a novel design of three-dimensional mhd flow of second...

Research ArticleA Novel Design of Three-Dimensional MHD Flow ofSecond-Grade Fluid past a Porous Plate

Muhammad Shoaib12 Rizwan Akhtar 1 Muhammad Abdul Rehman Khan3

Muhammad Afzal Rana 4 Abdul Majeed Siddiqui5 Zhu Zhiyu1

and Muhammad Asif Zahoor Raja 3

1School of Electronics and Information Jiangsu University of Science and Technology Zhenjiang China2Department of Mathematics COMSATS University Islamabad Attock Campus Kamra Road Attock Pakistan3Department of Electrical and Computer Engineering COMSATS University Islamabad Attock CampusAttock 43600 Pakistan4Department of Mathematics Riphah International University Sector I-14 Islamabad Pakistan5Department of Mathematics Pennsylvania State University York Campus1031 Edgecomb Avenue York PA 17403 USA

Correspondence should be addressed to Rizwan Akhtar rizwanjusteducn

Received 11 May 2019 Revised 19 June 2019 Accepted 4 August 2019 Published 28 August 2019

Academic Editor Francisco R Villatoro

Copyright copy 2019 Muhammad Shoaib et al +is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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 suctionvelocity +e suction velocity involves a steady distribution with a low superimposed perpendicularly varying dispersion +estrength of the uniform magnetic field is incorporated in the normal direction to the wall +e variable suction transforms thefluidic problem into a 3D flow problem because of variable suction velocity in the normal direction to the plane wall+e proposedmathematical modeling and its dynamical analysis are prescribed for the boundary layer flow keeping the magnetic effects withouttaking into consideration the induced magnetic field An analytical perturbation technique is employed for the series solutions ofthe system of ordinary differential equations of velocity profile and pressure Graphical illustrations are used to analyze thebehavior of different proficient parameters of interest +e magnetic parameter is responsible for accelerating the main flowvelocity while controlling the cross flow velocities

1 Introduction

+e electrically conducting fluid flows due to the effect of thetransverse magnetic field have received attention of a largesection of research community because of their use in en-gineering astrophysics and geophysics and in the field ofaerodynamics to control the boundary layer In industriesthe induced magnetic field-based procedures are used forexcursion heat exchanger pump and levitate liquid metalsGersten and Gross [1] investigated the heat transfer effects onthe three-dimensional fluid flows based on main and crossflow velocity components along a porous plane wall based onthe transverse sinusoidal suction velocity distribution Singh

et al [2] analyzed the problem of the 3D porous mediumbased on convective flow+e authors also incorporated heattransfer effects in the fluidic problem Furthermore Singh[3 4] analyzed the hydromagnetic and magnetohydrody-namic impacts on 3D free convective fluid flow +e authorsconsidered a vertical porous wall for the impact of porosityon the flow problem Furthermore the effect of the magneticfield with unvarying strength is incorporated perpendicularto the free convective fluid flow which is electrically con-ducting through a semi-infinite plate [5] Later on Helmy [6]discussed the heat transfer impact on the flow of an elec-trically conducting fluid across an infinite plane wall withvariable suction

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 2584397 11 pageshttpsdoiorg10115520192584397

Researchers have extensively exploited the field ofmagnetohydrodynamics with reference to geometricalconfiguration various types of formulations and dif-ferent dimensions using analytical and numericalmethods [7ndash19] +e authors investigated the influence ofthe magnetic field on different structures of flow di-mensions for distributions of concentrations and tem-peratures Moreover the magnetic parameter is proved tobe a controlling parameter to restrain the fluid and heatflows under the closed spaces Different types of non-Newtonian fluids bargain applications in many magne-tohydrodynamic devices and in power generation as wellMagnetohydrodynamics with different types of convec-tive flows heat generationabsorption and transferanalysis [20ndash25] have been investigated in detail

Gupta and Johari [26] presented the analysis of 3Dmagnetohydrodynamic flow across a porous plane walland the fluid taken into account is laminar having thepower of conducting heat +e authors considered amagnetic flux in the normal direction to the plateMoreover Guria and Jana [27] considered a verticalporous wall for the three-dimensional hydrodynamicfluid flow problem Furthermore Greenspan and Carrier[28] Rossow [29] and Singh [30] presented their studiesextensively on the magnetohydrodynamic impacts on theflow across a plane wall based on injection or suction+ere are some non-Newtonian models presenting thethree-dimensional fluidic problems though a porous wallwith variable suction [31ndash35] Periodic suction has re-ceived very much attention in the field of aerodynamics[36ndash38] +e porous medium has very much importancein the fluid dynamics [39ndash45] Abbas et al [46] studied3D peristaltic flow fluid with hyperbolic tangent in thenonuniform channel along flexible walls Bhatti and Lu[47] studied analytical analysis of the head-on collisionmechanism among hydroelastic solitary waves withuniform current Jhorar et al [48] analyzed the micro-fluid in the channel for the electroosmosis-modulatedbiomechanical transport

+e motivation of this study is to analyze the 3D MHDflow of the simplest non-Newtonian second-grade fluidicmodel through a semi-infinite wall which is based on suc-cession of waves or curves with fluctuating velocity distri-bution A uniform suction velocity along the surface of theplane wall transforms the problem into a 2D asymptoticsuction velocity solution [49] however because of thevariable suction velocity distribution in the normal di-rection the fluidic system turns out to be 3D +e analyticalperturbation technique is incorporated for finding the seriessolution of this problem +e proposed outcomes are esti-mated for different parameters of interest such as the suctionparameter α second-grade parameter K Reynolds numberRe and Hartmann number M +is article is organized asfollows Section 2 consists of the statement of the problemSection 3 presents the perturbation method Section 4specifies the design of the problem Sections 5 and 6 describeanalytical solutions Section 7 integrates results and dis-cussion and Section 8 contains conclusions In addition tothese appendices and nomenclature are given thereafter

2 Statement of the Problem

In this problem the 3D steady laminar magnetohydrody-namic flow of an incompressible non-Newtonian second-grade fluid passing through an infinite plate is considered+e xlowastzlowast plane is considered where ylowast-axis is normal to theplane (see Figure 1) A time-independent distribution is abasic steady distribution [1] in which l and ε indicate thewavelength and amplitude of the variable suction velocityrespectively+us variable suction velocity has the followingform

v zlowast

( 1113857 minus v0 1 + ε cos πzlowast

l1113888 1113889 (1)

+e fluidic system becomes two-dimensional because ofconstant suction velocity whereas it is three-dimensional incase of variable suction velocity [49] A constant magneticfield B0 in the normal direction to the xlowastzlowast wall is appliedAlso the following are considered (i) the fluid has electricconduction (ii) the fluid has steady and laminar flow (iii)the fluid has uniform free stream velocity (iv) the magneticReynolds number is at small scale and also the inducedmagnetic field is inconsiderable (v) Hall and polarizationeffects are neglected and (vi) all physical properties of theparameters are independent of xlowast because of the infiniteextended length of the plate in the xlowast direction but the flowremains 3D because of the variable suction velocity (1)

3 Perturbation Method

Perturbation methods [50ndash52] are strong mathematical tools tofind the seriesapproximate solutions of those problems whoseanalytic or exact solutions are not possible or hard to find+esetechniques have frequently been used for the problems arisingin the fields of engineering and science +e function u(y z ε)is convoluted in physical problems and then it can be shownmathematically by the differential equation

L(u y z ε) 0 (2)

subject to the boundary condition

B(u ε) 0 (3)

where y is a vector or scalar independent variable and ε is aparameter One cannot solve this problem exactly in generalHowever if there exists an ε ε0 (ε can be scaled so that ε 0)for which the above problem can be solved exactly then oneexplores to obtain the solution for small ε in the form

u(y z ε) u0(y) + εu1(y z) + ε2u2(y z) + middot middot middot (4)

where un (n 0 1 2 ) does not depend on ε and u0(y) isthe solution of the problem for ε 0 One then substitutesthis expansion into equations (2) and (3) which expands forsmall ε and collects coefficients of each power of ε Sincethese equations must hold for all values of ε each coefficientof ϵ must vanish independently because sequences of ϵ arelinearly independent +ese usually are simpler equationsgoverning un (n 0 1 2 ) which can be solvedsuccessively

2 Mathematical Problems in Engineering

4 Design of the Problem

+e equations of continuity andmomentum are presented inthe following way

zwlowast

zzlowast+

zvlowast

zylowast 0 (5)

ρ wlowastzulowast

zzlowast+ vlowastzulowast

zylowast1113888 1113889 μ

z2ulowast

zylowast2 +

z2ulowast

zzlowast21113890 1113891

+ α1 vlowastz

3ulowast

zylowast3 + wlowast z3ulowast

zylowast2

zzlowast1113890

+ vlowast z3ulowast

zylowast zzlowast2 + wlowastz

3ulowast

zzlowast3 1113891 minus ρB

20ulowast

(6)

ρ wlowastzvlowast

zzlowast+ vlowastzvlowast

zylowast1113888 1113889 minus

zplowast

zy+ μ

z2vlowast

zylowast2 +

z2vlowast

zzlowast21113890 1113891

+ α1 vlowastz

3vlowast

zylowast3 + wlowast z3vlowast

zylowast2zzlowast

1113890

+ vlowast z3vlowast


3vlowast

zzlowast3 +

zvlowast

zzlowastz2vlowast

zylowast zzlowast

+zulowast

zzlowastz2ulowast

zylowast zzlowast+ 5

zvlowast

zylowastz2vlowast

zylowast2

+zvlowast

zzlowastz2wlowast

zylowast2 + 2

zulowast

zylowastz2ulowast

zylowast2

+ 2zwlowast

zylowastz2wlowast

zylowast2 +

zulowast

zylowastz2ulowast

zzlowast2

+zvlowast

zylowastz2vlowast

zzlowast21113891

(7)

ρ vlowastzwlowast

zylowast+ wlowastzwlowast

zzlowast1113888 1113889 minus

zplowast

zzlowast+ μ

z2wlowast

zylowast2 +

z2wlowast

zzlowast21113890 1113891

+ α1 wlowastz

3vlowast

zylowast3 + vlowastz

3wlowast

zylowast31113890

+ vlowast z3wlowast


3wlowast

zzlowast3

+zwlowast

zylowastz2wlowast

zylowast zzlowast+

zulowast

zylowastz2ulowast

zylowast zzlowast

+ 5zwlowast

zzlowastz2wlowast

zzlowast2 +

zwlowast

zylowastz2vlowast

zzlowast2

+ 2zulowast

zzlowastz2ulowast

zzlowast2 + 2

zvlowast

zzlowastz2vlowast

zzlowast2

+zulowast

zzlowastz2ulowast

zylowast2 +

zwlowast

zzlowastz2wlowast

zylowast2 1113891 minus ρB

20wlowast

(8)

with the boundary conditions [7]

ulowast

0 vlowast

minus v0 1 + ε cos πzlowast

l1113888 1113889 w

lowast 0 at y

lowast 0

ulowast

U vlowast

minus v0 wlowast

0 plowast

plowastinfin as y

lowast ⟶infin

(9)

