hydrothermal analysis on mhd squeezing nanofluid …...allel plates using homotopy perturbation...

ORIGINAL PAPER Open Access

Hydrothermal analysis on MHD squeezingnanofluid flow in parallel plates byanalytical methodKh. Hosseinzadeh, M. Alizadeh and D. D. Ganji*

Abstract

Background: In this paper, the heat and mass transfer of MHD nanofluid squeezing flow between two parallelplates are investigated. In squeezing flows, a material is compressed between two parallel plates and thensqueezed out radially. The significance of this study is the hydrothermal investigation of MHD nanofluid duringsqueezing flow. The affecting parameters on the flow and heat transfer are Brownian motion, Thermophoresisparameter, Squeezing parameter and the magnetic field.

Methods: By applying the proper similarity parameters, the governing equations of the problem are converted tonondimensional forms and are solved analytically using the Homotopy Perturbation Method (HPM) and theCollocation Method (CM). Moreover, the analytical solution is compared with numerical Finite Element Method(FEM) and a good agreement is obtained.

Results: The results indicated that increasing the Brownian motion parameter causes an increase in thetemperature profile, while an inverse treatment is observed for the concentration profile. Also, it was found thatenhancing the thermophoresis parameter results in decreasing the temperature profile and augmenting theconcentration profile.

Conclusions: Effects of active parameters have been considered for the flow, heat and mass transfer. The resultsindicated that temperature boundary layer thickness will increases by augmentation of Brownian motion parameterand Thermophoresis parameter, while it decreases by raising the other active parameters.

Keywords: Squeezing flow, MHD, Nanofluid, Brownian motion, Thermophoresis phenomenon, Collocation Method(CM), HPM

BackgroundInvestigation of heat and mass transfer of viscous flowbetween two parallel plates is one of the most importantand well-known academic research topics because of thewide range of its scientific and engineering approaches,such as polymer processing, systems of lubricating, cool-ing towers, and food processing.Utilizing nano-scale particles in the base fluid seems

to be a creative technique to improve the heat transferrate. As known, nano-particle-containing fluids arecalled nanofluids. Choi 1995 was the first one ever toname the nano-particle-containing fluids as nanofluids.

Sheikholeslami and Ganji (2013a) studied analytically theheat transfer of a nanofluid flow compressed between par-allel plates using Homotopy Perturbation Method (HPM).They indicated that the Nusselt number is directly relatedto nanoparticle volume fraction along with Squeeze num-ber and Eckert number for two separated plates, whilethere is an inverse relationship between the Nusselt num-ber and Squeeze number when two plates are squeezed.Sheikholeslami and Bhatti (2017a) studied the heat trans-

fer enhancement of nanofluid flow by using EHD. Resultsindicated that the effect of Coulomb force is more consider-able in lower values of Reynolds number. The shape effectsof nanoparticles on the natural convection nanofluid flow ina porous semi-annulus were studied by Sheikholeslami andBhatti (2017b). Results illustrated that the maximum rate of

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* Correspondence: [email protected] of Mechanical Engineering, Babol Noshirvani University ofTechnology, Babol, Iran

Hosseinzadeh et al. International Journal of Mechanical and Materials Engineering (2018) 13:4 https://doi.org/10.1186/s40712-018-0089-7

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heat transfer is obtained at platelet shape. Bhatti and Rashidi(2016) investigated the influences of thermos-diffusion andthermal radiation on nanofluid flow over a stretching sheet.The authors showed that the temperature profile is an in-creasing function of thermal radiation and thermophoresisparameters. Ghadikolaei et al. (2017) reviewed the micropo-lar nanofluid flow over a porous stretching sheet. It wasfound that raising the radiation parameter enhances theboundary layer thickness. Ghadikolaei et al. (2017) analyzedthe stagnation-point flow of hybrid nanofluid over a stretch-ing sheet. They concluded that using hybrid nanofluid in-stead of conventional nanofluid results in higher Nusseltnumbers. Dogonchi et al. (2017) investigated the influenceof thermal radiation on MHD nanofluid flow in a porouschannel. Results illustrated that there is a direct relationshipbetween the Nusselt number and nanofluid volume fraction.Recently, many authors analyzed the effect of nanoparticlesin theirs studies (Abbas et al. 2017; Bhatti et al. 2017b; SaediArdahaie et al. 2018; Abbas et al. 2017).The results of the time-dependent chemical reaction on

