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Research ArticleA Simplified Finite Difference Method (SFDM) Solution viaTridiagonal Matrix Algorithm for MHD Radiating NanofluidFlow over a Slippery Sheet Submerged in a Permeable Medium

M Asif Farooq 1 A Salahuddin1 Asif Mushtaq 2 and M Razzaq3

1Department of Mathematics School of Natural Sciences (SNS) National University of Sciences and Technology (NUST)Sector H-12 Islamabad 44000 Pakistan2Seksjon for Matematikk Nord Universitet Bodoslash 8026 Norway3Department of Mathematics Lahore University of Management Sciences (LUMS) Lahore 54792 Pakistan

Correspondence should be addressed to Asif Mushtaq asifmushtaqnordno

Received 9 October 2020 Revised 4 January 2021 Accepted 8 January 2021 Published 27 January 2021

Academic Editor Muhammad mubashir bhatti muhammad09shueducn

Copyright copy 2021M Asif Farooq et al)is is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper we turn our attention to the mathematical model to simulate steady hydromagnetic and radiating nanofluid flowpast an exponentially stretching sheet A numerical modeling technique simplified finite difference method (SFDM) has beenapplied to the flow model that is based on partial differential equations (PDEs) which is converted to nonlinear ordinarydifferential equations (ODEs) by using similarity variables For the resultant algebraic system the SFDM uses the tridiagonalmatrix algorithm (TDMA) in computing the solution )e effectiveness of numerical scheme is verified by comparing it withsolution from the literature However where reference solution is not available one can compare its numerical results with theresults of MATLAB built-in package bvp4c )e velocity temperature and concentration profiles are graphed for a variety ofparameters ie Prandtl number Grashof number thermal radiation parameter Darcy number Eckert number Lewis numberand Brownian and thermophoresis parameters )e significant effects of the associated emerging thermophysical parameters ieskin friction coefficient local Nusselt number and local Sherwood numbers are analyzed and discussed in detail Numericalresults are compared from the available literature and found a close agreement with each other It is found that the Eckert numberupsurges the velocity curve However the dimensionless temperature declines with the Grashof number It is also shown that theSFDM gives good results when compared with the results obtained from bvp4c and results from the literature

1 Introduction

)e stretching sheet flow has several interesting engineeringapplications such as in a chemical engineering plantrsquospolymer handling unit and in metallurgy for the metalworking system Crane [1] researched the continuous two-dimensional boundary layer flow induced by stretching thesheet moving in its own plane at a velocity linearly varyingfrom a fixed point on the sheet Immediately after Crane [1]abundant work in this direction is reported and discussed

Makinde and Aziz [2] explored the effect of boundarylayer flow over linearly stretching nanofluid while Mustafaet al [3] concentrated on boundary layer flow for an ex-ponentially stretching sheet and solved the issue using the

technique of the homotopy analysis method to calculateanalytical solutions Realistically as discussed by Gupta andGupta [4] stretching a plastic sheet may not necessarily belinear

Since Choi and Eastmanrsquos pioneering research [5]surveys associated to nanofluid dynamics have risen sig-nificantly in contemporary times due to the low thermalconductivity of prevalent heat transfer liquids which causesthe device to function inefficiently and consume additionalenergy A new method has been introduced to optimizemachine operation by dispersing solid particles with a basefluid Nanofluid defines the suspension in standard baseliquids such as water ethylene glycol andmotor oil of strongparticles of a nanometer size References from [2 3 5] give a

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 6628009 17 pageshttpsdoiorg10115520216628009

thorough overview of the nanofluid literature Innovativefluid types are needed in these days to achieve more effectiveoutput Sheikholeslami and Bhatti [6] studied forced con-vection of nanofluid considering nanoparticlesrsquo shape im-pacts )e Brownian motion impact on nanofluid flowwithin a porous cavity was recently regarded by Sheikho-leslami [7] He found that convective flow enhances withincrease in Darcy number

)e research of electrically conductive fluid flow hasmany applications in engineering issues such as MHDgenerators plasma research nuclear reactors geothermalenergy extraction and aerodynamic boundary layer control[8] Reddy et al [9] studied the effects of frictional andirregular temperature on non-Newtonian MHD fluid flowsowing to stretched surface

Mishra and Singh [10] addressed dual solutions of mixedconvection flow with momentum and heat slip over apermeable shrinking cylinder When studying the flowmodels in nanoscales or microscales the interaction of thefluid surface is mostly controlled by models of slip flow)ese models were checked from asymptotic solution usingthe Boltzmann technique where the internal kinetic solutionmatches the outer (ie bulk) NavierndashStokes solution andmatching is only achieved when the slipjump coefficient isregarded at the border or surface (Hadjiconstantinou[11 12]))erefore the slip coefficients are the result of theseassessments Due to its simplicity the slip flow phenomenonis always preferred to no-slip situations )e NavierndashStokesequations are still valid here and only the boundary con-ditions change in compliance with the slip flow model Withthe newly suggested second-order slip flowmodel Fang et al[13] evaluated the slip flow over a permeable shrinkingsurface Ullah et al [14] examined the two-dimensional flowof ReinerndashPhilippoff fluid thin films over an unstablestretching sheet in the variable heat distribution andradiation

Khan et al [15] provided thermal radiation and viscousdissipation impacts on the unstable nanofluid boundarylayer flow over a stretching sheet In this research theyaccounted for the viscous dissipation impact and discoveredthat the heat boundary layer thickness is increased by in-creasing the values of Eckert number Ibrahim and Shankar[16] evaluated the impact of thermophoresis on Brownianfluid movement owing to stretching sheet

Several technological systems depend on the impact ofbuoyancy Makinde et al [17] examined combined impactsof buoyancy force convective warming Brownian move-ment thermophoresis and magnetic field on stagnationpoint stream and heat exchange due to nanofluid streamtowards an extending sheet Ali and Yousef [18] analyzedlaminar mixed convection heat transfer from continuouslystretching vertical surface with energy functional form forwall temperature by considering the impact of buoyancyMixed convection heat transfer from an exponentiallystretching sheet was explored by Partha et al [19] )ey alsoanalyzed influence of buoyancy along with viscous dissi-pation and the flow is governed by the mixed convectionparameter (GrRe2) )e effect of viscous dissipation innatural convection process has been investigated by Gebhart

[20] and Gebhart and Mollendorf [21] Magyari and Keller[22] analytically as well as numerically evaluated the con-tinuous free fluid flow and thermal transfer from an ex-ponentially stretching vertical surface with an exponentialtemperature distribution Unsteady flow of thermally ra-diating nanofluid over nonlinearly stretching sheet wasdiscussed by Seth et al [23] )ey noted that the nanofluidrsquosvelocity curve depends on the unsteadiness velocity slip andstretching velocity nonlinearity Makinde et al [24] reportedthe two-dimensional unsteady MHD radiating electricallyconducting fluid past a slippery stretching sheet embeddedin a porous medium Using the explicit finite differencescheme they solved the system of higher-order nonlinearPDEs Hamid and Khan [25] have discussed the thermo-physical properties of the flow of Williamson nanofluid andsolved their problem numerically )ey concluded that thestronger the magnetic field resulted in decreasing ofboundary layer thickness Some other references in thisdirection can be consulted in [26ndash28]

Qing et al [29] researched the entropy generation ofnanofluid owing to a magnetic field over a stretching surfaceHosseini et al [30] discussed heat transfer of nanofluid flow inmicrochannel heat sink (MCHS) in the presence of a magneticfield )e influence of chemical reaction and heat generationabsorption on mixed convective flow of nanofluid past anexponentially stretched surface has been examined by Eid [31]and numerical solutions have been obtained by utilizing theshooting technique along with the RungendashKuttandashFehlbergmethod Afify and Elgazery [32] investigated numerically theboundary layer flow of Maxwell nanofluid with convectiveboundary condition and heat absorption )e result showedthat nanoparticle concentration reduces with higher chemicalreaction parameter whereas a reverse pattern is noted fortemperature Reviews of viscous fluid flow problems fornonlinear stretching sheet have been presented by Prasad et al[33] Afzal [34] and Nandeppanavar et al [35] Nadeem andLee [36] studied analytically the problem of steady boundarylayer flow of nanofluid over an exponentially stretching surfaceincluding the effects of Brownian motion and thermophoresisparameters )e influence of solar energy radiations in thetime-dependent Hiemenz flow of nanofluid over a wedge wasdiscussed byMohamad et al [37] In [38] Sheikholeslami et aldiscussed natural convection inside a sinusoidal annulusTripathi et al [39] reported shape effects of nanoparticle onblood flow in a microvascular flow Bhatti et al [40] havediscussed the movement of gyrotactic microorganism in amagnetized nanofluid over a plate Ibrahim and Anbessa [41]discussed Casson nanofluid with Hall and Ion slip effectsIbrahim and Negera [42] investigated Williamson nanofluidover a stretching cylinder with activation energy For similarwork in this direction the reader referred to [43]

In all previous studies a usual course is followed in one wayor the other and discussion is intended towards linearly ornonlinearly stretching sheets in the absence of some importantemerging parameters)e aim of this work is to add numericalmethodology SFDM in the literature so that it can be ap-plicable to many problems containing coupled ODEs To thebest of our knowledge the current mathematical model alongwith numerical consideration has not been discussed before

2 Mathematical Problems in Engineering

)e paper is planned in the following order Section 2commences by laying out the mathematical model of thephysical problem Numerical procedure is opted and dis-cussed in Section 3 In the same section the detailed de-scription of the SFDM is given As a consequence ofnumerical calculations results and discussion are followedin Section 4 At the end of the paper the conclusions arepresented in Section 5

2 Mathematical Formulation

We deliberate a two dimensional steady incompressiblelaminar and MHD flow of an electrically conducting

nanofluid occupied over a slippery stretching sheet sub-merged in a porous medium )e geometrical description offluid flow over a sheet is shown in Figure 1 In the figurex-axis has been chosen along the sheet and y-axis normal toit

After making use of these assumptions the set of con-tinuity momentum energy and concentration equationsincorporating the Buongiorno model is written as follows[44]

zx(u) + zy(v) 0 (1)

uux + vuy ]uyy minusσB

2ou

ρminus]u

K+ gβ T minus Tinfin( 1113857 (2)

uTx + vTy k

ρCp

Tyy1113872 1113873 +]

Cp

uy1113872 11138732

+σB

2ou

2

ρCp

+]u

2

CpKminus

1ρCp

qry +Q T minus Tinfin( 1113857

ρCp

+ τ DB CyTy1113872 1113873 +DT

TinfinTy1113872 1113873

21113890 1113891 (3)

uCx + vCy DB Cyy1113872 1113873 +DT

TinfinTyy1113872 1113873 (4)

here the velocity components (u v) are considered along andnormal of the sheet μ is the coefficient of viscosity ρ is thedensity of the fluid σ is the electrical conductivity of the fluidTis fluidrsquos temperature K is the permeability β is the thermalexpansion coefficient k is the thermal conductivity Cp is thespecific heat capacity at constant pressure qr is the radioactiveheat flux Q is the heat source coefficient C is the concen-tration and τ (ρC)p(ρC)f where (ρC)p and (ρC)f areheat capacities of the nanofluid and base fluid respectivelyAlso DB and DT are Brownian and thermophoretic diffusioncoefficients respectively Tinfin is the ambient fluid temperatureand Cinfin is the ambient fluid concentration

