Research ArticleHeat and Mass Transfer Analysis of Nanofluid Flow Based on CuAl2O3 and TiO2 over a Moving Rotating Plate and Impact ofVarious Nanoparticle Shapes

W Abbas 1 and M M Magdy2

1Basic and Applied Science Department College of Engineering and TechnologyArab Academy for Science Technology and Maritime Transport Cairo Egypt2Maintenance and Operation Department Safety Aviation Sector Cairo Airport Cairo Egypt

Correspondence should be addressed to W Abbas wael_abassaastedu

Received 3 January 2020 Revised 28 February 2020 Accepted 4 March 2020 Published 7 May 2020

Academic Editor Mariano Torrisi

Copyright copy 2020W Abbas and M M Magdy )is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the original work is properly cited

)e study of rotating nanofluid flows has a vital role in several applications such as in food processing rotating machinery coolingsystems and chemical fluid )e aims of the present work are to improve the thermophysical properties of convective flow and heattransfer for unsteady nanofluid past a moving rotating plate in the presence of ohmic viscous dissipations Brownian andthermophoresis diffusion )e system is strained under the effect of strong magnetic field and then the Hall current is consideredFor this investigation three different types of the nanoparticles Cu (copper) Al2O3 (aluminium oxide) and TiO2 (titanium dioxide)with various shapes (spherical cylindrical and brick) are considered and water is used as a base nanofluid )e system governingequations are solved semianalytically using homotopy perturbation technique In order to validate the present work differentcomparisons are made under some special cases with previously published results and found an excellent agreement It is observedthat the shape of nanoparticles plays a substantial role to significantly determine the flow behaviour Also it can be found that the useof the cylindrical nanoparticle shape has better improvement for heat transfer rate compared with the other nanoparticle shapes

1 Introduction

In recent years investigation of nanofluid flow has gainedconsiderable interest by the researchers owing to increase inthe implementations in different fields of technology sci-ence biomechanics chemical and nuclear industriesNanofluid can be applied to engineering problems such assolar energy collection heat exchangers and engine cooling[1] )e term of nanofluids means the addition of smallquantity of nanometer-sized particles nominally less than100 nm into base fluids like oil water biofluids ethyleneand lubricants)is termwas first introduced by Choi [2] headded small nanoparticles in a base fluid to increase thethermal conductivity Eastmann et al [3] discussed thethermal conductivity in ethylene glycol nanofluid-basedcopper nanoparticles Since then the growth of nanofluidmathematical modelling advances by demonstrating thecharacteristics of thermophoresis and Brownian diffusion

Chamkha et al [4] studied the thermal radiation effect onnanofluid boundary layer flow past a vertical cone Time-dependent viscosity effect on free convection heat transferof magnetohydrodynamic nanofluid was investigated bySheikholeslami et al [5] Hayat et al [1] investigated theflow with heat and mass transfer characteristics in mag-netohydrodynamic (MHD) squeezing flow between twosurfaces )e effect of Joule heating and Hall current onnanofluid flow along a vertical cone with heat and masstransfer was investigated by Abbas and Sayed [6]Mahanthesh et al [7] analyzed the effect of the novel ex-ponential space-dependent heat source on magnetohy-drodynamic nanofluid flow past a stretchable rotating diskwith cross-diffusion and the convective condition Latermany researchers have been engaged in recent develop-ments on nanofluids [8ndash11]

Study of convective heat transfer and flow throughmagnetohydrodynamic nanofluid in the recent years has

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 9606382 12 pageshttpsdoiorg10115520209606382

received significant attraction by many researchers for theiruseful applications in both engineering and sciences such asreactors cooling or heating processes and solar energyShehzad et al [12] studied the convective heat transfer effectsof nanofluid flow on a wavy channel with Brownian andthermophoresis diffusion effect Heat transfer of a nanofluidover a stretching sheet with considering the effects of tem-perature jump slip velocity and radiation was investigated byShen et al [13] Jahan et al [14] reported the nanofluid flowwith stability and regression analysis of a permeablestretching sheet over the heated surface In other studies theconvective heat transfer through nanofluid bounded by amoving rotating frame is investigated due to the increase inengineering and technological process applications Hamadand Pop [15] studied heat transfer of unsteady nanofluid flowfor a moving oscillatory plate with constant heat sourceNanofluid flow through two plates with rotating system inpresence of magnetic field was reported by Sheikholeslamiand Ganji [16] Hussain et al [17] presented the analyticalsolution of unsteady flow past a moving rotating plateconsidering the effects of Hall current and chemical reactionMany investigators studied the thermocapillary convectivewith heat transfer on the fluidnanofluid flow field underdifferent categories and conditions [18ndash21]

)e thermal radiation effect has attractedmuch attentionfor researchers for its enormous area of scientific and en-gineering applications such as electric power food solar cellpanels and medical industry [22] Mahanthesh et al [23]presented effects of nonlinear thermal radiation and sus-pended nanoparticles on mixed convection boundary layernanofluid flow past a melting vertical surface Kumar et al[24] studied thermal analysis of bioconvection magneto-nanofluid flow containing gyrotactic microorganisms withsecond-order velocity slip and convective conditions Razaet al [25] presented the combined effects of thermal radi-ation and magnetic field of molybdenum disulfide nanofluidon a channel with shape effects

Based on published literature it is observed that themotion of nanofluid flow through amoving rotating plate has asignificant potential role in several applications such as foodprocessing polymer processing rotating machinery cooling ofmetallic electronic component surfaces and chemical fluid)erefore the objective of the present study is to investigate theeffects of the nanoparticle shape Brownian and thermopho-resis diffusion on the motion ofMHD unsteady nanofluid flow

through past a moving rotating plate in the presence of ohmicviscous dissipations and thermal radiation )ree differenttypes of the nanoparticles are considered )ese types arecopper aluminium oxide and titanium dioxide with variousshapes (spherical cylindrical and brick) and water is used as abase nanofluid )e problem is formulated and solved semi-analytical using homotopy perturbation technique Physicalparameters are displayed and analyzed and pertinent resultsare discussed in detail using graphs Different comparisons aremade under some special cases with previously publishedresults which are conducted to validate the current results andprove efficiency of the methodology

2 Mathematical Model and Formulation

21 Problem Description Consider unsteady MHD nano-fluid flow of ambient temperature Tinfin and concentration Cinfinwith heat and mass transfer through the porous mediumpast a moving flat plate in a rotating frame )e coordinatesystem (x y z) is selected in such a way that x minus axis is takenalong the plate z minus axis is perpendicular to it and y minus axis isnormal to the (xz) plane )e physical model is described inFigure 1)ree different kinds of nanoparticles Cu (copper)Al2O3 (aluminium oxide) and TiO2 (titanium dioxide) withwater used as a base nanofluid are considered )e fluid andnanoparticles are assumed to have no slip between them andboth are in thermal equilibrium Table 1 clearly lists thethermophysical properties of the base fluid and differentnanoparticles [26 27] In this work there are three shapes ofnanoparticles which are spherical cylindrical and brickemployed into account Also the model is assumed rotatingabout z minus axis with constant angular velocity Ω )e fluid ispermeated by applied time-dependent magnetic field whichtakes the form B(t) G(eminus αt) which is taken to be actingalong the positive z-direction where G and α are constantsIt is assumed that the magnetic Reynolds number is to bevery small so that the induced magnetic field can beneglected and there is no applied external electric field It isalso assumed that the frequency of electron-atom collision tobe relatively high so that the Hall effect cannot be neglected

22 Governing Equations With above assumptions thenanofluid flow is governed by conservations of continuitymomentum energy and concentration equations as

nabla middot V 0

zV

zt+(V middot nabla)V minusΩ times V

1ρnf

nabla(μ middot nablaV) + J times B(t) + FT + FC + Fp1113960 1113961

zT

zt+(V middot nabla)T

1(ρCp)nf

nabla knf middot nablaT1113872 1113873 +J middot J

σnf

+DmkTρnf

Cs

nabla middot (nablaC)ρsCps DB(nablaT middot nablaC) +DT

Tinfin(nablaT middot nablaT)1113888 11138891113890

+ Q0 T minus Tinfin( 1113857 + φ minus nabla middot qr⎤⎦

zC

zt+(V middot nablaC) nabla middot DnablaC +

DT

TinfinnablaT1113890 1113891 + kc C minus Cinfin( 1113857

(1)

2 Mathematical Problems in Engineering

where ρnf is the nanofluid density V denotes the velocityvector which has the following componentsV (u(z t) v(z t) w0) μ is the nanofluid viscosity FT isthe thermal expansion term due to temperature differenceFC denotes the thermal expansion term due to concentrationdifference FP is the porous medium term Cpnf denotes thenanofluid specific heat at constant pressure knf denotes thenanofluid thermal conductivity σnf is the electric con-ductivity Dm denotes the mass diffusivity kT is the thermaldiffusivity Cs denotes the susceptibility of concentrationρsCps denotes the nanoparticle heat capacity and DB andDT denote the Brownian and thermophoresis coefficientrespectively Q0 is the source constant and φ is the viscousdissipation term qr is the radiative heat flux Tm is the meantemperature and kc is the chemical reaction coefficient J

denotes the electric current density which is formulated bygeneralized Ohmrsquos law including Hall current as [28ndash30]

J σnf[V times B minus ξ(J times B)] (2)

where ξ is the Hall factorFurthermore radiative heat flux is formulated by Ros-

seland approximation as

qr minus4σlowast

3αlowastzT4

zz (3)

σlowast and αlowast denote the constant StefanndashBoltzman andmean absorption coefficient respectively Consider tem-perature differences are sufficiently small and the term T4

can be expressed as a linear function into the Taylor seriesabout Tinfin and after neglecting higher-order terms it yields

T4

4T3infinT minus 3T

4infin (4)

Under the assumptions made above the governingequations are as follows

ρnf

zu

zt+ w0

zu

zz+ 2Ωv1113888 1113889 μnf

z2u

zz2 minusσnfB2(t)

1 + m2 (u + mv) minusμnf

kp

u + ρnfβlowastTnf

g T minus Tinfin( 1113857 + ρnfβlowastCnf

g C minus Cinfin( 1113857 (5)

ρnf

zv

zt+ w0

zv

zzminus 2Ωu1113888 1113889 μnf

z2v

zz2 minusσnfB2(t)

1 + m2 (v minus mu) minusμnf

kp

v (6)

ρnfCPnf

zT

zt+ w0

zT

zz1113888 1113889 knf

z2T

zz2 + μnf

zu

zz1113888 1113889

2

+σnfB2(t)

1 + m2 u2

+ v2

1113872 1113873 + Q T minus Tinfin( 1113857 +DmkTρnf

Cs

z2C

zz2

+ ρsCPsDB

zC

zz

zT

zz+

DT

Tinfin

zT

zz1113888 1113889

2⎡⎣ ⎤⎦ minus

16σlowastT3infin

3αlowastz2T

zz2

(7)

zC

zt+ w0

zC

zz D

z2C

zz2 + kc C minus Cinfin( 1113857 +DT

Tinfin

z2T

zz2 (8)

U0

B0

Cw

Tw

Tinfin Cinfin

x

v

zNanofluid

Ω

Figure 1 Physical model of the problem

Table 1 )ermophysical properties [26 27]

)ermophysical Water Copper Aluminium oxide Titanium dioxideproperties H2O Cu Al2O3 TiO2

Cp(JkgK) 417900 38500 76500 68600ρ(kgm3) 99710 893300 397000 425000K(WmK) 0613 40000 4600 890βlowast times 10minus 5 (1K) 21 167 063 105σ(Sm) 550 times 10minus 6 116310 times 107 13170 times 107 238 times 106

Mathematical Problems in Engineering 3

where βlowastTnf denotes the nanofluid thermal expansion co-efficient due to temperature difference βlowastCnf denotes thenanofluid thermal expansion coefficient due to concentra-tion difference and m σnfξB0 is the Hall parameter [31]

)e boundary conditions are

tle 0 u 0

w 0

T Tinfin

C Cinfin

z 0

tgt 0 u U0

w 0

T Tw

C Cw

z⟶infin

tgt 0 u⟶ 0

w⟶ 0

T⟶ Tinfin

C⟶ Cinfin

(9)

23 ermophysical Properties of Nanofluid )e nanofluidthermophysical properties were calculated using the fol-lowing forms [6 26]

Nanofluid effective dynamic viscosity effective densityand heat capacity are given as

μnf μf

(1 minus ϕ)25

J1 ρf

ρnf

1

(1 minus ϕ) + ϕ ρsρf1113872 1113873

J2 (ρCp)f

(ρCp)nf

1

(1 minus ϕ) + ϕ (ρCp)s(ρCp)f1113872 1113873

(10)

where ϕ denotes the nanofluid volume fraction Sub-scripts f nf and s denote the base fluid nanofluidand solid particle respectivelyAlso nanofluid effective electric conductivity and ef-fective thermal expansion coefficient are given as

J3 σnf

σf

1 +3 σsσf1113872 1113873 minus 11113872 1113873ϕ

σsσf1113872 1113873 + 21113872 1113873 minus σsσf1113872 1113873 minus 11113872 1113873ϕ

J4 ρβlowast( 1113857nf

ρβlowast( 1113857f

(1 minus ϕ) + ϕρβlowast( 1113857s

ρβlowast( 1113857f

(11)

Effective thermal conductivity and effect of nano-particle shapes are considered by using the Hamiltonand Crosser model as [26]

J5 knf

kf

ks + np minus 11113872 1113873kf minus np minus 11113872 1113873ϕ kf minus ks1113872 1113873

ks + np minus 11113872 1113873kf + ϕ kf minus ks1113872 1113873 (12)

where np indicates the empirical nanoparticle shapefactor which is given by

np 3Ψs

(13)

Here Ψs is the sphericity of the nanoparticles which isdefined as the ratio of the surface area of a sphere to thearea of the particle with a volume equal to that of a particleand has value 1 (np 3) for spherical-shaped nano-particles and 05 (np 6) for spherical-shaped nano-particles)e characteristics of shape factor np for variousnanoparticle shapes have been displayed in Table 2 [32]

Introducing the following nondimensional variables

(1113954x 1113954y 1113954z) (x y z)U0

]f

1113954t U2

0]f

t

1113954u u

U0

1113954w w

U0

θ T minus Tinfin

Tw minus Tinfin

Γ C minus Cinfin

Cw minus Cinfin

(14)

Governing equations (5)ndash(8) are reduced to nondi-mensional form after dropping the cap as

zu

zt+ s

zu

zz+ Rv

J1

(1minus ϕ)25z2u

zz2 minusHa2J1J3

1+ m2 (u + mv)

+ GrJ1J4θ+ GcJ1J4ΓminusMJ1

(1minus ϕ)25 u

(15)

zv

zt+ s

zv

zzminus Ru

J1

(1minus ϕ)25z2v

zz2 minusHa2J1J31+ m2 (v minus mu)

minusMJ1

(1minus ϕ)25 v

(16)

zθzt

+ szθzz

1Pr

J5J2z2θzz2 +

Ha2Ec

1+ m2 J3J2 u2

+ v2

1113872 1113873 + QsJ2θ

minus QrJ2z2θzz2 + DuJ2

z2Γzz2 + NBJ2

zθzz

zΓzz

+ NTJ2zθzz

1113888 1113889

2

+Ec

(1minus ϕ)25J2zu

zz1113888 1113889

2

+zv

zz1113888 1113889

2⎛⎝ ⎞⎠

(17)

zΓzt

+ szΓzz

1

SC

z2Γzz2 + ccΓ+ Sr

z2θzz2

(18)

4 Mathematical Problems in Engineering

where R (2Ω]f)U20 is the rotation parameter Ha2

μfB20σfρ2fU2

0 is the Hartmann number squared Gr

g]fβlowastTf(Tw minus Tinfin)U3

0 is the Grashof number Gc

g]fβlowastCf(Cw minus Cinfin)U3

0 is the modified Grashof numberM μ2fU

20kρ

2f is the porous medium parameter

Pr CPfμfkf is the Prandtl number Ec U20CPf(Tw minus

Tinfin) is the Eckart number Qs Q]fU20(ρCp)f is the heat

source parameter Qr 16σT3infin3αlowast]f(ρCp)f is the radiation

parameter Du (DmkT(Cw minus Cinfin))(CsCPf]f(Tw minus Tinfin)) isthe Dufour number Brownian parameter NB DBτ(Cw minus Cinfin)]f NT DTτ(Tw minus Tinfin)]fTinfin denotes ther-mophoresis parameter nanoparticles and base fluid heatcapacity ratio is τ ρsCps

ρfCpf Sc ]fD is the Schmidt

number cc kc]fU20 is the chemical reaction parameter and

Sr DT(Tw minus Tinfin)Tinfin(Cw minus Cinfin) denotes the Soret number)e boundary conditions reduce to

t 0 u 0

w 0

θ⟶ 0

Γ⟶ 0

z 0

tgt 0 u 1

w 0

θ 1

Γ 1

z⟶infin

tgt 0 u⟶ 0

w⟶ 0

θ⟶ 0

Γ⟶ 0

(19)

24Engineering InterestPhysicalQuantities For engineeringinterest the local skin friction (Cf) local Nusselt number(Nux) and local Sherwood number (Shx) are defined as[33ndash35]

Cf τw

ρfU20

Nux xqw

kf Tw minus Tinfin( 1113857

Shx xqm

DB Tw minus Tinfin( 1113857

(20)

where τw qw and qm denote the wall shear stress and heatand mass fluxes respectively which are given by

τw μnf

zu

zz1113888 1113889

z0

qw minusknf

zT

zz+ qr1113888 1113889

z0

qm minusDB

zC

zz1113888 1113889

z0

(21)

By substituting equation (14) into equations (20) and(21)

Nux minusknf

kf

Rex 1 + Qr( 1113857θprime(0) (22)

where Rex U0x]f is the local Reynolds number

3 Methodology

Here the homotopy perturbation method that was firstdeveloped by He [36 37] and improved by Wu and He [38]to solve governing equations (15)ndash(18) subject to boundarycondition (19) is used To illustrate the fundamental ideas ofthis method we first consider the time-dependent differ-ential equation as follows

Υ(ζ(λ t)) minus f(λ t) 0 λ isin R (23)

where Υ is the operator ζ(λ t) denotes the unknownfunction f(λ t) is a known function λ are spatial variablest denote temporal independent variables and R is thedomain

)e operator Υ can be decomposed into the linear partL(λ t) and N(λ t) is the remaining part )ereforeequation (23) can be rewritten as

L(λ t) + N(λ t) minus f(λ t) 0 λ isin R (24)

Homotopy traditional technique constructs a homotopyfunction Ψ(λ t p) as follows

H [Ψ(λ t p)] (1 minus p) (LΨ(λ t p)) minus L f0(λ t)( 11138571113858 1113859

+ p [Υ(Ψ(λ t p)) minus f(λ t)] 0

(25)

where p isin [0 1] is an embedding parameter and f0(λ t) isan initial guess approximation of equation (23) which sat-isfies boundary and initial condition

Consequently the solution of equation (23) is expressedas

Ψ((λ t p) 1113944n

i0Ψi(λ t)p

i (26)

Ideal approximation solution can be obtained when p

1 as

ζ((λ t) 1113944n

i0ζ i(λ t) (27)

Table 2 Shape factor for various nanoparticle shapes [32]

Spherical Cylindrical Brick

Shapes of nanoparticles

Shape factor (np) 3 6 37

Mathematical Problems in Engineering 5

According to equation (24) the appropriate initial ar-bitrary guess approximation and linear operators for eachequation are proposed as

u0 y 1 + eminus t

1113872 1113873

v0 y 1 + eminus t

1113872 1113873

θ0 y 1 minus eminus t

1113872 1113873

Γ0 y 1 minus eminus t

1113872 1113873

L(u) z2u

zz2

L(v) z2v

zz2

L(θ) z2θzz2

L(Γ) z2Γzz2

(28)

Now HPM constructed equations which satisfy equa-tion (25) to governing nonlinear differential equations(15)ndash(18) as follows

H(u p) pJ1

(1 minus ϕ)25 L(u)1113888 1113889 +(1 minus p) minuszu

ztminus s

zu

zzminus Rv minus

Ha2J1J31 + m2 (u + mv) + GrJ1J4θ + GcJ1J4Γ minus

MJ1

(1 minus ϕ)25 u1113888 1113889 0

(29)

H(v p) pJ1

(1 minus ϕ)25 L(v)1113888 1113889 +(1 minus p) minuszv

ztminus s

zv

zzminus Ru minus

Ha2J1J3

1 + m2 (v minus mu) minusMJ1

(1 minus ϕ)25 v1113888 1113889 0 (30)