the viscous fluid flow over an unsteady stretching sheetwere considered by Abd-El Aziz (2010). Furthermore, for acertain viscous fluid between parallel disks, the magnetohydrodynamic squeezed flow was investigated by Domairryand Aziz (2009). Also, for the fluid flow between parallelplates, the Homotopy Analysis Method (HAM (Domairryand Ziabakhsh 2009a; Domairry and Ziabakhsh 2009b; Zia-bakhsh et al. 2009)) was utilized by Mustafa et al. (2012).The majority of the problems in the field of engineering

especially heat transfer equations contain nonlinear equa-tions. In this case, some of these nonlinear equations canbe solved by numerical approaches, while some others aresolvable using various analytical methods such as Perturb-ation method (PM) (Bhatti and Lu 2017), Collocationmethod (CM) (Rahimi et al. 2017; Atouei et al. 2015),Homotopy perturbation method (HPM), and Variational it-eration method (VIM). Hence, the elimination of small par-ameter has been the critical issue for the scientistsnowadays which has led to introduction of various ways tosolve these special problems. One of these ways is the im-plementation of the semi-exact method called “HPM”which does not need any small parameters. The Homotopyperturbation method was suggested and modified by He(2004). This method results in a rapid convergence of thesolution series in most cases. Including both the efficiencyand the accuracy in solving a large number of nonlinearproblems, the HPM proved itself capable in dealing withsuch problems. Ijaz et al. (2018) applied the HPM to inves-tigate the effect of liquid-solid particles interaction in awavy channel. Dogonchi et al. (2015) analyzed the sedimen-tation of non-spherical particles in Newtonain media usingDTM-Pade approximation. It was found that enhancingthe sphericity of particles results in augmenting the velocityprofile.

Mosayebidorcheh et al. (2016) studied the analysis of tur-bulent MHD Couette nanofluid flow and heat transferusing hybrid DTM–FDM. Sheikholeslami et al. (2011) in-vestigated the rotation of MHD viscous flow along with theheat transfer between stretching and porous surfaces usingHPM. Results showed that an increase in the rotation par-ameter along with the increase in the blowing velocity par-ameter and Prandtl number would cause an increase in theNusselt number. The profiles of the variables such as thevelocity, temperature, and the concentration of the nano-fluids affected by the magnetic field are investigated byUddin et al. (2014). They comprehended that the presenceof magnetic field would cause a decrease in the velocityfield and an increase in temperature and concentration pro-files. Also, it was found that the convective heating param-eter leads to augment the velocity, temperature, andconcentration profiles. Uddin et al. (2014) also studied non-Newtonian nanofluid slip flow over a permeable stretchingsheet. Their result indicated that the skin friction factorplays a key role in the characteristics of nanofluid flow. Inaddition, the chemical reaction of nanofluid in free convec-tion in the presence of magnetic field was investigated byUddin et al. (2015). During the study of Jing et al. (2015),they discovered that the presence of nanoparticles withinthe fluid can extremely increase the effective thermal con-ductivity of the fluid, and as a result, the heat transfer char-acteristics will be improved. Sheikholeslami and Ganji(2013b) examined the nanofluid flow squeezed betweenparallel plates utilizing Homotopy perturbation method(HPM). They reported that the Nusselt number has directrelationship with nanoparticle volume fraction, the Squeezenumber and the Eckert number in the case of separatedplates, while its relationship with the Squeeze number inthe case of squeezed plates is vice versa. Paying attention tothe nanoparticle migration, the mixed convection of alu-mina–water nanofluid inside a concentric annulus was in-vestigated by Malvandi and Ganji (2016). Sulochana et al.(2016) examined the effect of transpiration on the mag-netohydrodynamic stagnation-point flow of a Carreaunanofluid toward a stretching/shrinking sheet in the pres-ence of thermophoresis and Brownian motion, numerically.They discovered that by raising the thermophoresis param-eter, both the heat and mass transfer rates will be increased,whereas the Weissenberg number enlarges the momentumboundary layer thickness along with the heat and masstransfer rate. Sheikholeslami et al. (2016) studied the effectof Lorentz forces on forced-convection nanofluid flow overa stretched surface. Their results indicated that the skinfriction coefficient increases by amplifying the magneticfield, while it decreases by enhancing the velocity ratio par-ameter. Sudarsana Reddy and Chamkha (2016) analyzedthe influence of size, shape, and type of nanoparticles alongwith the type and temperature of the base fluid on the nat-ural convection MHD nanofluid flow. Their results revealed

Hosseinzadeh et al. International Journal of Mechanical and Materials Engineering (2018) 13:4 of 13