Now the following dimensionless parameters are in-troduced [20]

x xlowast

l

y ylowast

l

z zlowast

l

u ulowast

U

v vlowast

U

w wlowast

U

p plowast

ρU2

(10)

+en equations (5)ndash(8) become

υ (z )

x

y

z

Free stream velocity U

B0

Figure 1 Geometrical representation of the problem

Mathematical Problems in Engineering 3

zv

zy+

zw

zz 0 (11)

wzu

zz+ v

zu

zy

1Re

z2u

zy2z2u

zz21113890 1113891

+ K1113890wz3u

zy2 zz+ v

z3u

zy3 + vz3u

zy zz2

+ wz3u

zz31113891 minus Mu

(12)

wzv

zz+ v

zv

zy minus

zp

zy+

1Re

z2v

zy2 +z2v

zz21113890 1113891

+ K vz3v

zy3 + wz3v

zy2 zz+ v

z3v

zy zz21113890

+ wz3v

zz3 +zv

zz

z2v

zy zz+

zu

zz

z2u

zy zz

+ 5zv

zy

z2v

zy2 +zv

zz

z2w

zy2 + 2zu

zy

z2u

zy2

+ 2zw

zy

z2w

zy2 +zu

zy

z2u

zz2 +zv

zy

z2v

zz21113891

(13)

wzw

zz+ v

zw

zy minus

zp

zz+

1Re

z2w

zy2 +z2w

zz21113890 1113891

+ K wz3v

zy3 + vz3w

zy3 + vz3w

zy zz2 + wz3w

zz31113890

+zw

zy

z2w

zy zz+

zu

zy

z2u

zy zz+ 5

zw

zz

z2w

zz2

+zw

zy

z2v

zz2 + 2zu

zz

z2u

zz2 + 2zv

zz

z2v

zz2 +zu

zz

z2u

zy2

+zw

zz

z2w

zy21113891 minus Mw

(14)

and the boundary conditions (9) take the forms

u 0 v v(z) minus α(1 + ε cos π(z)) w 0 aty 0

u 1 v minus α w 0 asy⟶infin

(15)

where

Re Ul

]

α v0U

M σB2

0l

ρU

K α1ρl2

(16)

Since ε is a small number solutions are assumed asfollows

u(y z) u0 + εu1 + ε2u2 + middot middot middot

v(y z) v0 + εv1 + ε2v2 + middot middot middot

w(y z) w0 + εw1 + ε2w2 + middot middot middot

p(y z) p0 + εp1 + ε2p2 + middot middot middot

(17)

For ε 0 the problem becomes two-dimensional be-cause of constant suction velocity given in equation (1)which is resulted as follows

KαRe

d3u0

dy3 minusd2u0

dy21113888 1113889 minus αRe

du0

dy+ MReu0 0 (18)

subject to boundary conditions

u0 0 aty 0

u0 1 asy⟶infin(19)

Consider the following form of the solution

u0 u00 + Ku01 + O K2

1113872 1113873 (20)

where K is a small elastic parameter Using equation (20) inequations (18) and (19) and correlating the coefficients of K0

and K the following boundary value problems are obtained

d2u00

dy2 + αRe

du00

dyminus MReu00 0

u00(0) 0

u00(infin) 1

(21)

d2u01

dy2 minus αRe

d3u00

dy3 + αRe

du01

dyminus MReu01 0

u01(0) 0

u01(infin) 0

(22)

Solving equations (21) and (22) we get

u00(y) 1 minus eminus S1y

(23)

u01(y) αReS

31

αRe minus 2S1( 1113857ye

minus S1y (24)

+erefore in the light of equations (23) and (24)equation (20) gives


+ KαReS

31


minus S1y (25)

When εne 0 equation (17) is substituted into equations(11)ndash(14) to get the system of partial differential equationscorresponding to terms of first order

zv1

zy+

zw1

zz 0 (26)


minus αzu1

zy+ v1

zu0

zy

1Re

z2u1

zy2 +z2u1

zz21113888 1113889

+ K1113888 minus αz3u1

zy3 minus αz3u1

zy zz2

+ v1z3u0

zy3 1113889 minus Mu1

(27)

minus αzv1

zy minus

zp1

zy+

1Re

z2v1

zy2 +z2v1

zz21113888 1113889

minus Kαz3v1

zy3 +z3v1

zy zz21113888 1113889

(28)

minus αzw1

zy minus

zp1

zz+

1Re

z2w1

zy2 +z2w1

zz21113888 1113889

minus Kαz3w1

zy3 +z3w1

zy zz21113888 1113889 minus Mw1

(29)

+e corresponding conditions on the boundary (15) takethe form

u1 0 v1 v(z) minus α cos πz w1 0 aty 0

u1 0 v1 0 w1 0 asy⟶infin(30)

5 Cross Flow Solution

+e cross flow velocity components v1 and w1 along withpressure p1 are considered and presented in the followingway

v1(y z) v11(y)cos π(z) (31)

w1(y z) minus1π

v11prime (y)sin π(z) (32)

p1(y z) p11(y)cos π(z) (33)

Substituting equations (31) and (32) in equations (28)and (29) we obtain

KαRe vPrimeprime11 minus π2v11prime1113874 1113875 minus v Prime11 + π2v11 minus αRev11prime minus Rep11prime

(34)

KαRe minus vPrimePrime11 + π2v Prime111113874 1113875 + v

Primeprime11 minus π2v11prime + αRev

Prime11

minus MRev11prime π2Rep11

(35)

Eliminating the terms p11 and p11prime from equations (34)and (35) we get

KαRe minus vPrimePrimeprime11 + 2π2vPrimeprime11 minus π4

v11prime1113874 1113875 + vPrimePrime11 + αRevPrimeprime11 minus 2π2v Prime11

minus MRevPrime11 minus π2αRev11prime + π4

v11 0

(36)

+e conditions on the boundary of the plate become

v11(0) minus α

v11prime (0) 0(37)

We assume that

v11 v110 + Kv111 + O K2

1113872 1113873

p11 p110 + Kp111 + O K2

1113872 1113873(38)

+en the corresponding conditions on the boundarytake the form

v110(0) minus α

v111prime (0) v111(0) v110prime (0) 0(39)

From equations (36) and (38) with the boundary con-ditions (39) we obtain

v11 α

λ1 minus λ2( 1113857λ2e

minus λ1yminus λ1e

minus λ2y1113872 1113873

minus Kα2Re

A1 minus A2

λ1 minus λ21113888 1113889e

minus λ1y1113888

minusA1 minus A2

λ1 minus λ21113888 1113889e

minus λ2y+ y A1e

minus λ1yminus A2e

minus λ2y1113872 11138731113889

(40)

+e expression of p11 is not presented here for thepurpose of saving space Substituting equation (40) inequations (31) and (32) we get

v1(y z) α

λ1 minus λ2( 1113857λ2e


minus λ2y1113872 1113873

minus Kα2Re1113888A1 minus A2

λ1 minus λ21113888 1113889e

minus λ1y

minusA1 minus A2

λ1 minus λ21113888 1113889e

minus λ2y

+ y A1eminus λ1y

minus A2eminus λ2y

1113872 11138731113889cos πz

(41)

w1(y z) αλ1λ2

π λ1 minus λ2( 1113857minus e

minus λ2y+ e

minus λ1y1113872 1113873 +

Kα2Re

π

minus λ1A1 minus A2

λ1 minus λ21113888 1113889eminus λ1y + λ2

A1 minus A2

λ1 minus λ21113888 1113889eminus λ2y

+A1eminus λ1y minus A2e

minus λ2y + y minus λ1A1eminus λ1y + A2λ2eminus λ2y( 1113857

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠sin πz (42)


6 Main Flow Solution

+e solution of equation (27) with conditions on theboundary (30) is considered in this section +e main flowvelocity component u1 is assumed as

u1(y z) u11(y)cos π(z) (43)

+en the conditions on the boundary of the plate arereduced to

u11 0 aty 0


Furthermore it is assumed thatu11 u110 + Ku111 + O K

21113872 1113873 (45)

+en the analogous boundary conditions (30) are

u111 u110 0 aty 0

u111 u110 0 asy⟶infin(46)

In view of equations (25) (41) and (43)ndash(46) equation(27) yields

u(y z) 1 minus eminus S1y

+KαReS

31

αRe minus 2S1ye

minus S1y+ ε

αReS1

λ1 minus λ2( 1113857

A3eminus λ1+m( )y minus A4e

minus λ2+m( )y

+ A4 minus A3( 1113857eminus λy

⎛⎜⎜⎝ ⎞⎟⎟⎠cos πz

+ εKαReS1

λ2 minus λ1( 1113857F2C16 + F3C15 + F4C13 + F5C11( 1113857e

minus λy1113872 1113873cos πz

+ εKαReS1

λ2 minus λ1( 1113857

minusF1

αRe minus 2λ1113888 1113889yeminus λy minus F2C16e

minus λ1+m( )y minus F3C15eminus λ2+m( )y

minus F4 C13eminus λ1+m( )y + C14yeminus λ1+m( )y1113872 1113873


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

cos πz

(47)

It should be noted that the limiting velocityu1 asM 0 andK⟶ 0 differs from that computed by Gersten and Gross [1]+is is because of some calculation mistakes in their work

7 Results and Discussion

+e 3D steady laminar MHD flow of incompressible sec-ond-grade fluid across a horizontal plate with infinite lengthsubjected to variable suction is analyzed A well-knownperturbation technique is employed to solve the governingequations for the velocity profile and pressure Graphicaland tabular illustrations are used to analyze the behavior ofdifferent proficient parameters of interest

71 Main Flow Velocity Field +e velocity profiles are pre-sented for dimensionless parameters for the dynamics of thepresent flow problem such as the suction parameter α second-grade parameter K Reynolds number Re and magnetic pa-rameter M +e ranges of the parameters of interest appearingin the model are considered according to the adjustment ofphysical quantities in the present fluidic problem+e values ofthe suction parameter are small because of the boundary layerregion which is close to the plane wall Since the holes in thesemi-infinite plate vary in size and shape variable suctionvelocity distribution is considered close to the region of theplate but the value of suction velocity becomes uniform whenone moves in the region away from the plane wall +eseproposed variations are presented in Figures 2ndash5+e impact of

the suction variable α on the main velocity component u isshown in Figure 2 +e component of velocity u decreases withthe increase of α Figure 3 shows the influence of the second-grade parameter K on the velocity in the main flow direction uIt is shown that the magnitude of this flow velocity increasesnear the plate but a reverse trend is noticedwhen one goes awayfrom the planewall Figure 4 exhibits the impact of themagneticparameter M on the velocity component based on the mainflow direction u In Figure 4 it can be seen that the velocitybased on the main flow direction is accelerating function of themagnetic parameter M Figure 5 depicts that the main flowvelocity component u retards in the neighborhood of the plateas Re increases and a reverse trend is seen as its position movesaway from the plate Furthermore u⟶ 1 as y⟶infin