H(θ p) p1Pr

J5J2 minus QrJ21113888 1113889L(θ)1113888 1113889 +(1 minus p)zθzt

+ szθzz

+Ha2Ec

1 + m2J3J2 u2

+ v2

1113872 1113873 + QsJ2θ + DuJ2z2Γzz21113888

+ NBJ2zθzz

zΓzz

+ NTJ2zθzz

1113888 1113889

2

+Ec

(1 minus ϕ)25J2zu

zz1113888 1113889

2

+zv

zz1113888 1113889

2⎛⎝ ⎞⎠⎞⎠ 0

(31)

H(Γ p) p1SC

L(Γ)1113888 1113889 +(1 minus p) minuszΓzt

minus szΓzz

+ ccΓ + Sr

z2θzz21113888 1113889 0 (32)

And assuming the solution of equations (29)ndash(32) it canbe written in the following forms

u(z t) u1(t)z + u2(t)z2

+ u3(t)z3

+ middot middot middot

v(z t) v1(t)z + v2(t)z2

+ v3(t)z3

+ middot middot middot

θ(z t) θ1(t)z + θ2(t)z2

+ θ3(t)z3

+ middot middot middot

Γ(z t) Γ1(t)z + Γ2(t)z2

+ Γ3(t)z3

+ middot middot middot (33)

Substituting equation (26) into equations (27)ndash(31) aftersome simplifications and rearranging depending on powersof p-terms and solved using conditions (19) the followingapproximations can be obtained

6 Mathematical Problems in Engineering

u1(t) 1590J1

eminusαt

(1minus ϕ)52

MJ1 1+ eαt

1113872 1113873 + Ha2J1J4A1 1+ e

αt1113872 1113873 + GcJ1J3 1minus e

αt1113872 1113873 minus α(1minus s)GrJ1J2 1minus e

αt1113872 1113873 + R 1+ e

αt1113872 11138731113872 1113873

u2(t) 323J1

eminusαt

(1minus ϕ)52

MJ1 1+ eαt

1113872 1113873 + Ha2J1J4A1 1+ e

αt1113872 1113873 + GcJ1J3 1minus e

αt1113872 1113873 minus α(1+ s) minus GrJ1J2 1minus e

αt1113872 1113873 + R 1+ e

αt1113872 11138731113872 1113873

u3(t) 1590J1

(1minus ϕ)52

Ha2J1J4(1minus m) minus R minus αminus e

αt1113872 1113873

v1(t) 1590

(1minus ϕ)52

minuseminusαt

(R + αs) + Ha2J4J1

9015

eminusαt

+ m2

minus m3

+ A1eminusαt

1113874 11138751113874 1113875

v2(t) minus323

Ha2J1J4(1minus ϕ)

52+ R(1minus ϕ)

52+ Ha

2J1J4m(1minus ϕ)

52+323αe

minusαt(1minus ϕ)

52minus323

Ha2J1J4e

minusαt(1minus ϕ)

52

+323Reminusαt

(1minus ϕ)52

v3(t) minusMJ1(1minus ϕ)52

+ +Ha2J4m

3e

minusαt(1minus ϕ)

52323

minus 0125

θ1 minus32Pre

minusαt α+ J6Qr 1minus eαt( 1113857( 1113857

3J5J6+4Pre

minus2αt eαt minus 1( 1113857 seαt + DNBJ6 1minus eαt( 1113857( 1113857

J5J6minus256EcHa2J4Pre

minus2αt m2 + 1( 1113857 eαt + 1( 11138572

3J5+18

θ2 Pre

minus2αt eαt minus 1( 1113857 seαt + DNBJ6 1minus eαt( 1113857( 1113857

2J5J6

θ3 Pre

minusαt α+ J6Qr 1minus eαt( 1113857( 1113857

6J5J6

Γ1(t) Sce

minusαt αminus cc 1+ eαt( 1113857( 1113857

6

Γ2(t) 32Sce

minusαt αminus cc 1+ eαt( 1113857( 1113857

3minus 4Sce

minusαt(Sr + s) e

αtminus 11113872 1113873 minus 0125

Γ3(t) Sce

minusαt(Sr + s) eαt minus 1( 1113857

2

(34)

where A1 m3 + m2 + m + 1 and A2 minusm3 + m2 minus m

4 Solution Validation

To check the accuracy and efficiency of the methodology thecurrent results are verified by comparing with previouslypublished results in some special cases which are tabulated inTables 3ndash5 )e present results of different skin frictioncoefficient values Cf and Nusselt number NuRex for dif-ferent values of nanoparticle volume fraction ϕ and rotationparameter R are compared with those reported by Das [39]and Reddy et al [40] with both nanoparticles Cu and Al2O3which are illustrated in Tables 3 and 4 )e results obtainedcome with excellent agreement of previous studies

Furthermore the comparison for different values of theSherwood number minusShx against time t and chemical reac-tion cc with obtained results by Hussain et al [17] at ϕ 00is listed in Table 5 Also it can be seen that the presentobtained results are in an excellent agreement with the

previously published results )is proves the validity of thepresent work and shows how powerful the solution reachedby homotopy perturbation technique

5 Results and Discussion

)e influence of various physical parameters such asnanoparticle volume fraction ϕ shape of nanoparticles npHall current m Brownian NB thermophoretic NT andchemical reaction cc which are computed from the ap-proximate solution reported in the previous section is dis-played graphically and discussed )e graphical results areillustrated in Figures 2ndash11 )e numerical results are per-formed by taking values of some constant parameters as s

05 M 05 Pr 07 Ec 02 Gc 1 Gr 1 Sc 03

Du 05 G 5 and Sr 05Figures 2 and 3 display the effect of nanoparticle

volume fraction parameter ϕ on the fluid velocity u andtemperature profiles θ for either Cu minus water Al2O3 minus water

Mathematical Problems in Engineering 7

Table 3 Result comparison for Cf

Cf

ϕ RCuminuswater Al2O3minuswater

Present Das [39] Reddy et al [40] Present Das [39]

000 03 20355 20354 203542 20355 20354004 23158 23154 231535 05134 21831008 26111 26107 261068 21832 23384012 29120 29196 291953 25030 25020014 30792 30789 307891 10885 10872005 00 23790 23786 237862 22144 22143

05 24042 24040 240401 22333 2233315 25755 25755 257546 23669 2366625 28236 28235 28251 25697 25695

Table 4 Result comparison for NuxRex

NuxRex

ϕ RCuminuswater Al2O3minuswater

Present Das [39] Reddy et al [40] Present Das [39]

000 03 04607 04607 04607 04607 04607004 05135 05134 05134 05115 05114008 05695 05698 05698 05656 05655012 06304 06303 06303 06234 06233014 06622 06622 06622 06536 06537005 00 05272 05271 05271 05271 05271

05 mdash mdash mdash mdash mdash15 mdash mdash mdash mdash mdash25 05271 05271 05271 05272 05271

Table 5 Result comparison for minusShx

minusShx

ccdarr t⟶ 03 05 07 03 05 07Present Hussain et al [17]

02 0535254 0438632 0389005 054907 044765 0386382 0849652 0815633 0767225 085729 0820922 0781565 12252 116674 114632 124260 116967 1143922

ϕ = 0 007 01

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

Cu-waterAI2O3-waterTiO2-water

Figure 2 )e effect of volume fraction of nanoparticles on the velocity profile

8 Mathematical Problems in Engineering

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θϕ = 0 007 012

1

08

06

04

02

00 2 3 4 5 6 7 81

Cu-waterAI2O3-waterTiO2-water

z

Figure 3 )e effect of volume fraction nanoparticle on thetemperature profile

Cylindrical shape

Brick shape

Spherical shape

1

08

06

04

02

00 2 3 4 5 6 7 81

z

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θ

Cu-waterAI2O3-waterTiO2-water

Figure 4 )e effect of nanoparticle shape on the temperatureprofile

R = 1 2 4

Cu-waterTiO2-water

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

Figure 5 )e effect of rotation parameter on the velocity profile

m = 1 15 2

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 6 )e effect of Hall current on the velocity profile

NB = 02 05 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07m = 1γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 7 )e effect of Brownian motion parameter on the tem-perature profile

ϕ = 005NB = 07m = 1γc = 05Qs = 1Qr = 09R = 1

NT = 03 06 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 8 )e effect of thermophoresis parameter on the tem-perature profile

Mathematical Problems in Engineering 9

and TiO2 minus water nanofluid It is clear that increase innanoparticle volume fraction parameter ϕ decreases the fluidvelocity but the temperature profiles increase It is due to thefact that nanofluid density increases with increasing thevolume fraction of nanoparticles and it causes to slow downthe nanofluid flow velocity and increases the thickness of thethermal boundary layer which improves thermal conduc-tivity and increases surface heat transfer rate Also it isobserved that nanofluid Cu minus water heat transfer rate isgreater than Al2O3 and TiO2 rates So nanoparticles havinghigh thermal conductivity enhance heat transfer more thanthe particles having low thermal conductivity On thecontrary it is seen from Tables 2 and 3 that wall skin frictioncoefficient Cf and the Nusselt number NuRex increase withincreasing in nanoparticle volume fraction for both Cu minus

water and Al2O3 minus water)e influence of the nanoparticle shape on the tem-

perature distribution of nanofluid is studied through Fig-ure 4 )is figure indicates that the heat transfer rate ishigher for the cylindrical nanoparticle shape than thespherical and brick nanoparticle shapes for eitherCu minus water Al2O3 minus water and TiO2 minus water nanofluid)erefore the spherical nanoparticle shape tends to dragmore heat from the boundary layer because it has a greatersurface area while this effect is less evident for othernanoparticle shapes )us the use of the cylindrical nano-particle shape has better improvement in effective thermalconductivity when compared with other nanoparticle shapes(spherical and brick)

)e influence of rotation parameter R on the velocityprofiles against z is displayed in Figure 5 It observed that anincrease in rotation parameter leads to decreasing theboundary layer momentum so it decreases the values ofnanofluid velocity From Tables 2 and 3 it can be noted thatthe wall skin friction coefficient Cf increases with an in-crease in the rotation parameter but the Nusselt numberNuRex remains unchanged with both Cu minus water andAl2O3 minus water

Figure 6 presents the influence of Hall current parameterm on main velocity u It is clear that the velocity distributionof nanofluid reduces with an increase in Hall current pa-rameter)is happens naturally due to the fact that when themagnetic field is strengthened it enhances the opposingforce which is represented by the well-known Lorentz force

Figures 7 and 8 show the Brownian and thermophoresiseffects on the temperature profile θ It is seen that thetemperature distribution increases with increasing for theseparameters )is is due to the fact that the thermophoreticand Brownian parameters assist to improve the thermalboundary layer thickness Also it is observed that the heattransfer rate of Cu minus water nanofluid is greater thanTiO2 minus water

Figure 9 depicts the influence of chemical parameter cc

on the concentration profile Γ It is noticed that the increasein concentration profile is accompanied by a decrease in thechemical parameter On the contrary it is seen from Table 4that the Sherwood number minusShx increases due to increasingthe chemical parameter but the Sherwood number decreaseswhile increasing the time t )is implies that the chemical

Cu-waterTiO2-water

ϕ = 005NT = 07NB = 07m = 1

γc = 01 05 1

Qs = 1Qr = 09R = 1

0 2 3 4 5 6 7 81z

1

08

06

04

02

0

12

14

Γ

Figure 9 )e effect of chemical reaction parameter on the con-centration profile

Cu-waterTiO2-water

ϕ = 005

NB = 07

m = 1γc = 05

Qs = 1 2 3

Qr = 09R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

NT = 07

Figure 10 )e effect of heat source parameter on the temperatureprofile

ϕ = 005

NB = 07NT = 07

m = 1

γc = 05Qs = 1

Qr = 05 09 13

R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 11 )e effect of radiation parameter on the temperatureprofile

10 Mathematical Problems in Engineering

reaction has tendency to increase the mass transfer rate atthe surface plate and it is reduced with the progress of time

Figures 10 and 11 present the heat source parameter Qs

and thermal radiation parameter Qr effects on the tem-perature profiles It is clear from Figure 10 that with anincrease in the heat generation parameter the boundarylayer temperature decreases and as a consequence to de-crease the thermal boundary layer thickness In addition it isclear from Figure 11 that with an increase in the thermalradiation the temperature profiles of the nanofluid increaseIt is well known that the increase in the radiation parameterenhances the energy transport inside the fluid and as aconsequence to increase the heat transfer rate

6 Conclusion

An analysis is done to solve the magnetohydro dynamicsunsteady nanofluid flow through the porous medium past amoving plate in a rotating frame with considering the heatand mass transfer )ree different types of the nanoparticlesnamely copper aluminium oxide and titanium dioxide andwater used as a base nanofluid are considered with variousshapes (spherical cylindrical and brick) )e important ef-fects of nanoparticle shape Brownian and thermophoresisdiffusion thermal radiation ohmic and viscous dissipationshave been included Homotopy perturbation method is ap-plied to solve the governing nonlinear partial differentialequations )e efficiency of the methodology has been vali-dated through comparison with the previously publishedresults and observed a very close agreement It is found that

)e main velocity component decreases with increaseof nanoparticle volume fraction On the contrarythermal properties increase with increase in nano-particle volume fraction)e enhancement of heat transfer rate and mechanicalproperty improvement is high in copper as comparedwith both aluminium oxide and titanium dioxide)ermal conductivity of cylindrical-shaped nanoparticlesis more efficient and possesses better improvement onheat transfer rate than the other shaped onesIncrease in the Brownian and thermophoresis pa-rameter is accompanied by an increase in the thermalboundary layer thicknessIncreasing Hall current and rotation parameters de-creases the main fluid velocity componentHPM has a close agreement and is recommended formany nonlinear problems

Data Availability

)e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] T Hayat R Sajjad A Alsaedi T Muhammad and R EllahildquoOn squeezed flow of couple stress nanofluid between twoparallel platesrdquo Results in Physics vol 7 pp 553ndash561 2017

[2] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition pp 99ndash105San Francisco CA USA 1995

[3] J A Eastman S U S Choi S Li W Yu and L J )ompsonldquoAnomalously increased effective thermal conductivities of eth-ylene glycol-based nanofluids containing copper nanoparticlesrdquoApplied Physics Letters vol 78 no 6 pp 718ndash720 2001

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

[5] M Sheikholeslami M M Rashidi T Hayat and D D GanjildquoFree convection of magnetic nanofluid considering MFDviscosity effectrdquo Journal of Molecular Liquids vol 218pp 393ndash399 2016

[6] W Abbas and E A Sayed ldquoHall current and joule heatingeffects on free convection flow of a nanofluid over a verticalcone in presence of thermal radiationrdquo ermal Sciencevol 21 pp 2603ndash2614 2017

[7] B Mahanthesh B Gireesha I Animasaun T Muhammadand N Shashikumar ldquoMHD flow of SWCNT and MWCNTnanoliquids past a rotating stretchable disk with thermal andexponential space dependent heat sourcerdquo Physica Scriptavol 94 no 8 Article ID 085214 2019

[8] I L Animasaun O K Koriko K S Adegbie et al ldquoCom-parative analysis between 36 nm and 47 nm alumina-waternanofluid flows in the presence of Hall effectrdquo Journal ofermal Analysis and Calorimetry vol 135 no 2 pp 873ndash886 2019

[9] I Ullah S Shafie I Khan and K L Hsiao ldquoBrownian dif-fusion and thermophoresis mechanisms in Casson fluid over amoving wedgerdquo Results in Physics vol 9 pp 183ndash194 2018

[10] P B S Kumar B J Gireesha R S R Gorla andB Mahanthesh ldquoMagnetohydrodynamic flow of Williamsonnanofluid due to an exponentially stretching surface in thepresence of thermal radiation and chemical reactionrdquo Journalof Nanofluids vol 6 no 2 pp 264ndash272 2017

[11] X Zhou Y Jiang X Li et al ldquoNumerical investigation of heattransfer enhancement and entropy generation of naturalconvection in a cavity containing nano liquid-metal fluidrdquoInternational Communications in Heat and Mass Transfervol 106 pp 46ndash54 2019

[12] N Shehzad A Zeeshan R Ellahi and K Vafai ldquoConvectiveheat transfer of nanofluid in a wavy channel Buongiornorsquosmathematical modelrdquo Journal of Molecular Liquids vol 222pp 446ndash455 2016

[13] B Shen L Zheng C Zhang and X Zhang ldquoBioconvectionheat transfer of a nanofluid over a stretching sheet with ve-locity slip and temperature jumprdquo ermal Science vol 21no 6 pp 2347ndash2356 2017

[14] S Jahan H Sakidin R Nazar and I Pop ldquoAnalysis of heattransfer in nanofluid past a convectively heated permeablestretchingshrinking sheet with regression and stability ana-lysesrdquo Results in Physics vol 10 pp 395ndash405 2018

[15] M A A Hamad and I Pop ldquoUnsteady MHD free convectionflow past a vertical permeable flat plate in a rotating frame ofreference with constant heat source in a nanofluidrdquo Heat andMass Transfer vol 47 no 12 pp 1517ndash1524 2011

Mathematical Problems in Engineering 11

[16] M Sheikholeslami and D D Ganji ldquo)ree dimensional heatand mass transfer in a rotating system using nanofluidrdquoPowder Technology vol 253 pp 789ndash796 2014

[17] S M Hussain J Jain G S Seth and M M Rashidi ldquoFreeconvective heat transfer with Hall effects heat absorption andchemical reaction over an accelerated moving plate in a ro-tating systemrdquo Journal of Magnetism and Magnetic Materialsvol 422 pp 112ndash123 2017

[18] X Zhou Y Jiang Y Hou and M Du ldquo)ermocapillaryconvection instability in annular two-layer system undervarious gravity levelsrdquo Microgravity Science and Technologyvol 31 no 5 pp 641ndash648 2019

[19] Y Jiang and X Zhou ldquoHeat transfer and entropy generationanalysis of nanofluids thermocapillary convection around abubble in a cavityrdquo International Communications in Heatand Mass Transfer vol 105 pp 37ndash45 2019

[20] Y Jiang X Zhou and Y Wang ldquoEffects of nanoparticleshapes on heat andmass transfer of nanofluid thermocapillaryconvection around a gas bubblerdquo Microgravity Science andTechnology pp 1ndash11 2019 In press

[21] Y Jiang X Zhou and Y Wang ldquoComprehensive heattransfer performance analysis of nanofluid mixed forced andthermocapillary convection around a gas bubble in mini-channelrdquo International Communications in Heat and MassTransfer vol 110 Article ID 104386 2020

[22] B Mahanthesh B J Gireesha R S R Gorla F M Abbasiand S A Shehzad ldquoNumerical solutions for magnetohy-drodynamic flow of nanofluid over a bidirectional non-linearstretching surface with prescribed surface heat flux bound-aryrdquo Journal of Magnetism and Magnetic Materials vol 417pp 189ndash196 2016

[23] B Mahanthesh B J Gireesha and I L Animasaun ldquoEx-ploration of non-linear thermal radiation and suspendednanoparticles effects on mixed convection boundary layerflow of nanoliquids on a melting vertical surfacerdquo Journal ofNanofluids vol 7 no 5 pp 833ndash843 2018

[24] P S Kumar B Mahanthesh B Gireesha and S ShehzadldquoQuadratic convective flow of radiated nano-jeffrey liquidsubject to multiple convective conditions and Cattaneo-Christov double diffusionrdquo Applied Mathematics and Me-chanics vol 39 no 9 pp 1311ndash1326 2018

[25] J Raza F Mebarek-Oudina and A Chamkha ldquoMagneto-hydrodynamic flow of molybdenum disulfide nanofluid in achannel with shape effectsrdquo Multidiscipline Modeling inMaterials and Structures vol 15 no 4 pp 737ndash757 2019

[26] T Hayat B Ahmed F M Abbasi and A Alsaedi ldquoHy-dromagnetic peristalsis of water based nanofluids with tem-perature dependent viscosity a comparative studyrdquo Journal ofMolecular Liquids vol 234 pp 324ndash329 2017

[27] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filledwith nanofluidsrdquo International Journal of Heat and FluidFlow vol 29 no 5 pp 1326ndash1336 2008

[28] M A El Kot and W Abbas ldquoNumerical technique of bloodflow through catheterized arteries with overlapping stenosisrdquoComputer Methods in Biomechanics and Biomedical Engi-neering vol 20 no 1 pp 45ndash58 2017

[29] H A Attia W Abbas A El-Din Abdin andM A M Abdeen ldquoEffects of ion slip and Hall current onunsteady Couette flow of a dusty fluid through porous mediawith heat transferrdquo High Temperature vol 53 no 6pp 891ndash898 2015