21 Boundary Conditions )e preceding mathematicalmodel allows the following boundary condition

u(x 0) Uw +μL1

uy

v(x 0) 0

T(x 0) Tw

C(x 0) Cw

u⟶ 0

T⟶ Tinfin

C⟶ Cinfin asy⟶infin

(5)

where L1 is the slip length Here Uw U0exL is the

stretching velocity where U0 is the reference velocity AndTw Tinfin + T0e

x(2L) is the variable temperature at the sheetwith T0 being a reference temperature Also Cw Cinfin +

C0ex(2L) is the variable concentration at the sheet with C0

being a constant

22 Method of Solution By introducing similarity variablesη ψ(η) θ(η) and ϕ(η) as dimensionless independentvariable stream function temperature and concentrationfor the momentum energy and concentration equations(1)ndash(4) and in the boundary conditions (5)

η

U0

2]L

1113970

ex(2L)

y

ψ(η) 2U0]L

1113968e

x(2L)f(η)

u U0fprimeexL

v minus

U0]2L

1113970

ex(2L)

fprimeη + f( 1113857

θ(η) T minus Tinfin

T0e

minus x(2L)

ϕ(η) C minus Cinfin

C0e

minus x(2L)

(6)

gives the following nonlinear ordinary differential equations

Mathematical Problems in Engineering 3

f

+ ffPrime minus 2 fprime( 11138572

minus M +1

Da1113874 1113875fprime + 2Grθ 0 (7)

Pr θfprime minus fθprime( 1113857 minus (1 + Nr)θPrime minus PrEc fPrime( 11138572

minus PrEc M +1

Da1113874 1113875 fprime( 1113857

2

minus 2PrSθ minus PrNbϕprimeθprime minus PrNtθprime2 0

(8)

ϕPrime +Nt

NbθPrime + Le fϕprime minus fprimeϕ( 1113857 0 (9)

f(η) 0

fprime(η) 1 + λfPrime(η)

θ(η) 1

ϕ(η) 1 as η⟶ 0

(10)

fprime(η)⟶ 0

θ(η)⟶ 0

ϕ(η)⟶ 0 as η⟶infin

(11)

In the above equations various parameters appear which areM PrNtNbGrDaNr Ec S Le and λ In order these are themagnetic parameter Prandtl number thermophoresis param-eter Brownian parameter Grashof number Darcyrsquos numberthermal radiation effect heat source or sink Lewis number andthe slip parameter )eir expressions are grouped as follows

Pr μCp

k

M 2LσB

20

Uwρ

Nt τDT Tw minus Tinfin( 1113857

Tinfin]

Nb τDB Cw minus Cinfin( 1113857

]

Le ]

DB

Gr βgL Tw minus Tinfin( 1113857

U2w

Da KUw

2]L

Ec U

2w

TwCp

S QL

UwρCp

(12)

23 Physical Quantities Now that the flow equations areknown the physical quantities that measures roughnessheat transfer rate and concentration rate at the sheet canbe obtained First the skin friction coefficient Cf is givenby

Cf ]

U2w

zu

zy1113888 1113889

y0 (13)

Second the local Nusselt number Nux is written as

Nux minus (1 + Nr)x

Tw minus Tinfin( 1113857

zT

zy1113888 1113889

y0 (14)

)ird the local Sherwood number Shx is defined as

Shx minusx

Cw minus Cinfin( 1113857

zC

zy1113888 1113889

y0 (15)

After substituting similarity variables in (13)ndash(15) thisyields the expressions as follows

Cf 1

2Rex

1113968 fPrime(0)

Nux minus (1 + Nr)

xRex

2L

1113970

θprime(0)

Shx minus

xRex

2L

1113970

ϕprime(0)

(16)

Here Rex Ux] is a local Reynolds number

3 Numerical Procedures

In search of solution for the above problem given inequations (7)ndash(9) the only plausible way to compute so-lution is numerically We find numerical solutions by using

y

v

u T

Tinfin

B0 (magnetic field)

g

x

U = Uw

T = Tw

Figure 1 Schematic diagram of the problem


two numerical techniques )e first numerical method weuse is the SFDM and the second one is the famous algorithmwritten in MATLAB and commonly known as bvp4c )usdue details on the SFDM will be presented first followed bybrief description on bvp4c

31 SFDM )is work is influenced by Na [45] in whichsome numerical results are displayed for linear ODEs Forcoupled nonlinear ODEs we expand these ideas theoreti-cally and execute them in MATLAB )e algorithm withnecessary details for the SFDM is as follows

(1) Reduce third-order ODEs to a pair of ODEs of thefirst and second order

(2) Use Taylor series to linearize the system of nonlinearODEs

(3) Substitute finite difference formulas in thederivatives

(4) Finally solve the algebraic system by TDMA

)e results are shown for N 1000 grid points Gen-erally the domain length varies with different parametersHowever the domain value η 7 seems enough to showsteady state results To initiate the SFDM procedure weassume fprime F in equation (7) and we get

d2F

dη2 minus f

dF

dη+ 2F

2+ M +

1Da

1113874 1113875F minus 2Grθ (17)

Define a new variable as

ξ1 η F Fprime( 1113857 minus fdF

dη+ 2F

2+ M +

1Da

1113874 1113875F minus 2Grθ (18)

and approximate (dFdη) by forward difference approxi-mation with constant width h

ξ1 η F Fprime( 1113857 minus fi

Fi+1 minus Fi

h1113888 1113889 + 2F

2i + M +

1Da

1113874 1113875Fi minus 2Grθi

(19)

)e coefficients are written as

An minuszf

zFprime minus (minus f) f fi

Bn minuszf

zF minus 4F minus M +

1Da

1113874 1113875

Bn minus 4Fi minus M +1

Da1113874 1113875

Dn ξ1 η F Fprime( 1113857 + BnFi + An

Fi+1 minus Fi

h

(20)

Simplifying the above we reach at

aiFiminus 1 + biFi + ciFi+1 ri i 1 2 3 N (21)

where

ai 2 minus hAn

bi 2h2Bn minus 4

ci 2 + hAn

ri 2h2Dn

(22)

In the matrix-vector form it is written in compact as

A F s (23)

where

A

b1 c1

a2 b2 c2

aNminus 2 bNminus 2 cNminus 2

aNminus 1 bNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F

F1

F2

middot

middot

FNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

s

s1

s2

middot

middot

sNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

)ematrix A is a tridiagonal matrix and is written in LUfactorization as

A LU (25)

where

L

β1a2 β2

aNminus 2 βNminus 2


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

U

1 c1

1 c2

1 cNminus 2

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(26)


where L and U are the lower and upper triangular matricesrespectively Here the unknowns (βi ci) i 1 2 N minus 1are to be related as

β1 minus 1 minusλh

c1 λβ1h

βi bi minus aiciminus 1 i 2 3 N minus 1

βici ci i 2 3 N minus 2

(27)

After defining these relations (23) becomes

LUF s

UF z

Lz s

(28)

and we have

β1a2 β2


aNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

z3

middot

middot

middot

zNminus 2

zNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

s1

s2

s3

middot

middot

middot

sNminus 2

sNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)

)e unknown elements of s are written as

z1 s1

β1 si

si minus aiziminus 1

βi

i 2 3 N minus 1

1 c1

1 c2

1 cNminus 2

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F1

F2

middot

middot

middot

FNminus 2

FNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

middot

middot

middot

zNminus 2

zNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(30)

We get

Fiminus 1 ziminus 1

Fi zi minus ciFi+1 i N minus 2 N minus 3 3 2 1(31)

which is a solution of (17) We can easily find f from fprime F

which is in the discretization form written as follows

fi+1 minus fi

h Fi (32)

which gives a required solution of (7) A similar procedurecan also be opted for solutions θ and ϕ For the sake ofbrevity we only present coefficients for these ODEs andleave the details which follows on the same line as presentedabove For example we have the energy and concentrationequation as follows


d2θ

dη2

11 + Nr

Pr θF minus fdθdη

1113888 1113889 minus PrEcdF

dη1113888 1113889

2

minus PrEc M +1

Da1113874 1113875F

2minus 2PrSθ minus PrNb

dϕdη

dθdη

minus PrNtdθdη

1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

ξ2 η θ θprime( 1113857 1

1 + Nr

Pr θiFi minus fi

θi minus θiminus 1

h1113888 1113889 minus PrEc

Fi minus Fiminus 1

dη1113888 1113889

2

minus PrEc M +1

Da1113874 1113875F

2i minus 2PrSθi

⎧⎨

⎩

⎫⎬

⎭

minus1

1 + Nr

PrNbϕi minus ϕiminus 1

h


hminus PrNt


dη1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

Ann minuszf

zθprime minus

11 + Nr

minus Prf minus PrNbdϕdη

minus 2PrNtdθdη

1113896 1113897

Ann minuszf

zθprime

11 + Nr

Prfi + PrNbdϕi minus ϕiminus 1

h+ 2PrNt

dθi minus θiminus 1

h1113896 1113897

Bnn minuszf

zθ

minus 11 + Nr

PrF minus 2PrS1113864 1113865

Bnn minuszf

zθ

minus 11 + Nr

PrFi minus 2PrS1113864 1113865

d2ϕ

dη2

minus Nt

Nb

d2θ

dη2minus Le f

dϕdη

minus Fϕ1113888 1113889

ξ3 ηϕϕprime( 1113857 minus Nt

Nb

θiminus 1 minus 2θi + θi+1

dη2minus Le fi

ϕi minus ϕiminus 1

hminus Fiϕi1113888 1113889

(33)

Similarly the coefficients for (9) are written as

Annn Lefi

Bnnn minus LeFi(34)

Boundary conditions are discretized as

F1 1 + λF2 minus F1

h1113888 1113889 (35)

32 bvp4c )is section presents the second numericalmethod of the studied problem given in (7)ndash(9) whichsubject to the boundary conditions (10) and (11) We useMATLAB built-in function bvp4c for this purpose For

description and details of this method one can refer to [46]Let us define the variables as

y1 f

y2 fprime

y3 fPrime

y4 θ

y5 θprime

y6 ϕ

y7 ϕprime

(36)


)e system of first-order equations is given as follows

y1prime fprime y2

y2prime fPrime y3

y3prime f

minus y1y3 + 2 y2( 11138572

+ M +1

Da1113874 1113875y2 minus 2Gry4

y4prime θprime y5

y5prime θPrime 1

1 + NrPry4y2 minus Pry1y5 minus PrEcy

23 minus M +

1Da

1113874 1113875EcPry22 minus 2PrSy4 minus PrNby7y5 minus NtPry2

51113874 1113875

y6prime ϕprime y7

y7prime ϕPrime Ley6y2 minus Ley1y7 minusNt

Nby5prime

(37)

and boundary conditions are given as follows

y0(1) 0

y0(2) minus 1 minus λy0(3) 0

y0(4) minus 1 0

y0(6) minus 1 0

yinf(2) 0

yinf(4) 0

yinf(6) 0

(38)

4 Results and Discussion

In this section the focus is to analyze the role of embeddedparameters on the velocity temperature and concentrationResults of the current study are displayed in the tabular aswell as graphical form

For minus fPrime(0) the results are compared with the solutionspublished in the literature and this comparison is listed inTable 1)e results demonstrate that the numerical values ofSFDM are accurate and closely agreed with one another