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that decreasing the size of the nanoparticles leads to a sig-nificant natural convection heat transfer rate. Moreover,types of nanoparticles and the base fluid also impressed thenatural convection heat transfer. Mishra and Bhatti (2017)investigated the MHD stagnation-point flow over a shrink-ing sheet, numerically. The authors compared the accuracyof their solution with previous studies and found that agood agreement was obtained. Newly, the study of MHDflow in different geometries has attracted many attentions(Bhatti et al. 2017a; Ghadikolaei et al. 2017; Hatami et al.2014; Bhatti et al. 2018).The main goal of the present study is to investigate

the effect of Brownian motion and thermophoresisphenomenon on the squeezing nanofluid flow and heattransfer between two parallel flat plates in the presenceof variable magnetic field. Both the flow and heat trans-fer characteristics have been examined under the effectsof Squeeze number, suction parameter, Hartmann num-ber, Brownian motion parameter, Thermophoretic par-ameter, and Lewis number.

Problem description and governing equationsThis study is concerned with incompressible two-dimensional flow of squeezing nanofluid between twoparallel and movable plates at distance of h(t) =H(1 −at)1/2 from each other. The schematic model of theproblem is depicted in Fig. 1. As shown in Fig. 1, B(t) =B0(1 − at)−1/2 is the variable magnetic field that appliedperpendicular to the plates. To simplify the problem,only the flow patterns on the left part of the channelhave been mentioned. It should be noted that the flowpatterns of squeezing flow are axisymmetric.The “×” marks show that magnetic field is perpendicu-

lar to the illustrated plane. “+” and “−” indicate the posi-tive and negative charges, respectively. Tw and Cw

represent the temperature and concentration of nano-particles at the bottom disk, while the temperature andconcentration of nanoparticles at the upper disk are de-noted by TH and CH. The upper disk at Z = h(t) can

move toward or away from the motionless bottom disk

with the velocity of dh.dt

.

For a > 0 and a < 0, two plates are squeezed and sepa-rated, respectively. The viscous dissipation effect alongwith the generated heat remained intact due to the fric-tion caused by shear forces in the flow. It should benoted that when the fluid is largely viscous or is flowingat a high speed, the dissipation effect is quite important.Knowing that the nanofluid is a two-component mix-ture, the following assumption has been considered:Incompressible; no-chemical reaction; with negligible-

ness of viscous dissipation and radiative heat transfer;nano-solid-particles and the base fluid are in thermalequilibrium and without any slip between them. Theequations which govern the flow, heat, and mass transferin viscous fluid are as follows (Hashmi et al. 2012; Tur-kyilmazoglu 2017):

∇!:V!¼ 0 ð1Þ

ρ∂ ν!∂t

� �þ ν!:∇

!� �ν!¼ −∇

!pþ μ∇ 2 ν!

þ σ J!� B

!� �ð2Þ

ρcp� � ∂T

∂t

� �þ ν!:∇

!� �T

¼ K∇ 2T

þ τ DB þ ∇!T :∇

!C

� �þ DT

Tm∇!T :∇

!T

� �� ð3Þ

ρcp� � ∂C

∂t

� �þ ν!:∇

!� �C ¼ DB∇ 2C þ DT

Tm∇ 2T ð4Þ

where J!¼ E

!þ ðV!� B!Þ, E is neglected due to small

magnetic Reynolds, so J!¼ ðV!� B

!Þ V!¼ ðU ;V ;W Þ

is the velocity vector; T, P, ρ, μ, CP,K are thetemperature, pressure, density, viscosity, heat capaci-tance, and thermal conductivity of nanofluid, respect-

ively. Also, the operation of ∇!

can be defined as:

∇!¼ ∂

∂X;∂∂y

;∂∂z

� �ð5Þ

The boundary conditions are as follows:

z ¼ h tð Þ : u ¼ 0w ¼ dhdt

T ¼ ThC ¼ Ch

z ¼ 0 : u ¼ 0;w ¼ −w0ffiffiffiffiffiffiffiffiffiffi1−at

p ;T ¼ TW ;C ¼ CW

ð6Þ

where u and v represent velocity components in the r-and z-directions, respectively, ρ is the density, μ is thedynamic viscosity, p is the pressure, T is thetemperature, C is the nanoparticles concentration, α is

Fig. 1 Schematic of the problem. Nanofluid between parallel platesin the presence of variable magnetic field


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Table

1HPM

andCM

solutio

nwhe

nA=1,M=1,S=1,Le

=1,Nb=1,Nt=

1,Pr=6.2.