[30] H A Attia W Abbas and M A M Abdeen ldquoIon slip effecton unsteady Couette flow of a dusty fluid in the presence of

uniform suction and injection with heat transferrdquo Journal ofthe Brazilian Society of Mechanical Sciences and Engineeringvol 38 no 8 pp 2381ndash2391 2016

[31] H A Attia W Abbas M A M Abdeen and A A M SaidldquoHeat transfer between two parallel porous plates for Couetteflow under pressure gradient and Hall currentrdquo Sadhanavol 40 no 1 pp 183ndash197 2015

[32] M Sheikholeslami A Shafee M Ramzan and Z Li ldquoIn-vestigation of Lorentz forces and radiation impacts onnanofluid treatment in a porous semi annulus via Darcy lawrdquoJournal of Molecular Liquids vol 272 pp 8ndash14 2018

[33] P S Kumar B Gireesha B Mahanthesh and A J Chamkhaldquo)ermal analysis of nanofluid flow containing gyrotacticmicroorganisms in bioconvection and second-order slip withconvective conditionrdquo Journal of ermal Analysis andCalorimetry vol 136 no 5 pp 1947ndash1957 2019

[34] B Mahanthesh F Mabood B Gireesha and R Gorla ldquoEffectsof chemical reaction and partial slip on the three-dimensionalflow of a nanofluid impinging on an exponentially stretchingsurfacerdquo e European Physical Journal Plus vol 132 no 3p 113 2017

[35] B J Gireesha R S R Gorla and B Mahanthesh ldquoEffect ofsuspended nanoparticles on three-dimensional mhd flowheat and mass transfer of radiating eyring-powell fluid over astretching sheetrdquo Journal of Nanofluids vol 4 no 4pp 474ndash484 2015

[36] J-H He ldquoHomotopy perturbation techniquerdquo ComputerMethods in Applied Mechanics and Engineering vol 178no 3-4 pp 257ndash262 1999

[37] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo Interna-tional Journal of Non-Linear Mechanics vol 35 no 1pp 37ndash43 2000

[38] Y Wu and J-H He ldquoHomotopy perturbation method fornonlinear oscillators with coordinate-dependent massrdquo Re-sults in Physics vol 10 pp 270-271 2018

[39] K Das ldquoFlow and heat transfer characteristics of nanofluids ina rotating framerdquo Alexandria Engineering Journal vol 53no 3 pp 757ndash766 2014

[40] J V R Reddy V Sugunamma N Sandeep and C SulochanaldquoInfluence of chemical reaction radiation and rotation onMHD nanofluid flow past a permeable flat plate in porousmediumrdquo Journal of the Nigerian Mathematical Societyvol 35 no 1 pp 48ndash65 2016

12 Mathematical Problems in Engineering

where ρnf is the nanofluid density V denotes the velocityvector which has the following componentsV (u(z t) v(z t) w0) μ is the nanofluid viscosity FT isthe thermal expansion term due to temperature differenceFC denotes the thermal expansion term due to concentrationdifference FP is the porous medium term Cpnf denotes thenanofluid specific heat at constant pressure knf denotes thenanofluid thermal conductivity σnf is the electric con-ductivity Dm denotes the mass diffusivity kT is the thermaldiffusivity Cs denotes the susceptibility of concentrationρsCps denotes the nanoparticle heat capacity and DB andDT denote the Brownian and thermophoresis coefficientrespectively Q0 is the source constant and φ is the viscousdissipation term qr is the radiative heat flux Tm is the meantemperature and kc is the chemical reaction coefficient J

denotes the electric current density which is formulated bygeneralized Ohmrsquos law including Hall current as [28ndash30]

J σnf[V times B minus ξ(J times B)] (2)

where ξ is the Hall factorFurthermore radiative heat flux is formulated by Ros-

seland approximation as

qr minus4σlowast

3αlowastzT4

zz (3)

σlowast and αlowast denote the constant StefanndashBoltzman andmean absorption coefficient respectively Consider tem-perature differences are sufficiently small and the term T4

can be expressed as a linear function into the Taylor seriesabout Tinfin and after neglecting higher-order terms it yields

T4

4T3infinT minus 3T

4infin (4)

Under the assumptions made above the governingequations are as follows

ρnf

zu

zt+ w0

zu

zz+ 2Ωv1113888 1113889 μnf

z2u

zz2 minusσnfB2(t)

1 + m2 (u + mv) minusμnf

kp

u + ρnfβlowastTnf

g T minus Tinfin( 1113857 + ρnfβlowastCnf

g C minus Cinfin( 1113857 (5)

ρnf

zv

zt+ w0

zv

zzminus 2Ωu1113888 1113889 μnf

z2v

zz2 minusσnfB2(t)

1 + m2 (v minus mu) minusμnf

kp

v (6)

ρnfCPnf

zT

zt+ w0

zT

zz1113888 1113889 knf

z2T

zz2 + μnf

zu

zz1113888 1113889

2

+σnfB2(t)

1 + m2 u2

+ v2

1113872 1113873 + Q T minus Tinfin( 1113857 +DmkTρnf

Cs

z2C

zz2

+ ρsCPsDB

zC

zz

zT

zz+

DT

Tinfin

zT

zz1113888 1113889

2⎡⎣ ⎤⎦ minus

16σlowastT3infin

3αlowastz2T

zz2

(7)

zC

zt+ w0

zC

zz D

z2C

zz2 + kc C minus Cinfin( 1113857 +DT

Tinfin

z2T

zz2 (8)

U0

B0

Cw

Tw

Tinfin Cinfin

x

v

zNanofluid

Ω

Figure 1 Physical model of the problem

Table 1 )ermophysical properties [26 27]

)ermophysical Water Copper Aluminium oxide Titanium dioxideproperties H2O Cu Al2O3 TiO2

Cp(JkgK) 417900 38500 76500 68600ρ(kgm3) 99710 893300 397000 425000K(WmK) 0613 40000 4600 890βlowast times 10minus 5 (1K) 21 167 063 105σ(Sm) 550 times 10minus 6 116310 times 107 13170 times 107 238 times 106

Mathematical Problems in Engineering 3

where βlowastTnf denotes the nanofluid thermal expansion co-efficient due to temperature difference βlowastCnf denotes thenanofluid thermal expansion coefficient due to concentra-tion difference and m σnfξB0 is the Hall parameter [31]

)e boundary conditions are

tle 0 u 0

w 0

T Tinfin

C Cinfin

z 0

tgt 0 u U0

w 0

T Tw

C Cw

z⟶infin

tgt 0 u⟶ 0

w⟶ 0

T⟶ Tinfin

C⟶ Cinfin

(9)

23 ermophysical Properties of Nanofluid )e nanofluidthermophysical properties were calculated using the fol-lowing forms [6 26]

Nanofluid effective dynamic viscosity effective densityand heat capacity are given as

μnf μf

(1 minus ϕ)25

J1 ρf

ρnf

1

(1 minus ϕ) + ϕ ρsρf1113872 1113873

J2 (ρCp)f

(ρCp)nf

1

(1 minus ϕ) + ϕ (ρCp)s(ρCp)f1113872 1113873

(10)

where ϕ denotes the nanofluid volume fraction Sub-scripts f nf and s denote the base fluid nanofluidand solid particle respectivelyAlso nanofluid effective electric conductivity and ef-fective thermal expansion coefficient are given as

J3 σnf

σf

1 +3 σsσf1113872 1113873 minus 11113872 1113873ϕ

σsσf1113872 1113873 + 21113872 1113873 minus σsσf1113872 1113873 minus 11113872 1113873ϕ

J4 ρβlowast( 1113857nf

ρβlowast( 1113857f

(1 minus ϕ) + ϕρβlowast( 1113857s

ρβlowast( 1113857f

(11)

Effective thermal conductivity and effect of nano-particle shapes are considered by using the Hamiltonand Crosser model as [26]

J5 knf

kf

ks + np minus 11113872 1113873kf minus np minus 11113872 1113873ϕ kf minus ks1113872 1113873

ks + np minus 11113872 1113873kf + ϕ kf minus ks1113872 1113873 (12)

where np indicates the empirical nanoparticle shapefactor which is given by

np 3Ψs

(13)

Here Ψs is the sphericity of the nanoparticles which isdefined as the ratio of the surface area of a sphere to thearea of the particle with a volume equal to that of a particleand has value 1 (np 3) for spherical-shaped nano-particles and 05 (np 6) for spherical-shaped nano-particles)e characteristics of shape factor np for variousnanoparticle shapes have been displayed in Table 2 [32]

Introducing the following nondimensional variables

(1113954x 1113954y 1113954z) (x y z)U0

]f

1113954t U2

0]f

t

1113954u u

U0

1113954w w

U0

θ T minus Tinfin

Tw minus Tinfin

Γ C minus Cinfin

Cw minus Cinfin

(14)

Governing equations (5)ndash(8) are reduced to nondi-mensional form after dropping the cap as

zu

zt+ s

zu

zz+ Rv

J1

(1minus ϕ)25z2u

zz2 minusHa2J1J3

1+ m2 (u + mv)

+ GrJ1J4θ+ GcJ1J4ΓminusMJ1

(1minus ϕ)25 u

(15)

zv

zt+ s

zv

zzminus Ru

J1

(1minus ϕ)25z2v

zz2 minusHa2J1J31+ m2 (v minus mu)

minusMJ1

(1minus ϕ)25 v

(16)

zθzt

+ szθzz

1Pr

J5J2z2θzz2 +

Ha2Ec

1+ m2 J3J2 u2

+ v2

1113872 1113873 + QsJ2θ

minus QrJ2z2θzz2 + DuJ2

z2Γzz2 + NBJ2

zθzz

zΓzz

+ NTJ2zθzz

1113888 1113889

2

+Ec

(1minus ϕ)25J2zu

zz1113888 1113889

2

+zv

zz1113888 1113889

2⎛⎝ ⎞⎠

(17)

zΓzt

+ szΓzz

1

SC

z2Γzz2 + ccΓ+ Sr

z2θzz2

(18)

4 Mathematical Problems in Engineering

where R (2Ω]f)U20 is the rotation parameter Ha2

μfB20σfρ2fU2

0 is the Hartmann number squared Gr

g]fβlowastTf(Tw minus Tinfin)U3

0 is the Grashof number Gc

g]fβlowastCf(Cw minus Cinfin)U3

0 is the modified Grashof numberM μ2fU

20kρ

2f is the porous medium parameter

Pr CPfμfkf is the Prandtl number Ec U20CPf(Tw minus

Tinfin) is the Eckart number Qs Q]fU20(ρCp)f is the heat

source parameter Qr 16σT3infin3αlowast]f(ρCp)f is the radiation

parameter Du (DmkT(Cw minus Cinfin))(CsCPf]f(Tw minus Tinfin)) isthe Dufour number Brownian parameter NB DBτ(Cw minus Cinfin)]f NT DTτ(Tw minus Tinfin)]fTinfin denotes ther-mophoresis parameter nanoparticles and base fluid heatcapacity ratio is τ ρsCps

ρfCpf Sc ]fD is the Schmidt

number cc kc]fU20 is the chemical reaction parameter and

Sr DT(Tw minus Tinfin)Tinfin(Cw minus Cinfin) denotes the Soret number)e boundary conditions reduce to

t 0 u 0

w 0

θ⟶ 0

Γ⟶ 0

z 0

tgt 0 u 1

w 0

θ 1

Γ 1

z⟶infin

tgt 0 u⟶ 0

w⟶ 0

θ⟶ 0

Γ⟶ 0

(19)

24Engineering InterestPhysicalQuantities For engineeringinterest the local skin friction (Cf) local Nusselt number(Nux) and local Sherwood number (Shx) are defined as[33ndash35]

Cf τw

ρfU20

Nux xqw

kf Tw minus Tinfin( 1113857

Shx xqm

DB Tw minus Tinfin( 1113857

(20)

where τw qw and qm denote the wall shear stress and heatand mass fluxes respectively which are given by

τw μnf

zu

zz1113888 1113889

z0

qw minusknf

zT

zz+ qr1113888 1113889

z0

qm minusDB

zC

zz1113888 1113889

z0

(21)

By substituting equation (14) into equations (20) and(21)

Nux minusknf

kf

Rex 1 + Qr( 1113857θprime(0) (22)

where Rex U0x]f is the local Reynolds number

3 Methodology

Here the homotopy perturbation method that was firstdeveloped by He [36 37] and improved by Wu and He [38]to solve governing equations (15)ndash(18) subject to boundarycondition (19) is used To illustrate the fundamental ideas ofthis method we first consider the time-dependent differ-ential equation as follows

Υ(ζ(λ t)) minus f(λ t) 0 λ isin R (23)

where Υ is the operator ζ(λ t) denotes the unknownfunction f(λ t) is a known function λ are spatial variablest denote temporal independent variables and R is thedomain

)e operator Υ can be decomposed into the linear partL(λ t) and N(λ t) is the remaining part )ereforeequation (23) can be rewritten as

L(λ t) + N(λ t) minus f(λ t) 0 λ isin R (24)

Homotopy traditional technique constructs a homotopyfunction Ψ(λ t p) as follows

H [Ψ(λ t p)] (1 minus p) (LΨ(λ t p)) minus L f0(λ t)( 11138571113858 1113859

+ p [Υ(Ψ(λ t p)) minus f(λ t)] 0

(25)

where p isin [0 1] is an embedding parameter and f0(λ t) isan initial guess approximation of equation (23) which sat-isfies boundary and initial condition

Consequently the solution of equation (23) is expressedas

Ψ((λ t p) 1113944n

i0Ψi(λ t)p

i (26)

Ideal approximation solution can be obtained when p

1 as

ζ((λ t) 1113944n

i0ζ i(λ t) (27)

Table 2 Shape factor for various nanoparticle shapes [32]

Spherical Cylindrical Brick

Shapes of nanoparticles

Shape factor (np) 3 6 37

Mathematical Problems in Engineering 5

According to equation (24) the appropriate initial ar-bitrary guess approximation and linear operators for eachequation are proposed as

u0 y 1 + eminus t

1113872 1113873

v0 y 1 + eminus t

1113872 1113873

θ0 y 1 minus eminus t

1113872 1113873

Γ0 y 1 minus eminus t

1113872 1113873

L(u) z2u

zz2

L(v) z2v

zz2

L(θ) z2θzz2

L(Γ) z2Γzz2

(28)

Now HPM constructed equations which satisfy equa-tion (25) to governing nonlinear differential equations(15)ndash(18) as follows

H(u p) pJ1

(1 minus ϕ)25 L(u)1113888 1113889 +(1 minus p) minuszu

ztminus s

zu

zzminus Rv minus

Ha2J1J31 + m2 (u + mv) + GrJ1J4θ + GcJ1J4Γ minus

MJ1

(1 minus ϕ)25 u1113888 1113889 0

(29)

H(v p) pJ1

(1 minus ϕ)25 L(v)1113888 1113889 +(1 minus p) minuszv

ztminus s

zv

zzminus Ru minus

Ha2J1J3

1 + m2 (v minus mu) minusMJ1

(1 minus ϕ)25 v1113888 1113889 0 (30)

H(θ p) p1Pr

J5J2 minus QrJ21113888 1113889L(θ)1113888 1113889 +(1 minus p)zθzt

+ szθzz

+Ha2Ec

1 + m2J3J2 u2

+ v2

1113872 1113873 + QsJ2θ + DuJ2z2Γzz21113888

+ NBJ2zθzz

zΓzz

+ NTJ2zθzz

1113888 1113889

2

+Ec

(1 minus ϕ)25J2zu

zz1113888 1113889

2

+zv

zz1113888 1113889

2⎛⎝ ⎞⎠⎞⎠ 0

(31)

H(Γ p) p1SC

L(Γ)1113888 1113889 +(1 minus p) minuszΓzt

minus szΓzz

+ ccΓ + Sr

z2θzz21113888 1113889 0 (32)

And assuming the solution of equations (29)ndash(32) it canbe written in the following forms

u(z t) u1(t)z + u2(t)z2

+ u3(t)z3

+ middot middot middot

v(z t) v1(t)z + v2(t)z2

+ v3(t)z3

+ middot middot middot

θ(z t) θ1(t)z + θ2(t)z2

+ θ3(t)z3

+ middot middot middot

Γ(z t) Γ1(t)z + Γ2(t)z2

+ Γ3(t)z3

+ middot middot middot (33)

Substituting equation (26) into equations (27)ndash(31) aftersome simplifications and rearranging depending on powersof p-terms and solved using conditions (19) the followingapproximations can be obtained

6 Mathematical Problems in Engineering

u1(t) 1590J1

eminusαt

(1minus ϕ)52

MJ1 1+ eαt

1113872 1113873 + Ha2J1J4A1 1+ e

αt1113872 1113873 + GcJ1J3 1minus e

αt1113872 1113873 minus α(1minus s)GrJ1J2 1minus e

αt1113872 1113873 + R 1+ e

αt1113872 11138731113872 1113873

u2(t) 323J1

eminusαt

(1minus ϕ)52

MJ1 1+ eαt

1113872 1113873 + Ha2J1J4A1 1+ e

αt1113872 1113873 + GcJ1J3 1minus e

αt1113872 1113873 minus α(1+ s) minus GrJ1J2 1minus e

αt1113872 1113873 + R 1+ e

αt1113872 11138731113872 1113873

u3(t) 1590J1

(1minus ϕ)52

Ha2J1J4(1minus m) minus R minus αminus e

αt1113872 1113873

v1(t) 1590

(1minus ϕ)52

minuseminusαt

(R + αs) + Ha2J4J1

9015

eminusαt

+ m2

minus m3

+ A1eminusαt

1113874 11138751113874 1113875

v2(t) minus323

Ha2J1J4(1minus ϕ)

52+ R(1minus ϕ)

52+ Ha

2J1J4m(1minus ϕ)

52+323αe

minusαt(1minus ϕ)

52minus323

Ha2J1J4e

minusαt(1minus ϕ)

52

+323Reminusαt

(1minus ϕ)52

v3(t) minusMJ1(1minus ϕ)52

+ +Ha2J4m

3e

minusαt(1minus ϕ)

52323

minus 0125

θ1 minus32Pre

minusαt α+ J6Qr 1minus eαt( 1113857( 1113857

3J5J6+4Pre

minus2αt eαt minus 1( 1113857 seαt + DNBJ6 1minus eαt( 1113857( 1113857

J5J6minus256EcHa2J4Pre

minus2αt m2 + 1( 1113857 eαt + 1( 11138572

3J5+18

θ2 Pre

minus2αt eαt minus 1( 1113857 seαt + DNBJ6 1minus eαt( 1113857( 1113857

2J5J6

θ3 Pre

minusαt α+ J6Qr 1minus eαt( 1113857( 1113857

6J5J6

Γ1(t) Sce

minusαt αminus cc 1+ eαt( 1113857( 1113857

6

Γ2(t) 32Sce

minusαt αminus cc 1+ eαt( 1113857( 1113857

3minus 4Sce

minusαt(Sr + s) e

αtminus 11113872 1113873 minus 0125

Γ3(t) Sce

minusαt(Sr + s) eαt minus 1( 1113857

2

(34)

where A1 m3 + m2 + m + 1 and A2 minusm3 + m2 minus m

4 Solution Validation

To check the accuracy and efficiency of the methodology thecurrent results are verified by comparing with previouslypublished results in some special cases which are tabulated inTables 3ndash5 )e present results of different skin frictioncoefficient values Cf and Nusselt number NuRex for dif-ferent values of nanoparticle volume fraction ϕ and rotationparameter R are compared with those reported by Das [39]and Reddy et al [40] with both nanoparticles Cu and Al2O3which are illustrated in Tables 3 and 4 )e results obtainedcome with excellent agreement of previous studies

Furthermore the comparison for different values of theSherwood number minusShx against time t and chemical reac-tion cc with obtained results by Hussain et al [17] at ϕ 00is listed in Table 5 Also it can be seen that the presentobtained results are in an excellent agreement with the

previously published results )is proves the validity of thepresent work and shows how powerful the solution reachedby homotopy perturbation technique

5 Results and Discussion

)e influence of various physical parameters such asnanoparticle volume fraction ϕ shape of nanoparticles npHall current m Brownian NB thermophoretic NT andchemical reaction cc which are computed from the ap-proximate solution reported in the previous section is dis-played graphically and discussed )e graphical results areillustrated in Figures 2ndash11 )e numerical results are per-formed by taking values of some constant parameters as s

05 M 05 Pr 07 Ec 02 Gc 1 Gr 1 Sc 03

Du 05 G 5 and Sr 05Figures 2 and 3 display the effect of nanoparticle

volume fraction parameter ϕ on the fluid velocity u andtemperature profiles θ for either Cu minus water Al2O3 minus water