In Table 2 when the admissible values of the magneticparameter increase resultantly the skin friction coefficientalso increases However reduction in both temperature andconcentration gradients is observed One can also observethat the magnitude of the local Nusselt and the localSherwood numbers increases and the skin friction coefficientdecreases with the rise of values of Darcyrsquos number Grashofnumber enhances local Nusselt number and local Sherwoodnumber whereas this reduces the skin friction coefficientHowever Lewis number Le causes slight change in skinfriction coefficient while concentration gradient and walltemperature gradient reduce

It is also evident from Table 2 that local Sherwoodnumber increases by increasing Nt but the effect is seen tobe reverse on skin friction coefficient while local Nusselt

number remains constant )e skin friction coefficientlocal Nusselt number and local Sherwood number de-crease with respect to thermophoretic parameter Nb (seeTable 3) From Table 3 one can observe an increase inlocal Sherwood number along the range of Ec )is alsocauses a surge in local Sherwood number whereas its effecton the skin friction coefficient and local Nusselt number isopposite

41 Effect of Magnetic Parameter M Figure 2 shows a de-creasing trend in velocity profiles against M(2leMle 8)tothe point where η asymp 250 After this point the boundarylayer thickness demonstrates the opposite behaviour Fig-ure 3 illustrates an increase in thermal boundary layerthickness due to an increase in a magnetic parameterHowever minor increase in concentration profile is pre-sented in Figure 4 )e reduction of the momentumboundary layer is strongly influenced by the magnetic pa-rameter strength which produces Lorentz force and thatoffer resistance to the flow

42 Effects of Darcy Number Da Darcy number Da char-acterizes the strength of permeability of the porous mediumFigures 5 and 6 depict increasing values of Darcy number(05leDa le 155) that increase the velocity profile whileconcentration profile decreases However the temperaturedecreases in the boundary layer region )us the thicknessof the thermal boundary layer decreases as shown inFigure 7

43 Effects of Lewis Number Le Figure 8 displays the vari-ations of velocity profiles due to the variations in the valuesof Lewis number (05leLele 2) It is observed that the ve-locity profile decreases with an increase in Le For Lege 1 themass transport is dominant that resists the flow One canalso observe that in Figure 9 the temperature profile as well


as the thickness of the boundary layer initially increases andthen decreases with an increase due to Lewis number )isimplies that the momentum boundary layer thickness de-creases when a ratio of thermal diffusivity to a mass dif-fusivity increases Figure 10 demonstrates the nanoparticlevolume fraction for several values of Lewis number Le ac-companying reduction in concentration boundary layerthickness

44 Effects of Grashof Number Gr )e Gr approximates theratio of buoyancy to viscous forces and represents how domi-nant is buoyancy force which is responsible for the convictioncomparing to viscous forces Either convection or viscous forcesare dominant and the results are displayed in Figures 11ndash13 Itcan be observed that temperature and concentration decreasewith the Grashof number Gr(3leGrle 65) but there is anabrupt change in a velocity profile

Table 1 )e comparison of skin friction coefficient to previous data for λ S Ec Nr Gr 0 and Da infin and for various values PrM Nb Nt and Le

Present resultPr M Nb Nt Le Sharif et al [47] bvp4c SFDM07 0 05 05 1 128183 12818089 12646694mdash 01 mdash mdash mdash 132104 13210148 13030810mdash 02 mdash mdash mdash 135895 13589575 13402296mdash 03 mdash mdash mdash 139581 13957745 13762525

Table 2 Results for minus fPrime(0) minus θprime(0) and minus ϕprime(0) obtained by fixing values of parameters Pr 62 Nt 2 Ec 02 Nb 8 S 01 Nr 5and λ 3

bvp4c SFDMM Da Gr Le minus fPrime(0) minus θprime(0) minus ϕprime(0) minus fPrime(0) minus θprime(0) minus ϕprime(0)

1 4 03 8 01945 00100 22070 01945175 00099219 219674011 mdash mdash mdash 01977 00094 21769 01977045 00093481 2166990912 mdash mdash mdash 02008 00088 21477 02007573 00087879 2138091613 mdash mdash mdash 02037 00083 21193 02036818 00082440 211001721 5 03 8 01929 00103 22223 01928713 00102230 22119414mdash 6 mdash mdash 01918 00105 22327 01917538 0010423 22221985mdash 7 mdash mdash 01910 00106 22401 01909458 00105674 22295847mdash 8 mdash mdash 01903 00107 22458 01903340 00106763 223515981 5 04 8 01754 00121 23805 01753267 00121091 23689312minus 2 mdash 05 mdash 01592 00136 25161 01591568 00136077 25033457mdash mdash 06 mdash 01441 00149 26354 01440571 00148417 26216478mdash mdash 07 mdash 01299 00159 27426 01298239 00158828 272777981 5 07 9 01302 00159 29079 01301768 00149605 28912397mdash mdash mdash 13 01313 00124 34900 01312932 00124113 34660959mdash mdash mdash 17 01321 00108 39863 01321069 00108305 39552333mdash mdash mdash 21 01328 00097 44263 01327388 00097292 43880280

Table 3 Results for minus fPrime(0) minus θprime(0) and minus ϕprime(0) obtained by various values of parameters Pr 62 M 2 Da 5 Gr 07 Nr 5Le 8 and λ 3

bvp4c Simplified FDMNb Nt Ec minus fPrime(0) minus θprime(0) minus ϕprime(0) minus fPrime(0) minus θprime(0) minus ϕprime(0)

2 2 02 01660 00270 24876 01682158 00746442 249738306 mdash mdash 01648 00148 24800 01647564 00148202 246796918 mdash mdash 01641 00104 24777 01640238 00104038 2465646110 mdash mdash 01636 00081 24774 01635227 00081378 2465257010 05 01 01656 00106 24477 01655962 00105904 24350691mdash 1 mdash 01650 00106 24580 01649419 00106063 24455417mdash 15 mdash 01644 00106 24668 01643695 00106205 24546071mdash 2 mdash 01639 00106 24745 0163872 00106326 2462460515 2 0 01634 00086 24739 01632997 00086251 24618319mdash mdash 01 01631 00070 24763 01630272 00069807 24641318mdash mdash 02 01628 00053 24787 01627443 00053296 24665070mdash mdash 03 01625 00037 24812 01624500 00036714 24689664


45 Effects of Nt Figure 14 shows that the velocity profileincreases with Nt in the range (05leNtle 2) Figure 15 il-lustrates the variations of thermophoretic parameter ontemperature profile It validated the fact that thermophoreticparameter enhances the temperature profile Since the

thermophoretic phenomenon transferred nanoparticlesfrom hot surface to the cold region it resulted in increasingthe temperature of the fluid Figure 16 suggests that astronger thermophoretic parameter produces minor changein nanoparticle volume fraction

Pr = 62 Da = 5 Gr = 2 Nr = 5 Nb = 2 Nt = 05Ec = 01 Le = 5 λ = 05 S = 01

M = 2M = 4

M = 6M = 8

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 5 6 7 80η

Figure 2 Velocity profiles for different M

M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Figure 3 Temperature profiles for different M

M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Figure 4 Concentration profiles for different M

Da = 08Da = 1

Da = 12Da = 14

Pr = 62 M = 2 Gr = 2 Nr = 5 Nb = 2 Nt = 05Ec = 01 Le = 5 λ = 3 S = 01

0

01

02

03

04

05

06

07

08

09

fprime(η)

1 2 3 4 5 60η

Figure 5 Velocity profiles for different Da


46 Effects of Nb Incremental Brownian parameterNb(5leNble 20) causes slight change in nanoparticlesvolume fraction which increases the velocity profile aspresented in Figures 17 and 18 Figure 19 suggests that astronger Brownian motion is responsible for an increase inthermal boundary layer thickness

47 Effects of Eckert Number Ec Eckert number plays animportant role in high speed flows for which viscous dissi-pation is significant It gives relative importance of the kineticenergy in heat transfer flows For Ec≪ 1 the energy equationgives the balance between conduction and convection FromFigures 20ndash22 the effects of this dissipation on velocity


Da = 02Da = 04

Da = 06Da = 08

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η

Figure 6 Concentration profiles for different Da


0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Da = 05Da = 55

Da = 105Da = 155

Figure 7 Temperature profiles for different Da

Pr = 62 M = 2 Gr = 2 Nr = 5 Nb = 10 Nt = 2Ec = 01 Da = 5 λ = 3 S = 01

Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 50η

Figure 8 Velocity profiles for different Le


Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 50η

Figure 9 Temperature profiles for different Le


temperature and concentration profile have been shown Itdepicts that in the absence of Ec the dimensionless velocity islowest at the surface and then increases with increasing Ec)e

dimensionless temperature is lowest inside the thermalboundary layer and increases with Ec while the effect of aviscous dissipation is insignificant on concentration profile

Pr = 62 M = 2 Gr = 2 Nr = 5 Nb = 10 Nt = 2 Ec = 01 Da = 5 λ = 3 S = 01

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Le = 05Le = 1

Le = 15Le = 2

Figure 10 Concentration profiles for different Le

Pr = 62 M = 2 Da = 5 Nr = 5 Nb = 10 Nt = 05 Ec = 01 Le = 5 λ = 3 S = 01

0

02

04

06

08

1

12

14

16

18

fprime(η)

1 2 3 4 50η

Gr = 3Gr = 4

Gr = 5Gr = 6

Figure 11 Velocity profiles for different Gr

Pr = 62 M = 2 Da = 5 Nr = 5 Nb = 2 Nt = 05Ec = 01 Le = 5 λ = 3 S = 01

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Gr = 05Gr = 25

Gr = 45Gr = 65

Figure 12 Temperature profiles for different Gr


0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Gr = 05Gr = 25

Gr = 45Gr = 65

Figure 13 Concentration profiles for different Gr


Pr = 62 M = 2 Gr = 2 Nr = 5 Nb = 8 Da = 5 Ec = 01 Le = 5 λ = 3 S = 01

Nt = 05Nt = 1

Nt = 15Nt = 2

minus01

0

01

02

03

04

05

06

07

fprime(η)

1 2 3 4 5 6 7 80η

Figure 14 Velocity profiles for different Nt


Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η

Figure 15 Temperature profiles for different Nt

Pr = 62 M = 2 Gr = 2 Nr = 5 Nb = 2 Da = 5Ec = 01 Le = 10 λ = 2 S = 01

Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 20η

Figure 16 Concentration profiles for different Nt

Pr = 62 M = 2 Gr = 5 Nr = 5 Da = 5 Nt = 05 Ec = 01 Le = 5 λ = 3 S = 01

Nb = 5Nb = 10

Nb = 15Nb = 20

0

02

04

06

08

1

12

14

16

fprime(η

)

05 1 15 2 25 3 35 40η

Figure 17 Velocity profiles for different Nb


Pr = 62 M = 2 Gr = 5 Nr = 5 Da = 5 Nt = 2Ec = 01 Le = 2 λ = 3 S = 01

Nb = 6Nb = 7

Nb = 8Nb = 9

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 30η

Figure 18 Concentration profiles for different Nb


Nb = 5Nb = 10

Nb = 15Nb = 20

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η

Figure 19 Temperature profiles for different Nb

Pr = 62 M = 2 Gr = 9 Nr = 5 Nb = 2 Nt = 05 Da = 5 Le = 5 λ = 3 S = 01

Ec = 0Ec = 01

Ec = 02Ec = 03

0

05

1

15

2

25

fprime(η)