ηf(η

)θ(η)

ϕ(η)

HPM

CM

FEM

%Error

HPM

CM

FEM

%Error

HPM

CM

FEM

%Error

01

11

01

11

01

11

0

0.1

0.9828509855

0.9824627697

0.9824361313

0.0000266384

0.9346923870

0.8308268138

0.830835128

0.0000083142

0.7793705558

0.9000934056

0.9112683052

−0.0111748996

0.2

0.9392513782

0.9375660234

0.9375269886

0.00003900384

0.8640178590

0.6904013197

0.6910783899

−0.0006770702

0.6012650162

0.7989509826

0.8108534964

−0.0119025138

0.3

0.8788058754

0.8752398967

0.8752157845

0.0000241122

0.7884934362

0.5714471980

0.5725408493

−0.0010936513

0.4559056726

0.6970283829

0.7064057452

−0.0093773623

0.4

0.8090072079

0.8036034623

0.8035984828

0.0000049795

0.7076406847

0.4683468174

0.4695239098

−0.0011770924

0.3359299240

0.5947812584

0.6016217396

−0.0068404812

0.5

0.7359775756

0.7293903870

0.7293979521

0.0000075651

0.6202061754

0.3767871883

0.37785019

−0.0010630017

0.2360877164

0.4926652610

0.4980783825

−0.0054131215

0.6

0.6650106060

0.6583174081

0.658336159

− 0.0000187509

0.5243799637

0.2934637222

0.2943510479

−0.0008873257

0.1530318820

0.3911360429

0.3962207368

−0.0050846939

0.7

0.6009852669

0.5953956265

0.5954289745

−0.000033348

0.4179140248

0.2158417967

0.2165647056

−0.0007229089

0.08537940058

0.2906492557

0.2959127913

−0.0052635356

0.8

0.5487041816

0.5451846184

0.5452260815

− 0.0000414609

0.2979942644

0.1419761291

0.1425481689

−0.0005720398

0.03409554226

0.1916605515

0.196753453

−0.0050929015

0.9

0.5131945405

0.5119893652

0.5120144707

− 0.0000251055

0.1607055638

0.07038795247

0.0707603109

− 0.00037235843

0.003119131100

0.09462558233

0.0982635139

− 0.00363793157

10.4999999999

0.5000000008

0.5

0.0000000008

00

00

00

00


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the thermal diffusivity, DB is the Brownian motion coef-ficient, Tm is the mean fluid temperature, and k is thethermal conductivity. The total diffusion mass flux fornanoparticles is the last term in the energy equationwhich is given as a sum of the Brownian motion andthermophoresis terms. In addition, τ is the dimension-less parameter which can calculate the ratio of effectiveheat capacity of the nanoparticles to heat capacity of thefluid. The parameters of similarity solution are asfollows:

u ¼ ar2 1‐atð Þ f

0ηð Þ;w ¼ ‐

aHffiffiffiffiffiffiffiffi1‐at

p f ηð Þ; η ¼ z

Hffiffiffiffiffiffiffiffi1‐at

p

B tð Þ ¼ B0ffiffiffiffiffiffiffiffi1‐at

p ; θ ¼ T‐Th

TW‐Th; f ¼ C‐Ch

CW‐Ch

ð7Þ

By removing the pressure gradient from Eqs. (2) and(3), then rewriting Eqs. (4) and (5), the final nonlinearequations can be obtained as follows (Turkyilmazoglu2016):

f‴0‐S η f‴ þ 3f″‐2f f‴� �

‐M2 f″ ¼ 0

θ″ þ prS 2fθ0‐ηθ

0� �þ prNbθ

0f0 þ prNtθ

02 ¼ 0

f″ þ LeS 2f f0‐η f

0� �þ NtNb

θ″ ¼ 0

8>>><>>>:

ð8Þ

Boundary conditions are described as follows:

f 0ð Þ ¼ A; f00ð Þ ¼ 0; θ 0ð Þ ¼ ϕ 0ð Þ ¼ 1

f01ð Þ ¼ 1

2; f

01ð Þ ¼ 0; θ 1ð Þ ¼ f 1ð Þ ¼ 0

ð9Þ

where S is the Squeeze number, A is the suction/blow-ing parameter, M is the Hartmann number, Nb is theBrownian motion parameter, Nt is the Thermophoreticparameter, Pr is the Prandtl number, and Le is the Lewisnumber and are defined as follows:

A ¼ W0

aH; S ¼ aH2

2υ;M ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiσB2

0H2

υ

s; Pr ¼ υ

α;

Nb ¼ ρcð ÞpDB CW‐Chð Þρcð Þ fυ

;Nt ¼ ρcð ÞpDT Tw‐Thð Þρcð Þ fTmυ

ð10Þ

The continuity equation is identically satisfied. Itshould be noted that A > 0 indicates the suction of fluidfrom the lower disk, while A < 0 represents the injectionflow.