72 Cross Flow Velocity Field +e velocity profile in the di-rection of cross flow is presented for dimensionless parametersfor the dynamics of the present flowproblem such as the suctionparameter α second-grade parameter K Reynolds number Reand magnetic parameter M +ese proposed variants arepresented in Figures 6ndash9 +e impact of the suction variable αon the cross flow velocity component v is shown in Figure 6+e component of velocity v decreases near the surface of theplate but a reverse impact is observed when one enters theregion away from the plate because of the suction velocityparameter α Figure 7 shows the impact of the elastic parameterK on the velocity in the cross flow direction v It is shown thatthe dominant impact of the second-grade parameter K in the


region close to the plate is seen and it is also observed that crossflow velocity is decreasing function of the non-Newtonianparameter K It is interesting to see that Figures 8 and 9 reflectalmost a similar impact of the magnetic parameter M andReynolds number Re on the cross velocity component In bothfigures the cross flow velocity accelerates as one moves in theregion away from the plate+e impact of the suction parameterα second-grade parameterK andHartmann numberM on thevelocity component w based on the cross flow direction ispresented in Table 1 It depicts that w increases as α increasesAlso the effect of K on w is noted It decreases in the regionclose to the wall but increases away from the plate and oppositebehavior of cross flow velocity is observed for different values ofHartmann number However it decreases in the y direction

8 Concluding Remarks

+e 3D steady laminar magnetohydrodynamic flow of anincompressible non-Newtonian second-grade fluid sub-jected to variable suction velocity is investigated +e keyoutcomes of this analysis are as follows

(i) +e velocity component based on the main flowdirection u decreases with the increase of the suctionparameter α

(ii) It is shown that the magnitude of the velocitycomponent based on the main flow direction in-creases near the plate but the main flow velocitydecreases when one goes away from the plate

(iii) +emain flow velocity is increasing function of themagnetic parameter M

(iv) +e limiting result of the velocity components asM 0 and K⟶ 0 is look-alike to that observedby Gersten and Gross [1] and also that computedby Singh [4] in the case of time independence

(v) +e Newtonian outcomes [1] are retrieved whenM 0 and K⟶ 0

00 02

00

05

10

ndash0504 06 08 10

y

u(y)

α = 01α = 02α = 05

Figure 2 Effects of M 2 K 01 Re 10 ε 01 and z 0 onu for the number of variants of α

00 02 04 06 08 10y

ndash4

ndash3

ndash2

ndash1

0

1

u(y)

K = 01K = 03K = 05

Figure 3 Effects of α 05 M 2 Re 10 ε 01 and z 0 onu for the number of variants of K

00 02 04 06 08 10y

00

02

04

06

08

10

u(y)

M = 2M = 4M = 8

Figure 4 Effects of α 01 K 01 Re 10 ε 01 and z 0 onu for the number of variants of M

00 02 04 06 08 10y

00

02

04

06

08

10

u(y)

Re = 10Re = 20Re = 30

Figure 5 Effects of M 2 K 01 α 01 ε 01 and z 0 onu for the number of variants of Re


(vi) +emain flow velocity of the fluidic system u declinesnear the plane wall as Re increases and it acceleratesas one moves away at a distance from the wallFurthermore u⟶ 1 as y⟶infin

(vii) +e component of velocity v decreases near thesurface of the plate but a reverse effect is seen whenone enters the region away from the plate because ofthe suction velocity parameter α

(viii) A similar impact of the magnetic parameter andReynolds number on the velocity component v

based on cross flow is observed

Nomenclature

L Wavelength of suction velocity distributionRe Reynolds numberM Hartmann numberK Second-grade parameterα Suction parameterB0 Uniform magnetic field applied in the ylowast

directionU Free stream velocityv0 Suction velocity(ulowast vlowast wlowast) +e dimensional velocity components along

xlowast ylowast and zlowast directions

00 05 10 15 20y

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

v(y)

α = 01α = 05α = 09

Figure 6 Effects of M 2 K 01 Re 10 ε 01 and z 0 onv for the number of variants of α

00

00

05 10 15

ndash10

ndash05

ndash15

20y

v(y)

K = 01K = 05K = 09

Figure 7 Effects of α 05 M 2 Re 10 ε 01 and z 0 onv for the number of variants of K

00ndash020

ndash015

ndash010

ndash005

02 04 06 08 10y

v(y)

M = 2M = 4M = 6

Figure 8 Effects of α 01 K 01 Re 10 ε 01 and z 0 onv for the number of variants of M

00

00

05 10 15 20y

ndash020

ndash015

ndash010

ndash005

v(y)

Re = 10Re = 20Re = 30

Figure 9 Effects of M 2 K 01 α 01 ε 01 and z 0 onv for the number of variants of Re

Table 1 Impacts of K and α on the velocity w based on cross flowfor fixing the values ε 01 z minus 05 and Re 10

yK 01α 01M 2

K 01α 05M 2

K 05α 05M 2

K 05α 05M 4

00 00 00 00 0005 0003216 0016261 0016228 001607110 0001352 0006597 0006889 000701115 0000442 0002076 0002201 000232220 0000133 0000599 0000630 0000698


Greek symbols

μ Coefficient of viscosityv Kinematic viscosityρ Densityσ Electrical conductivity

Appendix

S1 αRe +

αRe( 11138572

+ 4ReM

1113969

2

S2 αRe minus

αRe( 11138572

+ 4ReM

1113969

2

λ1 S1 +

S1( 11138572

+ 4π21113969

2

λ2 S2 +

S2( 11138572

+ 4π21113969

2

λ3 S1 minus

S1( 11138572

+ 4π21113969

2

λ4 S2 minus

S2( 11138572

+ 4π21113969

2

A1 λ1λ2 λ21 minus π21113872 1113873

2

λ1 minus λ2( 1113857 λ3 minus λ1( 11138572 λ4 minus λ1( 1113857

A2 λ1λ2 λ22 minus π21113872 1113873


A3 λ2

λ1 + S1( 11138572

minus αRe λ1 + S1( 1113857 minus π2 minus MRe

A4 λ1

λ2 + S1( 11138572


X1 minus α2Reλ2

A4 minus A3( 1113857

X4 minus S1 λ1 minus λ2( 1113857A1

αminus

S31λ2α2Re

αRe minus 2S11113888 1113889

X2 λ2S21α

2Re

αRe minus 2λ( 1113857minus

S1 A1 minus A2( 1113857

α

minus 2λ2αS21 minus α2ReS1 λ1 + S1( 1113857

2A3

X3 minusλ1S21α

2Re

αRe minus 2λ( 1113857+

S1 A1 minus A2( 1113857

α

+ 2λ1αS21 + α2ReS1 λ2 + S1( 1113857

2A4

X5 S1 λ1 minus λ2( 1113857A2

α+

S31λ1α2Re

αRe minus 2S11113888 1113889

X6 minusαλπ λ1 minus λ2( 1113857S1

π minus λ

(A1)

Data Availability

All the data used to support the findings of this researchwork are included in this article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

References

[1] K Gersten and J F Gross ldquoFlow and heat transfer along aplane wall with periodic suctionrdquo Zeitschrift fur AngewandteMathematik und Physik ZAMP vol 25 no 3 pp 399ndash4081974

[2] P Singh J K Mishra and K A Narayan ldquo+ree-dimensionalconvective flow and heat transfer in a porous mediumrdquo In-dian Journal of Pure and Applied Mathematics vol 19 no 11pp 1130ndash1135 1988

[3] K D Singh ldquo+ree-dimensional MHD free convection flowalong a vertical porous platerdquo Proceedings-Indian NationalScience Academy Part A Physical Sciences vol 57 no 4pp 547ndash552 1991

[4] K D Singh ldquoHydromagnetic effects on the three-dimensionalflow past a porous platerdquo ZAMP Zeitschrift fur AngewandteMathematik und Physik vol 41 no 3 pp 441ndash446 1990

[5] K D Singh ldquoHydromagnetic free convective flow past aporous platerdquo Indian Journal of Pure and Applied Mathe-matics vol 22 no 7 pp 591ndash599 1991

[6] K A Helmy ldquoOn the flow of an electrically conducting fluidand heat transfer along a plane wall with periodic suctionrdquoMeccanica vol 28 no 3 pp 227ndash232 1993

[7] CMaatki L Kolsi H F Oztop et al ldquoEffects of magnetic fieldon 3D double diffusive convection in a cubic cavity filled witha binary mixturerdquo International Communications in Heat andMass Transfer vol 49 pp 86ndash95 2013

[8] L Kolsi A Abidi N Borjini and B Aıssia ldquo+e effect of anexternal magnetic field on the entropy generation in three-dimensional natural convectionrdquo Aermal Science vol 14no 2 pp 341ndash352 2010

[9] A A A A Al-Rashed K Kalidasan L Kolsi et al ldquo+ree-dimensional investigation of the effects of external magneticfield inclination on laminar natural convection heat transferin CNT-water nanofluid filled cavityrdquo Journal of MolecularLiquids vol 252 pp 454ndash468 2018

[10] A A A A Al-Rashed L Kolsi H F Oztop et al ldquo3Dmagneto-convective heat transfer in CNT-nanofluid filledcavity under partially active magnetic fieldrdquo Physica E Low-Dimensional Systems and Nanostructures vol 99 pp 294ndash303 2018

[11] A J Chamkha and A R A Khaled ldquoHydromagnetic com-bined heat and mass transfer by natural convection from apermeable surface embedded in a fluid-saturated porousmediumrdquo International Journal of Numerical Methods forHeat amp Fluid Flow vol 10 no 5 pp 455ndash477 2000

[12] P S Reddy P Sreedevi and A J Chamkha ldquoMHD boundarylayer flow heat and mass transfer analysis over a rotating diskthrough porous medium saturated by Cu-water and Ag-waternanofluid with chemical reactionrdquo Powder Technologyvol 307 pp 46ndash55 2017

[13] R Tajammal M A Rana N Z Khan and M Shoaib ldquoSlipeffect on combined heat and mass transfer in three di-mensional MHD porous flow having heatrdquo in Proceedings ofthe 2018 15th International Bhurban Conference on Applied


Sciences and Technology (IBCAST) pp 635ndash644 IEEEIslamabad Pakistan January 2018

[14] S Das B Tarafdar and R N Jana ldquoHall effects on unsteadyMHD rotating flow past a periodically accelerated porousplate with slippagerdquo European Journal of MechanicsmdashBFluids vol 72 pp 135ndash143 2018

[15] R Gayathri A Govindarajan and R Sasikala ldquo+ree-di-mensional Couette flow of dusty fluid with heat transfer in thepresence of magnetic fieldrdquo Journal of Physics ConferenceSeries vol 1000 no 1 Article ID 012147 2018