05 1 15 2 25 3 35 40η

Figure 20 Velocity profiles for different Ec


Ec = 0Ec = 01

Ec = 02Ec = 03

0

01

02

03

04

05

06

07

08

09

1

θ(η)

1 2 3 4 50η

Figure 21 Temperature profiles for different Ec


5 Conclusions

)is study focuses on two objectives as follows (1) toproduce a mathematical model for nanofluid flow consid-ering MHD radiation porosity and slippery exponentiallystretching sheet which is immersed in a porous medium and(2) to develop a new numerical scheme for the solution of ageneral nonlinear ODEs which is applicable not only for thecurrent problem but many others )e key observations ofpresent work are as follows

(i) )e skin friction coefficient grows with a rise of thevalues of magnetic parameter but local Nusselt andSherwood numbers reduces )e rise of magneticparameter causes decrease in velocity initially afterthat the reverse effects can be seen It enhancestemperature and nanoparticle concentration of theboundary layer regime

(ii) )e Darcy number causes the thermal boundarylayer and solute concentration to reduce whereas itenhances the momentum boundary layer )e skinfriction coefficient decreases yet the wall temper-ature gradient and nanoparticle concentrationincrease with an increase in Darcy number

(iii) )e Grashof number enhances local Nusseltnumber and local Sherwood number whereas thisreduces the skin friction coefficient

(iv) )e Lewis number increases momentum andconcentration boundary layers Although it causesa slight change in the skin friction coefficient walltemperature gradient and boosts nanoparticleconcentration are reduced

(v) )e thermophoresis parameter causes both thethermal and momentum boundary layer to in-crease while its effect on nanoparticle volumefraction is insignificant )e skin friction coeffi-cient and local Sherwood number increase whilelocal Nusselt number remains unaltered

(vi) )e Brownian parameter increases thickness of thethermal boundary layer and momentum boundarylayer while there is a slight change in the con-centration boundary layer It causes a reduction inwall temperature gradient the skin friction coef-ficient the local Nusselt number and the localSherwood number

(vii) )e Eckert number Ec reduces the skin frictioncoefficient local Nusselt number and concentra-tion boundary layer while momentum boundarylayer thermal boundary layer and local Sherwoodnumber increase

(viii) )e SFDM has been successfully developed andapplied in the current problem One can show thatthe SFDM is easy to implement and convergesquickly

(ix) To validate one compares the SFDM results withbvp4c and those with the literature which gives agood account of agreement with each other

Data Availability

)ere are no experiment data involved in this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

Asif Mushtaq would like to acknowledge the support ofMathematics Teaching and Learning Research Groupswithin the Department of Mathematics FLU Nord Uni-versity Bodoslash

References

[1] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[2] O D Makinde and A Aziz ldquoBoundary layer flow of ananofluid past a stretching sheet with a convective boundaryconditionrdquo International Journal of Iermal Sciences vol 50no 7 pp 1326ndash1332 2011

[3] M Mustafa T Hayat and S Obaidat ldquoBoundary layer flow ofa nanofluid over an exponentially stretching sheet withconvective boundary conditionsrdquo International Journal ofNumerical Methods for Heat and Fluid Flow vol 23 no 6pp 945ndash959 2013

[4] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Ie CanadianJournal of Chemical Engineering vol 55 no 6 pp 744ndash7461977


0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η

Ec = 0Ec = 01

Ec = 02Ec = 03

Figure 22 Concentration profiles for different Ec


[5] S U Choi and J Eastman Enhancing thermal conductivity offluids with nanoparticles ASME International MechanicalEngineering Congress and Exhibition American Society ofMechanical Engineers (ASME) San Francisco CL USA1995

[6] M Sheikholeslami and M M Bhatti ldquoForced convection ofnanofluid in presence of constant magnetic field consideringshape effects of nanoparticlesrdquo International Journal of Heatand Mass Transfer vol 111 pp 1039ndash1049 2017

[7] M Sheikholeslami ldquoCuO-water nanofluid flow due tomagnetic field inside a porous media considering Brownianmotionrdquo Journal of Molecular Liquids vol 249 pp 921ndash9292018

[8] M Xenos N Kafoussias and G Karahalios ldquoMagnetohy-drodynamic compressible laminar boundary-layer adiabaticflow with adverse pressure gradient and continuous or lo-calized mass transferrdquo Canadian Journal of Physics vol 79no 10 pp 1247ndash1263 2001

[9] J R Reddy K A Kumar V Sugunamma and N SandeepldquoEffect of cross diffusion on MHD non-Newtonian fluids flowpast a stretching sheet with non-uniform heat sourcesink acomparative studyrdquo Alexandria Engineering Journal vol 57no 3 pp 1829ndash1838 2018

[10] U Mishra and G Singh ldquoDual solutions of mixed convectionflow with momentum and thermal slip flow over a permeableshrinking cylinderrdquo Computers amp Fluids vol 93 pp 107ndash1152014

[11] N G Hadjiconstantinou ldquoComment on Cercignanirsquos second-order slip coefficientrdquo Physics of Fluids vol 15 no 8pp 2352ndash2354 2003

[12] N G Hadjiconstantinou ldquo)e limits of Navier-Stokes theoryand kinetic extensions for describing small-scale gaseoushydrodynamicsrdquo Physics of Fluids vol 18 no 11 Article ID111301 2006

[13] T Fang S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 15 no 7 pp 1831ndash1842 2010

[14] A Ullah E Alzahrani Z Shah M Ayaz and S IslamldquoNanofluids thin film flow of reiner-philippoff fluid over anunstable stretching surface with brownian motion andthermophoresis effectsrdquo Coatings vol 9 no 1 p 21 2019

[15] M S Khan I Karim L E Ali and A Islam ldquoUnsteady MHDfree convection boundary-layer flow of a nanofluid along astretching sheet with thermal radiation and viscous dissipa-tion effectsrdquo International Nano Letters vol 2 no 1 p 242012

[16] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013

[17] O D Makinde W A Khan and Z H Khan ldquoBuoyancyeffects on MHD stagnation point flow and heat transfer of ananofluid past a convectively heated stretchingshrinkingsheetrdquo International Journal of Heat and Mass Transfervol 62 pp 526ndash533 2013

[18] M Ali and F Al-Yousef ldquoLaminar mixed convection from acontinuously moving vertical surface with suction or injec-tionrdquo Heat and Mass Transfer vol 33 no 4 pp 301ndash3061998

[19] M Partha P Murthy and G Rajasekhar ldquoEffect of viscousdissipation on the mixed convection heat transfer from anexponentially stretching surfacerdquo Heat and Mass Transfervol 41 no 4 pp 360ndash366 2005

[20] B Gebhart ldquoEffects of viscous dissipation in natural con-vectionrdquo Journal of Fluid Mechanics vol 14 no 2pp 225ndash232 1962

[21] B Gebhart and J Mollendorf ldquoViscous dissipation in externalnatural convection flowsrdquo Journal of Fluid Mechanics vol 38no 1 pp 97ndash107 1969

[22] E Magyari and B Keller ldquoHeat and mass transfer in theboundary layers on an exponentially stretching continuoussurfacerdquo Journal of Physics D Applied Physics vol 32 no 5p 577 1999

[23] G S Seth A Bhattacharyya R Kumar and A J ChamkhaldquoEntropy generation in hydromagnetic nanofluid flow over anon-linear stretching sheet with Navierrsquos velocity slip andconvective heat transferrdquo Physics of Fluids vol 30 no 12Article ID 122003 2018

[24] O D Makinde Z H Khan R Ahmad and W A KhanldquoNumerical study of unsteady hydromagnetic radiating fluidflow past a slippery stretching sheet embedded in a porousmediumrdquo Physics of Fluids vol 30 no 8 Article ID 0836012018

[25] A Hamid and M Khan ldquoUnsteady mixed convective flow ofWilliamson nanofluid with heat transfer in the presence ofvariable thermal conductivity and magnetic fieldrdquo Journal ofMolecular Liquids vol 260 pp 436ndash446 2018

[26] M Jafaryar M Sheikholeslami and Z Li ldquoCuO-waternanofluid flow and heat transfer in a heat exchanger tube withtwisted tape turbulatorrdquo Powder Technology vol 336pp 131ndash143 2018

[27] A Hamid M Hashim and M Khan ldquoImpacts of binarychemical reaction with activation energy on unsteady flow ofmagneto-Williamson nanofluidrdquo Journal of Molecular Liq-uids vol 262 pp 435ndash442 2018

[28] M Khan A Hashim and A Hafeez ldquoA review on slip-flowand heat transfer performance of nanofluids from a permeableshrinking surface with thermal radiation dual solutionsrdquoChemical Engineering Science vol 173 pp 1ndash11 2017

[29] J Qing M Bhatti M Abbas M Rashidi andM Ali ldquoEntropygeneration on MHD Casson nanofluid flow over a porousstretchingshrinking surfacerdquo Entropy vol 18 no 4 p 1232016

[30] S R Hosseini M Sheikholeslami M Ghasemian andD D Ganji ldquoNanofluid heat transfer analysis in a micro-channel heat sink (MCHS) under the effect of magnetic fieldby means of KKL modelrdquo Powder Technology vol 324pp 36ndash47 2018

[31] M R Eid ldquoChemical reaction effect on MHD boundary-layerflow of two-phase nanofluid model over an exponentiallystretching sheet with a heat generationrdquo Journal of MolecularLiquids vol 220 pp 718ndash725 2016

[32] A A Afify and N S Elgazery ldquoEffect of a chemical reactionon magnetohydrodynamic boundary layer flow of a Maxwellfluid over a stretching sheet with nanoparticlesrdquo Particuologyvol 29 pp 154ndash161 2016

[33] K V Prasad K Vajravelu and P S Datti ldquoMixed convectionheat transfer over a non-linear stretching surface with variablefluid propertiesrdquo International Journal of Non-linear Me-chanics vol 45 no 3 pp 320ndash330 2010

[34] N Afzal ldquoMomentum and thermal boundary layers over atwo-dimensional or axisymmetric non-linear stretchingsurface in a stationary fluidrdquo International Journal of Heatand Mass Transfer vol 53 no 1ndash3 pp 540ndash547 2010

[35] MM Nandeppanavar K Vajravelu M S Abel and C O NgldquoHeat transfer over a nonlinearly stretching sheet with non-uniform heat source and variable wall temperaturerdquo


International Journal of Heat and Mass Transfer vol 54no 23-24 pp 4960ndash4965 2011

[36] S Nadeem and C Lee ldquoBoundary layer flow of nanofluid overan exponentially stretching surfacerdquo Nanoscale ResearchLetters vol 7 no 1 p 94 2012

[37] R B Mohamad R Kandasamy and I Muhaimin ldquoEnhanceof heat transfer on unsteady Hiemenz flow of nanofluid over aporous wedge with heat sourcesink due to solar energy ra-diation with variable stream conditionrdquo Heat and MassTransfer vol 49 no 9 pp 1261ndash1269 2013

[38] M Sheikholeslami R Ellahi and C Fetecau ldquoCuOndashWaternanofluid magnetohydrodynamic natural convection inside asinusoidal annulus in presence of melting heat transferrdquoMathematical Problems in Engineering vol 2017 19 pagesArticle ID 5830279 2017

[39] D Tripathi J Prakash A K Tiwari and R Ellahi ldquo)ermalmicrorotation electromagnetic field and nanoparticle shapeeffects on Cu-CuOblood flow in microvascular vesselsrdquoMicrovascular Research vol 132 Article ID 104065 2020