MethodsCollocation method (CM)Suppose we have a differential operator D acting on afunction u to produce a function p (Hatami et al. 2013).

D u xð Þð Þ ¼ p xð Þ ð11ÞFunction u can be considered as a function ~u, which is

a linear combination of basic functions chosen from alinearly independent as follows:

u≅~u ¼Xni¼1

ciφi ð12Þ

Now, we can substitute ~u from Eq. (12) into the Eq.(11), generally p (x) is not the result of the operations.Therefore an error or residual will exist:

E xð Þ ¼ R xð Þ ¼ D ~u xð Þð Þ−p xð Þ≠0 ð13ÞThe basic principle of the Collocation method is to

lead an error or the residual to zero in some averagesense over the domain as follows:

ZXR xð Þ Wi xð Þ ¼ 0 i ¼ 1; 2;…; n ð14Þ

So that the number of weight functions Wi and thenumber of unknown constants ci (Eq.(13)) are exactlyequal. The result is a set of n algebraic equations for theunknown constants ci. In collocation method, theweighting functions are obtained from the family of Di-rac δ functions in the domain. That is, Wi(x) = δ(x − xi).The Dirac function is defined as follows:

δ x−xið Þ ¼ 1 if x ¼ xi0 Otherwise

�ð15Þ

Application of CMTo obtain an approximate solution for Eq. (8) in the do-main 0 < η < 1, we consider the basic function to polyno-mial in η. The trial solution contains threeundetermined coefficients and satisfies the condition forall values of c as follows:

f ηð Þ ¼ C0 þ C1ηþ C2η2 þ C3η3 þ C4η4 þ C5η5 þ C6η

θ ηð Þ ¼ C7 þ C8ηþ C9η2 þ C10η

3 þ C11η4 þ C12η

5 þ C13η6

ϕ ηð Þ ¼ C14 þ C15ηþ C16η2 þ C17η3

ð16Þwhere Eq. (16) satisfies the boundary conditions of Eq.

(9). The residual function (R (c1, c2, c3, η)) can be ob-tained by substituting Eq. (16) into Eq. (13). The residualis equal to zero only by exact solution of the problem.Here, the problem is solved by the approximate solution


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so that the residual stays close to zero throughout thedomain 0 < η < 1. Three points are needed to find thethree unknown parameters, so three specific points withapproximately equal distance should be chosen in thedomain. These points are:

η1 ¼14; η2 ¼

24; η3 ¼

34

ð17Þ

Eventually, by substitutions values of Eq. (17) into re-sidual function R (c1, c2, c3, η), a set of three long equa-tions with three unknown coefficients are obtained.After solving these unknown parameters (c1, c2, c3), thetemperature distribution equation, Eq. (16), will bedetermined.To find the solution, the parameters can be considered

as Pr = 6.2, A = 1, Nt = 1, and M = S = Le = Nb = 1, andbased on Eq. (10) (Sabbaghi et al. 2011), the f(η), θ(η)and ϕ(η) formulation are obtained as follows:

f ηð Þ ¼ 1−1:967401394η2 þ 2:249807551η3−1:176830107η4

þ:4738431393η5−0:7941918846e−x6

θ ηð Þ ¼ 1−1:864675877ηþ 1:894175562η2−1:748719915η3

þ1:054402097η4−:4154680758η5 þ 0:8028620820e−η6

ϕ ηð Þ ¼ 1−:9913679607η−0:8457402944e−η2 þ 0:7594199010e−η3

ð18Þ

Homotopy Perturbation Method (HPM)In order to show the main idea of this method, we con-sider the following equation (Turkyilmazoglu 2015):

A uð Þ− f rð Þ ¼ 0 r∈Ω ð19Þ

Considering the boundary conditions:

B u;∂u∂n

� �¼ 0; r∈Γ; ð20Þ

Fig. 2 a Comparison velocity profile of the solutions via CM, HPM, and numerical solution for A = 1, M = 1, S = 1, Le = 1, Nb = 1, Nt = 0.1, Pr = 6.2.b Comparison temperature profile of the solutions via CM, HPM, and numerical solution for A = 1, M = 1, S = 1, Le = 1, Nb = 1, Nt = 0.1, Pr = 6.2. cComparison concentration profile of the solutions via CM, HPM, and numerical solution for A = 1, M = 1, S = 1, Le = 1, Nb = 1, Nt = 0.1, Pr = 6.2