[16] R Nandkeolyar M Narayana S S Motsa and P SibandaldquoMagnetohydrodynamic mixed convective flow due to avertical plate with induced magnetic fieldrdquo Journal of AermalScience and Engineering Applications vol 10 no 6 Article ID061005 2018

[17] Y Swapna M C Raju R P Sharma and S V K VarmaldquoChemical reaction thermal radiation and injectionsuctioneffects on MHD mixed convective oscillatory flow through aporous medium bounded by two vertical porous platesrdquoBulletin of the Calcutta Mathematical Society vol 109 no 3pp 189ndash210 2017

[18] S Agarwalla and N Ahmed ldquoMHDmass transfer flow past aninclined plate with variable temperature and plate velocityembedded in a porous mediumrdquo Heat Transfer-Asian Re-search vol 47 no 1 pp 27ndash41 2018

[19] K Chand and N +akur ldquoEffects of rotation radiation andHall current on MHD flow of A viscoelastic fluid past aninfinite vertical porous plate through porous medium withheat absorption chemical reaction and variable suctionrdquo AeJournal of the Indian Mathematical Society vol 85 no 1-2pp 16ndash31 2018

[20] A Al-Mudhaf and A J Chamkha ldquoSimilarity solutions forMHD thermosolutal Marangoni convection over a flat surfacein the presence of heat generation or absorption effectsrdquoHeatand Mass Transfer vol 42 no 2 pp 112ndash121 2005

[21] A Chamkha ldquoMHD flow of a micropolar fluid past astretched permeable surface with heat generation or ab-sorptionrdquo Nonlinear Analysis Modelling and Control vol 14no 1 pp 27ndash40 2009

[22] H S Takhar A J Chamkha and G Nath ldquoUnsteady flow andheat transfer on a semi-infinite flat plate with an alignedmagnetic fieldrdquo International Journal of Engineering Sciencevol 37 no 13 pp 1723ndash1736 1999

[23] A J Chamkha ldquoCoupled heat and mass transfer by naturalconvection about a truncated cone in the presence of magneticfield and radiation effectsrdquo Numerical Heat Transfer Appli-cations vol 39 no 5 pp 511ndash530 2001

[24] M M Bhatti M A Abbas and M M Rashidi ldquoA robustnumerical method for solving stagnation point flow over apermeable shrinking sheet under the influence of MHDrdquoApplied Mathematics and Computation vol 316 pp 381ndash3892018

[25] M M Bhatti R Ellahi and A Zeeshan ldquoStudy of variablemagnetic field on the peristaltic flow of Jeffrey fluid in a non-uniform rectangular duct having compliant wallsrdquo Journal ofMolecular Liquids vol 222 pp 101ndash108 2016

[26] G D Gupta and R Johari ldquoMHD three dimensional flow pasta porous platerdquo Indian Journal of Pure and Applied Mathe-matics vol 32 no 3 pp 377ndash386 2001

[27] M Guria and R N Jana ldquoHydrodynamic effect on the three-dimensional flow past a vertical porous platerdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2005no 20 pp 3359ndash3372 2005

[28] H P Greenspan and G F Carrier ldquo+e magnetohydrody-namic flow past a flat platerdquo Journal of Fluid Mechanics vol 6no 1 pp 77ndash96 1959

[29] V J Rossow ldquoOn flow of electrically conducting fluids over aflat plate in the presence of a transverse magnetic fieldrdquoNACA Report No 1358 1958

[30] K D Singh ldquo+ree dimensional MHD oscillatory flow past aporous platerdquo ZAMM Journal of Applied Mathematics andMechanicsZeitschrift fr Angewandte Mathematik undMechanik vol 71 no 3 pp 192ndash195 1991

[31] A M Siddiqui M Shoaib and M A Rana ldquo+ree-di-mensional flow of Jeffrey fluid along an infinite plane wallwith periodic suctionrdquo Meccanica vol 52 no 11-12pp 2705ndash2714 2017

[32] M Shoaib M A Rana and A M Siddiqui ldquo+e effect of slipcondition on the three-dimensional flow of Jeffrey fluid alonga plane wall with periodic suctionrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 39 no 7pp 2495ndash2503 2017

[33] M A Rana Y Ali M Shoaib and M Numan ldquoMagneto-hydrodynamic three-dimensional Couette flow of a second-grade fluid with sinusoidal injectionsuctionrdquo Journal ofEngineering Aermophysics vol 28 no 1 pp 138ndash162 2019

[34] Y Ali M A Rana and M Shoaib ldquoMagnetohydrodynamicthree-dimensional Couette flow of a maxwell fluid with pe-riodic injectionsuctionrdquo Mathematical Problems in Engi-neering vol 2017 Article ID 1859693 19 pages 2017

[35] M Umar R Akhtar Z Sabir et al ldquoNumerical treatment forthe three-dimensional eyring-powell fluid flow over astretching sheet with velocity slip and activation energyrdquoAdvances in Mathematical Physics vol 2019 Article ID9860471 12 pages 2019

[36] K Yousefi and R Saleh ldquo+ree-dimensional suction flowcontrol and suction jet length optimization of NACA 0012wingrdquo Meccanica vol 50 no 6 pp 1481ndash1494 2015

[37] H Zhang S Chen Q Meng and S Wang ldquoFlow separationcontrol using unsteady pulsed suction through endwallbleeding holes in a highly loaded compressor cascaderdquoAerospace Science and Technology vol 72 pp 455ndash464 2018

[38] S Koganezawa A Mitsuishi T Shimura K IwamotoH Mamori and A Murata ldquoPathline analysis of travelingwavy blowing and suction control in turbulent pipe flow fordrag reductionrdquo International Journal of Heat and Fluid Flowvol 77 pp 388ndash401 2019

[39] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded in aporous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Aermophysical Engineering vol 15 no 2pp 81ndash94 2011

[40] P S Reddy and A J Chamkha ldquoSoret and Dufour effects onMHD convective flow of Al2O3ndashwater and TiO2ndashwaternanofluids past a stretching sheet in porous media with heatgenerationabsorptionrdquo Advanced Powder Technologyvol 27 no 4 pp 1207ndash1218 2016

[41] A J Chamkha S Abbasbandy A M Rashad andK Vajravelu ldquoRadiation effects on mixed convection about acone embedded in a porous medium filled with a nanofluidrdquoMeccanica vol 48 no 2 pp 275ndash285 2013

[42] A J Chamkha C Issa and K Khanafer ldquoNatural convectionfrom an inclined plate embedded in a variable porosity porousmedium due to solar radiationrdquo International Journal ofAermal Sciences vol 41 no 1 pp 73ndash81 2002

[43] A J Chamkha R A Mohamed and S E Ahmed ldquoUnsteadyMHD natural convection from a heated vertical porous plate


in a micropolar fluid with Joule heating chemical reactionand radiation effectsrdquo Meccanica vol 46 no 2 pp 399ndash4112011

[44] A J Chamkha and A R A Khaled ldquoSimilarity solutions forhydromagnetic mixed convection heat and mass transfer forHiemenz flow through porous mediardquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 10 no 1pp 94ndash115 2000

[45] A Chamkha ldquoSolar radiation assisted natural convection inuniform porous medium supported by a vertical flat platerdquoJournal of Heat Transfer vol 119 no 1 pp 89ndash96 1997

[46] M A Abbas Y Q Bai M M Bhatti and M M Rashidildquo+ree dimensional peristaltic flow of hyperbolic tangent fluidin non-uniform channel having flexible wallsrdquo AlexandriaEngineering Journal vol 55 no 1 pp 653ndash662 2016

[47] M Bhatti and D Lu ldquoAnalytical study of the head-on collisionprocess between hydroelastic solitary waves in the presence ofa uniform currentrdquo Symmetry vol 11 no 3 p 333 2019

[48] R Jhorar D Tripathi M M Bhatti and R Ellahi ldquoElec-troosmosis modulated biomechanical transport throughasymmetric microfluidics channelrdquo Indian Journal of Physicsvol 92 no 10 pp 1229ndash1238 2018

[49] H Schlichting Boundary Layer Aeory Mcgraw-Hill BookCo New York NY USA 1968

[50] AW Bush PerturbationMethods for Engineers and ScientistsCRC Library of Engineering Mathematics Boca Raton FLUSA 1992

[51] D Armbruster Perturbation Methods Bifurcation Aeory andComputer Algebraic Springer Berlin Germany 1987

[52] M H Holmes Introduction to Perturbation MethodsSpringer-Verlag Berlin Germany 2013


Hindawiwwwhindawicom Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Researchers have extensively exploited the field ofmagnetohydrodynamics with reference to geometricalconfiguration various types of formulations and dif-ferent dimensions using analytical and numericalmethods [7ndash19] +e authors investigated the influence ofthe magnetic field on different structures of flow di-mensions for distributions of concentrations and tem-peratures Moreover the magnetic parameter is proved tobe a controlling parameter to restrain the fluid and heatflows under the closed spaces Different types of non-Newtonian fluids bargain applications in many magne-tohydrodynamic devices and in power generation as wellMagnetohydrodynamics with different types of convec-tive flows heat generationabsorption and transferanalysis [20ndash25] have been investigated in detail

Gupta and Johari [26] presented the analysis of 3Dmagnetohydrodynamic flow across a porous plane walland the fluid taken into account is laminar having thepower of conducting heat +e authors considered amagnetic flux in the normal direction to the plateMoreover Guria and Jana [27] considered a verticalporous wall for the three-dimensional hydrodynamicfluid flow problem Furthermore Greenspan and Carrier[28] Rossow [29] and Singh [30] presented their studiesextensively on the magnetohydrodynamic impacts on theflow across a plane wall based on injection or suction+ere are some non-Newtonian models presenting thethree-dimensional fluidic problems though a porous wallwith variable suction [31ndash35] Periodic suction has re-ceived very much attention in the field of aerodynamics[36ndash38] +e porous medium has very much importancein the fluid dynamics [39ndash45] Abbas et al [46] studied3D peristaltic flow fluid with hyperbolic tangent in thenonuniform channel along flexible walls Bhatti and Lu[47] studied analytical analysis of the head-on collisionmechanism among hydroelastic solitary waves withuniform current Jhorar et al [48] analyzed the micro-fluid in the channel for the electroosmosis-modulatedbiomechanical transport

+e motivation of this study is to analyze the 3D MHDflow of the simplest non-Newtonian second-grade fluidicmodel through a semi-infinite wall which is based on suc-cession of waves or curves with fluctuating velocity distri-bution A uniform suction velocity along the surface of theplane wall transforms the problem into a 2D asymptoticsuction velocity solution [49] however because of thevariable suction velocity distribution in the normal di-rection the fluidic system turns out to be 3D +e analyticalperturbation technique is incorporated for finding the seriessolution of this problem +e proposed outcomes are esti-mated for different parameters of interest such as the suctionparameter α second-grade parameter K Reynolds numberRe and Hartmann number M +is article is organized asfollows Section 2 consists of the statement of the problemSection 3 presents the perturbation method Section 4specifies the design of the problem Sections 5 and 6 describeanalytical solutions Section 7 integrates results and dis-cussion and Section 8 contains conclusions In addition tothese appendices and nomenclature are given thereafter