Mathematical Problems in Engineering 7

Table 3 Result comparison for Cf

Cf

ϕ RCuminuswater Al2O3minuswater

Present Das [39] Reddy et al [40] Present Das [39]

000 03 20355 20354 203542 20355 20354004 23158 23154 231535 05134 21831008 26111 26107 261068 21832 23384012 29120 29196 291953 25030 25020014 30792 30789 307891 10885 10872005 00 23790 23786 237862 22144 22143

05 24042 24040 240401 22333 2233315 25755 25755 257546 23669 2366625 28236 28235 28251 25697 25695

Table 4 Result comparison for NuxRex

NuxRex

ϕ RCuminuswater Al2O3minuswater

Present Das [39] Reddy et al [40] Present Das [39]

000 03 04607 04607 04607 04607 04607004 05135 05134 05134 05115 05114008 05695 05698 05698 05656 05655012 06304 06303 06303 06234 06233014 06622 06622 06622 06536 06537005 00 05272 05271 05271 05271 05271

05 mdash mdash mdash mdash mdash15 mdash mdash mdash mdash mdash25 05271 05271 05271 05272 05271

Table 5 Result comparison for minusShx

minusShx

ccdarr t⟶ 03 05 07 03 05 07Present Hussain et al [17]

02 0535254 0438632 0389005 054907 044765 0386382 0849652 0815633 0767225 085729 0820922 0781565 12252 116674 114632 124260 116967 1143922

ϕ = 0 007 01

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

Cu-waterAI2O3-waterTiO2-water

Figure 2 )e effect of volume fraction of nanoparticles on the velocity profile

8 Mathematical Problems in Engineering

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θϕ = 0 007 012

1

08

06

04

02

00 2 3 4 5 6 7 81

Cu-waterAI2O3-waterTiO2-water

z

Figure 3 )e effect of volume fraction nanoparticle on thetemperature profile

Cylindrical shape

Brick shape

Spherical shape

1

08

06

04

02

00 2 3 4 5 6 7 81

z

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θ

Cu-waterAI2O3-waterTiO2-water

Figure 4 )e effect of nanoparticle shape on the temperatureprofile

R = 1 2 4

Cu-waterTiO2-water

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

Figure 5 )e effect of rotation parameter on the velocity profile

m = 1 15 2

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 6 )e effect of Hall current on the velocity profile

NB = 02 05 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07m = 1γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 7 )e effect of Brownian motion parameter on the tem-perature profile

ϕ = 005NB = 07m = 1γc = 05Qs = 1Qr = 09R = 1

NT = 03 06 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 8 )e effect of thermophoresis parameter on the tem-perature profile

Mathematical Problems in Engineering 9

and TiO2 minus water nanofluid It is clear that increase innanoparticle volume fraction parameter ϕ decreases the fluidvelocity but the temperature profiles increase It is due to thefact that nanofluid density increases with increasing thevolume fraction of nanoparticles and it causes to slow downthe nanofluid flow velocity and increases the thickness of thethermal boundary layer which improves thermal conduc-tivity and increases surface heat transfer rate Also it isobserved that nanofluid Cu minus water heat transfer rate isgreater than Al2O3 and TiO2 rates So nanoparticles havinghigh thermal conductivity enhance heat transfer more thanthe particles having low thermal conductivity On thecontrary it is seen from Tables 2 and 3 that wall skin frictioncoefficient Cf and the Nusselt number NuRex increase withincreasing in nanoparticle volume fraction for both Cu minus

water and Al2O3 minus water)e influence of the nanoparticle shape on the tem-

perature distribution of nanofluid is studied through Fig-ure 4 )is figure indicates that the heat transfer rate ishigher for the cylindrical nanoparticle shape than thespherical and brick nanoparticle shapes for eitherCu minus water Al2O3 minus water and TiO2 minus water nanofluid)erefore the spherical nanoparticle shape tends to dragmore heat from the boundary layer because it has a greatersurface area while this effect is less evident for othernanoparticle shapes )us the use of the cylindrical nano-particle shape has better improvement in effective thermalconductivity when compared with other nanoparticle shapes(spherical and brick)

)e influence of rotation parameter R on the velocityprofiles against z is displayed in Figure 5 It observed that anincrease in rotation parameter leads to decreasing theboundary layer momentum so it decreases the values ofnanofluid velocity From Tables 2 and 3 it can be noted thatthe wall skin friction coefficient Cf increases with an in-crease in the rotation parameter but the Nusselt numberNuRex remains unchanged with both Cu minus water andAl2O3 minus water

Figure 6 presents the influence of Hall current parameterm on main velocity u It is clear that the velocity distributionof nanofluid reduces with an increase in Hall current pa-rameter)is happens naturally due to the fact that when themagnetic field is strengthened it enhances the opposingforce which is represented by the well-known Lorentz force

Figures 7 and 8 show the Brownian and thermophoresiseffects on the temperature profile θ It is seen that thetemperature distribution increases with increasing for theseparameters )is is due to the fact that the thermophoreticand Brownian parameters assist to improve the thermalboundary layer thickness Also it is observed that the heattransfer rate of Cu minus water nanofluid is greater thanTiO2 minus water

Figure 9 depicts the influence of chemical parameter cc

on the concentration profile Γ It is noticed that the increasein concentration profile is accompanied by a decrease in thechemical parameter On the contrary it is seen from Table 4that the Sherwood number minusShx increases due to increasingthe chemical parameter but the Sherwood number decreaseswhile increasing the time t )is implies that the chemical

Cu-waterTiO2-water

ϕ = 005NT = 07NB = 07m = 1

γc = 01 05 1

Qs = 1Qr = 09R = 1

0 2 3 4 5 6 7 81z

1

08

06

04

02

0

12

14

Γ

Figure 9 )e effect of chemical reaction parameter on the con-centration profile

Cu-waterTiO2-water

ϕ = 005

NB = 07

m = 1γc = 05

Qs = 1 2 3

Qr = 09R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

NT = 07

Figure 10 )e effect of heat source parameter on the temperatureprofile

ϕ = 005

NB = 07NT = 07

m = 1

γc = 05Qs = 1

Qr = 05 09 13

R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 11 )e effect of radiation parameter on the temperatureprofile

10 Mathematical Problems in Engineering

reaction has tendency to increase the mass transfer rate atthe surface plate and it is reduced with the progress of time

Figures 10 and 11 present the heat source parameter Qs

and thermal radiation parameter Qr effects on the tem-perature profiles It is clear from Figure 10 that with anincrease in the heat generation parameter the boundarylayer temperature decreases and as a consequence to de-crease the thermal boundary layer thickness In addition it isclear from Figure 11 that with an increase in the thermalradiation the temperature profiles of the nanofluid increaseIt is well known that the increase in the radiation parameterenhances the energy transport inside the fluid and as aconsequence to increase the heat transfer rate

6 Conclusion

An analysis is done to solve the magnetohydro dynamicsunsteady nanofluid flow through the porous medium past amoving plate in a rotating frame with considering the heatand mass transfer )ree different types of the nanoparticlesnamely copper aluminium oxide and titanium dioxide andwater used as a base nanofluid are considered with variousshapes (spherical cylindrical and brick) )e important ef-fects of nanoparticle shape Brownian and thermophoresisdiffusion thermal radiation ohmic and viscous dissipationshave been included Homotopy perturbation method is ap-plied to solve the governing nonlinear partial differentialequations )e efficiency of the methodology has been vali-dated through comparison with the previously publishedresults and observed a very close agreement It is found that

)e main velocity component decreases with increaseof nanoparticle volume fraction On the contrarythermal properties increase with increase in nano-particle volume fraction)e enhancement of heat transfer rate and mechanicalproperty improvement is high in copper as comparedwith both aluminium oxide and titanium dioxide)ermal conductivity of cylindrical-shaped nanoparticlesis more efficient and possesses better improvement onheat transfer rate than the other shaped onesIncrease in the Brownian and thermophoresis pa-rameter is accompanied by an increase in the thermalboundary layer thicknessIncreasing Hall current and rotation parameters de-creases the main fluid velocity componentHPM has a close agreement and is recommended formany nonlinear problems

Data Availability

)e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] T Hayat R Sajjad A Alsaedi T Muhammad and R EllahildquoOn squeezed flow of couple stress nanofluid between twoparallel platesrdquo Results in Physics vol 7 pp 553ndash561 2017

[2] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition pp 99ndash105San Francisco CA USA 1995

[3] J A Eastman S U S Choi S Li W Yu and L J )ompsonldquoAnomalously increased effective thermal conductivities of eth-ylene glycol-based nanofluids containing copper nanoparticlesrdquoApplied Physics Letters vol 78 no 6 pp 718ndash720 2001

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

[5] M Sheikholeslami M M Rashidi T Hayat and D D GanjildquoFree convection of magnetic nanofluid considering MFDviscosity effectrdquo Journal of Molecular Liquids vol 218pp 393ndash399 2016

[6] W Abbas and E A Sayed ldquoHall current and joule heatingeffects on free convection flow of a nanofluid over a verticalcone in presence of thermal radiationrdquo ermal Sciencevol 21 pp 2603ndash2614 2017

[7] B Mahanthesh B Gireesha I Animasaun T Muhammadand N Shashikumar ldquoMHD flow of SWCNT and MWCNTnanoliquids past a rotating stretchable disk with thermal andexponential space dependent heat sourcerdquo Physica Scriptavol 94 no 8 Article ID 085214 2019

[8] I L Animasaun O K Koriko K S Adegbie et al ldquoCom-parative analysis between 36 nm and 47 nm alumina-waternanofluid flows in the presence of Hall effectrdquo Journal ofermal Analysis and Calorimetry vol 135 no 2 pp 873ndash886 2019

[9] I Ullah S Shafie I Khan and K L Hsiao ldquoBrownian dif-fusion and thermophoresis mechanisms in Casson fluid over amoving wedgerdquo Results in Physics vol 9 pp 183ndash194 2018

[10] P B S Kumar B J Gireesha R S R Gorla andB Mahanthesh ldquoMagnetohydrodynamic flow of Williamsonnanofluid due to an exponentially stretching surface in thepresence of thermal radiation and chemical reactionrdquo Journalof Nanofluids vol 6 no 2 pp 264ndash272 2017

[11] X Zhou Y Jiang X Li et al ldquoNumerical investigation of heattransfer enhancement and entropy generation of naturalconvection in a cavity containing nano liquid-metal fluidrdquoInternational Communications in Heat and Mass Transfervol 106 pp 46ndash54 2019

[12] N Shehzad A Zeeshan R Ellahi and K Vafai ldquoConvectiveheat transfer of nanofluid in a wavy channel Buongiornorsquosmathematical modelrdquo Journal of Molecular Liquids vol 222pp 446ndash455 2016

[13] B Shen L Zheng C Zhang and X Zhang ldquoBioconvectionheat transfer of a nanofluid over a stretching sheet with ve-locity slip and temperature jumprdquo ermal Science vol 21no 6 pp 2347ndash2356 2017

[14] S Jahan H Sakidin R Nazar and I Pop ldquoAnalysis of heattransfer in nanofluid past a convectively heated permeablestretchingshrinking sheet with regression and stability ana-lysesrdquo Results in Physics vol 10 pp 395ndash405 2018

[15] M A A Hamad and I Pop ldquoUnsteady MHD free convectionflow past a vertical permeable flat plate in a rotating frame ofreference with constant heat source in a nanofluidrdquo Heat andMass Transfer vol 47 no 12 pp 1517ndash1524 2011

Mathematical Problems in Engineering 11

[16] M Sheikholeslami and D D Ganji ldquo)ree dimensional heatand mass transfer in a rotating system using nanofluidrdquoPowder Technology vol 253 pp 789ndash796 2014

[17] S M Hussain J Jain G S Seth and M M Rashidi ldquoFreeconvective heat transfer with Hall effects heat absorption andchemical reaction over an accelerated moving plate in a ro-tating systemrdquo Journal of Magnetism and Magnetic Materialsvol 422 pp 112ndash123 2017

[18] X Zhou Y Jiang Y Hou and M Du ldquo)ermocapillaryconvection instability in annular two-layer system undervarious gravity levelsrdquo Microgravity Science and Technologyvol 31 no 5 pp 641ndash648 2019

[19] Y Jiang and X Zhou ldquoHeat transfer and entropy generationanalysis of nanofluids thermocapillary convection around abubble in a cavityrdquo International Communications in Heatand Mass Transfer vol 105 pp 37ndash45 2019

[20] Y Jiang X Zhou and Y Wang ldquoEffects of nanoparticleshapes on heat andmass transfer of nanofluid thermocapillaryconvection around a gas bubblerdquo Microgravity Science andTechnology pp 1ndash11 2019 In press

[21] Y Jiang X Zhou and Y Wang ldquoComprehensive heattransfer performance analysis of nanofluid mixed forced andthermocapillary convection around a gas bubble in mini-channelrdquo International Communications in Heat and MassTransfer vol 110 Article ID 104386 2020

[22] B Mahanthesh B J Gireesha R S R Gorla F M Abbasiand S A Shehzad ldquoNumerical solutions for magnetohy-drodynamic flow of nanofluid over a bidirectional non-linearstretching surface with prescribed surface heat flux bound-aryrdquo Journal of Magnetism and Magnetic Materials vol 417pp 189ndash196 2016

[23] B Mahanthesh B J Gireesha and I L Animasaun ldquoEx-ploration of non-linear thermal radiation and suspendednanoparticles effects on mixed convection boundary layerflow of nanoliquids on a melting vertical surfacerdquo Journal ofNanofluids vol 7 no 5 pp 833ndash843 2018

[24] P S Kumar B Mahanthesh B Gireesha and S ShehzadldquoQuadratic convective flow of radiated nano-jeffrey liquidsubject to multiple convective conditions and Cattaneo-Christov double diffusionrdquo Applied Mathematics and Me-chanics vol 39 no 9 pp 1311ndash1326 2018

[25] J Raza F Mebarek-Oudina and A Chamkha ldquoMagneto-hydrodynamic flow of molybdenum disulfide nanofluid in achannel with shape effectsrdquo Multidiscipline Modeling inMaterials and Structures vol 15 no 4 pp 737ndash757 2019

[26] T Hayat B Ahmed F M Abbasi and A Alsaedi ldquoHy-dromagnetic peristalsis of water based nanofluids with tem-perature dependent viscosity a comparative studyrdquo Journal ofMolecular Liquids vol 234 pp 324ndash329 2017

[27] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filledwith nanofluidsrdquo International Journal of Heat and FluidFlow vol 29 no 5 pp 1326ndash1336 2008

[28] M A El Kot and W Abbas ldquoNumerical technique of bloodflow through catheterized arteries with overlapping stenosisrdquoComputer Methods in Biomechanics and Biomedical Engi-neering vol 20 no 1 pp 45ndash58 2017

[29] H A Attia W Abbas A El-Din Abdin andM A M Abdeen ldquoEffects of ion slip and Hall current onunsteady Couette flow of a dusty fluid through porous mediawith heat transferrdquo High Temperature vol 53 no 6pp 891ndash898 2015

[30] H A Attia W Abbas and M A M Abdeen ldquoIon slip effecton unsteady Couette flow of a dusty fluid in the presence of

uniform suction and injection with heat transferrdquo Journal ofthe Brazilian Society of Mechanical Sciences and Engineeringvol 38 no 8 pp 2381ndash2391 2016

[31] H A Attia W Abbas M A M Abdeen and A A M SaidldquoHeat transfer between two parallel porous plates for Couetteflow under pressure gradient and Hall currentrdquo Sadhanavol 40 no 1 pp 183ndash197 2015

[32] M Sheikholeslami A Shafee M Ramzan and Z Li ldquoIn-vestigation of Lorentz forces and radiation impacts onnanofluid treatment in a porous semi annulus via Darcy lawrdquoJournal of Molecular Liquids vol 272 pp 8ndash14 2018

[33] P S Kumar B Gireesha B Mahanthesh and A J Chamkhaldquo)ermal analysis of nanofluid flow containing gyrotacticmicroorganisms in bioconvection and second-order slip withconvective conditionrdquo Journal of ermal Analysis andCalorimetry vol 136 no 5 pp 1947ndash1957 2019

[34] B Mahanthesh F Mabood B Gireesha and R Gorla ldquoEffectsof chemical reaction and partial slip on the three-dimensionalflow of a nanofluid impinging on an exponentially stretchingsurfacerdquo e European Physical Journal Plus vol 132 no 3p 113 2017

[35] B J Gireesha R S R Gorla and B Mahanthesh ldquoEffect ofsuspended nanoparticles on three-dimensional mhd flowheat and mass transfer of radiating eyring-powell fluid over astretching sheetrdquo Journal of Nanofluids vol 4 no 4pp 474ndash484 2015

[36] J-H He ldquoHomotopy perturbation techniquerdquo ComputerMethods in Applied Mechanics and Engineering vol 178no 3-4 pp 257ndash262 1999

[37] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo Interna-tional Journal of Non-Linear Mechanics vol 35 no 1pp 37ndash43 2000

[38] Y Wu and J-H He ldquoHomotopy perturbation method fornonlinear oscillators with coordinate-dependent massrdquo Re-sults in Physics vol 10 pp 270-271 2018

[39] K Das ldquoFlow and heat transfer characteristics of nanofluids ina rotating framerdquo Alexandria Engineering Journal vol 53no 3 pp 757ndash766 2014

[40] J V R Reddy V Sugunamma N Sandeep and C SulochanaldquoInfluence of chemical reaction radiation and rotation onMHD nanofluid flow past a permeable flat plate in porousmediumrdquo Journal of the Nigerian Mathematical Societyvol 35 no 1 pp 48ndash65 2016

12 Mathematical Problems in Engineering

where βlowastTnf denotes the nanofluid thermal expansion co-efficient due to temperature difference βlowastCnf denotes thenanofluid thermal expansion coefficient due to concentra-tion difference and m σnfξB0 is the Hall parameter [31]

)e boundary conditions are

tle 0 u 0

w 0

T Tinfin

C Cinfin

z 0

tgt 0 u U0

w 0

T Tw

C Cw

z⟶infin

tgt 0 u⟶ 0

w⟶ 0

T⟶ Tinfin

C⟶ Cinfin

(9)

23 ermophysical Properties of Nanofluid )e nanofluidthermophysical properties were calculated using the fol-lowing forms [6 26]

Nanofluid effective dynamic viscosity effective densityand heat capacity are given as

μnf μf

(1 minus ϕ)25

J1 ρf

ρnf

1

(1 minus ϕ) + ϕ ρsρf1113872 1113873

J2 (ρCp)f

(ρCp)nf

1

(1 minus ϕ) + ϕ (ρCp)s(ρCp)f1113872 1113873

(10)

where ϕ denotes the nanofluid volume fraction Sub-scripts f nf and s denote the base fluid nanofluidand solid particle respectivelyAlso nanofluid effective electric conductivity and ef-fective thermal expansion coefficient are given as

J3 σnf

σf

1 +3 σsσf1113872 1113873 minus 11113872 1113873ϕ

σsσf1113872 1113873 + 21113872 1113873 minus σsσf1113872 1113873 minus 11113872 1113873ϕ

J4 ρβlowast( 1113857nf

ρβlowast( 1113857f

(1 minus ϕ) + ϕρβlowast( 1113857s

ρβlowast( 1113857f

(11)

Effective thermal conductivity and effect of nano-particle shapes are considered by using the Hamiltonand Crosser model as [26]

J5 knf

kf

ks + np minus 11113872 1113873kf minus np minus 11113872 1113873ϕ kf minus ks1113872 1113873

ks + np minus 11113872 1113873kf + ϕ kf minus ks1113872 1113873 (12)

where np indicates the empirical nanoparticle shapefactor which is given by

np 3Ψs

(13)

Here Ψs is the sphericity of the nanoparticles which isdefined as the ratio of the surface area of a sphere to thearea of the particle with a volume equal to that of a particleand has value 1 (np 3) for spherical-shaped nano-particles and 05 (np 6) for spherical-shaped nano-particles)e characteristics of shape factor np for variousnanoparticle shapes have been displayed in Table 2 [32]

Introducing the following nondimensional variables

(1113954x 1113954y 1113954z) (x y z)U0

]f

1113954t U2

0]f

t

1113954u u

U0

1113954w w

U0

θ T minus Tinfin

Tw minus Tinfin

Γ C minus Cinfin

Cw minus Cinfin

(14)