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where A is a general differential operator, B is aboundary operator, f (r) is a known analytical function,and Γ is the boundary of the domainΩ.A can be divided into linear and nonlinear parts, where

L and N represent the linear and nonlinear parts, re-spectively. Therefore, Eq. (21) can be rewritten as follow:

L uð Þ þ N uð Þ− f rð Þ ¼ 0; r∈Ω; ð21Þ

The structure of homotopy perturbation is shown asfollow:

H v; pð Þ ¼ 1−pð Þ L vð Þ−L u0ð Þ½ � þ p A vð Þ− f rð Þ½ �¼ 0 ð22Þ

where,

v r; pð Þ : Ω� 0; 1½ �→R ð23Þ

Here, P ∈ [0, 1] is an embedding parameter and u0 is thefirst approximation that satisfies the boundary condition.The solution of Eq. (22) can be defined as a power seriesin p as following:

V ¼ V 0 þ PV 1 þ P2V 2 þ…… ð24ÞFinally, the best approximation for solution is written

as:

u ¼ limp→lv ¼ v0 þ v1 þ v2 þ ::… ð25Þ

Application of HPMIn order to solve a problem with Homotopy Perturb-ation Method (HPM), we can construct a homotopy ofEq. (8) as follows:

H f ; pð Þ ¼ 1−pð Þ f iv− f iv0� �

þ p f‴0−s η f‴ þ 3 f ″−2 f f ‴� �

−M2 f ″� �

¼ 0;

ð26Þ

H θ;Pð Þ ¼ 1−Pð Þ θ″−θ″0� �

þP θ″ þ PrS 2 f θ0−ηθ

0� �þ prNbθ

0ϕ

0 þ PrNtθ02� �

¼ 0

ð27Þ

Fig. 3 a Effect of A parameter on velocity distribution when M = 1, S = 1, Le = 1, Nb = 1, Nt = 1, Pr = 6.2. b Effect of S parameter on velocitydistribution when M = 1, A = 1, Le = 1, Nb = 1, Nt = 1, Pr = 6.2. c Effect of M parameter on velocity distribution when Nt = 1, A = 1, Le = 1, S = 1,Nb = 1, Pr = 6.2


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H ϕ;Pð Þ ¼ 1−Pð Þ ϕ″−ϕ″0

� �þ P ϕ″ þ LeSð2 f ϕ 0

−ηϕ0 þ Nt

Nbθ″

� �¼ 0; ð28Þ

f and θ can be defined as follows:

f ηð Þ ¼ f 0 ηð Þ þ f 1 ηð Þ þ ::…Xni¼0

f i ηð Þ ð29Þ

θ ηð Þ ¼ θ0 ηð Þ þ θ1 ηð Þ þ :…Xni¼0

θi ηð Þ ð30Þ

ϕ ηð Þ ¼ ϕ0 ηð Þ þ ϕ1 ηð Þ þ :…Xni¼0

ϕi ηð Þ ð31Þ

By substituting f, θ, ϕ from Eqs. (29–31) into Eqs. (26–28)and some simplification, then rearranging the equations interms of powers of p, the following equations are achieved:

p0 :f 0

iv ¼ 0;θ0

0 0 ¼ 0;ϕ″ ¼ 0;

ð32Þ

And boundary conditions are:

f 0 0ð Þ ¼ A; f000ð Þ ¼ 0; θ0 0ð Þ ¼ ϕ0 0ð Þ ¼ 1

f 1 0ð Þ ¼ 12; f0

01ð Þ ¼ 0; θ0 1ð Þ ¼ ϕ0 1ð Þ ¼ 0

ð33Þ

pl :

f1iv−Sη f 0

0 00−3Sf 00 0 þ 2sf 0 f 0

0 00−M2 f 00 0 ¼ 0

θ10 0 þ prNbθ0

0ϕ0

0 þ prNt θ00� �2

−prSηθ00 þ 2 PrSf 0θ0

0 ¼ 0

ϕ″1 þ Nt=Nbθ0

0 0 þ 2LeSf 0ϕ00−LeSηϕ0

0 ¼ 0

ð34ÞAnd boundary conditions are:

f 1 0ð Þ ¼ 0; f01 0ð Þ ¼ 0; θ1 0ð Þ ¼ ϕ1 0ð Þ ¼ 0

f 1 1ð Þ ¼ 0; f01 1ð Þ ¼ 0; θ1 1ð Þ ¼ ϕ1 1ð Þ ¼ 0

ð35Þ

By solving Eqs. (32) and (34) with boundary conditionsand then substituting their answers into Eqs. (29–31), f,θ, ϕ are obtained as follows:

f ηð Þ ¼ 1−:25890η9 þ :24844η8−0:21932e−2η15 þ 0:47762e−3η14

þ0:36385e−4η17 þ 0:53870e−3η16 þ 1:3516η5 þ 0:15284e

−1η13−0:38072e−η12 þ 0:32282e−5η19−0:30668e−4η18

−1:9412η2 þ 2:4363η3−1:8661η4 þ 0:72865e−η7−:64135η6

þ0:13957e−η11 þ :10822η10θ ηð Þ ¼ 1−:62167η−0:53182e

−1η9 þ :21219η8−0:33031e−2η15−0:27890e−2η14

−0:18401e−3η17 þ 0:15641e−2η16−0:22944e−η5

þ0:10149e−η13−0:23106e−2η12−:34694η2 þ :38050η3

−:51379η4−:27422η7 þ :29031η6 þ 0:46540e−η11

−0:99914e−η10ϕ ηð Þ ¼ 1−2:4579η−0:36η9−0:21724η8

−0:0033031η15−0:0091460η14−0:00018401η17

þ0:0015641η16 þ 0:84535η5 þ 0:049556η13−0:041270η12

þ2:7314η2−2:2496η3 þ 0:88028η4 þ 1:3372η7−1:7434η6

−0:14292η11 þ 0:37952η10

ð36Þ

Numerical Finite Element Method (FEM)Some several methods can be useful to find a solution offluid flow and heat transfer problems such as the Finite Dif-ference Method, the Finite Volume method (FVM), and theFinite Element Method (FEM). The control volume FiniteElement Method (CVFEM) contains interesting featuresfrom both the FVM and FEM. The CVFEM benefits from

Fig. 4 a Effect of A parameter on temperature distribution whenM = 1, S = 1, Le = 1, Nb = 1, Nt = 1, Pr = 6.2. b Effect of A parameteron concentration distribution when M = 1, S = 1, Le = 1, Nb = 1,Nt = 1, Pr = 6.2


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the flexibility of the FEMs to discretize complex geometrywith conservative formulation of the FVMs, in which thevariables can be easily interpreted physically in terms offluxes, forces, and sources (Kandelousi and Ganji n.d.).FlexPDE is a scripted Finite Element model builder

and numerical solver. This software performs the essen-tial operations to turn a description of a partial differen-tial equations system into a Finite Element model andfinally solve the system, and present graphical and tabu-lar output of the results (Table 1).

Results and discussionIn the present study, the effect of Brownian motion andThermophoresis phenomenon on the heat and mass

transfer of MHD nanofluid flow between parallel platesis investigated, and Collocation Method (CM), Homo-topy Perturbation Method (HPM) along with the finiteelement Method (FEM) are applied to solve this problemusing Maple 16 and FlexPDE 5 softwares. The influenceof certain active parameters such as Squeeze number,suction parameter, Hartmann number, Prandtl number,Brownian motion parameter, Thermophoretic parameterand Lewis number on the flow and heat transfer charac-teristics are examined. The presented code is validatedby comparing the obtained results with the results of fi-nite element method (FEM) (Fig. 2). The comparisonwell showed that by implementing this code, a highly ac-curate solution is obtained to solve the problem.

Fig. 6 a Effect of Nb parameter on temperature distribution whenM = 1, A = 1, Le = 1, S = 1, Nt = 1, Pr = 6.2. b Effect of Nb parameteron concentration distribution when M = 1, A = 1, Le = 1, S = 1,Nt = 1, Pr = 6.2

Fig. 5 a Effect of S parameter on temperature distribution whenM = 1, A = 1, Le = 1, Nb = 1, Nt = 1, Pr = 6.2. b Effect of S parameteron concentration distribution when M = 1, A = 1, Le = 1, Nb = 1,Nt = 1, Pr = 6.2


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The effect of suction parameter, Squeeze number, andHartmann number on velocity profile is shown in Fig. 3.Increasing the suction parameter would cause an in-crease in velocity profile due to the increase of the tur-bulence in the flow. It can be seen that the velocityvalues drop by enhancing the Squeeze number becausethe plate remained close to each other and limits the vel-ocity. Also, it can be found that enhancing the Hart-mann number in the flow results in augmenting thevelocity profile.Figures 4 and 5 represent the influences of suction

parameter and Squeeze number on the temperature andconcentration profiles, respectively. Figures depicted that

increasing the suction parameter would cause a decreasein thermal boundary layer thickness and concentrationprofiles. Effect of Brownian motion parameter ontemperature and concentration profiles is shown inFig. 6, while the effect of Thermophoretic parameter onthe mentioned profiles is examined in Fig. 7. It can beobserved that increasing the Brownian motion parameterresults in increasing the temperature profile, while theinfluence of Thermophoretic parameter on temperatureprofile is vice versa compared to Brownian motion par-ameter, whereas increasing both the Brownian motionparameter and Thermophoretic parameter individuallywould cause a decrease in concentration profiles which