2 Statement of the Problem

In this problem the 3D steady laminar magnetohydrody-namic flow of an incompressible non-Newtonian second-grade fluid passing through an infinite plate is considered+e xlowastzlowast plane is considered where ylowast-axis is normal to theplane (see Figure 1) A time-independent distribution is abasic steady distribution [1] in which l and ε indicate thewavelength and amplitude of the variable suction velocityrespectively+us variable suction velocity has the followingform

v zlowast

( 1113857 minus v0 1 + ε cos πzlowast

l1113888 1113889 (1)

+e fluidic system becomes two-dimensional because ofconstant suction velocity whereas it is three-dimensional incase of variable suction velocity [49] A constant magneticfield B0 in the normal direction to the xlowastzlowast wall is appliedAlso the following are considered (i) the fluid has electricconduction (ii) the fluid has steady and laminar flow (iii)the fluid has uniform free stream velocity (iv) the magneticReynolds number is at small scale and also the inducedmagnetic field is inconsiderable (v) Hall and polarizationeffects are neglected and (vi) all physical properties of theparameters are independent of xlowast because of the infiniteextended length of the plate in the xlowast direction but the flowremains 3D because of the variable suction velocity (1)

3 Perturbation Method

Perturbation methods [50ndash52] are strong mathematical tools tofind the seriesapproximate solutions of those problems whoseanalytic or exact solutions are not possible or hard to find+esetechniques have frequently been used for the problems arisingin the fields of engineering and science +e function u(y z ε)is convoluted in physical problems and then it can be shownmathematically by the differential equation

L(u y z ε) 0 (2)

subject to the boundary condition

B(u ε) 0 (3)

where y is a vector or scalar independent variable and ε is aparameter One cannot solve this problem exactly in generalHowever if there exists an ε ε0 (ε can be scaled so that ε 0)for which the above problem can be solved exactly then oneexplores to obtain the solution for small ε in the form

u(y z ε) u0(y) + εu1(y z) + ε2u2(y z) + middot middot middot (4)

where un (n 0 1 2 ) does not depend on ε and u0(y) isthe solution of the problem for ε 0 One then substitutesthis expansion into equations (2) and (3) which expands forsmall ε and collects coefficients of each power of ε Sincethese equations must hold for all values of ε each coefficientof ϵ must vanish independently because sequences of ϵ arelinearly independent +ese usually are simpler equationsgoverning un (n 0 1 2 ) which can be solvedsuccessively




zwlowast

zzlowast+

zvlowast

zylowast 0 (5)

ρ wlowastzulowast


zylowast1113888 1113889 μ

z2ulowast

zylowast2 +

z2ulowast

zzlowast21113890 1113891

+ α1 vlowastz

3ulowast


zylowast2

zzlowast1113890

+ vlowast z3ulowast


3ulowast


20ulowast

(6)

ρ wlowastzvlowast



zplowast

zy+ μ

z2vlowast

zylowast2 +

z2vlowast

zzlowast21113890 1113891

+ α1 vlowastz

3vlowast


zylowast2zzlowast

1113890

+ vlowast z3vlowast


3vlowast

zzlowast3 +

zvlowast

zzlowastz2vlowast

zylowast zzlowast

+zulowast

zzlowastz2ulowast


zvlowast

zylowastz2vlowast

zylowast2

+zvlowast

zzlowastz2wlowast

zylowast2 + 2

zulowast

zylowastz2ulowast

zylowast2

+ 2zwlowast

zylowastz2wlowast

zylowast2 +

zulowast

zylowastz2ulowast

zzlowast2

+zvlowast

zylowastz2vlowast

zzlowast21113891

(7)

ρ vlowastzwlowast



zplowast

zzlowast+ μ

z2wlowast

zylowast2 +

z2wlowast

zzlowast21113890 1113891

+ α1 wlowastz

3vlowast


3wlowast

zylowast31113890

+ vlowast z3wlowast


3wlowast

zzlowast3

+zwlowast

zylowastz2wlowast

zylowast zzlowast+

zulowast

zylowastz2ulowast

zylowast zzlowast

+ 5zwlowast

zzlowastz2wlowast

zzlowast2 +

zwlowast

zylowastz2vlowast

zzlowast2

+ 2zulowast

zzlowastz2ulowast

zzlowast2 + 2

zvlowast

zzlowastz2vlowast

zzlowast2

+zulowast

zzlowastz2ulowast

zylowast2 +

zwlowast

zzlowastz2wlowast


20wlowast

(8)


ulowast

0 vlowast


l1113888 1113889 w

lowast 0 at y

lowast 0

ulowast

U vlowast

minus v0 wlowast

0 plowast

plowastinfin as y

lowast ⟶infin

(9)


x xlowast

l

y ylowast

l

z zlowast

l

u ulowast

U

v vlowast

U

w wlowast

U

p plowast

ρU2

(10)


υ (z )

x

y

z


B0



zv

zy+

zw

zz 0 (11)

wzu

zz+ v

zu

zy

1Re

z2u

zy2z2u

zz21113890 1113891

+ K1113890wz3u

zy2 zz+ v

z3u

zy3 + vz3u

zy zz2

+ wz3u

zz31113891 minus Mu

(12)

wzv

zz+ v

zv

zy minus

zp

zy+

1Re

z2v

zy2 +z2v

zz21113890 1113891

+ K vz3v

zy3 + wz3v

zy2 zz+ v

z3v

zy zz21113890

+ wz3v

zz3 +zv

zz

z2v

zy zz+

zu

zz

z2u

zy zz

+ 5zv

zy

z2v

zy2 +zv

zz

z2w

zy2 + 2zu

zy

z2u

zy2

+ 2zw

zy

z2w

zy2 +zu

zy

z2u

zz2 +zv

zy

z2v

zz21113891

(13)

wzw

zz+ v

zw

zy minus

zp

zz+

1Re

z2w

zy2 +z2w

zz21113890 1113891

+ K wz3v

zy3 + vz3w

zy3 + vz3w

zy zz2 + wz3w

zz31113890

+zw

zy

z2w

zy zz+

zu

zy

z2u

zy zz+ 5

zw

zz

z2w

zz2

+zw

zy

z2v

zz2 + 2zu

zz

z2u

zz2 + 2zv

zz

z2v

zz2 +zu

zz

z2u

zy2

+zw

zz

z2w

zy21113891 minus Mw

(14)




(15)

where

Re Ul

]

α v0U

M σB2

0l

ρU

K α1ρl2

(16)






(17)


KαRe

d3u0

dy3 minusd2u0

dy21113888 1113889 minus αRe

du0

dy+ MReu0 0 (18)


u0 0 aty 0



u0 u00 + Ku01 + O K2

1113872 1113873 (20)



d2u00

dy2 + αRe

du00

dyminus MReu00 0

u00(0) 0

u00(infin) 1

(21)

d2u01

dy2 minus αRe

d3u00

dy3 + αRe

du01

dyminus MReu01 0

u01(0) 0

u01(infin) 0

(22)



(23)

u01(y) αReS

31


minus S1y (24)



+ KαReS

31


minus S1y (25)


zv1

zy+

zw1

zz 0 (26)


minus αzu1

zy+ v1

zu0

zy

1Re

z2u1

zy2 +z2u1

zz21113888 1113889


zy3 minus αz3u1

zy zz2

+ v1z3u0


(27)

minus αzv1

zy minus

zp1

zy+

1Re

z2v1

zy2 +z2v1

zz21113888 1113889

minus Kαz3v1

zy3 +z3v1

zy zz21113888 1113889

(28)

minus αzw1

zy minus

zp1

zz+

1Re

z2w1

zy2 +z2w1

zz21113888 1113889

minus Kαz3w1

zy3 +z3w1

zy zz21113888 1113889 minus Mw1

(29)



u1 0 v1 0 w1 0 asy⟶infin(30)



v1(y z) v11(y)cos π(z) (31)

w1(y z) minus1π


p1(y z) p11(y)cos π(z) (33)



(34)



Prime11


(35)





v11 0

(36)


v11(0) minus α

v11prime (0) 0(37)

We assume that

v11 v110 + Kv111 + O K2

1113872 1113873

p11 p110 + Kp111 + O K2

1113872 1113873(38)


v110(0) minus α

v111prime (0) v111(0) v110prime (0) 0(39)


v11 α

λ1 minus λ2( 1113857λ2e


minus λ2y1113872 1113873

minus Kα2Re

A1 minus A2

λ1 minus λ21113888 1113889e

minus λ1y1113888

minusA1 minus A2

λ1 minus λ21113888 1113889e

minus λ2y+ y A1e

minus λ1yminus A2e

minus λ2y1113872 11138731113889

(40)


v1(y z) α

λ1 minus λ2( 1113857λ2e


minus λ2y1113872 1113873


λ1 minus λ21113888 1113889e

minus λ1y

minusA1 minus A2

λ1 minus λ21113888 1113889e

minus λ2y

+ y A1eminus λ1y

minus A2eminus λ2y

1113872 11138731113889cos πz

(41)

w1(y z) αλ1λ2


minus λ2y+ e

minus λ1y1113872 1113873 +

Kα2Re

π



A1 minus A2




⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠sin πz (42)




u1(y z) u11(y)cos π(z) (43)


u11 0 aty 0



21113872 1113873 (45)


u111 u110 0 aty 0




+KαReS

31

αRe minus 2S1ye

minus S1y+ ε

αReS1

λ1 minus λ2( 1113857


minus λ2+m( )y


⎛⎜⎜⎝ ⎞⎟⎟⎠cos πz

+ εKαReS1



+ εKαReS1

λ2 minus λ1( 1113857

minusF1





⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

cos πz

(47)
















00 02

00

05

10

ndash0504 06 08 10

y

u(y)

α = 01α = 02α = 05


00 02 04 06 08 10y

ndash4

ndash3

ndash2

ndash1

0

1

u(y)

K = 01K = 03K = 05


00 02 04 06 08 10y

00

02

04

06

08

10

u(y)

M = 2M = 4M = 8


00 02 04 06 08 10y

00

02

04

06

08

10

u(y)

Re = 10Re = 20Re = 30







Nomenclature




00 05 10 15 20y

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

v(y)

α = 01α = 05α = 09


00

00

05 10 15

ndash10

ndash05

ndash15

20y

v(y)

K = 01K = 05K = 09


00ndash020

ndash015

ndash010

ndash005

02 04 06 08 10y

v(y)

M = 2M = 4M = 6


00

00

05 10 15 20y

ndash020

ndash015

ndash010

ndash005

v(y)

Re = 10Re = 20Re = 30



yK 01α 01M 2

K 01α 05M 2

K 05α 05M 2

K 05α 05M 4

00 00 00 00 0005 0003216 0016261 0016228 001607110 0001352 0006597 0006889 000701115 0000442 0002076 0002201 000232220 0000133 0000599 0000630 0000698