[40] M M Bhatti A Shahid T Abbas S Z Alamri and R EllahildquoStudy of activation energy on the movement of gyrotacticmicroorganism in a magnetized nanofluids past a porousplaterdquo Processes vol 8 no 3 p 328 2020

[41] W Ibrahim and T Anbessa ldquo)ree-dimensional MHDmixedconvection flow of Casson nanofluid with Hall and Ion slipeffectsrdquo Mathematical Problems in Engineering vol 2020Article ID 8656147 16 pages 2020

[42] W Ibrahim and M Negera ldquo)e investigation of MHDWilliamson nanofluid over stretching cylinder with the effectof activation energyrdquo Advances in Mathematical Physicsvol 2020 Article ID 9523630 18 pages 2020

[43] A F Elelamy N S Elgazery and R Ellahi ldquoBlood flow ofMHD non-Newtonian nanofluid with heat transfer and slipeffectsrdquo International Journal of Numerical Methods for Heatamp Fluid Flow vol 62 no 3 2020

[44] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journalof Heat Transfer vol 128 no 3 pp 240ndash250 2006

[45] T Y Na Computational methods in engineering boundaryvalue problems Academic Press Cambridge MA USA 1980

[46] L F Shampine J Kierzenka and M W Reichelt ldquoSolvingboundary value problems for ordinary differential equationsin MATLAB with bvp4crdquo Tutorial notes vol 16 pp 1ndash272000

[47] R Sharif M A Farooq and A Mushtaq ldquoMagnetohydro-dynamic study of variable fluid properties and their impact onnanofluid over an exponentially stretching sheetrdquo Journal ofNanofluids vol 8 no 6 pp 1249ndash1259 2019


thorough overview of the nanofluid literature Innovativefluid types are needed in these days to achieve more effectiveoutput Sheikholeslami and Bhatti [6] studied forced con-vection of nanofluid considering nanoparticlesrsquo shape im-pacts )e Brownian motion impact on nanofluid flowwithin a porous cavity was recently regarded by Sheikho-leslami [7] He found that convective flow enhances withincrease in Darcy number

)e research of electrically conductive fluid flow hasmany applications in engineering issues such as MHDgenerators plasma research nuclear reactors geothermalenergy extraction and aerodynamic boundary layer control[8] Reddy et al [9] studied the effects of frictional andirregular temperature on non-Newtonian MHD fluid flowsowing to stretched surface

Mishra and Singh [10] addressed dual solutions of mixedconvection flow with momentum and heat slip over apermeable shrinking cylinder When studying the flowmodels in nanoscales or microscales the interaction of thefluid surface is mostly controlled by models of slip flow)ese models were checked from asymptotic solution usingthe Boltzmann technique where the internal kinetic solutionmatches the outer (ie bulk) NavierndashStokes solution andmatching is only achieved when the slipjump coefficient isregarded at the border or surface (Hadjiconstantinou[11 12]))erefore the slip coefficients are the result of theseassessments Due to its simplicity the slip flow phenomenonis always preferred to no-slip situations )e NavierndashStokesequations are still valid here and only the boundary con-ditions change in compliance with the slip flow model Withthe newly suggested second-order slip flowmodel Fang et al[13] evaluated the slip flow over a permeable shrinkingsurface Ullah et al [14] examined the two-dimensional flowof ReinerndashPhilippoff fluid thin films over an unstablestretching sheet in the variable heat distribution andradiation

Khan et al [15] provided thermal radiation and viscousdissipation impacts on the unstable nanofluid boundarylayer flow over a stretching sheet In this research theyaccounted for the viscous dissipation impact and discoveredthat the heat boundary layer thickness is increased by in-creasing the values of Eckert number Ibrahim and Shankar[16] evaluated the impact of thermophoresis on Brownianfluid movement owing to stretching sheet

Several technological systems depend on the impact ofbuoyancy Makinde et al [17] examined combined impactsof buoyancy force convective warming Brownian move-ment thermophoresis and magnetic field on stagnationpoint stream and heat exchange due to nanofluid streamtowards an extending sheet Ali and Yousef [18] analyzedlaminar mixed convection heat transfer from continuouslystretching vertical surface with energy functional form forwall temperature by considering the impact of buoyancyMixed convection heat transfer from an exponentiallystretching sheet was explored by Partha et al [19] )ey alsoanalyzed influence of buoyancy along with viscous dissi-pation and the flow is governed by the mixed convectionparameter (GrRe2) )e effect of viscous dissipation innatural convection process has been investigated by Gebhart

[20] and Gebhart and Mollendorf [21] Magyari and Keller[22] analytically as well as numerically evaluated the con-tinuous free fluid flow and thermal transfer from an ex-ponentially stretching vertical surface with an exponentialtemperature distribution Unsteady flow of thermally ra-diating nanofluid over nonlinearly stretching sheet wasdiscussed by Seth et al [23] )ey noted that the nanofluidrsquosvelocity curve depends on the unsteadiness velocity slip andstretching velocity nonlinearity Makinde et al [24] reportedthe two-dimensional unsteady MHD radiating electricallyconducting fluid past a slippery stretching sheet embeddedin a porous medium Using the explicit finite differencescheme they solved the system of higher-order nonlinearPDEs Hamid and Khan [25] have discussed the thermo-physical properties of the flow of Williamson nanofluid andsolved their problem numerically )ey concluded that thestronger the magnetic field resulted in decreasing ofboundary layer thickness Some other references in thisdirection can be consulted in [26ndash28]

Qing et al [29] researched the entropy generation ofnanofluid owing to a magnetic field over a stretching surfaceHosseini et al [30] discussed heat transfer of nanofluid flow inmicrochannel heat sink (MCHS) in the presence of a magneticfield )e influence of chemical reaction and heat generationabsorption on mixed convective flow of nanofluid past anexponentially stretched surface has been examined by Eid [31]and numerical solutions have been obtained by utilizing theshooting technique along with the RungendashKuttandashFehlbergmethod Afify and Elgazery [32] investigated numerically theboundary layer flow of Maxwell nanofluid with convectiveboundary condition and heat absorption )e result showedthat nanoparticle concentration reduces with higher chemicalreaction parameter whereas a reverse pattern is noted fortemperature Reviews of viscous fluid flow problems fornonlinear stretching sheet have been presented by Prasad et al[33] Afzal [34] and Nandeppanavar et al [35] Nadeem andLee [36] studied analytically the problem of steady boundarylayer flow of nanofluid over an exponentially stretching surfaceincluding the effects of Brownian motion and thermophoresisparameters )e influence of solar energy radiations in thetime-dependent Hiemenz flow of nanofluid over a wedge wasdiscussed byMohamad et al [37] In [38] Sheikholeslami et aldiscussed natural convection inside a sinusoidal annulusTripathi et al [39] reported shape effects of nanoparticle onblood flow in a microvascular flow Bhatti et al [40] havediscussed the movement of gyrotactic microorganism in amagnetized nanofluid over a plate Ibrahim and Anbessa [41]discussed Casson nanofluid with Hall and Ion slip effectsIbrahim and Negera [42] investigated Williamson nanofluidover a stretching cylinder with activation energy For similarwork in this direction the reader referred to [43]

In all previous studies a usual course is followed in one wayor the other and discussion is intended towards linearly ornonlinearly stretching sheets in the absence of some importantemerging parameters)e aim of this work is to add numericalmethodology SFDM in the literature so that it can be ap-plicable to many problems containing coupled ODEs To thebest of our knowledge the current mathematical model alongwith numerical consideration has not been discussed before







zx(u) + zy(v) 0 (1)


2ou

ρminus]u


uTx + vTy k

ρCp

Tyy1113872 1113873 +]

Cp

uy1113872 11138732

+σB

2ou

2

ρCp

+]u

2

CpKminus

1ρCp


ρCp

+ τ DB CyTy1113872 1113873 +DT

TinfinTy1113872 1113873

21113890 1113891 (3)

uCx + vCy DB Cyy1113872 1113873 +DT

TinfinTyy1113872 1113873 (4)



u(x 0) Uw +μL1

uy

v(x 0) 0

T(x 0) Tw

C(x 0) Cw

u⟶ 0

T⟶ Tinfin


(5)





being a constant


η

U0

2]L

1113970

ex(2L)

y

ψ(η) 2U0]L

1113968e

x(2L)f(η)

u U0fprimeexL

v minus

U0]2L

1113970

ex(2L)



T0e

minus x(2L)


C0e

minus x(2L)

(6)



f


minus M +1

Da1113874 1113875fprime + 2Grθ 0 (7)


minus PrEc M +1

Da1113874 1113875 fprime( 1113857

2


(8)

ϕPrime +Nt


f(η) 0


θ(η) 1

ϕ(η) 1 as η⟶ 0

(10)

fprime(η)⟶ 0

θ(η)⟶ 0


(11)


Pr μCp

k

M 2LσB

20

Uwρ


Tinfin]


]

Le ]

DB


U2w

Da KUw

2]L

Ec U

2w

TwCp

S QL

UwρCp

(12)


Cf ]

U2w

zu

zy1113888 1113889

y0 (13)


Nux minus (1 + Nr)x


zT

zy1113888 1113889

y0 (14)


Shx minusx


zC

zy1113888 1113889

y0 (15)


Cf 1

2Rex

1113968 fPrime(0)

Nux minus (1 + Nr)

xRex

2L

1113970

θprime(0)

Shx minus

xRex

2L

1113970

ϕprime(0)

(16)




y

v

u T

Tinfin

B0 (magnetic field)

g

x

U = Uw

T = Tw










d2F

dη2 minus f

dF

dη+ 2F

2+ M +

1Da

1113874 1113875F minus 2Grθ (17)



dη+ 2F

2+ M +

1Da

1113874 1113875F minus 2Grθ (18)



Fi+1 minus Fi

h1113888 1113889 + 2F

2i + M +

1Da

1113874 1113875Fi minus 2Grθi

(19)


An minuszf


Bn minuszf


1Da

1113874 1113875


Da1113874 1113875


Fi+1 minus Fi

h

(20)



where

ai 2 minus hAn

bi 2h2Bn minus 4

ci 2 + hAn

ri 2h2Dn

(22)


A F s (23)

where

A

b1 c1

a2 b2 c2


aNminus 1 bNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F

F1

F2

middot

middot

FNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

s

s1

s2

middot

middot

sNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


A LU (25)

where

L

β1a2 β2



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

U

1 c1

1 c2

1 cNminus 2

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(26)




c1 λβ1h



(27)


LUF s

UF z

Lz s

(28)

and we have

β1a2 β2


aNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

z3

middot

middot

middot

zNminus 2

zNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

s1

s2

s3

middot

middot

middot

sNminus 2

sNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


z1 s1

β1 si


βi

i 2 3 N minus 1

1 c1

1 c2

1 cNminus 2

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F1

F2

middot

middot

middot

FNminus 2

FNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

middot

middot

middot

zNminus 2

zNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(30)