Governing equations (5)ndash(8) are reduced to nondi-mensional form after dropping the cap as

zu

zt+ s

zu

zz+ Rv

J1

(1minus ϕ)25z2u

zz2 minusHa2J1J3

1+ m2 (u + mv)

+ GrJ1J4θ+ GcJ1J4ΓminusMJ1

(1minus ϕ)25 u

(15)

zv

zt+ s

zv

zzminus Ru

J1

(1minus ϕ)25z2v

zz2 minusHa2J1J31+ m2 (v minus mu)

minusMJ1

(1minus ϕ)25 v

(16)

zθzt

+ szθzz

1Pr

J5J2z2θzz2 +

Ha2Ec

1+ m2 J3J2 u2

+ v2

1113872 1113873 + QsJ2θ

minus QrJ2z2θzz2 + DuJ2

z2Γzz2 + NBJ2

zθzz

zΓzz

+ NTJ2zθzz

1113888 1113889

2

+Ec

(1minus ϕ)25J2zu

zz1113888 1113889

2

+zv

zz1113888 1113889

2⎛⎝ ⎞⎠

(17)

zΓzt

+ szΓzz

1

SC

z2Γzz2 + ccΓ+ Sr

z2θzz2

(18)

4 Mathematical Problems in Engineering

where R (2Ω]f)U20 is the rotation parameter Ha2

μfB20σfρ2fU2

0 is the Hartmann number squared Gr

g]fβlowastTf(Tw minus Tinfin)U3

0 is the Grashof number Gc

g]fβlowastCf(Cw minus Cinfin)U3

0 is the modified Grashof numberM μ2fU

20kρ

2f is the porous medium parameter

Pr CPfμfkf is the Prandtl number Ec U20CPf(Tw minus

Tinfin) is the Eckart number Qs Q]fU20(ρCp)f is the heat

source parameter Qr 16σT3infin3αlowast]f(ρCp)f is the radiation

parameter Du (DmkT(Cw minus Cinfin))(CsCPf]f(Tw minus Tinfin)) isthe Dufour number Brownian parameter NB DBτ(Cw minus Cinfin)]f NT DTτ(Tw minus Tinfin)]fTinfin denotes ther-mophoresis parameter nanoparticles and base fluid heatcapacity ratio is τ ρsCps

ρfCpf Sc ]fD is the Schmidt

number cc kc]fU20 is the chemical reaction parameter and

Sr DT(Tw minus Tinfin)Tinfin(Cw minus Cinfin) denotes the Soret number)e boundary conditions reduce to

t 0 u 0

w 0

θ⟶ 0

Γ⟶ 0

z 0

tgt 0 u 1

w 0

θ 1

Γ 1

z⟶infin

tgt 0 u⟶ 0

w⟶ 0

θ⟶ 0

Γ⟶ 0

(19)

24Engineering InterestPhysicalQuantities For engineeringinterest the local skin friction (Cf) local Nusselt number(Nux) and local Sherwood number (Shx) are defined as[33ndash35]

Cf τw

ρfU20

Nux xqw

kf Tw minus Tinfin( 1113857

Shx xqm

DB Tw minus Tinfin( 1113857

(20)

where τw qw and qm denote the wall shear stress and heatand mass fluxes respectively which are given by

τw μnf

zu

zz1113888 1113889

z0

qw minusknf

zT

zz+ qr1113888 1113889

z0

qm minusDB

zC

zz1113888 1113889

z0

(21)

By substituting equation (14) into equations (20) and(21)

Nux minusknf

kf

Rex 1 + Qr( 1113857θprime(0) (22)

where Rex U0x]f is the local Reynolds number

3 Methodology

Here the homotopy perturbation method that was firstdeveloped by He [36 37] and improved by Wu and He [38]to solve governing equations (15)ndash(18) subject to boundarycondition (19) is used To illustrate the fundamental ideas ofthis method we first consider the time-dependent differ-ential equation as follows

Υ(ζ(λ t)) minus f(λ t) 0 λ isin R (23)

where Υ is the operator ζ(λ t) denotes the unknownfunction f(λ t) is a known function λ are spatial variablest denote temporal independent variables and R is thedomain

)e operator Υ can be decomposed into the linear partL(λ t) and N(λ t) is the remaining part )ereforeequation (23) can be rewritten as

L(λ t) + N(λ t) minus f(λ t) 0 λ isin R (24)

Homotopy traditional technique constructs a homotopyfunction Ψ(λ t p) as follows

H [Ψ(λ t p)] (1 minus p) (LΨ(λ t p)) minus L f0(λ t)( 11138571113858 1113859

+ p [Υ(Ψ(λ t p)) minus f(λ t)] 0

(25)

where p isin [0 1] is an embedding parameter and f0(λ t) isan initial guess approximation of equation (23) which sat-isfies boundary and initial condition

Consequently the solution of equation (23) is expressedas

Ψ((λ t p) 1113944n

i0Ψi(λ t)p

i (26)

Ideal approximation solution can be obtained when p

1 as

ζ((λ t) 1113944n

i0ζ i(λ t) (27)

Table 2 Shape factor for various nanoparticle shapes [32]

Spherical Cylindrical Brick

Shapes of nanoparticles

Shape factor (np) 3 6 37

Mathematical Problems in Engineering 5

According to equation (24) the appropriate initial ar-bitrary guess approximation and linear operators for eachequation are proposed as

u0 y 1 + eminus t

1113872 1113873

v0 y 1 + eminus t

1113872 1113873

θ0 y 1 minus eminus t

1113872 1113873

Γ0 y 1 minus eminus t

1113872 1113873

L(u) z2u

zz2

L(v) z2v

zz2

L(θ) z2θzz2

L(Γ) z2Γzz2

(28)

Now HPM constructed equations which satisfy equa-tion (25) to governing nonlinear differential equations(15)ndash(18) as follows

H(u p) pJ1

(1 minus ϕ)25 L(u)1113888 1113889 +(1 minus p) minuszu

ztminus s

zu

zzminus Rv minus

Ha2J1J31 + m2 (u + mv) + GrJ1J4θ + GcJ1J4Γ minus

MJ1

(1 minus ϕ)25 u1113888 1113889 0

(29)

H(v p) pJ1

(1 minus ϕ)25 L(v)1113888 1113889 +(1 minus p) minuszv

ztminus s

zv

zzminus Ru minus

Ha2J1J3

1 + m2 (v minus mu) minusMJ1

(1 minus ϕ)25 v1113888 1113889 0 (30)

H(θ p) p1Pr

J5J2 minus QrJ21113888 1113889L(θ)1113888 1113889 +(1 minus p)zθzt

+ szθzz

+Ha2Ec

1 + m2J3J2 u2

+ v2

1113872 1113873 + QsJ2θ + DuJ2z2Γzz21113888

+ NBJ2zθzz

zΓzz

+ NTJ2zθzz

1113888 1113889

2

+Ec

(1 minus ϕ)25J2zu

zz1113888 1113889

2

+zv

zz1113888 1113889

2⎛⎝ ⎞⎠⎞⎠ 0

(31)

H(Γ p) p1SC

L(Γ)1113888 1113889 +(1 minus p) minuszΓzt

minus szΓzz

+ ccΓ + Sr

z2θzz21113888 1113889 0 (32)

And assuming the solution of equations (29)ndash(32) it canbe written in the following forms

u(z t) u1(t)z + u2(t)z2

+ u3(t)z3

+ middot middot middot

v(z t) v1(t)z + v2(t)z2

+ v3(t)z3

+ middot middot middot

θ(z t) θ1(t)z + θ2(t)z2

+ θ3(t)z3

+ middot middot middot

Γ(z t) Γ1(t)z + Γ2(t)z2

+ Γ3(t)z3

+ middot middot middot (33)

Substituting equation (26) into equations (27)ndash(31) aftersome simplifications and rearranging depending on powersof p-terms and solved using conditions (19) the followingapproximations can be obtained

6 Mathematical Problems in Engineering

u1(t) 1590J1

eminusαt

(1minus ϕ)52

MJ1 1+ eαt

1113872 1113873 + Ha2J1J4A1 1+ e

αt1113872 1113873 + GcJ1J3 1minus e

αt1113872 1113873 minus α(1minus s)GrJ1J2 1minus e

αt1113872 1113873 + R 1+ e

αt1113872 11138731113872 1113873

u2(t) 323J1

eminusαt

(1minus ϕ)52

MJ1 1+ eαt

1113872 1113873 + Ha2J1J4A1 1+ e

αt1113872 1113873 + GcJ1J3 1minus e

αt1113872 1113873 minus α(1+ s) minus GrJ1J2 1minus e

αt1113872 1113873 + R 1+ e

αt1113872 11138731113872 1113873

u3(t) 1590J1

(1minus ϕ)52

Ha2J1J4(1minus m) minus R minus αminus e

αt1113872 1113873

v1(t) 1590

(1minus ϕ)52

minuseminusαt

(R + αs) + Ha2J4J1

9015

eminusαt

+ m2

minus m3

+ A1eminusαt

1113874 11138751113874 1113875

v2(t) minus323

Ha2J1J4(1minus ϕ)

52+ R(1minus ϕ)

52+ Ha

2J1J4m(1minus ϕ)

52+323αe

minusαt(1minus ϕ)

52minus323

Ha2J1J4e

minusαt(1minus ϕ)

52

+323Reminusαt

(1minus ϕ)52

v3(t) minusMJ1(1minus ϕ)52

+ +Ha2J4m

3e

minusαt(1minus ϕ)

52323

minus 0125

θ1 minus32Pre

minusαt α+ J6Qr 1minus eαt( 1113857( 1113857

3J5J6+4Pre

minus2αt eαt minus 1( 1113857 seαt + DNBJ6 1minus eαt( 1113857( 1113857

J5J6minus256EcHa2J4Pre

minus2αt m2 + 1( 1113857 eαt + 1( 11138572

3J5+18

θ2 Pre

minus2αt eαt minus 1( 1113857 seαt + DNBJ6 1minus eαt( 1113857( 1113857

2J5J6

θ3 Pre

minusαt α+ J6Qr 1minus eαt( 1113857( 1113857

6J5J6

Γ1(t) Sce

minusαt αminus cc 1+ eαt( 1113857( 1113857

6

Γ2(t) 32Sce

minusαt αminus cc 1+ eαt( 1113857( 1113857

3minus 4Sce

minusαt(Sr + s) e

αtminus 11113872 1113873 minus 0125

Γ3(t) Sce

minusαt(Sr + s) eαt minus 1( 1113857

2

(34)

where A1 m3 + m2 + m + 1 and A2 minusm3 + m2 minus m

4 Solution Validation

To check the accuracy and efficiency of the methodology thecurrent results are verified by comparing with previouslypublished results in some special cases which are tabulated inTables 3ndash5 )e present results of different skin frictioncoefficient values Cf and Nusselt number NuRex for dif-ferent values of nanoparticle volume fraction ϕ and rotationparameter R are compared with those reported by Das [39]and Reddy et al [40] with both nanoparticles Cu and Al2O3which are illustrated in Tables 3 and 4 )e results obtainedcome with excellent agreement of previous studies

Furthermore the comparison for different values of theSherwood number minusShx against time t and chemical reac-tion cc with obtained results by Hussain et al [17] at ϕ 00is listed in Table 5 Also it can be seen that the presentobtained results are in an excellent agreement with the

previously published results )is proves the validity of thepresent work and shows how powerful the solution reachedby homotopy perturbation technique

5 Results and Discussion

)e influence of various physical parameters such asnanoparticle volume fraction ϕ shape of nanoparticles npHall current m Brownian NB thermophoretic NT andchemical reaction cc which are computed from the ap-proximate solution reported in the previous section is dis-played graphically and discussed )e graphical results areillustrated in Figures 2ndash11 )e numerical results are per-formed by taking values of some constant parameters as s

05 M 05 Pr 07 Ec 02 Gc 1 Gr 1 Sc 03

Du 05 G 5 and Sr 05Figures 2 and 3 display the effect of nanoparticle

volume fraction parameter ϕ on the fluid velocity u andtemperature profiles θ for either Cu minus water Al2O3 minus water

Mathematical Problems in Engineering 7

Table 3 Result comparison for Cf

Cf

ϕ RCuminuswater Al2O3minuswater

Present Das [39] Reddy et al [40] Present Das [39]

000 03 20355 20354 203542 20355 20354004 23158 23154 231535 05134 21831008 26111 26107 261068 21832 23384012 29120 29196 291953 25030 25020014 30792 30789 307891 10885 10872005 00 23790 23786 237862 22144 22143

05 24042 24040 240401 22333 2233315 25755 25755 257546 23669 2366625 28236 28235 28251 25697 25695

Table 4 Result comparison for NuxRex

NuxRex

ϕ RCuminuswater Al2O3minuswater

Present Das [39] Reddy et al [40] Present Das [39]

000 03 04607 04607 04607 04607 04607004 05135 05134 05134 05115 05114008 05695 05698 05698 05656 05655012 06304 06303 06303 06234 06233014 06622 06622 06622 06536 06537005 00 05272 05271 05271 05271 05271

05 mdash mdash mdash mdash mdash15 mdash mdash mdash mdash mdash25 05271 05271 05271 05272 05271

Table 5 Result comparison for minusShx

minusShx

ccdarr t⟶ 03 05 07 03 05 07Present Hussain et al [17]

02 0535254 0438632 0389005 054907 044765 0386382 0849652 0815633 0767225 085729 0820922 0781565 12252 116674 114632 124260 116967 1143922

ϕ = 0 007 01

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

Cu-waterAI2O3-waterTiO2-water

Figure 2 )e effect of volume fraction of nanoparticles on the velocity profile

8 Mathematical Problems in Engineering

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θϕ = 0 007 012

1

08

06

04

02

00 2 3 4 5 6 7 81

Cu-waterAI2O3-waterTiO2-water

z

Figure 3 )e effect of volume fraction nanoparticle on thetemperature profile

Cylindrical shape

Brick shape

Spherical shape

1

08

06

04

02

00 2 3 4 5 6 7 81

z

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θ

Cu-waterAI2O3-waterTiO2-water

Figure 4 )e effect of nanoparticle shape on the temperatureprofile

R = 1 2 4

Cu-waterTiO2-water

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

Figure 5 )e effect of rotation parameter on the velocity profile

m = 1 15 2

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 6 )e effect of Hall current on the velocity profile

NB = 02 05 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07m = 1γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 7 )e effect of Brownian motion parameter on the tem-perature profile

ϕ = 005NB = 07m = 1γc = 05Qs = 1Qr = 09R = 1

NT = 03 06 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 8 )e effect of thermophoresis parameter on the tem-perature profile

Mathematical Problems in Engineering 9

and TiO2 minus water nanofluid It is clear that increase innanoparticle volume fraction parameter ϕ decreases the fluidvelocity but the temperature profiles increase It is due to thefact that nanofluid density increases with increasing thevolume fraction of nanoparticles and it causes to slow downthe nanofluid flow velocity and increases the thickness of thethermal boundary layer which improves thermal conduc-tivity and increases surface heat transfer rate Also it isobserved that nanofluid Cu minus water heat transfer rate isgreater than Al2O3 and TiO2 rates So nanoparticles havinghigh thermal conductivity enhance heat transfer more thanthe particles having low thermal conductivity On thecontrary it is seen from Tables 2 and 3 that wall skin frictioncoefficient Cf and the Nusselt number NuRex increase withincreasing in nanoparticle volume fraction for both Cu minus

water and Al2O3 minus water)e influence of the nanoparticle shape on the tem-

perature distribution of nanofluid is studied through Fig-ure 4 )is figure indicates that the heat transfer rate ishigher for the cylindrical nanoparticle shape than thespherical and brick nanoparticle shapes for eitherCu minus water Al2O3 minus water and TiO2 minus water nanofluid)erefore the spherical nanoparticle shape tends to dragmore heat from the boundary layer because it has a greatersurface area while this effect is less evident for othernanoparticle shapes )us the use of the cylindrical nano-particle shape has better improvement in effective thermalconductivity when compared with other nanoparticle shapes(spherical and brick)

)e influence of rotation parameter R on the velocityprofiles against z is displayed in Figure 5 It observed that anincrease in rotation parameter leads to decreasing theboundary layer momentum so it decreases the values ofnanofluid velocity From Tables 2 and 3 it can be noted thatthe wall skin friction coefficient Cf increases with an in-crease in the rotation parameter but the Nusselt numberNuRex remains unchanged with both Cu minus water andAl2O3 minus water

Figure 6 presents the influence of Hall current parameterm on main velocity u It is clear that the velocity distributionof nanofluid reduces with an increase in Hall current pa-rameter)is happens naturally due to the fact that when themagnetic field is strengthened it enhances the opposingforce which is represented by the well-known Lorentz force

Figures 7 and 8 show the Brownian and thermophoresiseffects on the temperature profile θ It is seen that thetemperature distribution increases with increasing for theseparameters )is is due to the fact that the thermophoreticand Brownian parameters assist to improve the thermalboundary layer thickness Also it is observed that the heattransfer rate of Cu minus water nanofluid is greater thanTiO2 minus water

Figure 9 depicts the influence of chemical parameter cc

on the concentration profile Γ It is noticed that the increasein concentration profile is accompanied by a decrease in thechemical parameter On the contrary it is seen from Table 4that the Sherwood number minusShx increases due to increasingthe chemical parameter but the Sherwood number decreaseswhile increasing the time t )is implies that the chemical

Cu-waterTiO2-water

ϕ = 005NT = 07NB = 07m = 1

γc = 01 05 1

Qs = 1Qr = 09R = 1

0 2 3 4 5 6 7 81z

1

08

06

04

02

0

12

14

Γ

Figure 9 )e effect of chemical reaction parameter on the con-centration profile

Cu-waterTiO2-water

ϕ = 005

NB = 07

m = 1γc = 05

Qs = 1 2 3

Qr = 09R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

NT = 07

Figure 10 )e effect of heat source parameter on the temperatureprofile

ϕ = 005

NB = 07NT = 07

m = 1

γc = 05Qs = 1

Qr = 05 09 13

R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 11 )e effect of radiation parameter on the temperatureprofile

10 Mathematical Problems in Engineering

reaction has tendency to increase the mass transfer rate atthe surface plate and it is reduced with the progress of time

Figures 10 and 11 present the heat source parameter Qs

and thermal radiation parameter Qr effects on the tem-perature profiles It is clear from Figure 10 that with anincrease in the heat generation parameter the boundarylayer temperature decreases and as a consequence to de-crease the thermal boundary layer thickness In addition it isclear from Figure 11 that with an increase in the thermalradiation the temperature profiles of the nanofluid increaseIt is well known that the increase in the radiation parameterenhances the energy transport inside the fluid and as aconsequence to increase the heat transfer rate

6 Conclusion

An analysis is done to solve the magnetohydro dynamicsunsteady nanofluid flow through the porous medium past amoving plate in a rotating frame with considering the heatand mass transfer )ree different types of the nanoparticlesnamely copper aluminium oxide and titanium dioxide andwater used as a base nanofluid are considered with variousshapes (spherical cylindrical and brick) )e important ef-fects of nanoparticle shape Brownian and thermophoresisdiffusion thermal radiation ohmic and viscous dissipationshave been included Homotopy perturbation method is ap-plied to solve the governing nonlinear partial differentialequations )e efficiency of the methodology has been vali-dated through comparison with the previously publishedresults and observed a very close agreement It is found that

)e main velocity component decreases with increaseof nanoparticle volume fraction On the contrarythermal properties increase with increase in nano-particle volume fraction)e enhancement of heat transfer rate and mechanicalproperty improvement is high in copper as comparedwith both aluminium oxide and titanium dioxide)ermal conductivity of cylindrical-shaped nanoparticlesis more efficient and possesses better improvement onheat transfer rate than the other shaped onesIncrease in the Brownian and thermophoresis pa-rameter is accompanied by an increase in the thermalboundary layer thicknessIncreasing Hall current and rotation parameters de-creases the main fluid velocity componentHPM has a close agreement and is recommended formany nonlinear problems

Data Availability

)e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] T Hayat R Sajjad A Alsaedi T Muhammad and R EllahildquoOn squeezed flow of couple stress nanofluid between twoparallel platesrdquo Results in Physics vol 7 pp 553ndash561 2017

[2] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition pp 99ndash105San Francisco CA USA 1995

[3] J A Eastman S U S Choi S Li W Yu and L J )ompsonldquoAnomalously increased effective thermal conductivities of eth-ylene glycol-based nanofluids containing copper nanoparticlesrdquoApplied Physics Letters vol 78 no 6 pp 718ndash720 2001

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

[5] M Sheikholeslami M M Rashidi T Hayat and D D GanjildquoFree convection of magnetic nanofluid considering MFDviscosity effectrdquo Journal of Molecular Liquids vol 218pp 393ndash399 2016

