appearing in the model are considered according to the adjustment of physical quantities in the present fluidic problem. The values of the suction parameter are small because of the boundary layer region which is close to the plane wall. Since the holes in the semi-infinite plate vary in size and shape, variable suction velocity distribution is considered close to the region of the plate, but the value of suction velocity becomes uniform when one moves in the region away from the plane wall. These proposed variations are presented in Figures 2–5. The impact of the suction variable α on the main velocity component u is shown in Figure 2. The component of velocity u decreases with the increase of α. Figure 3 shows the influence of the second-grade parameter K on the velocity in the main flow direction u. It is shown that the magnitude of this flow velocity increases near the plate, but a reverse trend is noticed when one goes away from the plane wall. Figure 4 exhibits the impact of the magnetic parameter M on the velocity component based on the main flow direction u. In Figure 4, it can be seen that the velocity based on the main flow direction is accelerating function of the magnetic parameter M. Figure 5 depicts that the main flow velocity component u retards in the neighborhood of the plate as Re increases, and a reverse trend is seen as its position moves away from the plate. Furthermore, u⟶1 as y⟶∞.Figure 2
Effects ofM=2,K=0.1,Re=10,ε=0.1, and z=0 on u for the number of variants of α.Figure 3
Effects ofα=0.5,M=2,Re=10,ε=0.1, and z=0 on u for the number of variants of K.Figure 4
Effects ofα=0.1,K=0.1,Re=10,ε=0.1, and z=0 on u for the number of variants of M.Figure 5
Effects ofM=2,K=0.1,α=0.1,ε=0.1, and z=0 on u for the number of variants of Re.


### 7.2. Cross Flow Velocity Field

The velocity profile in the direction of cross flow is presented for dimensionless parameters for the dynamics of the present flow problem such as the suction parameterα, second-grade parameter K, Reynolds number Re, and magnetic parameter M. These proposed variants are presented in Figures 6–9. The impact of the suction variable α on the cross flow velocity component v is shown in Figure 6. The component of velocity v decreases near the surface of the plate, but a reverse impact is observed when one enters the region away from the plate because of the suction velocity parameter α. Figure 7 shows the impact of the elastic parameter K on the velocity in the cross flow direction v. It is shown that the dominant impact of the second-grade parameter K in the region close to the plate is seen, and it is also observed that cross flow velocity is decreasing function of the non-Newtonian parameter K. It is interesting to see that Figures 8 and 9 reflect almost a similar impact of the magnetic parameter M and Reynolds number Re on the cross velocity component. In both figures, the cross flow velocity accelerates as one moves in the region away from the plate. The impact of the suction parameter α, second-grade parameter K, and Hartmann number M on the velocity component w based on the cross flow direction is presented in Table 1. It depicts that w increases as α increases. Also, the effect of K on w is noted. It decreases in the region close to the wall but increases away from the plate, and opposite behavior of cross flow velocity is observed for different values of Hartmann number. However, it decreases in the y direction.Figure 6
Effects ofM=2,K=0.1,Re=10,ε=0.1, and z=0 on v for the number of variants of α.Figure 7
Effects ofα=0.5,M=2,Re=10,ε=0.1, and z=0 on v for the number of variants of K.Figure 8
Effects ofα=0.1,K=0.1,Re=10,ε=0.1, and z=0 on v for the number of variants of M.Figure 9
Effects ofM=2,K=0.1,α=0.1,ε=0.1, and z=0 on v for the number of variants of Re.Table 1
Impacts ofK and α on the velocity w based on cross flow for fixing the values ε=0.1, z=−0.5, and Re=10.
y
K = 0.1, α=0.1, M=2
K = 0.1, α=0.5, M=2
K = 0.5, α=0.5, M=2
K = 0.5, α=0.5, M=4
0.0
0.0
0.0
0.0
0.0
0.5
0.003216
0.016261
0.016228
0.016071
1.0
0.001352
0.006597
0.006889
0.007011
1.5
0.000442
0.002076
0.002201
0.002322
2.0
0.000133
0.000599
0.000630
0.000698