We get

Fiminus 1 ziminus 1




fi+1 minus fi

h Fi (32)



d2θ

dη2

11 + Nr


1113888 1113889 minus PrEcdF

dη1113888 1113889

2

minus PrEc M +1

Da1113874 1113875F


dϕdη

dθdη

minus PrNtdθdη

1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

ξ2 η θ θprime( 1113857 1

1 + Nr

Pr θiFi minus fi


h1113888 1113889 minus PrEc

Fi minus Fiminus 1

dη1113888 1113889

2

minus PrEc M +1

Da1113874 1113875F

2i minus 2PrSθi

⎧⎨

⎩

⎫⎬

⎭

minus1

1 + Nr


h


hminus PrNt


dη1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

Ann minuszf

zθprime minus

11 + Nr


minus 2PrNtdθdη

1113896 1113897

Ann minuszf

zθprime

11 + Nr


h+ 2PrNt


h1113896 1113897

Bnn minuszf

zθ

minus 11 + Nr

PrF minus 2PrS1113864 1113865

Bnn minuszf

zθ

minus 11 + Nr


d2ϕ

dη2

minus Nt

Nb

d2θ

dη2minus Le f

dϕdη

minus Fϕ1113888 1113889


Nb


dη2minus Le fi


hminus Fiϕi1113888 1113889

(33)


Annn Lefi

Bnnn minus LeFi(34)



h1113888 1113889 (35)



y1 f

y2 fprime

y3 fPrime

y4 θ

y5 θprime

y6 ϕ

y7 ϕprime

(36)



y1prime fprime y2

y2prime fPrime y3

y3prime f

minus y1y3 + 2 y2( 11138572

+ M +1

Da1113874 1113875y2 minus 2Gry4

y4prime θprime y5

y5prime θPrime 1


23 minus M +

1Da


51113874 1113875

y6prime ϕprime y7


Nby5prime

(37)


y0(1) 0


y0(4) minus 1 0

y0(6) minus 1 0

yinf(2) 0

yinf(4) 0

yinf(6) 0

(38)

























M = 2M = 4

M = 6M = 8

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η


Da = 08Da = 1

Da = 12Da = 14


0

01

02

03

04

05

06

07

08

09

fprime(η)

1 2 3 4 5 60η






Da = 02Da = 04

Da = 06Da = 08

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Da = 05Da = 55

Da = 105Da = 155



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 50η



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 50η






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Le = 05Le = 1

Le = 15Le = 2



0

02

04

06

08

1

12

14

16

18

fprime(η)

1 2 3 4 50η

Gr = 3Gr = 4

Gr = 5Gr = 6



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Gr = 05Gr = 25

Gr = 45Gr = 65



0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Gr = 05Gr = 25

Gr = 45Gr = 65




Nt = 05Nt = 1

Nt = 15Nt = 2

minus01

0

01

02

03

04

05

06

07

fprime(η)

1 2 3 4 5 6 7 80η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 20η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

02

04

06

08

1

12

14

16

fprime(η

)

05 1 15 2 25 3 35 40η




Nb = 6Nb = 7

Nb = 8Nb = 9

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 30η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

05

1

15

2

25

fprime(η)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

01

02

03

04

05

06

07

08

09

1

θ(η)

1 2 3 4 50η



5 Conclusions











Data Availability




Acknowledgments


References






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η

Ec = 0Ec = 01

Ec = 02Ec = 03






















































zx(u) + zy(v) 0 (1)


2ou

ρminus]u


uTx + vTy k

ρCp

Tyy1113872 1113873 +]

Cp

uy1113872 11138732

+σB

2ou

2

ρCp

+]u

2

CpKminus

1ρCp


ρCp

+ τ DB CyTy1113872 1113873 +DT

TinfinTy1113872 1113873

21113890 1113891 (3)

uCx + vCy DB Cyy1113872 1113873 +DT

TinfinTyy1113872 1113873 (4)



u(x 0) Uw +μL1

uy

v(x 0) 0

T(x 0) Tw

C(x 0) Cw

u⟶ 0

T⟶ Tinfin


(5)





being a constant


η

U0

2]L

1113970

ex(2L)

y

ψ(η) 2U0]L

1113968e

x(2L)f(η)

u U0fprimeexL

v minus

U0]2L

1113970

ex(2L)



T0e

minus x(2L)


C0e

minus x(2L)

(6)



f


minus M +1

Da1113874 1113875fprime + 2Grθ 0 (7)


minus PrEc M +1

Da1113874 1113875 fprime( 1113857

2


(8)

ϕPrime +Nt


f(η) 0


θ(η) 1

ϕ(η) 1 as η⟶ 0

(10)

fprime(η)⟶ 0

θ(η)⟶ 0


(11)


Pr μCp

k

M 2LσB

20

Uwρ


Tinfin]


]

Le ]

DB


U2w

Da KUw

2]L

Ec U

2w

TwCp

S QL

UwρCp

(12)


Cf ]

U2w

zu

zy1113888 1113889

y0 (13)


Nux minus (1 + Nr)x


zT

zy1113888 1113889

y0 (14)


Shx minusx


zC

zy1113888 1113889

y0 (15)


Cf 1

2Rex

1113968 fPrime(0)

Nux minus (1 + Nr)

xRex

2L

1113970

θprime(0)

Shx minus

xRex

2L

1113970

ϕprime(0)

(16)




y

v

u T

Tinfin

B0 (magnetic field)

g

x

U = Uw

T = Tw










d2F

dη2 minus f

dF

dη+ 2F

2+ M +

1Da

1113874 1113875F minus 2Grθ (17)



dη+ 2F

2+ M +

1Da

1113874 1113875F minus 2Grθ (18)



Fi+1 minus Fi

h1113888 1113889 + 2F

2i + M +

1Da

1113874 1113875Fi minus 2Grθi

(19)


An minuszf


Bn minuszf


1Da

1113874 1113875


Da1113874 1113875


Fi+1 minus Fi

h

(20)



where

ai 2 minus hAn

bi 2h2Bn minus 4

ci 2 + hAn

ri 2h2Dn

(22)


A F s (23)

where

A

b1 c1

a2 b2 c2


aNminus 1 bNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F

F1

F2

middot

middot

FNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

s

s1

s2

middot

middot

sNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


A LU (25)

where

L

β1a2 β2



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

U

1 c1

1 c2

1 cNminus 2

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(26)




c1 λβ1h



(27)


LUF s

UF z

Lz s

(28)

and we have

β1a2 β2


aNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

z3

middot

middot

middot

zNminus 2

zNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

s1

s2

s3

middot

middot

middot

sNminus 2

sNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


z1 s1

β1 si


βi

i 2 3 N minus 1

1 c1

1 c2

1 cNminus 2

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F1

F2

middot

middot

middot

FNminus 2

FNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

middot

middot

middot

zNminus 2

zNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(30)

We get

Fiminus 1 ziminus 1




fi+1 minus fi

h Fi (32)



d2θ

dη2

11 + Nr


1113888 1113889 minus PrEcdF

dη1113888 1113889

2

minus PrEc M +1

Da1113874 1113875F


dϕdη

dθdη

minus PrNtdθdη

1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

ξ2 η θ θprime( 1113857 1

1 + Nr

Pr θiFi minus fi


h1113888 1113889 minus PrEc

Fi minus Fiminus 1

dη1113888 1113889

2

minus PrEc M +1

Da1113874 1113875F

2i minus 2PrSθi

⎧⎨

⎩

⎫⎬

⎭

minus1

1 + Nr


h


hminus PrNt


dη1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

Ann minuszf

zθprime minus

11 + Nr


minus 2PrNtdθdη

1113896 1113897

Ann minuszf

zθprime

11 + Nr


h+ 2PrNt


h1113896 1113897

Bnn minuszf

zθ

minus 11 + Nr

PrF minus 2PrS1113864 1113865

Bnn minuszf

zθ

minus 11 + Nr


d2ϕ

dη2

minus Nt

Nb

d2θ

dη2minus Le f

dϕdη

minus Fϕ1113888 1113889


Nb


dη2minus Le fi


hminus Fiϕi1113888 1113889

(33)


Annn Lefi

Bnnn minus LeFi(34)



h1113888 1113889 (35)



y1 f

y2 fprime

y3 fPrime

y4 θ

y5 θprime

y6 ϕ

y7 ϕprime

(36)



y1prime fprime y2

y2prime fPrime y3

y3prime f

minus y1y3 + 2 y2( 11138572

+ M +1

Da1113874 1113875y2 minus 2Gry4

y4prime θprime y5

y5prime θPrime 1


23 minus M +

1Da


51113874 1113875

y6prime ϕprime y7


Nby5prime

(37)


y0(1) 0


y0(4) minus 1 0

y0(6) minus 1 0

yinf(2) 0

yinf(4) 0

yinf(6) 0

(38)

























M = 2M = 4

M = 6M = 8

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η


Da = 08Da = 1

Da = 12Da = 14


0

01

02

03

04

05

06

07

08

09

fprime(η)

1 2 3 4 5 60η






Da = 02Da = 04

Da = 06Da = 08

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Da = 05Da = 55

Da = 105Da = 155



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 50η



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 50η






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Le = 05Le = 1

Le = 15Le = 2



0

02

04

06

08

1

12

14

16

18

fprime(η)

1 2 3 4 50η

Gr = 3Gr = 4

Gr = 5Gr = 6



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Gr = 05Gr = 25

Gr = 45Gr = 65



0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Gr = 05Gr = 25

Gr = 45Gr = 65




Nt = 05Nt = 1

Nt = 15Nt = 2

minus01

0

01

02

03

04

05

06

07

fprime(η)

1 2 3 4 5 6 7 80η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 20η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

02

04

06

08

1

12

14

16

fprime(η

)

05 1 15 2 25 3 35 40η




Nb = 6Nb = 7

Nb = 8Nb = 9

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 30η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

05

1

15

2

25

fprime(η)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

01

02

03

04

05

06

07

08

09

1

θ(η)

1 2 3 4 50η



5 Conclusions











Data Availability




Acknowledgments


References






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η

Ec = 0Ec = 01

Ec = 02Ec = 03

















































f


minus M +1

Da1113874 1113875fprime + 2Grθ 0 (7)


minus PrEc M +1

Da1113874 1113875 fprime( 1113857

2


(8)

ϕPrime +Nt


f(η) 0


θ(η) 1

ϕ(η) 1 as η⟶ 0

(10)

fprime(η)⟶ 0

θ(η)⟶ 0


(11)


Pr μCp

k

M 2LσB

20

Uwρ


Tinfin]


]

Le ]

DB


U2w

Da KUw

2]L

Ec U

2w

TwCp

S QL

UwρCp

(12)


Cf ]

U2w

zu

zy1113888 1113889

y0 (13)


Nux minus (1 + Nr)x


zT

zy1113888 1113889

y0 (14)


Shx minusx


zC

zy1113888 1113889

y0 (15)


Cf 1

2Rex

1113968 fPrime(0)

Nux minus (1 + Nr)

xRex

2L

1113970

θprime(0)

Shx minus

xRex

2L

1113970

ϕprime(0)

(16)




y

v

u T

Tinfin

B0 (magnetic field)

g

x

U = Uw

T = Tw










d2F

dη2 minus f

dF

dη+ 2F

2+ M +

1Da

1113874 1113875F minus 2Grθ (17)



dη+ 2F

2+ M +

1Da

1113874 1113875F minus 2Grθ (18)



Fi+1 minus Fi

h1113888 1113889 + 2F

2i + M +

1Da

1113874 1113875Fi minus 2Grθi

(19)


An minuszf


Bn minuszf


1Da

1113874 1113875


Da1113874 1113875


Fi+1 minus Fi

h

(20)



where

ai 2 minus hAn

bi 2h2Bn minus 4

ci 2 + hAn

ri 2h2Dn

(22)