[6] W Abbas and E A Sayed ldquoHall current and joule heatingeffects on free convection flow of a nanofluid over a verticalcone in presence of thermal radiationrdquo ermal Sciencevol 21 pp 2603ndash2614 2017

[7] B Mahanthesh B Gireesha I Animasaun T Muhammadand N Shashikumar ldquoMHD flow of SWCNT and MWCNTnanoliquids past a rotating stretchable disk with thermal andexponential space dependent heat sourcerdquo Physica Scriptavol 94 no 8 Article ID 085214 2019

[8] I L Animasaun O K Koriko K S Adegbie et al ldquoCom-parative analysis between 36 nm and 47 nm alumina-waternanofluid flows in the presence of Hall effectrdquo Journal ofermal Analysis and Calorimetry vol 135 no 2 pp 873ndash886 2019

[9] I Ullah S Shafie I Khan and K L Hsiao ldquoBrownian dif-fusion and thermophoresis mechanisms in Casson fluid over amoving wedgerdquo Results in Physics vol 9 pp 183ndash194 2018

[10] P B S Kumar B J Gireesha R S R Gorla andB Mahanthesh ldquoMagnetohydrodynamic flow of Williamsonnanofluid due to an exponentially stretching surface in thepresence of thermal radiation and chemical reactionrdquo Journalof Nanofluids vol 6 no 2 pp 264ndash272 2017

[11] X Zhou Y Jiang X Li et al ldquoNumerical investigation of heattransfer enhancement and entropy generation of naturalconvection in a cavity containing nano liquid-metal fluidrdquoInternational Communications in Heat and Mass Transfervol 106 pp 46ndash54 2019

[12] N Shehzad A Zeeshan R Ellahi and K Vafai ldquoConvectiveheat transfer of nanofluid in a wavy channel Buongiornorsquosmathematical modelrdquo Journal of Molecular Liquids vol 222pp 446ndash455 2016

[13] B Shen L Zheng C Zhang and X Zhang ldquoBioconvectionheat transfer of a nanofluid over a stretching sheet with ve-locity slip and temperature jumprdquo ermal Science vol 21no 6 pp 2347ndash2356 2017

[14] S Jahan H Sakidin R Nazar and I Pop ldquoAnalysis of heattransfer in nanofluid past a convectively heated permeablestretchingshrinking sheet with regression and stability ana-lysesrdquo Results in Physics vol 10 pp 395ndash405 2018

[15] M A A Hamad and I Pop ldquoUnsteady MHD free convectionflow past a vertical permeable flat plate in a rotating frame ofreference with constant heat source in a nanofluidrdquo Heat andMass Transfer vol 47 no 12 pp 1517ndash1524 2011

Mathematical Problems in Engineering 11

[16] M Sheikholeslami and D D Ganji ldquo)ree dimensional heatand mass transfer in a rotating system using nanofluidrdquoPowder Technology vol 253 pp 789ndash796 2014

[17] S M Hussain J Jain G S Seth and M M Rashidi ldquoFreeconvective heat transfer with Hall effects heat absorption andchemical reaction over an accelerated moving plate in a ro-tating systemrdquo Journal of Magnetism and Magnetic Materialsvol 422 pp 112ndash123 2017

[18] X Zhou Y Jiang Y Hou and M Du ldquo)ermocapillaryconvection instability in annular two-layer system undervarious gravity levelsrdquo Microgravity Science and Technologyvol 31 no 5 pp 641ndash648 2019

[19] Y Jiang and X Zhou ldquoHeat transfer and entropy generationanalysis of nanofluids thermocapillary convection around abubble in a cavityrdquo International Communications in Heatand Mass Transfer vol 105 pp 37ndash45 2019

[20] Y Jiang X Zhou and Y Wang ldquoEffects of nanoparticleshapes on heat andmass transfer of nanofluid thermocapillaryconvection around a gas bubblerdquo Microgravity Science andTechnology pp 1ndash11 2019 In press

[21] Y Jiang X Zhou and Y Wang ldquoComprehensive heattransfer performance analysis of nanofluid mixed forced andthermocapillary convection around a gas bubble in mini-channelrdquo International Communications in Heat and MassTransfer vol 110 Article ID 104386 2020

[22] B Mahanthesh B J Gireesha R S R Gorla F M Abbasiand S A Shehzad ldquoNumerical solutions for magnetohy-drodynamic flow of nanofluid over a bidirectional non-linearstretching surface with prescribed surface heat flux bound-aryrdquo Journal of Magnetism and Magnetic Materials vol 417pp 189ndash196 2016

[23] B Mahanthesh B J Gireesha and I L Animasaun ldquoEx-ploration of non-linear thermal radiation and suspendednanoparticles effects on mixed convection boundary layerflow of nanoliquids on a melting vertical surfacerdquo Journal ofNanofluids vol 7 no 5 pp 833ndash843 2018

[24] P S Kumar B Mahanthesh B Gireesha and S ShehzadldquoQuadratic convective flow of radiated nano-jeffrey liquidsubject to multiple convective conditions and Cattaneo-Christov double diffusionrdquo Applied Mathematics and Me-chanics vol 39 no 9 pp 1311ndash1326 2018

[25] J Raza F Mebarek-Oudina and A Chamkha ldquoMagneto-hydrodynamic flow of molybdenum disulfide nanofluid in achannel with shape effectsrdquo Multidiscipline Modeling inMaterials and Structures vol 15 no 4 pp 737ndash757 2019

[26] T Hayat B Ahmed F M Abbasi and A Alsaedi ldquoHy-dromagnetic peristalsis of water based nanofluids with tem-perature dependent viscosity a comparative studyrdquo Journal ofMolecular Liquids vol 234 pp 324ndash329 2017

[27] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filledwith nanofluidsrdquo International Journal of Heat and FluidFlow vol 29 no 5 pp 1326ndash1336 2008

[28] M A El Kot and W Abbas ldquoNumerical technique of bloodflow through catheterized arteries with overlapping stenosisrdquoComputer Methods in Biomechanics and Biomedical Engi-neering vol 20 no 1 pp 45ndash58 2017

[29] H A Attia W Abbas A El-Din Abdin andM A M Abdeen ldquoEffects of ion slip and Hall current onunsteady Couette flow of a dusty fluid through porous mediawith heat transferrdquo High Temperature vol 53 no 6pp 891ndash898 2015

[30] H A Attia W Abbas and M A M Abdeen ldquoIon slip effecton unsteady Couette flow of a dusty fluid in the presence of

uniform suction and injection with heat transferrdquo Journal ofthe Brazilian Society of Mechanical Sciences and Engineeringvol 38 no 8 pp 2381ndash2391 2016

[31] H A Attia W Abbas M A M Abdeen and A A M SaidldquoHeat transfer between two parallel porous plates for Couetteflow under pressure gradient and Hall currentrdquo Sadhanavol 40 no 1 pp 183ndash197 2015

[32] M Sheikholeslami A Shafee M Ramzan and Z Li ldquoIn-vestigation of Lorentz forces and radiation impacts onnanofluid treatment in a porous semi annulus via Darcy lawrdquoJournal of Molecular Liquids vol 272 pp 8ndash14 2018

[33] P S Kumar B Gireesha B Mahanthesh and A J Chamkhaldquo)ermal analysis of nanofluid flow containing gyrotacticmicroorganisms in bioconvection and second-order slip withconvective conditionrdquo Journal of ermal Analysis andCalorimetry vol 136 no 5 pp 1947ndash1957 2019

[34] B Mahanthesh F Mabood B Gireesha and R Gorla ldquoEffectsof chemical reaction and partial slip on the three-dimensionalflow of a nanofluid impinging on an exponentially stretchingsurfacerdquo e European Physical Journal Plus vol 132 no 3p 113 2017

[35] B J Gireesha R S R Gorla and B Mahanthesh ldquoEffect ofsuspended nanoparticles on three-dimensional mhd flowheat and mass transfer of radiating eyring-powell fluid over astretching sheetrdquo Journal of Nanofluids vol 4 no 4pp 474ndash484 2015

[36] J-H He ldquoHomotopy perturbation techniquerdquo ComputerMethods in Applied Mechanics and Engineering vol 178no 3-4 pp 257ndash262 1999

[37] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo Interna-tional Journal of Non-Linear Mechanics vol 35 no 1pp 37ndash43 2000

[38] Y Wu and J-H He ldquoHomotopy perturbation method fornonlinear oscillators with coordinate-dependent massrdquo Re-sults in Physics vol 10 pp 270-271 2018

[39] K Das ldquoFlow and heat transfer characteristics of nanofluids ina rotating framerdquo Alexandria Engineering Journal vol 53no 3 pp 757ndash766 2014

[40] J V R Reddy V Sugunamma N Sandeep and C SulochanaldquoInfluence of chemical reaction radiation and rotation onMHD nanofluid flow past a permeable flat plate in porousmediumrdquo Journal of the Nigerian Mathematical Societyvol 35 no 1 pp 48ndash65 2016

12 Mathematical Problems in Engineering

where R (2Ω]f)U20 is the rotation parameter Ha2

μfB20σfρ2fU2

0 is the Hartmann number squared Gr

g]fβlowastTf(Tw minus Tinfin)U3

0 is the Grashof number Gc

g]fβlowastCf(Cw minus Cinfin)U3

0 is the modified Grashof numberM μ2fU

20kρ

2f is the porous medium parameter

Pr CPfμfkf is the Prandtl number Ec U20CPf(Tw minus

Tinfin) is the Eckart number Qs Q]fU20(ρCp)f is the heat

source parameter Qr 16σT3infin3αlowast]f(ρCp)f is the radiation

parameter Du (DmkT(Cw minus Cinfin))(CsCPf]f(Tw minus Tinfin)) isthe Dufour number Brownian parameter NB DBτ(Cw minus Cinfin)]f NT DTτ(Tw minus Tinfin)]fTinfin denotes ther-mophoresis parameter nanoparticles and base fluid heatcapacity ratio is τ ρsCps

ρfCpf Sc ]fD is the Schmidt

number cc kc]fU20 is the chemical reaction parameter and

Sr DT(Tw minus Tinfin)Tinfin(Cw minus Cinfin) denotes the Soret number)e boundary conditions reduce to

t 0 u 0

w 0

θ⟶ 0

Γ⟶ 0

z 0

tgt 0 u 1

w 0

θ 1

Γ 1

z⟶infin

tgt 0 u⟶ 0

w⟶ 0

θ⟶ 0

Γ⟶ 0

(19)

24Engineering InterestPhysicalQuantities For engineeringinterest the local skin friction (Cf) local Nusselt number(Nux) and local Sherwood number (Shx) are defined as[33ndash35]

Cf τw

ρfU20

Nux xqw

kf Tw minus Tinfin( 1113857

Shx xqm

DB Tw minus Tinfin( 1113857

(20)

where τw qw and qm denote the wall shear stress and heatand mass fluxes respectively which are given by

τw μnf

zu

zz1113888 1113889

z0

qw minusknf

zT

zz+ qr1113888 1113889

z0

qm minusDB

zC

zz1113888 1113889

z0

(21)

By substituting equation (14) into equations (20) and(21)

Nux minusknf

kf

Rex 1 + Qr( 1113857θprime(0) (22)

where Rex U0x]f is the local Reynolds number

3 Methodology

Here the homotopy perturbation method that was firstdeveloped by He [36 37] and improved by Wu and He [38]to solve governing equations (15)ndash(18) subject to boundarycondition (19) is used To illustrate the fundamental ideas ofthis method we first consider the time-dependent differ-ential equation as follows

Υ(ζ(λ t)) minus f(λ t) 0 λ isin R (23)

where Υ is the operator ζ(λ t) denotes the unknownfunction f(λ t) is a known function λ are spatial variablest denote temporal independent variables and R is thedomain

)e operator Υ can be decomposed into the linear partL(λ t) and N(λ t) is the remaining part )ereforeequation (23) can be rewritten as

L(λ t) + N(λ t) minus f(λ t) 0 λ isin R (24)

Homotopy traditional technique constructs a homotopyfunction Ψ(λ t p) as follows

H [Ψ(λ t p)] (1 minus p) (LΨ(λ t p)) minus L f0(λ t)( 11138571113858 1113859

+ p [Υ(Ψ(λ t p)) minus f(λ t)] 0

(25)

where p isin [0 1] is an embedding parameter and f0(λ t) isan initial guess approximation of equation (23) which sat-isfies boundary and initial condition

Consequently the solution of equation (23) is expressedas

Ψ((λ t p) 1113944n

i0Ψi(λ t)p

i (26)

Ideal approximation solution can be obtained when p

1 as

ζ((λ t) 1113944n

i0ζ i(λ t) (27)

Table 2 Shape factor for various nanoparticle shapes [32]

Spherical Cylindrical Brick

Shapes of nanoparticles

Shape factor (np) 3 6 37

Mathematical Problems in Engineering 5

According to equation (24) the appropriate initial ar-bitrary guess approximation and linear operators for eachequation are proposed as

u0 y 1 + eminus t

1113872 1113873

v0 y 1 + eminus t

1113872 1113873

θ0 y 1 minus eminus t

1113872 1113873

Γ0 y 1 minus eminus t

1113872 1113873

L(u) z2u

zz2

L(v) z2v

zz2

L(θ) z2θzz2

L(Γ) z2Γzz2

(28)

Now HPM constructed equations which satisfy equa-tion (25) to governing nonlinear differential equations(15)ndash(18) as follows

H(u p) pJ1

(1 minus ϕ)25 L(u)1113888 1113889 +(1 minus p) minuszu

ztminus s

zu

zzminus Rv minus

Ha2J1J31 + m2 (u + mv) + GrJ1J4θ + GcJ1J4Γ minus

MJ1

(1 minus ϕ)25 u1113888 1113889 0

(29)

H(v p) pJ1

(1 minus ϕ)25 L(v)1113888 1113889 +(1 minus p) minuszv

ztminus s

zv

zzminus Ru minus

Ha2J1J3

1 + m2 (v minus mu) minusMJ1

(1 minus ϕ)25 v1113888 1113889 0 (30)

H(θ p) p1Pr

J5J2 minus QrJ21113888 1113889L(θ)1113888 1113889 +(1 minus p)zθzt

+ szθzz

+Ha2Ec

1 + m2J3J2 u2

+ v2

1113872 1113873 + QsJ2θ + DuJ2z2Γzz21113888

+ NBJ2zθzz

zΓzz

+ NTJ2zθzz

1113888 1113889

2

+Ec

(1 minus ϕ)25J2zu

zz1113888 1113889

2

+zv

zz1113888 1113889

2⎛⎝ ⎞⎠⎞⎠ 0

(31)

H(Γ p) p1SC

L(Γ)1113888 1113889 +(1 minus p) minuszΓzt

minus szΓzz

+ ccΓ + Sr

z2θzz21113888 1113889 0 (32)

And assuming the solution of equations (29)ndash(32) it canbe written in the following forms

u(z t) u1(t)z + u2(t)z2

+ u3(t)z3

+ middot middot middot

v(z t) v1(t)z + v2(t)z2

+ v3(t)z3

+ middot middot middot

θ(z t) θ1(t)z + θ2(t)z2

+ θ3(t)z3

+ middot middot middot

Γ(z t) Γ1(t)z + Γ2(t)z2

+ Γ3(t)z3

+ middot middot middot (33)

Substituting equation (26) into equations (27)ndash(31) aftersome simplifications and rearranging depending on powersof p-terms and solved using conditions (19) the followingapproximations can be obtained

6 Mathematical Problems in Engineering

u1(t) 1590J1

eminusαt

(1minus ϕ)52

MJ1 1+ eαt

1113872 1113873 + Ha2J1J4A1 1+ e

αt1113872 1113873 + GcJ1J3 1minus e

αt1113872 1113873 minus α(1minus s)GrJ1J2 1minus e

αt1113872 1113873 + R 1+ e

αt1113872 11138731113872 1113873

u2(t) 323J1

eminusαt

(1minus ϕ)52

MJ1 1+ eαt

1113872 1113873 + Ha2J1J4A1 1+ e

αt1113872 1113873 + GcJ1J3 1minus e

αt1113872 1113873 minus α(1+ s) minus GrJ1J2 1minus e

αt1113872 1113873 + R 1+ e

αt1113872 11138731113872 1113873

u3(t) 1590J1

(1minus ϕ)52

Ha2J1J4(1minus m) minus R minus αminus e

αt1113872 1113873

v1(t) 1590

(1minus ϕ)52

minuseminusαt

(R + αs) + Ha2J4J1

9015

eminusαt

+ m2

minus m3

+ A1eminusαt

1113874 11138751113874 1113875

v2(t) minus323

Ha2J1J4(1minus ϕ)

52+ R(1minus ϕ)

52+ Ha

2J1J4m(1minus ϕ)

52+323αe

minusαt(1minus ϕ)

52minus323

Ha2J1J4e

minusαt(1minus ϕ)

52

+323Reminusαt

(1minus ϕ)52

v3(t) minusMJ1(1minus ϕ)52

+ +Ha2J4m

3e

minusαt(1minus ϕ)

52323

minus 0125

θ1 minus32Pre

minusαt α+ J6Qr 1minus eαt( 1113857( 1113857

3J5J6+4Pre

minus2αt eαt minus 1( 1113857 seαt + DNBJ6 1minus eαt( 1113857( 1113857

J5J6minus256EcHa2J4Pre

minus2αt m2 + 1( 1113857 eαt + 1( 11138572

3J5+18

θ2 Pre

minus2αt eαt minus 1( 1113857 seαt + DNBJ6 1minus eαt( 1113857( 1113857

2J5J6

θ3 Pre

minusαt α+ J6Qr 1minus eαt( 1113857( 1113857

6J5J6

Γ1(t) Sce

minusαt αminus cc 1+ eαt( 1113857( 1113857

6

Γ2(t) 32Sce

minusαt αminus cc 1+ eαt( 1113857( 1113857

3minus 4Sce

minusαt(Sr + s) e

αtminus 11113872 1113873 minus 0125

Γ3(t) Sce

minusαt(Sr + s) eαt minus 1( 1113857

2

(34)

where A1 m3 + m2 + m + 1 and A2 minusm3 + m2 minus m

4 Solution Validation

To check the accuracy and efficiency of the methodology thecurrent results are verified by comparing with previouslypublished results in some special cases which are tabulated inTables 3ndash5 )e present results of different skin frictioncoefficient values Cf and Nusselt number NuRex for dif-ferent values of nanoparticle volume fraction ϕ and rotationparameter R are compared with those reported by Das [39]and Reddy et al [40] with both nanoparticles Cu and Al2O3which are illustrated in Tables 3 and 4 )e results obtainedcome with excellent agreement of previous studies

Furthermore the comparison for different values of theSherwood number minusShx against time t and chemical reac-tion cc with obtained results by Hussain et al [17] at ϕ 00is listed in Table 5 Also it can be seen that the presentobtained results are in an excellent agreement with the

previously published results )is proves the validity of thepresent work and shows how powerful the solution reachedby homotopy perturbation technique

5 Results and Discussion

)e influence of various physical parameters such asnanoparticle volume fraction ϕ shape of nanoparticles npHall current m Brownian NB thermophoretic NT andchemical reaction cc which are computed from the ap-proximate solution reported in the previous section is dis-played graphically and discussed )e graphical results areillustrated in Figures 2ndash11 )e numerical results are per-formed by taking values of some constant parameters as s

05 M 05 Pr 07 Ec 02 Gc 1 Gr 1 Sc 03

Du 05 G 5 and Sr 05Figures 2 and 3 display the effect of nanoparticle

volume fraction parameter ϕ on the fluid velocity u andtemperature profiles θ for either Cu minus water Al2O3 minus water

Mathematical Problems in Engineering 7

Table 3 Result comparison for Cf

Cf

ϕ RCuminuswater Al2O3minuswater

Present Das [39] Reddy et al [40] Present Das [39]

000 03 20355 20354 203542 20355 20354004 23158 23154 231535 05134 21831008 26111 26107 261068 21832 23384012 29120 29196 291953 25030 25020014 30792 30789 307891 10885 10872005 00 23790 23786 237862 22144 22143

05 24042 24040 240401 22333 2233315 25755 25755 257546 23669 2366625 28236 28235 28251 25697 25695

Table 4 Result comparison for NuxRex

NuxRex

ϕ RCuminuswater Al2O3minuswater

Present Das [39] Reddy et al [40] Present Das [39]

000 03 04607 04607 04607 04607 04607004 05135 05134 05134 05115 05114008 05695 05698 05698 05656 05655012 06304 06303 06303 06234 06233014 06622 06622 06622 06536 06537005 00 05272 05271 05271 05271 05271