Fig. 7 a Effect of Nt parameter on temperature distribution whenM = 1, A = 1, Le = 1, S = 1, Nb = 1, Pr = 6.2. b Effect of Nt parameteron concentration distribution when M = 1, A = 1, Le = 1, S = 1,Nb = 1, Pr = 6.2

Fig. 8 a Effect of M parameter on temperature distribution whenNt = 1, A = 1, Le = 1, S = 1, Nb = 1, Pr = 6.2. b Effect of M parameteron concentration distribution when Nt = 1, A = 1, Le = 1, S = 1,Nb = 1, Pr = 6.2


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is depicted in Fig. 8. As shown in these figures, the nano-particle temperature and concentration profiles are de-creasing functions of the Hartmann number. Figure 9shows the effect of Lewis number on temperature andconcentration profiles. It is observed that an increase intemperature profile near the bottom plate and also a de-crease in temperature profile near the top plate are the re-sults of enhancing the Lewis number. Finally, it should bementioned that higher values of nanoparticle concentra-tion are obtained by enhancing the Lewis number.

ConclusionThe present study examines the effect of Brownian mo-tion and Thermophoresis phenomenon on the heat and

mass transfer of MHD nanofluid flow between parallelplates. To examine this problem, a number of methodssuch as the Collocation Method (CM), the HomotopyPerturbation Method (HPM), and the Finite ElementMethod (FEM) were applied. The results indicated thatthe outcomes of Collocation Method have the bestagreement with the numerical solutions. The crucial ef-fect of Brownian motion and thermophoresis parameterhas been included in the model of nanofluid. Effects ofactive parameters have been considered for the flow,heat, and mass transfer. The results indicated thattemperature boundary layer thickness will increase byaugmentation of Brownian motion parameter and Ther-mophoresis parameter, while it decreases by raising theother active parameters. Also, it can be concluded thatthe thickness of concentration boundary layer declinesby enhancing the Brownian motion parameter, while aninverse trend is observed by augmenting the Thermo-phoresis parameter.

AbbreviationsTw: Nanoparticle temperature (K); Cw: Nanoparticles concentration (%wt);A: Suction/blowing parameter; B: Magnetic field (T); FEM: Finite ElementMethod; H: Height between the plate (m); HPM: Homotopy PerturbationMethod; k: Thermal conductivity (w/m. k); Le: Lewis number; M: Hartmannnumber; Nb: Brownian motion parameter; Nt: Thermophoretic parameter;p: Pressure (Pa); P: Pressure term; Pr: Prandtl number; S: Squeeze number;t: Time (s); Z: Vertical direction

Greek symbolsα: Thermal diffusivity; σ: Stefan–Boltzmann constant (w/m2k4); ρ: Density (kg/m3); θ: Dimensionless temperature; ϕ: Nanoparticle volume fraction;η: Dimensionless variable

Subscriptsf: Base fluid; nf: Nanofluid; CM: Collocation Method

AcknowledgementsNot applicable

FundingThis research doesn’t have any funding.

Availability of data and materialsNot applicable

Authors’ contributionsKh. Hosseinzadeh has involved in conception and design of the problemand analysis and interpretation of data. M. Alizadeh made substantialcontributions to the study specifically in the conception, design and runningmost of the simulation analysis. D.D. Ganji contributed in the developmentof the concept and understood the theory behind the concept. All authorscontributed to the problem formulation, drafted the manuscript, and readand approved the final manuscript.

Authors’ informationKh. Hosseinzadeh is PhD student in mechanical engineering at BabolNoshirvani University of Technology.M. Alizadeh is PhD student in mechanical engineering at Babol NoshirvaniUniversity of Technology.D.D.Ganji is full professor in mechanical engineering faculty at BabolNoshirvani University of Technology.

Fig. 9 a Effect of Le parameter on temperature distribution whenNt = 1, A = 1, M = 1, S = 1, Nb = 1, Pr = 6.2. b Effect of Le parameteron concentration distribution when Nt = 1, A = 1, M = 1, S = 1,Nb = 1, Pr = 6.2


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Competing interestsThe authors declare that they have no competing interests.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional affiliations.

Received: 15 September 2017 Accepted: 19 January 2018

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