A F s (23)

where

A

b1 c1

a2 b2 c2


aNminus 1 bNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F

F1

F2

middot

middot

FNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

s

s1

s2

middot

middot

sNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


A LU (25)

where

L

β1a2 β2



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

U

1 c1

1 c2

1 cNminus 2

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(26)




c1 λβ1h



(27)


LUF s

UF z

Lz s

(28)

and we have

β1a2 β2


aNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

z3

middot

middot

middot

zNminus 2

zNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

s1

s2

s3

middot

middot

middot

sNminus 2

sNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


z1 s1

β1 si


βi

i 2 3 N minus 1

1 c1

1 c2

1 cNminus 2

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F1

F2

middot

middot

middot

FNminus 2

FNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

middot

middot

middot

zNminus 2

zNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(30)

We get

Fiminus 1 ziminus 1




fi+1 minus fi

h Fi (32)



d2θ

dη2

11 + Nr


1113888 1113889 minus PrEcdF

dη1113888 1113889

2

minus PrEc M +1

Da1113874 1113875F


dϕdη

dθdη

minus PrNtdθdη

1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

ξ2 η θ θprime( 1113857 1

1 + Nr

Pr θiFi minus fi


h1113888 1113889 minus PrEc

Fi minus Fiminus 1

dη1113888 1113889

2

minus PrEc M +1

Da1113874 1113875F

2i minus 2PrSθi

⎧⎨

⎩

⎫⎬

⎭

minus1

1 + Nr


h


hminus PrNt


dη1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

Ann minuszf

zθprime minus

11 + Nr


minus 2PrNtdθdη

1113896 1113897

Ann minuszf

zθprime

11 + Nr


h+ 2PrNt


h1113896 1113897

Bnn minuszf

zθ

minus 11 + Nr

PrF minus 2PrS1113864 1113865

Bnn minuszf

zθ

minus 11 + Nr


d2ϕ

dη2

minus Nt

Nb

d2θ

dη2minus Le f

dϕdη

minus Fϕ1113888 1113889


Nb


dη2minus Le fi


hminus Fiϕi1113888 1113889

(33)


Annn Lefi

Bnnn minus LeFi(34)



h1113888 1113889 (35)



y1 f

y2 fprime

y3 fPrime

y4 θ

y5 θprime

y6 ϕ

y7 ϕprime

(36)



y1prime fprime y2

y2prime fPrime y3

y3prime f

minus y1y3 + 2 y2( 11138572

+ M +1

Da1113874 1113875y2 minus 2Gry4

y4prime θprime y5

y5prime θPrime 1


23 minus M +

1Da


51113874 1113875

y6prime ϕprime y7


Nby5prime

(37)


y0(1) 0


y0(4) minus 1 0

y0(6) minus 1 0

yinf(2) 0

yinf(4) 0

yinf(6) 0

(38)

























M = 2M = 4

M = 6M = 8

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η


Da = 08Da = 1

Da = 12Da = 14


0

01

02

03

04

05

06

07

08

09

fprime(η)

1 2 3 4 5 60η






Da = 02Da = 04

Da = 06Da = 08

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Da = 05Da = 55

Da = 105Da = 155



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 50η



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 50η






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Le = 05Le = 1

Le = 15Le = 2



0

02

04

06

08

1

12

14

16

18

fprime(η)

1 2 3 4 50η

Gr = 3Gr = 4

Gr = 5Gr = 6



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Gr = 05Gr = 25

Gr = 45Gr = 65



0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Gr = 05Gr = 25

Gr = 45Gr = 65




Nt = 05Nt = 1

Nt = 15Nt = 2

minus01

0

01

02

03

04

05

06

07

fprime(η)

1 2 3 4 5 6 7 80η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 20η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

02

04

06

08

1

12

14

16

fprime(η

)

05 1 15 2 25 3 35 40η




Nb = 6Nb = 7

Nb = 8Nb = 9

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 30η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

05

1

15

2

25

fprime(η)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

01

02

03

04

05

06

07

08

09

1

θ(η)

1 2 3 4 50η



5 Conclusions











Data Availability




Acknowledgments


References






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η

Ec = 0Ec = 01

Ec = 02Ec = 03
























































d2F

dη2 minus f

dF

dη+ 2F

2+ M +

1Da

1113874 1113875F minus 2Grθ (17)



dη+ 2F

2+ M +

1Da

1113874 1113875F minus 2Grθ (18)



Fi+1 minus Fi

h1113888 1113889 + 2F

2i + M +

1Da

1113874 1113875Fi minus 2Grθi

(19)


An minuszf


Bn minuszf


1Da

1113874 1113875


Da1113874 1113875


Fi+1 minus Fi

h

(20)



where

ai 2 minus hAn

bi 2h2Bn minus 4

ci 2 + hAn

ri 2h2Dn

(22)


A F s (23)

where

A

b1 c1

a2 b2 c2


aNminus 1 bNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F

F1

F2

middot

middot

FNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

s

s1

s2

middot

middot

sNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


A LU (25)

where

L

β1a2 β2



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

U

1 c1

1 c2

1 cNminus 2

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(26)




c1 λβ1h



(27)


LUF s

UF z

Lz s

(28)

and we have

β1a2 β2


aNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

z3

middot

middot

middot

zNminus 2

zNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

s1

s2

s3

middot

middot

middot

sNminus 2

sNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


z1 s1

β1 si


βi

i 2 3 N minus 1

1 c1

1 c2

1 cNminus 2

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F1

F2

middot

middot

middot

FNminus 2

FNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

middot

middot

middot

zNminus 2

zNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(30)

We get

Fiminus 1 ziminus 1




fi+1 minus fi

h Fi (32)



d2θ

dη2

11 + Nr


1113888 1113889 minus PrEcdF

dη1113888 1113889

2

minus PrEc M +1

Da1113874 1113875F


dϕdη

dθdη

minus PrNtdθdη

1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

ξ2 η θ θprime( 1113857 1

1 + Nr

Pr θiFi minus fi


h1113888 1113889 minus PrEc

Fi minus Fiminus 1

dη1113888 1113889

2

minus PrEc M +1

Da1113874 1113875F

2i minus 2PrSθi

⎧⎨

⎩

⎫⎬

⎭

minus1

1 + Nr


h


hminus PrNt


dη1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

Ann minuszf

zθprime minus

11 + Nr


minus 2PrNtdθdη

1113896 1113897

Ann minuszf

zθprime

11 + Nr


h+ 2PrNt


h1113896 1113897

Bnn minuszf

zθ

minus 11 + Nr

PrF minus 2PrS1113864 1113865

Bnn minuszf

zθ

minus 11 + Nr


d2ϕ

dη2

minus Nt

Nb

d2θ

dη2minus Le f

dϕdη

minus Fϕ1113888 1113889


Nb


dη2minus Le fi


hminus Fiϕi1113888 1113889

(33)


Annn Lefi

Bnnn minus LeFi(34)



h1113888 1113889 (35)



y1 f

y2 fprime

y3 fPrime

y4 θ

y5 θprime

y6 ϕ

y7 ϕprime

(36)



y1prime fprime y2

y2prime fPrime y3

y3prime f

minus y1y3 + 2 y2( 11138572

+ M +1

Da1113874 1113875y2 minus 2Gry4

y4prime θprime y5

y5prime θPrime 1


23 minus M +

1Da


51113874 1113875

y6prime ϕprime y7


Nby5prime

(37)


y0(1) 0


y0(4) minus 1 0

y0(6) minus 1 0

yinf(2) 0

yinf(4) 0

yinf(6) 0

(38)

























M = 2M = 4

M = 6M = 8

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η


Da = 08Da = 1

Da = 12Da = 14


0

01

02

03

04

05

06

07

08

09

fprime(η)

1 2 3 4 5 60η






Da = 02Da = 04

Da = 06Da = 08

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Da = 05Da = 55

Da = 105Da = 155



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 50η



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 50η






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Le = 05Le = 1

Le = 15Le = 2



0

02

04

06

08

1

12

14

16

18

fprime(η)

1 2 3 4 50η

Gr = 3Gr = 4

Gr = 5Gr = 6



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Gr = 05Gr = 25

Gr = 45Gr = 65



0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Gr = 05Gr = 25

Gr = 45Gr = 65




Nt = 05Nt = 1

Nt = 15Nt = 2

minus01

0

01

02

03

04

05

06

07

fprime(η)

1 2 3 4 5 6 7 80η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 20η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

02

04

06

08

1

12

14

16

fprime(η

)

05 1 15 2 25 3 35 40η




Nb = 6Nb = 7

Nb = 8Nb = 9

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 30η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

05

1

15

2

25

fprime(η)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

01

02

03

04

05

06

07

08

09

1

θ(η)

1 2 3 4 50η



5 Conclusions











Data Availability




Acknowledgments


References






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η

Ec = 0Ec = 01

Ec = 02Ec = 03



















































c1 λβ1h



(27)


LUF s

UF z

Lz s

(28)

and we have

β1a2 β2


aNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

z3

middot

middot

middot

zNminus 2

zNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

s1

s2

s3

middot

middot

middot

sNminus 2

sNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


z1 s1

β1 si


βi

i 2 3 N minus 1

1 c1

1 c2

1 cNminus 2

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F1

F2

middot

middot

middot

FNminus 2

FNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

middot

middot

middot

zNminus 2

zNminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(30)

We get

Fiminus 1 ziminus 1




fi+1 minus fi

h Fi (32)



d2θ

dη2

11 + Nr


1113888 1113889 minus PrEcdF

dη1113888 1113889

2

minus PrEc M +1

Da1113874 1113875F


dϕdη

dθdη

minus PrNtdθdη

1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

ξ2 η θ θprime( 1113857 1

1 + Nr

Pr θiFi minus fi


h1113888 1113889 minus PrEc

Fi minus Fiminus 1

dη1113888 1113889

2

minus PrEc M +1

Da1113874 1113875F

2i minus 2PrSθi

⎧⎨

⎩

⎫⎬

⎭

minus1

1 + Nr


h


hminus PrNt


dη1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

Ann minuszf

zθprime minus

11 + Nr


minus 2PrNtdθdη

1113896 1113897

Ann minuszf

zθprime

11 + Nr


h+ 2PrNt


h1113896 1113897

Bnn minuszf

zθ

minus 11 + Nr

PrF minus 2PrS1113864 1113865

Bnn minuszf

zθ

minus 11 + Nr


d2ϕ

dη2

minus Nt

Nb

d2θ

dη2minus Le f

dϕdη

minus Fϕ1113888 1113889


Nb


dη2minus Le fi


hminus Fiϕi1113888 1113889

(33)


Annn Lefi

Bnnn minus LeFi(34)



h1113888 1113889 (35)



y1 f

y2 fprime

y3 fPrime

y4 θ

y5 θprime

y6 ϕ

y7 ϕprime

(36)



y1prime fprime y2

y2prime fPrime y3

y3prime f

minus y1y3 + 2 y2( 11138572

+ M +1

Da1113874 1113875y2 minus 2Gry4

y4prime θprime y5

y5prime θPrime 1


23 minus M +

1Da


51113874 1113875

y6prime ϕprime y7


Nby5prime

(37)


y0(1) 0


y0(4) minus 1 0

y0(6) minus 1 0

yinf(2) 0

yinf(4) 0

yinf(6) 0

(38)

























M = 2M = 4

M = 6M = 8

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η


Da = 08Da = 1

Da = 12Da = 14


0

01

02

03

04

05

06

07

08

09

fprime(η)