05 mdash mdash mdash mdash mdash15 mdash mdash mdash mdash mdash25 05271 05271 05271 05272 05271

Table 5 Result comparison for minusShx

minusShx

ccdarr t⟶ 03 05 07 03 05 07Present Hussain et al [17]

02 0535254 0438632 0389005 054907 044765 0386382 0849652 0815633 0767225 085729 0820922 0781565 12252 116674 114632 124260 116967 1143922

ϕ = 0 007 01

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

Cu-waterAI2O3-waterTiO2-water

Figure 2 )e effect of volume fraction of nanoparticles on the velocity profile

8 Mathematical Problems in Engineering

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θϕ = 0 007 012

1

08

06

04

02

00 2 3 4 5 6 7 81

Cu-waterAI2O3-waterTiO2-water

z

Figure 3 )e effect of volume fraction nanoparticle on thetemperature profile

Cylindrical shape

Brick shape

Spherical shape

1

08

06

04

02

00 2 3 4 5 6 7 81

z

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θ

Cu-waterAI2O3-waterTiO2-water

Figure 4 )e effect of nanoparticle shape on the temperatureprofile

R = 1 2 4

Cu-waterTiO2-water

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

Figure 5 )e effect of rotation parameter on the velocity profile

m = 1 15 2

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 6 )e effect of Hall current on the velocity profile

NB = 02 05 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07m = 1γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 7 )e effect of Brownian motion parameter on the tem-perature profile

ϕ = 005NB = 07m = 1γc = 05Qs = 1Qr = 09R = 1

NT = 03 06 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 8 )e effect of thermophoresis parameter on the tem-perature profile

Mathematical Problems in Engineering 9

and TiO2 minus water nanofluid It is clear that increase innanoparticle volume fraction parameter ϕ decreases the fluidvelocity but the temperature profiles increase It is due to thefact that nanofluid density increases with increasing thevolume fraction of nanoparticles and it causes to slow downthe nanofluid flow velocity and increases the thickness of thethermal boundary layer which improves thermal conduc-tivity and increases surface heat transfer rate Also it isobserved that nanofluid Cu minus water heat transfer rate isgreater than Al2O3 and TiO2 rates So nanoparticles havinghigh thermal conductivity enhance heat transfer more thanthe particles having low thermal conductivity On thecontrary it is seen from Tables 2 and 3 that wall skin frictioncoefficient Cf and the Nusselt number NuRex increase withincreasing in nanoparticle volume fraction for both Cu minus

water and Al2O3 minus water)e influence of the nanoparticle shape on the tem-

perature distribution of nanofluid is studied through Fig-ure 4 )is figure indicates that the heat transfer rate ishigher for the cylindrical nanoparticle shape than thespherical and brick nanoparticle shapes for eitherCu minus water Al2O3 minus water and TiO2 minus water nanofluid)erefore the spherical nanoparticle shape tends to dragmore heat from the boundary layer because it has a greatersurface area while this effect is less evident for othernanoparticle shapes )us the use of the cylindrical nano-particle shape has better improvement in effective thermalconductivity when compared with other nanoparticle shapes(spherical and brick)

)e influence of rotation parameter R on the velocityprofiles against z is displayed in Figure 5 It observed that anincrease in rotation parameter leads to decreasing theboundary layer momentum so it decreases the values ofnanofluid velocity From Tables 2 and 3 it can be noted thatthe wall skin friction coefficient Cf increases with an in-crease in the rotation parameter but the Nusselt numberNuRex remains unchanged with both Cu minus water andAl2O3 minus water

Figure 6 presents the influence of Hall current parameterm on main velocity u It is clear that the velocity distributionof nanofluid reduces with an increase in Hall current pa-rameter)is happens naturally due to the fact that when themagnetic field is strengthened it enhances the opposingforce which is represented by the well-known Lorentz force

Figures 7 and 8 show the Brownian and thermophoresiseffects on the temperature profile θ It is seen that thetemperature distribution increases with increasing for theseparameters )is is due to the fact that the thermophoreticand Brownian parameters assist to improve the thermalboundary layer thickness Also it is observed that the heattransfer rate of Cu minus water nanofluid is greater thanTiO2 minus water

Figure 9 depicts the influence of chemical parameter cc

on the concentration profile Γ It is noticed that the increasein concentration profile is accompanied by a decrease in thechemical parameter On the contrary it is seen from Table 4that the Sherwood number minusShx increases due to increasingthe chemical parameter but the Sherwood number decreaseswhile increasing the time t )is implies that the chemical

Cu-waterTiO2-water

ϕ = 005NT = 07NB = 07m = 1

γc = 01 05 1

Qs = 1Qr = 09R = 1

0 2 3 4 5 6 7 81z

1

08

06

04

02

0

12

14

Γ

Figure 9 )e effect of chemical reaction parameter on the con-centration profile

Cu-waterTiO2-water

ϕ = 005

NB = 07

m = 1γc = 05

Qs = 1 2 3

Qr = 09R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

NT = 07

Figure 10 )e effect of heat source parameter on the temperatureprofile

ϕ = 005

NB = 07NT = 07

m = 1

γc = 05Qs = 1

Qr = 05 09 13

R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 11 )e effect of radiation parameter on the temperatureprofile

10 Mathematical Problems in Engineering

reaction has tendency to increase the mass transfer rate atthe surface plate and it is reduced with the progress of time

Figures 10 and 11 present the heat source parameter Qs

and thermal radiation parameter Qr effects on the tem-perature profiles It is clear from Figure 10 that with anincrease in the heat generation parameter the boundarylayer temperature decreases and as a consequence to de-crease the thermal boundary layer thickness In addition it isclear from Figure 11 that with an increase in the thermalradiation the temperature profiles of the nanofluid increaseIt is well known that the increase in the radiation parameterenhances the energy transport inside the fluid and as aconsequence to increase the heat transfer rate

6 Conclusion

An analysis is done to solve the magnetohydro dynamicsunsteady nanofluid flow through the porous medium past amoving plate in a rotating frame with considering the heatand mass transfer )ree different types of the nanoparticlesnamely copper aluminium oxide and titanium dioxide andwater used as a base nanofluid are considered with variousshapes (spherical cylindrical and brick) )e important ef-fects of nanoparticle shape Brownian and thermophoresisdiffusion thermal radiation ohmic and viscous dissipationshave been included Homotopy perturbation method is ap-plied to solve the governing nonlinear partial differentialequations )e efficiency of the methodology has been vali-dated through comparison with the previously publishedresults and observed a very close agreement It is found that

)e main velocity component decreases with increaseof nanoparticle volume fraction On the contrarythermal properties increase with increase in nano-particle volume fraction)e enhancement of heat transfer rate and mechanicalproperty improvement is high in copper as comparedwith both aluminium oxide and titanium dioxide)ermal conductivity of cylindrical-shaped nanoparticlesis more efficient and possesses better improvement onheat transfer rate than the other shaped onesIncrease in the Brownian and thermophoresis pa-rameter is accompanied by an increase in the thermalboundary layer thicknessIncreasing Hall current and rotation parameters de-creases the main fluid velocity componentHPM has a close agreement and is recommended formany nonlinear problems

Data Availability

)e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] T Hayat R Sajjad A Alsaedi T Muhammad and R EllahildquoOn squeezed flow of couple stress nanofluid between twoparallel platesrdquo Results in Physics vol 7 pp 553ndash561 2017

[2] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition pp 99ndash105San Francisco CA USA 1995

[3] J A Eastman S U S Choi S Li W Yu and L J )ompsonldquoAnomalously increased effective thermal conductivities of eth-ylene glycol-based nanofluids containing copper nanoparticlesrdquoApplied Physics Letters vol 78 no 6 pp 718ndash720 2001

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

[5] M Sheikholeslami M M Rashidi T Hayat and D D GanjildquoFree convection of magnetic nanofluid considering MFDviscosity effectrdquo Journal of Molecular Liquids vol 218pp 393ndash399 2016

[6] W Abbas and E A Sayed ldquoHall current and joule heatingeffects on free convection flow of a nanofluid over a verticalcone in presence of thermal radiationrdquo ermal Sciencevol 21 pp 2603ndash2614 2017

[7] B Mahanthesh B Gireesha I Animasaun T Muhammadand N Shashikumar ldquoMHD flow of SWCNT and MWCNTnanoliquids past a rotating stretchable disk with thermal andexponential space dependent heat sourcerdquo Physica Scriptavol 94 no 8 Article ID 085214 2019

[8] I L Animasaun O K Koriko K S Adegbie et al ldquoCom-parative analysis between 36 nm and 47 nm alumina-waternanofluid flows in the presence of Hall effectrdquo Journal ofermal Analysis and Calorimetry vol 135 no 2 pp 873ndash886 2019

[9] I Ullah S Shafie I Khan and K L Hsiao ldquoBrownian dif-fusion and thermophoresis mechanisms in Casson fluid over amoving wedgerdquo Results in Physics vol 9 pp 183ndash194 2018

[10] P B S Kumar B J Gireesha R S R Gorla andB Mahanthesh ldquoMagnetohydrodynamic flow of Williamsonnanofluid due to an exponentially stretching surface in thepresence of thermal radiation and chemical reactionrdquo Journalof Nanofluids vol 6 no 2 pp 264ndash272 2017

[11] X Zhou Y Jiang X Li et al ldquoNumerical investigation of heattransfer enhancement and entropy generation of naturalconvection in a cavity containing nano liquid-metal fluidrdquoInternational Communications in Heat and Mass Transfervol 106 pp 46ndash54 2019

[12] N Shehzad A Zeeshan R Ellahi and K Vafai ldquoConvectiveheat transfer of nanofluid in a wavy channel Buongiornorsquosmathematical modelrdquo Journal of Molecular Liquids vol 222pp 446ndash455 2016

[13] B Shen L Zheng C Zhang and X Zhang ldquoBioconvectionheat transfer of a nanofluid over a stretching sheet with ve-locity slip and temperature jumprdquo ermal Science vol 21no 6 pp 2347ndash2356 2017

[14] S Jahan H Sakidin R Nazar and I Pop ldquoAnalysis of heattransfer in nanofluid past a convectively heated permeablestretchingshrinking sheet with regression and stability ana-lysesrdquo Results in Physics vol 10 pp 395ndash405 2018

[15] M A A Hamad and I Pop ldquoUnsteady MHD free convectionflow past a vertical permeable flat plate in a rotating frame ofreference with constant heat source in a nanofluidrdquo Heat andMass Transfer vol 47 no 12 pp 1517ndash1524 2011

Mathematical Problems in Engineering 11

[16] M Sheikholeslami and D D Ganji ldquo)ree dimensional heatand mass transfer in a rotating system using nanofluidrdquoPowder Technology vol 253 pp 789ndash796 2014

[17] S M Hussain J Jain G S Seth and M M Rashidi ldquoFreeconvective heat transfer with Hall effects heat absorption andchemical reaction over an accelerated moving plate in a ro-tating systemrdquo Journal of Magnetism and Magnetic Materialsvol 422 pp 112ndash123 2017

[18] X Zhou Y Jiang Y Hou and M Du ldquo)ermocapillaryconvection instability in annular two-layer system undervarious gravity levelsrdquo Microgravity Science and Technologyvol 31 no 5 pp 641ndash648 2019

[19] Y Jiang and X Zhou ldquoHeat transfer and entropy generationanalysis of nanofluids thermocapillary convection around abubble in a cavityrdquo International Communications in Heatand Mass Transfer vol 105 pp 37ndash45 2019

[20] Y Jiang X Zhou and Y Wang ldquoEffects of nanoparticleshapes on heat andmass transfer of nanofluid thermocapillaryconvection around a gas bubblerdquo Microgravity Science andTechnology pp 1ndash11 2019 In press

[21] Y Jiang X Zhou and Y Wang ldquoComprehensive heattransfer performance analysis of nanofluid mixed forced andthermocapillary convection around a gas bubble in mini-channelrdquo International Communications in Heat and MassTransfer vol 110 Article ID 104386 2020

[22] B Mahanthesh B J Gireesha R S R Gorla F M Abbasiand S A Shehzad ldquoNumerical solutions for magnetohy-drodynamic flow of nanofluid over a bidirectional non-linearstretching surface with prescribed surface heat flux bound-aryrdquo Journal of Magnetism and Magnetic Materials vol 417pp 189ndash196 2016

[23] B Mahanthesh B J Gireesha and I L Animasaun ldquoEx-ploration of non-linear thermal radiation and suspendednanoparticles effects on mixed convection boundary layerflow of nanoliquids on a melting vertical surfacerdquo Journal ofNanofluids vol 7 no 5 pp 833ndash843 2018

[24] P S Kumar B Mahanthesh B Gireesha and S ShehzadldquoQuadratic convective flow of radiated nano-jeffrey liquidsubject to multiple convective conditions and Cattaneo-Christov double diffusionrdquo Applied Mathematics and Me-chanics vol 39 no 9 pp 1311ndash1326 2018

[25] J Raza F Mebarek-Oudina and A Chamkha ldquoMagneto-hydrodynamic flow of molybdenum disulfide nanofluid in achannel with shape effectsrdquo Multidiscipline Modeling inMaterials and Structures vol 15 no 4 pp 737ndash757 2019

[26] T Hayat B Ahmed F M Abbasi and A Alsaedi ldquoHy-dromagnetic peristalsis of water based nanofluids with tem-perature dependent viscosity a comparative studyrdquo Journal ofMolecular Liquids vol 234 pp 324ndash329 2017

[27] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filledwith nanofluidsrdquo International Journal of Heat and FluidFlow vol 29 no 5 pp 1326ndash1336 2008

[28] M A El Kot and W Abbas ldquoNumerical technique of bloodflow through catheterized arteries with overlapping stenosisrdquoComputer Methods in Biomechanics and Biomedical Engi-neering vol 20 no 1 pp 45ndash58 2017

[29] H A Attia W Abbas A El-Din Abdin andM A M Abdeen ldquoEffects of ion slip and Hall current onunsteady Couette flow of a dusty fluid through porous mediawith heat transferrdquo High Temperature vol 53 no 6pp 891ndash898 2015

[30] H A Attia W Abbas and M A M Abdeen ldquoIon slip effecton unsteady Couette flow of a dusty fluid in the presence of

uniform suction and injection with heat transferrdquo Journal ofthe Brazilian Society of Mechanical Sciences and Engineeringvol 38 no 8 pp 2381ndash2391 2016

[31] H A Attia W Abbas M A M Abdeen and A A M SaidldquoHeat transfer between two parallel porous plates for Couetteflow under pressure gradient and Hall currentrdquo Sadhanavol 40 no 1 pp 183ndash197 2015

[32] M Sheikholeslami A Shafee M Ramzan and Z Li ldquoIn-vestigation of Lorentz forces and radiation impacts onnanofluid treatment in a porous semi annulus via Darcy lawrdquoJournal of Molecular Liquids vol 272 pp 8ndash14 2018

[33] P S Kumar B Gireesha B Mahanthesh and A J Chamkhaldquo)ermal analysis of nanofluid flow containing gyrotacticmicroorganisms in bioconvection and second-order slip withconvective conditionrdquo Journal of ermal Analysis andCalorimetry vol 136 no 5 pp 1947ndash1957 2019

[34] B Mahanthesh F Mabood B Gireesha and R Gorla ldquoEffectsof chemical reaction and partial slip on the three-dimensionalflow of a nanofluid impinging on an exponentially stretchingsurfacerdquo e European Physical Journal Plus vol 132 no 3p 113 2017

[35] B J Gireesha R S R Gorla and B Mahanthesh ldquoEffect ofsuspended nanoparticles on three-dimensional mhd flowheat and mass transfer of radiating eyring-powell fluid over astretching sheetrdquo Journal of Nanofluids vol 4 no 4pp 474ndash484 2015

[36] J-H He ldquoHomotopy perturbation techniquerdquo ComputerMethods in Applied Mechanics and Engineering vol 178no 3-4 pp 257ndash262 1999

[37] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo Interna-tional Journal of Non-Linear Mechanics vol 35 no 1pp 37ndash43 2000

[38] Y Wu and J-H He ldquoHomotopy perturbation method fornonlinear oscillators with coordinate-dependent massrdquo Re-sults in Physics vol 10 pp 270-271 2018

[39] K Das ldquoFlow and heat transfer characteristics of nanofluids ina rotating framerdquo Alexandria Engineering Journal vol 53no 3 pp 757ndash766 2014

[40] J V R Reddy V Sugunamma N Sandeep and C SulochanaldquoInfluence of chemical reaction radiation and rotation onMHD nanofluid flow past a permeable flat plate in porousmediumrdquo Journal of the Nigerian Mathematical Societyvol 35 no 1 pp 48ndash65 2016

12 Mathematical Problems in Engineering

According to equation (24) the appropriate initial ar-bitrary guess approximation and linear operators for eachequation are proposed as

u0 y 1 + eminus t

1113872 1113873

v0 y 1 + eminus t

1113872 1113873

θ0 y 1 minus eminus t

1113872 1113873

Γ0 y 1 minus eminus t

1113872 1113873

L(u) z2u

zz2

L(v) z2v

zz2

L(θ) z2θzz2

L(Γ) z2Γzz2

(28)

Now HPM constructed equations which satisfy equa-tion (25) to governing nonlinear differential equations(15)ndash(18) as follows

H(u p) pJ1

(1 minus ϕ)25 L(u)1113888 1113889 +(1 minus p) minuszu

ztminus s

zu

zzminus Rv minus

Ha2J1J31 + m2 (u + mv) + GrJ1J4θ + GcJ1J4Γ minus

MJ1

(1 minus ϕ)25 u1113888 1113889 0

(29)

H(v p) pJ1

(1 minus ϕ)25 L(v)1113888 1113889 +(1 minus p) minuszv

ztminus s

zv

zzminus Ru minus

Ha2J1J3

1 + m2 (v minus mu) minusMJ1

(1 minus ϕ)25 v1113888 1113889 0 (30)

H(θ p) p1Pr

J5J2 minus QrJ21113888 1113889L(θ)1113888 1113889 +(1 minus p)zθzt

+ szθzz

+Ha2Ec

1 + m2J3J2 u2

+ v2

1113872 1113873 + QsJ2θ + DuJ2z2Γzz21113888

+ NBJ2zθzz

zΓzz

+ NTJ2zθzz

1113888 1113889

2

+Ec

(1 minus ϕ)25J2zu

zz1113888 1113889

2

+zv

zz1113888 1113889

2⎛⎝ ⎞⎠⎞⎠ 0

(31)

H(Γ p) p1SC

L(Γ)1113888 1113889 +(1 minus p) minuszΓzt

minus szΓzz

+ ccΓ + Sr

z2θzz21113888 1113889 0 (32)

And assuming the solution of equations (29)ndash(32) it canbe written in the following forms

u(z t) u1(t)z + u2(t)z2

+ u3(t)z3

+ middot middot middot

v(z t) v1(t)z + v2(t)z2

+ v3(t)z3

+ middot middot middot

θ(z t) θ1(t)z + θ2(t)z2

+ θ3(t)z3

+ middot middot middot

Γ(z t) Γ1(t)z + Γ2(t)z2

+ Γ3(t)z3

+ middot middot middot (33)

Substituting equation (26) into equations (27)ndash(31) aftersome simplifications and rearranging depending on powersof p-terms and solved using conditions (19) the followingapproximations can be obtained

6 Mathematical Problems in Engineering

u1(t) 1590J1

eminusαt

(1minus ϕ)52

MJ1 1+ eαt

1113872 1113873 + Ha2J1J4A1 1+ e

αt1113872 1113873 + GcJ1J3 1minus e

αt1113872 1113873 minus α(1minus s)GrJ1J2 1minus e

αt1113872 1113873 + R 1+ e

αt1113872 11138731113872 1113873

u2(t) 323J1

eminusαt

(1minus ϕ)52

MJ1 1+ eαt

1113872 1113873 + Ha2J1J4A1 1+ e

αt1113872 1113873 + GcJ1J3 1minus e

αt1113872 1113873 minus α(1+ s) minus GrJ1J2 1minus e

αt1113872 1113873 + R 1+ e

αt1113872 11138731113872 1113873

u3(t) 1590J1

(1minus ϕ)52

Ha2J1J4(1minus m) minus R minus αminus e

αt1113872 1113873

v1(t) 1590

(1minus ϕ)52

minuseminusαt

(R + αs) + Ha2J4J1

9015

eminusαt

+ m2

minus m3

+ A1eminusαt

1113874 11138751113874 1113875

v2(t) minus323

Ha2J1J4(1minus ϕ)

52+ R(1minus ϕ)