Greek symbols


Appendix

S1 αRe +

αRe( 11138572

+ 4ReM

1113969

2

S2 αRe minus

αRe( 11138572

+ 4ReM

1113969

2

λ1 S1 +

S1( 11138572

+ 4π21113969

2

λ2 S2 +

S2( 11138572

+ 4π21113969

2

λ3 S1 minus

S1( 11138572

+ 4π21113969

2

λ4 S2 minus

S2( 11138572

+ 4π21113969

2

A1 λ1λ2 λ21 minus π21113872 1113873

2


A2 λ1λ2 λ22 minus π21113872 1113873


A3 λ2

λ1 + S1( 11138572


A4 λ1

λ2 + S1( 11138572


X1 minus α2Reλ2

A4 minus A3( 1113857


αminus

S31λ2α2Re

αRe minus 2S11113888 1113889

X2 λ2S21α

2Re


S1 A1 minus A2( 1113857

α


2A3

X3 minusλ1S21α

2Re


S1 A1 minus A2( 1113857

α

+ 2λ1αS21 + α2ReS1 λ2 + S1( 1113857

2A4

X5 S1 λ1 minus λ2( 1113857A2

α+

S31λ1α2Re

αRe minus 2S11113888 1113889


π minus λ

(A1)

Data Availability




References

































































Journal of












Journal of







Volume 2018






Volume 2018










zwlowast

zzlowast+

zvlowast

zylowast 0 (5)

ρ wlowastzulowast


zylowast1113888 1113889 μ

z2ulowast

zylowast2 +

z2ulowast

zzlowast21113890 1113891

+ α1 vlowastz

3ulowast


zylowast2

zzlowast1113890

+ vlowast z3ulowast


3ulowast


20ulowast

(6)

ρ wlowastzvlowast



zplowast

zy+ μ

z2vlowast

zylowast2 +

z2vlowast

zzlowast21113890 1113891

+ α1 vlowastz

3vlowast


zylowast2zzlowast

1113890

+ vlowast z3vlowast


3vlowast

zzlowast3 +

zvlowast

zzlowastz2vlowast

zylowast zzlowast

+zulowast

zzlowastz2ulowast


zvlowast

zylowastz2vlowast

zylowast2

+zvlowast

zzlowastz2wlowast

zylowast2 + 2

zulowast

zylowastz2ulowast

zylowast2

+ 2zwlowast

zylowastz2wlowast

zylowast2 +

zulowast

zylowastz2ulowast

zzlowast2

+zvlowast

zylowastz2vlowast

zzlowast21113891

(7)

ρ vlowastzwlowast



zplowast

zzlowast+ μ

z2wlowast

zylowast2 +

z2wlowast

zzlowast21113890 1113891

+ α1 wlowastz

3vlowast


3wlowast

zylowast31113890

+ vlowast z3wlowast


3wlowast

zzlowast3

+zwlowast

zylowastz2wlowast

zylowast zzlowast+

zulowast

zylowastz2ulowast

zylowast zzlowast

+ 5zwlowast

zzlowastz2wlowast

zzlowast2 +

zwlowast

zylowastz2vlowast

zzlowast2

+ 2zulowast

zzlowastz2ulowast

zzlowast2 + 2

zvlowast

zzlowastz2vlowast

zzlowast2

+zulowast

zzlowastz2ulowast

zylowast2 +

zwlowast

zzlowastz2wlowast


20wlowast

(8)


ulowast

0 vlowast


l1113888 1113889 w

lowast 0 at y

lowast 0

ulowast

U vlowast

minus v0 wlowast

0 plowast

plowastinfin as y

lowast ⟶infin

(9)


x xlowast

l

y ylowast

l

z zlowast

l

u ulowast

U

v vlowast

U

w wlowast

U

p plowast

ρU2

(10)


υ (z )

x

y

z


B0



zv

zy+

zw

zz 0 (11)

wzu

zz+ v

zu

zy

1Re

z2u

zy2z2u

zz21113890 1113891

+ K1113890wz3u

zy2 zz+ v

z3u

zy3 + vz3u

zy zz2

+ wz3u

zz31113891 minus Mu

(12)

wzv

zz+ v

zv

zy minus

zp

zy+

1Re

z2v

zy2 +z2v

zz21113890 1113891

+ K vz3v

zy3 + wz3v

zy2 zz+ v

z3v

zy zz21113890

+ wz3v

zz3 +zv

zz

z2v

zy zz+

zu

zz

z2u

zy zz

+ 5zv

zy

z2v

zy2 +zv

zz

z2w

zy2 + 2zu

zy

z2u

zy2

+ 2zw

zy

z2w

zy2 +zu

zy

z2u

zz2 +zv

zy

z2v

zz21113891

(13)

wzw

zz+ v

zw

zy minus

zp

zz+

1Re

z2w

zy2 +z2w

zz21113890 1113891

+ K wz3v

zy3 + vz3w

zy3 + vz3w

zy zz2 + wz3w

zz31113890

+zw

zy

z2w

zy zz+

zu

zy

z2u

zy zz+ 5

zw

zz

z2w

zz2

+zw

zy

z2v

zz2 + 2zu

zz

z2u

zz2 + 2zv

zz

z2v

zz2 +zu

zz

z2u

zy2

+zw

zz

z2w

zy21113891 minus Mw

(14)




(15)

where

Re Ul

]

α v0U

M σB2

0l

ρU

K α1ρl2

(16)






(17)


KαRe

d3u0

dy3 minusd2u0

dy21113888 1113889 minus αRe

du0

dy+ MReu0 0 (18)


u0 0 aty 0



u0 u00 + Ku01 + O K2

1113872 1113873 (20)



d2u00

dy2 + αRe

du00

dyminus MReu00 0

u00(0) 0

u00(infin) 1

(21)

d2u01

dy2 minus αRe

d3u00

dy3 + αRe

du01

dyminus MReu01 0

u01(0) 0

u01(infin) 0

(22)



(23)

u01(y) αReS

31


minus S1y (24)



+ KαReS

31


minus S1y (25)


zv1

zy+

zw1

zz 0 (26)


minus αzu1

zy+ v1

zu0

zy

1Re

z2u1

zy2 +z2u1

zz21113888 1113889


zy3 minus αz3u1

zy zz2

+ v1z3u0


(27)

minus αzv1

zy minus

zp1

zy+

1Re

z2v1

zy2 +z2v1

zz21113888 1113889

minus Kαz3v1

zy3 +z3v1

zy zz21113888 1113889

(28)

minus αzw1

zy minus

zp1

zz+

1Re

z2w1

zy2 +z2w1

zz21113888 1113889

minus Kαz3w1

zy3 +z3w1

zy zz21113888 1113889 minus Mw1

(29)



u1 0 v1 0 w1 0 asy⟶infin(30)



v1(y z) v11(y)cos π(z) (31)

w1(y z) minus1π


p1(y z) p11(y)cos π(z) (33)



(34)



Prime11


(35)





v11 0

(36)


v11(0) minus α

v11prime (0) 0(37)

We assume that

v11 v110 + Kv111 + O K2

1113872 1113873

p11 p110 + Kp111 + O K2

1113872 1113873(38)


v110(0) minus α

v111prime (0) v111(0) v110prime (0) 0(39)


v11 α

λ1 minus λ2( 1113857λ2e


minus λ2y1113872 1113873

minus Kα2Re

A1 minus A2

λ1 minus λ21113888 1113889e

minus λ1y1113888

minusA1 minus A2

λ1 minus λ21113888 1113889e

minus λ2y+ y A1e

minus λ1yminus A2e

minus λ2y1113872 11138731113889

(40)


v1(y z) α

λ1 minus λ2( 1113857λ2e


minus λ2y1113872 1113873


λ1 minus λ21113888 1113889e

minus λ1y

minusA1 minus A2

λ1 minus λ21113888 1113889e

minus λ2y

+ y A1eminus λ1y

minus A2eminus λ2y

1113872 11138731113889cos πz

(41)

w1(y z) αλ1λ2


minus λ2y+ e

minus λ1y1113872 1113873 +

Kα2Re

π



A1 minus A2




⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠sin πz (42)




u1(y z) u11(y)cos π(z) (43)


u11 0 aty 0



21113872 1113873 (45)


u111 u110 0 aty 0




+KαReS

31

αRe minus 2S1ye

minus S1y+ ε

αReS1

λ1 minus λ2( 1113857


minus λ2+m( )y


⎛⎜⎜⎝ ⎞⎟⎟⎠cos πz

+ εKαReS1



+ εKαReS1

λ2 minus λ1( 1113857

minusF1





⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

cos πz

(47)
















00 02

00

05

10

ndash0504 06 08 10

y

u(y)

α = 01α = 02α = 05


00 02 04 06 08 10y

ndash4

ndash3

ndash2

ndash1

0

1

u(y)

K = 01K = 03K = 05


00 02 04 06 08 10y

00

02

04

06

08

10

u(y)

M = 2M = 4M = 8


00 02 04 06 08 10y

00

02

04

06

08

10

u(y)

Re = 10Re = 20Re = 30







Nomenclature




00 05 10 15 20y

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

v(y)

α = 01α = 05α = 09


00

00

05 10 15

ndash10

ndash05

ndash15

20y

v(y)

K = 01K = 05K = 09


00ndash020

ndash015

ndash010

ndash005

02 04 06 08 10y

v(y)

M = 2M = 4M = 6


00

00

05 10 15 20y

ndash020

ndash015

ndash010

ndash005

v(y)

Re = 10Re = 20Re = 30



yK 01α 01M 2

K 01α 05M 2

K 05α 05M 2

K 05α 05M 4

00 00 00 00 0005 0003216 0016261 0016228 001607110 0001352 0006597 0006889 000701115 0000442 0002076 0002201 000232220 0000133 0000599 0000630 0000698


Greek symbols


Appendix

S1 αRe +

αRe( 11138572

+ 4ReM

1113969

2

S2 αRe minus

αRe( 11138572

+ 4ReM

1113969

2

λ1 S1 +

S1( 11138572

+ 4π21113969

2

λ2 S2 +

S2( 11138572

+ 4π21113969

2

λ3 S1 minus

S1( 11138572

+ 4π21113969

2

λ4 S2 minus

S2( 11138572

+ 4π21113969

2

A1 λ1λ2 λ21 minus π21113872 1113873

2


A2 λ1λ2 λ22 minus π21113872 1113873


A3 λ2

λ1 + S1( 11138572


A4 λ1

λ2 + S1( 11138572


X1 minus α2Reλ2

A4 minus A3( 1113857


αminus

S31λ2α2Re

αRe minus 2S11113888 1113889

X2 λ2S21α

2Re


S1 A1 minus A2( 1113857

α


2A3

X3 minusλ1S21α

2Re


S1 A1 minus A2( 1113857

α

+ 2λ1αS21 + α2ReS1 λ2 + S1( 1113857

2A4

X5 S1 λ1 minus λ2( 1113857A2

α+

S31λ1α2Re

αRe minus 2S11113888 1113889


π minus λ

(A1)