1 2 3 4 5 60η






Da = 02Da = 04

Da = 06Da = 08

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Da = 05Da = 55

Da = 105Da = 155



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 50η



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 50η






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Le = 05Le = 1

Le = 15Le = 2



0

02

04

06

08

1

12

14

16

18

fprime(η)

1 2 3 4 50η

Gr = 3Gr = 4

Gr = 5Gr = 6



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Gr = 05Gr = 25

Gr = 45Gr = 65



0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Gr = 05Gr = 25

Gr = 45Gr = 65




Nt = 05Nt = 1

Nt = 15Nt = 2

minus01

0

01

02

03

04

05

06

07

fprime(η)

1 2 3 4 5 6 7 80η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 20η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

02

04

06

08

1

12

14

16

fprime(η

)

05 1 15 2 25 3 35 40η




Nb = 6Nb = 7

Nb = 8Nb = 9

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 30η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

05

1

15

2

25

fprime(η)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

01

02

03

04

05

06

07

08

09

1

θ(η)

1 2 3 4 50η



5 Conclusions











Data Availability




Acknowledgments


References






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η

Ec = 0Ec = 01

Ec = 02Ec = 03

















































d2θ

dη2

11 + Nr


1113888 1113889 minus PrEcdF

dη1113888 1113889

2

minus PrEc M +1

Da1113874 1113875F


dϕdη

dθdη

minus PrNtdθdη

1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

ξ2 η θ θprime( 1113857 1

1 + Nr

Pr θiFi minus fi


h1113888 1113889 minus PrEc

Fi minus Fiminus 1

dη1113888 1113889

2

minus PrEc M +1

Da1113874 1113875F

2i minus 2PrSθi

⎧⎨

⎩

⎫⎬

⎭

minus1

1 + Nr


h


hminus PrNt


dη1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

Ann minuszf

zθprime minus

11 + Nr


minus 2PrNtdθdη

1113896 1113897

Ann minuszf

zθprime

11 + Nr


h+ 2PrNt


h1113896 1113897

Bnn minuszf

zθ

minus 11 + Nr

PrF minus 2PrS1113864 1113865

Bnn minuszf

zθ

minus 11 + Nr


d2ϕ

dη2

minus Nt

Nb

d2θ

dη2minus Le f

dϕdη

minus Fϕ1113888 1113889


Nb


dη2minus Le fi


hminus Fiϕi1113888 1113889

(33)


Annn Lefi

Bnnn minus LeFi(34)



h1113888 1113889 (35)



y1 f

y2 fprime

y3 fPrime

y4 θ

y5 θprime

y6 ϕ

y7 ϕprime

(36)



y1prime fprime y2

y2prime fPrime y3

y3prime f

minus y1y3 + 2 y2( 11138572

+ M +1

Da1113874 1113875y2 minus 2Gry4

y4prime θprime y5

y5prime θPrime 1


23 minus M +

1Da


51113874 1113875

y6prime ϕprime y7


Nby5prime

(37)


y0(1) 0


y0(4) minus 1 0

y0(6) minus 1 0

yinf(2) 0

yinf(4) 0

yinf(6) 0

(38)

























M = 2M = 4

M = 6M = 8

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η


Da = 08Da = 1

Da = 12Da = 14


0

01

02

03

04

05

06

07

08

09

fprime(η)

1 2 3 4 5 60η






Da = 02Da = 04

Da = 06Da = 08

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Da = 05Da = 55

Da = 105Da = 155



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 50η



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 50η






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Le = 05Le = 1

Le = 15Le = 2



0

02

04

06

08

1

12

14

16

18

fprime(η)

1 2 3 4 50η

Gr = 3Gr = 4

Gr = 5Gr = 6



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Gr = 05Gr = 25

Gr = 45Gr = 65



0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Gr = 05Gr = 25

Gr = 45Gr = 65




Nt = 05Nt = 1

Nt = 15Nt = 2

minus01

0

01

02

03

04

05

06

07

fprime(η)

1 2 3 4 5 6 7 80η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 20η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

02

04

06

08

1

12

14

16

fprime(η

)

05 1 15 2 25 3 35 40η




Nb = 6Nb = 7

Nb = 8Nb = 9

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 30η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

05

1

15

2

25

fprime(η)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

01

02

03

04

05

06

07

08

09

1

θ(η)

1 2 3 4 50η



5 Conclusions











Data Availability




Acknowledgments


References






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η

Ec = 0Ec = 01

Ec = 02Ec = 03


















































y1prime fprime y2

y2prime fPrime y3

y3prime f

minus y1y3 + 2 y2( 11138572

+ M +1

Da1113874 1113875y2 minus 2Gry4

y4prime θprime y5

y5prime θPrime 1


23 minus M +

1Da


51113874 1113875

y6prime ϕprime y7


Nby5prime

(37)


y0(1) 0


y0(4) minus 1 0

y0(6) minus 1 0

yinf(2) 0

yinf(4) 0

yinf(6) 0

(38)

























M = 2M = 4

M = 6M = 8

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η


Da = 08Da = 1

Da = 12Da = 14


0

01

02

03

04

05

06

07

08

09

fprime(η)

1 2 3 4 5 60η






Da = 02Da = 04

Da = 06Da = 08

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Da = 05Da = 55

Da = 105Da = 155



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 50η



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 50η






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Le = 05Le = 1

Le = 15Le = 2



0

02

04

06

08

1

12

14

16

18

fprime(η)

1 2 3 4 50η

Gr = 3Gr = 4

Gr = 5Gr = 6



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Gr = 05Gr = 25

Gr = 45Gr = 65



0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Gr = 05Gr = 25

Gr = 45Gr = 65




Nt = 05Nt = 1

Nt = 15Nt = 2

minus01

0

01

02

03

04

05

06

07

fprime(η)

1 2 3 4 5 6 7 80η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 20η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

02

04

06

08

1

12

14

16

fprime(η

)

05 1 15 2 25 3 35 40η




Nb = 6Nb = 7

Nb = 8Nb = 9

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 30η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

05

1

15

2

25

fprime(η)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

01

02

03

04

05

06

07

08

09

1

θ(η)

1 2 3 4 50η



5 Conclusions











Data Availability




Acknowledgments


References






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η

Ec = 0Ec = 01

Ec = 02Ec = 03































































M = 2M = 4

M = 6M = 8

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η


M = 2M = 4

M = 6M = 8


0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η


Da = 08Da = 1

Da = 12Da = 14


0

01

02

03

04

05

06

07

08

09

fprime(η)

1 2 3 4 5 60η






Da = 02Da = 04

Da = 06Da = 08

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Da = 05Da = 55

Da = 105Da = 155



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 50η



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 50η






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Le = 05Le = 1

Le = 15Le = 2



0

02

04

06

08

1

12

14

16

18

fprime(η)

1 2 3 4 50η

Gr = 3Gr = 4

Gr = 5Gr = 6



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Gr = 05Gr = 25

Gr = 45Gr = 65



0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Gr = 05Gr = 25

Gr = 45Gr = 65




Nt = 05Nt = 1

Nt = 15Nt = 2

minus01

0

01

02

03

04

05

06

07

fprime(η)

1 2 3 4 5 6 7 80η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 20η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

02

04

06

08

1

12

14

16

fprime(η

)

05 1 15 2 25 3 35 40η




Nb = 6Nb = 7

Nb = 8Nb = 9

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 30η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

05

1

15

2

25

fprime(η)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

01

02

03

04

05

06

07

08

09

1

θ(η)

1 2 3 4 50η



5 Conclusions











Data Availability




Acknowledgments


References






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η

Ec = 0Ec = 01

Ec = 02Ec = 03




















































Da = 02Da = 04

Da = 06Da = 08

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Da = 05Da = 55

Da = 105Da = 155



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

fprime(η)

1 2 3 4 50η



Le = 05Le = 1

Le = 15Le = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 50η






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Le = 05Le = 1

Le = 15Le = 2



0

02

04

06

08

1

12

14

16

18

fprime(η)

1 2 3 4 50η

Gr = 3Gr = 4

Gr = 5Gr = 6



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Gr = 05Gr = 25

Gr = 45Gr = 65



0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Gr = 05Gr = 25

Gr = 45Gr = 65




Nt = 05Nt = 1

Nt = 15Nt = 2

minus01

0

01

02

03

04

05

06

07

fprime(η)

1 2 3 4 5 6 7 80η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 20η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

02

04

06

08

1

12

14

16

fprime(η

)

05 1 15 2 25 3 35 40η




Nb = 6Nb = 7

Nb = 8Nb = 9

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 30η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

05

1

15

2

25

fprime(η)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

01

02

03

04

05

06

07

08

09

1

θ(η)

1 2 3 4 50η



5 Conclusions











Data Availability




Acknowledgments


References






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η

Ec = 0Ec = 01

Ec = 02Ec = 03




















































0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Le = 05Le = 1

Le = 15Le = 2



0

02

04

06

08

1

12

14

16

18

fprime(η)

1 2 3 4 50η

Gr = 3Gr = 4

Gr = 5Gr = 6



0

01

02

03

04

05

06

07

08

09

1

θ (η

)

1 2 3 4 5 6 7 80η

Gr = 05Gr = 25

Gr = 45Gr = 65



0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

1 2 3 4 50η

Gr = 05Gr = 25

Gr = 45Gr = 65




Nt = 05Nt = 1

Nt = 15Nt = 2

minus01

0

01

02

03

04

05

06

07

fprime(η)

1 2 3 4 5 6 7 80η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 20η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

02

04

06

08

1

12

14

16

fprime(η

)

05 1 15 2 25 3 35 40η




Nb = 6Nb = 7

Nb = 8Nb = 9

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 30η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

05

1

15

2

25

fprime(η)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

01

02

03

04

05

06

07

08

09

1

θ(η)

1 2 3 4 50η



5 Conclusions











Data Availability




Acknowledgments


References






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η

Ec = 0Ec = 01

Ec = 02Ec = 03


















































Nt = 05Nt = 1

Nt = 15Nt = 2

minus01

0

01

02

03

04

05

06

07

fprime(η)

1 2 3 4 5 6 7 80η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Nt = 05Nt = 1

Nt = 15Nt = 2

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 20η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

02

04

06

08

1

12

14

16

fprime(η

)

05 1 15 2 25 3 35 40η




Nb = 6Nb = 7

Nb = 8Nb = 9

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 30η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

05

1

15

2

25

fprime(η)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

01

02

03

04

05

06

07

08

09

1

θ(η)

1 2 3 4 50η



5 Conclusions











Data Availability




Acknowledgments


References






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η

Ec = 0Ec = 01

Ec = 02Ec = 03


















































Nb = 6Nb = 7

Nb = 8Nb = 9

0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 30η



Nb = 5Nb = 10

Nb = 15Nb = 20

0

01

02

03

04

05

06

07

08

09

1

θ (η

)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

05

1

15

2

25

fprime(η)

05 1 15 2 25 3 35 40η



Ec = 0Ec = 01

Ec = 02Ec = 03

0

01

02

03

04

05

06

07

08

09

1

θ(η)

1 2 3 4 50η



5 Conclusions











Data Availability




Acknowledgments


References






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η

Ec = 0Ec = 01

Ec = 02Ec = 03

















































5 Conclusions











Data Availability




Acknowledgments


References






0

01

02

03

04

05

06

07

08

09

1

ϕ (η

)

05 1 15 2 25 3 35 40η

Ec = 0Ec = 01

Ec = 02Ec = 03

















































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