## 7.1. Main Flow Velocity Field

The velocity profiles are presented for dimensionless parameters for the dynamics of the present flow problem such as the suction parameterα, second-grade parameter K, Reynolds number Re, and magnetic parameter M. The ranges of the parameters of interest appearing in the model are considered according to the adjustment of physical quantities in the present fluidic problem. The values of the suction parameter are small because of the boundary layer region which is close to the plane wall. Since the holes in the semi-infinite plate vary in size and shape, variable suction velocity distribution is considered close to the region of the plate, but the value of suction velocity becomes uniform when one moves in the region away from the plane wall. These proposed variations are presented in Figures 2–5. The impact of the suction variable α on the main velocity component u is shown in Figure 2. The component of velocity u decreases with the increase of α. Figure 3 shows the influence of the second-grade parameter K on the velocity in the main flow direction u. It is shown that the magnitude of this flow velocity increases near the plate, but a reverse trend is noticed when one goes away from the plane wall. Figure 4 exhibits the impact of the magnetic parameter M on the velocity component based on the main flow direction u. In Figure 4, it can be seen that the velocity based on the main flow direction is accelerating function of the magnetic parameter M. Figure 5 depicts that the main flow velocity component u retards in the neighborhood of the plate as Re increases, and a reverse trend is seen as its position moves away from the plate. Furthermore, u⟶1 as y⟶∞.Figure 2
Effects ofM=2,K=0.1,Re=10,ε=0.1, and z=0 on u for the number of variants of α.Figure 3
Effects ofα=0.5,M=2,Re=10,ε=0.1, and z=0 on u for the number of variants of K.Figure 4
Effects ofα=0.1,K=0.1,Re=10,ε=0.1, and z=0 on u for the number of variants of M.Figure 5
Effects ofM=2,K=0.1,α=0.1,ε=0.1, and z=0 on u for the number of variants of Re.


## 7.2. Cross Flow Velocity Field

The velocity profile in the direction of cross flow is presented for dimensionless parameters for the dynamics of the present flow problem such as the suction parameterα, second-grade parameter K, Reynolds number Re, and magnetic parameter M. These proposed variants are presented in Figures 6–9. The impact of the suction variable α on the cross flow velocity component v is shown in Figure 6. The component of velocity v decreases near the surface of the plate, but a reverse impact is observed when one enters the region away from the plate because of the suction velocity parameter α. Figure 7 shows the impact of the elastic parameter K on the velocity in the cross flow direction v. It is shown that the dominant impact of the second-grade parameter K in the region close to the plate is seen, and it is also observed that cross flow velocity is decreasing function of the non-Newtonian parameter K. It is interesting to see that Figures 8 and 9 reflect almost a similar impact of the magnetic parameter M and Reynolds number Re on the cross velocity component. In both figures, the cross flow velocity accelerates as one moves in the region away from the plate. The impact of the suction parameter α, second-grade parameter K, and Hartmann number M on the velocity component w based on the cross flow direction is presented in Table 1. It depicts that w increases as α increases. Also, the effect of K on w is noted. It decreases in the region close to the wall but increases away from the plate, and opposite behavior of cross flow velocity is observed for different values of Hartmann number. However, it decreases in the y direction.Figure 6
Effects ofM=2,K=0.1,Re=10,ε=0.1, and z=0 on v for the number of variants of α.Figure 7
Effects ofα=0.5,M=2,Re=10,ε=0.1, and z=0 on v for the number of variants of K.Figure 8
Effects ofα=0.1,K=0.1,Re=10,ε=0.1, and z=0 on v for the number of variants of M.Figure 9
Effects ofM=2,K=0.1,α=0.1,ε=0.1, and z=0 on v for the number of variants of Re.Table 1
Impacts ofK and α on the velocity w based on cross flow for fixing the values ε=0.1, z=−0.5, and Re=10.
y
K = 0.1, α=0.1, M=2
K = 0.1, α=0.5, M=2
K = 0.5, α=0.5, M=2
K = 0.5, α=0.5, M=4
0.0
0.0
0.0
0.0
0.0
0.5
0.003216
0.016261
0.016228
0.016071
1.0
0.001352
0.006597
0.006889
0.007011
1.5
0.000442
0.002076
0.002201
0.002322
2.0
0.000133
0.000599
0.000630
0.000698


## 8. Concluding Remarks

The 3D steady, laminar magnetohydrodynamic flow of an incompressible non-Newtonian second-grade fluid subjected to variable suction velocity is investigated. The key outcomes of this analysis are as follows:(i)
The velocity component based on the main flow directionu decreases with the increase of the suction parameter α.(ii)
It is shown that the magnitude of the velocity component based on the main flow direction increases near the plate, but the main flow velocity decreases when one goes away from the plate.(iii)
The main flow velocity is increasing function of the magnetic parameterM.(iv)
The limiting result of the velocity components asM=0 and K⟶0 is look-alike to that observed by Gersten and Gross [1] and also that computed by Singh [4] in the case of time independence.(v)
The Newtonian outcomes [1] are retrieved when M=0 and K⟶0.(vi)
The main flow velocity of the fluidic systemu declines near the plane wall as Re increases, and it accelerates as one moves away at a distance from the wall. Furthermore, u⟶1 as y⟶∞.(vii)
The component of velocityv decreases near the surface of the plate, but a reverse effect is seen when one enters the region away from the plate because of the suction velocity parameter α.(viii)
A similar impact of the magnetic parameter and Reynolds number on the velocity componentv based on cross flow is observed.



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*Source: 2584397-2019-08-28.xml*