Data Availability




References

































































Journal of












Journal of







Volume 2018






Volume 2018








zv

zy+

zw

zz 0 (11)

wzu

zz+ v

zu

zy

1Re

z2u

zy2z2u

zz21113890 1113891

+ K1113890wz3u

zy2 zz+ v

z3u

zy3 + vz3u

zy zz2

+ wz3u

zz31113891 minus Mu

(12)

wzv

zz+ v

zv

zy minus

zp

zy+

1Re

z2v

zy2 +z2v

zz21113890 1113891

+ K vz3v

zy3 + wz3v

zy2 zz+ v

z3v

zy zz21113890

+ wz3v

zz3 +zv

zz

z2v

zy zz+

zu

zz

z2u

zy zz

+ 5zv

zy

z2v

zy2 +zv

zz

z2w

zy2 + 2zu

zy

z2u

zy2

+ 2zw

zy

z2w

zy2 +zu

zy

z2u

zz2 +zv

zy

z2v

zz21113891

(13)

wzw

zz+ v

zw

zy minus

zp

zz+

1Re

z2w

zy2 +z2w

zz21113890 1113891

+ K wz3v

zy3 + vz3w

zy3 + vz3w

zy zz2 + wz3w

zz31113890

+zw

zy

z2w

zy zz+

zu

zy

z2u

zy zz+ 5

zw

zz

z2w

zz2

+zw

zy

z2v

zz2 + 2zu

zz

z2u

zz2 + 2zv

zz

z2v

zz2 +zu

zz

z2u

zy2

+zw

zz

z2w

zy21113891 minus Mw

(14)




(15)

where

Re Ul

]

α v0U

M σB2

0l

ρU

K α1ρl2

(16)






(17)


KαRe

d3u0

dy3 minusd2u0

dy21113888 1113889 minus αRe

du0

dy+ MReu0 0 (18)


u0 0 aty 0



u0 u00 + Ku01 + O K2

1113872 1113873 (20)



d2u00

dy2 + αRe

du00

dyminus MReu00 0

u00(0) 0

u00(infin) 1

(21)

d2u01

dy2 minus αRe

d3u00

dy3 + αRe

du01

dyminus MReu01 0

u01(0) 0

u01(infin) 0

(22)



(23)

u01(y) αReS

31


minus S1y (24)



+ KαReS

31


minus S1y (25)


zv1

zy+

zw1

zz 0 (26)


minus αzu1

zy+ v1

zu0

zy

1Re

z2u1

zy2 +z2u1

zz21113888 1113889


zy3 minus αz3u1

zy zz2

+ v1z3u0


(27)

minus αzv1

zy minus

zp1

zy+

1Re

z2v1

zy2 +z2v1

zz21113888 1113889

minus Kαz3v1

zy3 +z3v1

zy zz21113888 1113889

(28)

minus αzw1

zy minus

zp1

zz+

1Re

z2w1

zy2 +z2w1

zz21113888 1113889

minus Kαz3w1

zy3 +z3w1

zy zz21113888 1113889 minus Mw1

(29)



u1 0 v1 0 w1 0 asy⟶infin(30)



v1(y z) v11(y)cos π(z) (31)

w1(y z) minus1π


p1(y z) p11(y)cos π(z) (33)



(34)



Prime11


(35)





v11 0

(36)


v11(0) minus α

v11prime (0) 0(37)

We assume that

v11 v110 + Kv111 + O K2

1113872 1113873

p11 p110 + Kp111 + O K2

1113872 1113873(38)


v110(0) minus α

v111prime (0) v111(0) v110prime (0) 0(39)


v11 α

λ1 minus λ2( 1113857λ2e


minus λ2y1113872 1113873

minus Kα2Re

A1 minus A2

λ1 minus λ21113888 1113889e

minus λ1y1113888

minusA1 minus A2

λ1 minus λ21113888 1113889e

minus λ2y+ y A1e

minus λ1yminus A2e

minus λ2y1113872 11138731113889

(40)


v1(y z) α

λ1 minus λ2( 1113857λ2e


minus λ2y1113872 1113873


λ1 minus λ21113888 1113889e

minus λ1y

minusA1 minus A2

λ1 minus λ21113888 1113889e

minus λ2y

+ y A1eminus λ1y

minus A2eminus λ2y

1113872 11138731113889cos πz

(41)

w1(y z) αλ1λ2


minus λ2y+ e

minus λ1y1113872 1113873 +

Kα2Re

π



A1 minus A2




⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠sin πz (42)




u1(y z) u11(y)cos π(z) (43)


u11 0 aty 0



21113872 1113873 (45)


u111 u110 0 aty 0




+KαReS

31

αRe minus 2S1ye

minus S1y+ ε

αReS1

λ1 minus λ2( 1113857


minus λ2+m( )y


⎛⎜⎜⎝ ⎞⎟⎟⎠cos πz

+ εKαReS1



+ εKαReS1

λ2 minus λ1( 1113857

minusF1





⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

cos πz

(47)
















00 02

00

05

10

ndash0504 06 08 10

y

u(y)

α = 01α = 02α = 05


00 02 04 06 08 10y

ndash4

ndash3

ndash2

ndash1

0

1

u(y)

K = 01K = 03K = 05


00 02 04 06 08 10y

00

02

04

06

08

10

u(y)

M = 2M = 4M = 8


00 02 04 06 08 10y

00

02

04

06

08

10

u(y)

Re = 10Re = 20Re = 30







Nomenclature




00 05 10 15 20y

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

v(y)

α = 01α = 05α = 09


00

00

05 10 15

ndash10

ndash05

ndash15

20y

v(y)

K = 01K = 05K = 09


00ndash020

ndash015

ndash010

ndash005

02 04 06 08 10y

v(y)

M = 2M = 4M = 6


00

00

05 10 15 20y

ndash020

ndash015

ndash010

ndash005

v(y)

Re = 10Re = 20Re = 30



yK 01α 01M 2

K 01α 05M 2

K 05α 05M 2

K 05α 05M 4

00 00 00 00 0005 0003216 0016261 0016228 001607110 0001352 0006597 0006889 000701115 0000442 0002076 0002201 000232220 0000133 0000599 0000630 0000698


Greek symbols


Appendix

S1 αRe +

αRe( 11138572

+ 4ReM

1113969

2

S2 αRe minus

αRe( 11138572

+ 4ReM

1113969

2

λ1 S1 +

S1( 11138572

+ 4π21113969

2

λ2 S2 +

S2( 11138572

+ 4π21113969

2

λ3 S1 minus

S1( 11138572

+ 4π21113969

2

λ4 S2 minus

S2( 11138572

+ 4π21113969

2

A1 λ1λ2 λ21 minus π21113872 1113873

2


A2 λ1λ2 λ22 minus π21113872 1113873


A3 λ2

λ1 + S1( 11138572


A4 λ1

λ2 + S1( 11138572


X1 minus α2Reλ2

A4 minus A3( 1113857


αminus

S31λ2α2Re

αRe minus 2S11113888 1113889

X2 λ2S21α

2Re


S1 A1 minus A2( 1113857

α


2A3

X3 minusλ1S21α

2Re


S1 A1 minus A2( 1113857

α

+ 2λ1αS21 + α2ReS1 λ2 + S1( 1113857

2A4

X5 S1 λ1 minus λ2( 1113857A2

α+

S31λ1α2Re

αRe minus 2S11113888 1113889


π minus λ

(A1)

Data Availability




References

































































Journal of












Journal of







Volume 2018






Volume 2018








minus αzu1

zy+ v1

zu0

zy

1Re

z2u1

zy2 +z2u1

zz21113888 1113889


zy3 minus αz3u1

zy zz2

+ v1z3u0


(27)

minus αzv1

zy minus

zp1

zy+

1Re

z2v1

zy2 +z2v1

zz21113888 1113889

minus Kαz3v1

zy3 +z3v1

zy zz21113888 1113889

(28)

minus αzw1

zy minus

zp1

zz+

1Re

z2w1

zy2 +z2w1

zz21113888 1113889

minus Kαz3w1

zy3 +z3w1

zy zz21113888 1113889 minus Mw1

(29)



u1 0 v1 0 w1 0 asy⟶infin(30)



v1(y z) v11(y)cos π(z) (31)

w1(y z) minus1π


p1(y z) p11(y)cos π(z) (33)



(34)



Prime11


(35)





v11 0

(36)


v11(0) minus α

v11prime (0) 0(37)

We assume that

v11 v110 + Kv111 + O K2

1113872 1113873

p11 p110 + Kp111 + O K2

1113872 1113873(38)


v110(0) minus α

v111prime (0) v111(0) v110prime (0) 0(39)


v11 α

λ1 minus λ2( 1113857λ2e


minus λ2y1113872 1113873

minus Kα2Re

A1 minus A2

λ1 minus λ21113888 1113889e

minus λ1y1113888

minusA1 minus A2

λ1 minus λ21113888 1113889e

minus λ2y+ y A1e

minus λ1yminus A2e

minus λ2y1113872 11138731113889

(40)


v1(y z) α

λ1 minus λ2( 1113857λ2e


minus λ2y1113872 1113873


λ1 minus λ21113888 1113889e

minus λ1y

minusA1 minus A2

λ1 minus λ21113888 1113889e

minus λ2y

+ y A1eminus λ1y

minus A2eminus λ2y

1113872 11138731113889cos πz

(41)

w1(y z) αλ1λ2


minus λ2y+ e

minus λ1y1113872 1113873 +

Kα2Re

π



A1 minus A2




⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠sin πz (42)




u1(y z) u11(y)cos π(z) (43)


u11 0 aty 0



21113872 1113873 (45)


u111 u110 0 aty 0




+KαReS

31

αRe minus 2S1ye

minus S1y+ ε

αReS1

λ1 minus λ2( 1113857


minus λ2+m( )y


⎛⎜⎜⎝ ⎞⎟⎟⎠cos πz

+ εKαReS1



+ εKαReS1

λ2 minus λ1( 1113857

minusF1





⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

cos πz

(47)
















00 02

00

05

10

ndash0504 06 08 10

y

u(y)

α = 01α = 02α = 05


00 02 04 06 08 10y

ndash4

ndash3

ndash2

ndash1

0

1

u(y)

K = 01K = 03K = 05


00 02 04 06 08 10y

00

02

04

06

08

10

u(y)

M = 2M = 4M = 8


00 02 04 06 08 10y

00

02

04

06

08

10

u(y)

Re = 10Re = 20Re = 30







Nomenclature




00 05 10 15 20y

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

v(y)

α = 01α = 05α = 09


00

00

05 10 15

ndash10

ndash05

ndash15

20y

v(y)

K = 01K = 05K = 09


00ndash020

ndash015

ndash010

ndash005

02 04 06 08 10y

v(y)

M = 2M = 4M = 6


00

00

05 10 15 20y

ndash020

ndash015

ndash010

ndash005

v(y)

Re = 10Re = 20Re = 30



yK 01α 01M 2

K 01α 05M 2

K 05α 05M 2

K 05α 05M 4

00 00 00 00 0005 0003216 0016261 0016228 001607110 0001352 0006597 0006889 000701115 0000442 0002076 0002201 000232220 0000133 0000599 0000630 0000698


Greek symbols


Appendix

S1 αRe +

αRe( 11138572

+ 4ReM

1113969

2

S2 αRe minus

αRe( 11138572

+ 4ReM

1113969

2

λ1 S1 +

S1( 11138572

+ 4π21113969

2

λ2 S2 +

S2( 11138572

+ 4π21113969

2

λ3 S1 minus

S1( 11138572

+ 4π21113969

2

λ4 S2 minus

S2( 11138572

+ 4π21113969

2

A1 λ1λ2 λ21 minus π21113872 1113873

2


A2 λ1λ2 λ22 minus π21113872 1113873


A3 λ2

λ1 + S1( 11138572


A4 λ1

λ2 + S1( 11138572


X1 minus α2Reλ2

A4 minus A3( 1113857


αminus

S31λ2α2Re

αRe minus 2S11113888 1113889

X2 λ2S21α

2Re


S1 A1 minus A2( 1113857

α


2A3

X3 minusλ1S21α

2Re


S1 A1 minus A2( 1113857

α

+ 2λ1αS21 + α2ReS1 λ2 + S1( 1113857

2A4

X5 S1 λ1 minus λ2( 1113857A2

α+

S31λ1α2Re

αRe minus 2S11113888 1113889


π minus λ

(A1)

Data Availability




References

































































Journal of












Journal of







Volume 2018






Volume 2018










u1(y z) u11(y)cos π(z) (43)


u11 0 aty 0



21113872 1113873 (45)


u111 u110 0 aty 0




+KαReS

31

αRe minus 2S1ye

minus S1y+ ε

αReS1

λ1 minus λ2( 1113857


minus λ2+m( )y


⎛⎜⎜⎝ ⎞⎟⎟⎠cos πz

+ εKαReS1



+ εKαReS1

λ2 minus λ1( 1113857

minusF1





⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

cos πz

(47)
















00 02

00

05

10

ndash0504 06 08 10

y

u(y)

α = 01α = 02α = 05


00 02 04 06 08 10y

ndash4

ndash3

ndash2

ndash1

0

1

u(y)

K = 01K = 03K = 05


00 02 04 06 08 10y

00

02

04

06

08

10

u(y)

M = 2M = 4M = 8


00 02 04 06 08 10y

00

02

04

06

08

10

u(y)

Re = 10Re = 20Re = 30







Nomenclature




00 05 10 15 20y

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

v(y)

α = 01α = 05α = 09


00

00

05 10 15

ndash10

ndash05

ndash15

20y

v(y)

K = 01K = 05K = 09


00ndash020

ndash015

ndash010

ndash005

02 04 06 08 10y

v(y)

M = 2M = 4M = 6


00

00

05 10 15 20y

ndash020

ndash015

ndash010

ndash005

v(y)

Re = 10Re = 20Re = 30



yK 01α 01M 2

K 01α 05M 2

K 05α 05M 2

K 05α 05M 4

00 00 00 00 0005 0003216 0016261 0016228 001607110 0001352 0006597 0006889 000701115 0000442 0002076 0002201 000232220 0000133 0000599 0000630 0000698


Greek symbols


Appendix

S1 αRe +

αRe( 11138572

+ 4ReM

1113969

2

S2 αRe minus

αRe( 11138572

+ 4ReM

1113969

2

λ1 S1 +

S1( 11138572

+ 4π21113969

2

λ2 S2 +

S2( 11138572

+ 4π21113969

2

λ3 S1 minus

S1( 11138572

+ 4π21113969

2

λ4 S2 minus

S2( 11138572

+ 4π21113969

2

A1 λ1λ2 λ21 minus π21113872 1113873

2


A2 λ1λ2 λ22 minus π21113872 1113873


A3 λ2

λ1 + S1( 11138572


A4 λ1

λ2 + S1( 11138572


X1 minus α2Reλ2

A4 minus A3( 1113857


αminus

S31λ2α2Re

αRe minus 2S11113888 1113889

X2 λ2S21α

2Re


S1 A1 minus A2( 1113857

α


2A3

X3 minusλ1S21α

2Re


S1 A1 minus A2( 1113857

α

+ 2λ1αS21 + α2ReS1 λ2 + S1( 1113857

2A4

X5 S1 λ1 minus λ2( 1113857A2

α+

S31λ1α2Re

αRe minus 2S11113888 1113889


π minus λ

(A1)

Data Availability




References

































































Journal of












Journal of







Volume 2018






Volume 2018
















00 02

00

05

10

ndash0504 06 08 10

y

u(y)

α = 01α = 02α = 05


00 02 04 06 08 10y

ndash4

ndash3

ndash2

ndash1

0

1

u(y)

K = 01K = 03K = 05


00 02 04 06 08 10y

00

02

04

06

08

10

u(y)

M = 2M = 4M = 8


00 02 04 06 08 10y

00

02

04

06

08

10

u(y)

Re = 10Re = 20Re = 30







Nomenclature




00 05 10 15 20y

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

v(y)

α = 01α = 05α = 09


00

00

05 10 15

ndash10

ndash05

ndash15

20y

v(y)

K = 01K = 05K = 09


00ndash020

ndash015

ndash010

ndash005

02 04 06 08 10y

v(y)

M = 2M = 4M = 6


00

00

05 10 15 20y

ndash020

ndash015

ndash010

ndash005

v(y)

Re = 10Re = 20Re = 30



yK 01α 01M 2

K 01α 05M 2

K 05α 05M 2

K 05α 05M 4

00 00 00 00 0005 0003216 0016261 0016228 001607110 0001352 0006597 0006889 000701115 0000442 0002076 0002201 000232220 0000133 0000599 0000630 0000698


Greek symbols


Appendix

S1 αRe +

αRe( 11138572

+ 4ReM

1113969

2

S2 αRe minus

αRe( 11138572

+ 4ReM

1113969

2

λ1 S1 +

S1( 11138572

+ 4π21113969

2

λ2 S2 +

S2( 11138572

+ 4π21113969

2

λ3 S1 minus

S1( 11138572

+ 4π21113969

2

λ4 S2 minus

S2( 11138572

+ 4π21113969

2

A1 λ1λ2 λ21 minus π21113872 1113873

2


A2 λ1λ2 λ22 minus π21113872 1113873


A3 λ2

λ1 + S1( 11138572


A4 λ1

λ2 + S1( 11138572


X1 minus α2Reλ2

A4 minus A3( 1113857


αminus

S31λ2α2Re

αRe minus 2S11113888 1113889

X2 λ2S21α

2Re


S1 A1 minus A2( 1113857

α


2A3

X3 minusλ1S21α

2Re


S1 A1 minus A2( 1113857

α

+ 2λ1αS21 + α2ReS1 λ2 + S1( 1113857

2A4

X5 S1 λ1 minus λ2( 1113857A2

α+

S31λ1α2Re

αRe minus 2S11113888 1113889


π minus λ

(A1)

Data Availability




References

































































Journal of












Journal of







Volume 2018






Volume 2018












Nomenclature




00 05 10 15 20y

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

v(y)

α = 01α = 05α = 09


00

00

05 10 15

ndash10

ndash05

ndash15

20y

v(y)

K = 01K = 05K = 09


00ndash020

ndash015

ndash010

ndash005

02 04 06 08 10y

v(y)

M = 2M = 4M = 6


00

00

05 10 15 20y

ndash020

ndash015

ndash010

ndash005

v(y)

Re = 10Re = 20Re = 30



yK 01α 01M 2

K 01α 05M 2

K 05α 05M 2

K 05α 05M 4

00 00 00 00 0005 0003216 0016261 0016228 001607110 0001352 0006597 0006889 000701115 0000442 0002076 0002201 000232220 0000133 0000599 0000630 0000698


Greek symbols


Appendix

S1 αRe +

αRe( 11138572

+ 4ReM

1113969

2

S2 αRe minus

αRe( 11138572

+ 4ReM

1113969

2

λ1 S1 +

S1( 11138572

+ 4π21113969

2

λ2 S2 +

S2( 11138572

+ 4π21113969

2

λ3 S1 minus

S1( 11138572

+ 4π21113969

2

λ4 S2 minus

S2( 11138572

+ 4π21113969

2

A1 λ1λ2 λ21 minus π21113872 1113873

2


A2 λ1λ2 λ22 minus π21113872 1113873


A3 λ2

λ1 + S1( 11138572


A4 λ1

λ2 + S1( 11138572


X1 minus α2Reλ2

A4 minus A3( 1113857


αminus

S31λ2α2Re

αRe minus 2S11113888 1113889

X2 λ2S21α

2Re


S1 A1 minus A2( 1113857

α


2A3

X3 minusλ1S21α

2Re


S1 A1 minus A2( 1113857

α

+ 2λ1αS21 + α2ReS1 λ2 + S1( 1113857

2A4

X5 S1 λ1 minus λ2( 1113857A2

α+

S31λ1α2Re

αRe minus 2S11113888 1113889


π minus λ

(A1)

Data Availability




References

































































Journal of












Journal of







Volume 2018






Volume 2018








Greek symbols


Appendix

S1 αRe +

αRe( 11138572

+ 4ReM

1113969

2

S2 αRe minus

αRe( 11138572

+ 4ReM

1113969

2

λ1 S1 +

S1( 11138572

+ 4π21113969

2

λ2 S2 +

S2( 11138572

+ 4π21113969

2

λ3 S1 minus

S1( 11138572

+ 4π21113969

2

λ4 S2 minus

S2( 11138572

+ 4π21113969

2

A1 λ1λ2 λ21 minus π21113872 1113873

2


A2 λ1λ2 λ22 minus π21113872 1113873


A3 λ2

λ1 + S1( 11138572


A4 λ1

λ2 + S1( 11138572


X1 minus α2Reλ2

A4 minus A3( 1113857


αminus

S31λ2α2Re

αRe minus 2S11113888 1113889

X2 λ2S21α

2Re


S1 A1 minus A2( 1113857

α


2A3

X3 minusλ1S21α

2Re


S1 A1 minus A2( 1113857

α

+ 2λ1αS21 + α2ReS1 λ2 + S1( 1113857

2A4

X5 S1 λ1 minus λ2( 1113857A2

α+

S31λ1α2Re

αRe minus 2S11113888 1113889


π minus λ

(A1)

Data Availability




References

































































Journal of












Journal of







Volume 2018






Volume 2018


























































Journal of












Journal of







Volume 2018






Volume 2018








a novel design of three-dimensional mhd flow of second...

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