52+ Ha

2J1J4m(1minus ϕ)

52+323αe

minusαt(1minus ϕ)

52minus323

Ha2J1J4e

minusαt(1minus ϕ)

52

+323Reminusαt

(1minus ϕ)52

v3(t) minusMJ1(1minus ϕ)52

+ +Ha2J4m

3e

minusαt(1minus ϕ)

52323

minus 0125

θ1 minus32Pre

minusαt α+ J6Qr 1minus eαt( 1113857( 1113857

3J5J6+4Pre

minus2αt eαt minus 1( 1113857 seαt + DNBJ6 1minus eαt( 1113857( 1113857

J5J6minus256EcHa2J4Pre

minus2αt m2 + 1( 1113857 eαt + 1( 11138572

3J5+18

θ2 Pre

minus2αt eαt minus 1( 1113857 seαt + DNBJ6 1minus eαt( 1113857( 1113857

2J5J6

θ3 Pre

minusαt α+ J6Qr 1minus eαt( 1113857( 1113857

6J5J6

Γ1(t) Sce

minusαt αminus cc 1+ eαt( 1113857( 1113857

6

Γ2(t) 32Sce

minusαt αminus cc 1+ eαt( 1113857( 1113857

3minus 4Sce

minusαt(Sr + s) e

αtminus 11113872 1113873 minus 0125

Γ3(t) Sce

minusαt(Sr + s) eαt minus 1( 1113857

2

(34)

where A1 m3 + m2 + m + 1 and A2 minusm3 + m2 minus m

4 Solution Validation

To check the accuracy and efficiency of the methodology thecurrent results are verified by comparing with previouslypublished results in some special cases which are tabulated inTables 3ndash5 )e present results of different skin frictioncoefficient values Cf and Nusselt number NuRex for dif-ferent values of nanoparticle volume fraction ϕ and rotationparameter R are compared with those reported by Das [39]and Reddy et al [40] with both nanoparticles Cu and Al2O3which are illustrated in Tables 3 and 4 )e results obtainedcome with excellent agreement of previous studies

Furthermore the comparison for different values of theSherwood number minusShx against time t and chemical reac-tion cc with obtained results by Hussain et al [17] at ϕ 00is listed in Table 5 Also it can be seen that the presentobtained results are in an excellent agreement with the

previously published results )is proves the validity of thepresent work and shows how powerful the solution reachedby homotopy perturbation technique

5 Results and Discussion

)e influence of various physical parameters such asnanoparticle volume fraction ϕ shape of nanoparticles npHall current m Brownian NB thermophoretic NT andchemical reaction cc which are computed from the ap-proximate solution reported in the previous section is dis-played graphically and discussed )e graphical results areillustrated in Figures 2ndash11 )e numerical results are per-formed by taking values of some constant parameters as s

05 M 05 Pr 07 Ec 02 Gc 1 Gr 1 Sc 03

Du 05 G 5 and Sr 05Figures 2 and 3 display the effect of nanoparticle

volume fraction parameter ϕ on the fluid velocity u andtemperature profiles θ for either Cu minus water Al2O3 minus water

Mathematical Problems in Engineering 7

Table 3 Result comparison for Cf

Cf

ϕ RCuminuswater Al2O3minuswater

Present Das [39] Reddy et al [40] Present Das [39]

000 03 20355 20354 203542 20355 20354004 23158 23154 231535 05134 21831008 26111 26107 261068 21832 23384012 29120 29196 291953 25030 25020014 30792 30789 307891 10885 10872005 00 23790 23786 237862 22144 22143

05 24042 24040 240401 22333 2233315 25755 25755 257546 23669 2366625 28236 28235 28251 25697 25695

Table 4 Result comparison for NuxRex

NuxRex

ϕ RCuminuswater Al2O3minuswater

Present Das [39] Reddy et al [40] Present Das [39]

000 03 04607 04607 04607 04607 04607004 05135 05134 05134 05115 05114008 05695 05698 05698 05656 05655012 06304 06303 06303 06234 06233014 06622 06622 06622 06536 06537005 00 05272 05271 05271 05271 05271

05 mdash mdash mdash mdash mdash15 mdash mdash mdash mdash mdash25 05271 05271 05271 05272 05271

Table 5 Result comparison for minusShx

minusShx

ccdarr t⟶ 03 05 07 03 05 07Present Hussain et al [17]

02 0535254 0438632 0389005 054907 044765 0386382 0849652 0815633 0767225 085729 0820922 0781565 12252 116674 114632 124260 116967 1143922

ϕ = 0 007 01

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

Cu-waterAI2O3-waterTiO2-water

Figure 2 )e effect of volume fraction of nanoparticles on the velocity profile

8 Mathematical Problems in Engineering

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θϕ = 0 007 012

1

08

06

04

02

00 2 3 4 5 6 7 81

Cu-waterAI2O3-waterTiO2-water

z

Figure 3 )e effect of volume fraction nanoparticle on thetemperature profile

Cylindrical shape

Brick shape

Spherical shape

1

08

06

04

02

00 2 3 4 5 6 7 81

z

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θ

Cu-waterAI2O3-waterTiO2-water

Figure 4 )e effect of nanoparticle shape on the temperatureprofile

R = 1 2 4

Cu-waterTiO2-water

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

Figure 5 )e effect of rotation parameter on the velocity profile

m = 1 15 2

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 6 )e effect of Hall current on the velocity profile

NB = 02 05 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07m = 1γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 7 )e effect of Brownian motion parameter on the tem-perature profile

ϕ = 005NB = 07m = 1γc = 05Qs = 1Qr = 09R = 1

NT = 03 06 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 8 )e effect of thermophoresis parameter on the tem-perature profile

Mathematical Problems in Engineering 9

and TiO2 minus water nanofluid It is clear that increase innanoparticle volume fraction parameter ϕ decreases the fluidvelocity but the temperature profiles increase It is due to thefact that nanofluid density increases with increasing thevolume fraction of nanoparticles and it causes to slow downthe nanofluid flow velocity and increases the thickness of thethermal boundary layer which improves thermal conduc-tivity and increases surface heat transfer rate Also it isobserved that nanofluid Cu minus water heat transfer rate isgreater than Al2O3 and TiO2 rates So nanoparticles havinghigh thermal conductivity enhance heat transfer more thanthe particles having low thermal conductivity On thecontrary it is seen from Tables 2 and 3 that wall skin frictioncoefficient Cf and the Nusselt number NuRex increase withincreasing in nanoparticle volume fraction for both Cu minus

water and Al2O3 minus water)e influence of the nanoparticle shape on the tem-

perature distribution of nanofluid is studied through Fig-ure 4 )is figure indicates that the heat transfer rate ishigher for the cylindrical nanoparticle shape than thespherical and brick nanoparticle shapes for eitherCu minus water Al2O3 minus water and TiO2 minus water nanofluid)erefore the spherical nanoparticle shape tends to dragmore heat from the boundary layer because it has a greatersurface area while this effect is less evident for othernanoparticle shapes )us the use of the cylindrical nano-particle shape has better improvement in effective thermalconductivity when compared with other nanoparticle shapes(spherical and brick)

)e influence of rotation parameter R on the velocityprofiles against z is displayed in Figure 5 It observed that anincrease in rotation parameter leads to decreasing theboundary layer momentum so it decreases the values ofnanofluid velocity From Tables 2 and 3 it can be noted thatthe wall skin friction coefficient Cf increases with an in-crease in the rotation parameter but the Nusselt numberNuRex remains unchanged with both Cu minus water andAl2O3 minus water

Figure 6 presents the influence of Hall current parameterm on main velocity u It is clear that the velocity distributionof nanofluid reduces with an increase in Hall current pa-rameter)is happens naturally due to the fact that when themagnetic field is strengthened it enhances the opposingforce which is represented by the well-known Lorentz force

Figures 7 and 8 show the Brownian and thermophoresiseffects on the temperature profile θ It is seen that thetemperature distribution increases with increasing for theseparameters )is is due to the fact that the thermophoreticand Brownian parameters assist to improve the thermalboundary layer thickness Also it is observed that the heattransfer rate of Cu minus water nanofluid is greater thanTiO2 minus water

Figure 9 depicts the influence of chemical parameter cc

on the concentration profile Γ It is noticed that the increasein concentration profile is accompanied by a decrease in thechemical parameter On the contrary it is seen from Table 4that the Sherwood number minusShx increases due to increasingthe chemical parameter but the Sherwood number decreaseswhile increasing the time t )is implies that the chemical

Cu-waterTiO2-water

ϕ = 005NT = 07NB = 07m = 1

γc = 01 05 1

Qs = 1Qr = 09R = 1

0 2 3 4 5 6 7 81z

1

08

06

04

02

0

12

14

Γ

Figure 9 )e effect of chemical reaction parameter on the con-centration profile

Cu-waterTiO2-water

ϕ = 005

NB = 07

m = 1γc = 05

Qs = 1 2 3

Qr = 09R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

NT = 07

Figure 10 )e effect of heat source parameter on the temperatureprofile

ϕ = 005

NB = 07NT = 07

m = 1

γc = 05Qs = 1

Qr = 05 09 13

R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 11 )e effect of radiation parameter on the temperatureprofile

10 Mathematical Problems in Engineering

reaction has tendency to increase the mass transfer rate atthe surface plate and it is reduced with the progress of time

Figures 10 and 11 present the heat source parameter Qs

and thermal radiation parameter Qr effects on the tem-perature profiles It is clear from Figure 10 that with anincrease in the heat generation parameter the boundarylayer temperature decreases and as a consequence to de-crease the thermal boundary layer thickness In addition it isclear from Figure 11 that with an increase in the thermalradiation the temperature profiles of the nanofluid increaseIt is well known that the increase in the radiation parameterenhances the energy transport inside the fluid and as aconsequence to increase the heat transfer rate

6 Conclusion

An analysis is done to solve the magnetohydro dynamicsunsteady nanofluid flow through the porous medium past amoving plate in a rotating frame with considering the heatand mass transfer )ree different types of the nanoparticlesnamely copper aluminium oxide and titanium dioxide andwater used as a base nanofluid are considered with variousshapes (spherical cylindrical and brick) )e important ef-fects of nanoparticle shape Brownian and thermophoresisdiffusion thermal radiation ohmic and viscous dissipationshave been included Homotopy perturbation method is ap-plied to solve the governing nonlinear partial differentialequations )e efficiency of the methodology has been vali-dated through comparison with the previously publishedresults and observed a very close agreement It is found that

)e main velocity component decreases with increaseof nanoparticle volume fraction On the contrarythermal properties increase with increase in nano-particle volume fraction)e enhancement of heat transfer rate and mechanicalproperty improvement is high in copper as comparedwith both aluminium oxide and titanium dioxide)ermal conductivity of cylindrical-shaped nanoparticlesis more efficient and possesses better improvement onheat transfer rate than the other shaped onesIncrease in the Brownian and thermophoresis pa-rameter is accompanied by an increase in the thermalboundary layer thicknessIncreasing Hall current and rotation parameters de-creases the main fluid velocity componentHPM has a close agreement and is recommended formany nonlinear problems

Data Availability

)e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] T Hayat R Sajjad A Alsaedi T Muhammad and R EllahildquoOn squeezed flow of couple stress nanofluid between twoparallel platesrdquo Results in Physics vol 7 pp 553ndash561 2017

[2] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition pp 99ndash105San Francisco CA USA 1995

[3] J A Eastman S U S Choi S Li W Yu and L J )ompsonldquoAnomalously increased effective thermal conductivities of eth-ylene glycol-based nanofluids containing copper nanoparticlesrdquoApplied Physics Letters vol 78 no 6 pp 718ndash720 2001

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

[5] M Sheikholeslami M M Rashidi T Hayat and D D GanjildquoFree convection of magnetic nanofluid considering MFDviscosity effectrdquo Journal of Molecular Liquids vol 218pp 393ndash399 2016

[6] W Abbas and E A Sayed ldquoHall current and joule heatingeffects on free convection flow of a nanofluid over a verticalcone in presence of thermal radiationrdquo ermal Sciencevol 21 pp 2603ndash2614 2017

[7] B Mahanthesh B Gireesha I Animasaun T Muhammadand N Shashikumar ldquoMHD flow of SWCNT and MWCNTnanoliquids past a rotating stretchable disk with thermal andexponential space dependent heat sourcerdquo Physica Scriptavol 94 no 8 Article ID 085214 2019

[8] I L Animasaun O K Koriko K S Adegbie et al ldquoCom-parative analysis between 36 nm and 47 nm alumina-waternanofluid flows in the presence of Hall effectrdquo Journal ofermal Analysis and Calorimetry vol 135 no 2 pp 873ndash886 2019

[9] I Ullah S Shafie I Khan and K L Hsiao ldquoBrownian dif-fusion and thermophoresis mechanisms in Casson fluid over amoving wedgerdquo Results in Physics vol 9 pp 183ndash194 2018

[10] P B S Kumar B J Gireesha R S R Gorla andB Mahanthesh ldquoMagnetohydrodynamic flow of Williamsonnanofluid due to an exponentially stretching surface in thepresence of thermal radiation and chemical reactionrdquo Journalof Nanofluids vol 6 no 2 pp 264ndash272 2017

[11] X Zhou Y Jiang X Li et al ldquoNumerical investigation of heattransfer enhancement and entropy generation of naturalconvection in a cavity containing nano liquid-metal fluidrdquoInternational Communications in Heat and Mass Transfervol 106 pp 46ndash54 2019

[12] N Shehzad A Zeeshan R Ellahi and K Vafai ldquoConvectiveheat transfer of nanofluid in a wavy channel Buongiornorsquosmathematical modelrdquo Journal of Molecular Liquids vol 222pp 446ndash455 2016

[13] B Shen L Zheng C Zhang and X Zhang ldquoBioconvectionheat transfer of a nanofluid over a stretching sheet with ve-locity slip and temperature jumprdquo ermal Science vol 21no 6 pp 2347ndash2356 2017

[14] S Jahan H Sakidin R Nazar and I Pop ldquoAnalysis of heattransfer in nanofluid past a convectively heated permeablestretchingshrinking sheet with regression and stability ana-lysesrdquo Results in Physics vol 10 pp 395ndash405 2018

[15] M A A Hamad and I Pop ldquoUnsteady MHD free convectionflow past a vertical permeable flat plate in a rotating frame ofreference with constant heat source in a nanofluidrdquo Heat andMass Transfer vol 47 no 12 pp 1517ndash1524 2011

Mathematical Problems in Engineering 11

[16] M Sheikholeslami and D D Ganji ldquo)ree dimensional heatand mass transfer in a rotating system using nanofluidrdquoPowder Technology vol 253 pp 789ndash796 2014

[17] S M Hussain J Jain G S Seth and M M Rashidi ldquoFreeconvective heat transfer with Hall effects heat absorption andchemical reaction over an accelerated moving plate in a ro-tating systemrdquo Journal of Magnetism and Magnetic Materialsvol 422 pp 112ndash123 2017

[18] X Zhou Y Jiang Y Hou and M Du ldquo)ermocapillaryconvection instability in annular two-layer system undervarious gravity levelsrdquo Microgravity Science and Technologyvol 31 no 5 pp 641ndash648 2019

[19] Y Jiang and X Zhou ldquoHeat transfer and entropy generationanalysis of nanofluids thermocapillary convection around abubble in a cavityrdquo International Communications in Heatand Mass Transfer vol 105 pp 37ndash45 2019

[20] Y Jiang X Zhou and Y Wang ldquoEffects of nanoparticleshapes on heat andmass transfer of nanofluid thermocapillaryconvection around a gas bubblerdquo Microgravity Science andTechnology pp 1ndash11 2019 In press

[21] Y Jiang X Zhou and Y Wang ldquoComprehensive heattransfer performance analysis of nanofluid mixed forced andthermocapillary convection around a gas bubble in mini-channelrdquo International Communications in Heat and MassTransfer vol 110 Article ID 104386 2020

[22] B Mahanthesh B J Gireesha R S R Gorla F M Abbasiand S A Shehzad ldquoNumerical solutions for magnetohy-drodynamic flow of nanofluid over a bidirectional non-linearstretching surface with prescribed surface heat flux bound-aryrdquo Journal of Magnetism and Magnetic Materials vol 417pp 189ndash196 2016

[23] B Mahanthesh B J Gireesha and I L Animasaun ldquoEx-ploration of non-linear thermal radiation and suspendednanoparticles effects on mixed convection boundary layerflow of nanoliquids on a melting vertical surfacerdquo Journal ofNanofluids vol 7 no 5 pp 833ndash843 2018

[24] P S Kumar B Mahanthesh B Gireesha and S ShehzadldquoQuadratic convective flow of radiated nano-jeffrey liquidsubject to multiple convective conditions and Cattaneo-Christov double diffusionrdquo Applied Mathematics and Me-chanics vol 39 no 9 pp 1311ndash1326 2018

[25] J Raza F Mebarek-Oudina and A Chamkha ldquoMagneto-hydrodynamic flow of molybdenum disulfide nanofluid in achannel with shape effectsrdquo Multidiscipline Modeling inMaterials and Structures vol 15 no 4 pp 737ndash757 2019

[26] T Hayat B Ahmed F M Abbasi and A Alsaedi ldquoHy-dromagnetic peristalsis of water based nanofluids with tem-perature dependent viscosity a comparative studyrdquo Journal ofMolecular Liquids vol 234 pp 324ndash329 2017

[27] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filledwith nanofluidsrdquo International Journal of Heat and FluidFlow vol 29 no 5 pp 1326ndash1336 2008

[28] M A El Kot and W Abbas ldquoNumerical technique of bloodflow through catheterized arteries with overlapping stenosisrdquoComputer Methods in Biomechanics and Biomedical Engi-neering vol 20 no 1 pp 45ndash58 2017

[29] H A Attia W Abbas A El-Din Abdin andM A M Abdeen ldquoEffects of ion slip and Hall current onunsteady Couette flow of a dusty fluid through porous mediawith heat transferrdquo High Temperature vol 53 no 6pp 891ndash898 2015

[30] H A Attia W Abbas and M A M Abdeen ldquoIon slip effecton unsteady Couette flow of a dusty fluid in the presence of

uniform suction and injection with heat transferrdquo Journal ofthe Brazilian Society of Mechanical Sciences and Engineeringvol 38 no 8 pp 2381ndash2391 2016

[31] H A Attia W Abbas M A M Abdeen and A A M SaidldquoHeat transfer between two parallel porous plates for Couetteflow under pressure gradient and Hall currentrdquo Sadhanavol 40 no 1 pp 183ndash197 2015

[32] M Sheikholeslami A Shafee M Ramzan and Z Li ldquoIn-vestigation of Lorentz forces and radiation impacts onnanofluid treatment in a porous semi annulus via Darcy lawrdquoJournal of Molecular Liquids vol 272 pp 8ndash14 2018

[33] P S Kumar B Gireesha B Mahanthesh and A J Chamkhaldquo)ermal analysis of nanofluid flow containing gyrotacticmicroorganisms in bioconvection and second-order slip withconvective conditionrdquo Journal of ermal Analysis andCalorimetry vol 136 no 5 pp 1947ndash1957 2019

[34] B Mahanthesh F Mabood B Gireesha and R Gorla ldquoEffectsof chemical reaction and partial slip on the three-dimensionalflow of a nanofluid impinging on an exponentially stretchingsurfacerdquo e European Physical Journal Plus vol 132 no 3p 113 2017

[35] B J Gireesha R S R Gorla and B Mahanthesh ldquoEffect ofsuspended nanoparticles on three-dimensional mhd flowheat and mass transfer of radiating eyring-powell fluid over astretching sheetrdquo Journal of Nanofluids vol 4 no 4pp 474ndash484 2015

[36] J-H He ldquoHomotopy perturbation techniquerdquo ComputerMethods in Applied Mechanics and Engineering vol 178no 3-4 pp 257ndash262 1999

[37] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo Interna-tional Journal of Non-Linear Mechanics vol 35 no 1pp 37ndash43 2000

[38] Y Wu and J-H He ldquoHomotopy perturbation method fornonlinear oscillators with coordinate-dependent massrdquo Re-sults in Physics vol 10 pp 270-271 2018

[39] K Das ldquoFlow and heat transfer characteristics of nanofluids ina rotating framerdquo Alexandria Engineering Journal vol 53no 3 pp 757ndash766 2014

[40] J V R Reddy V Sugunamma N Sandeep and C SulochanaldquoInfluence of chemical reaction radiation and rotation onMHD nanofluid flow past a permeable flat plate in porousmediumrdquo Journal of the Nigerian Mathematical Societyvol 35 no 1 pp 48ndash65 2016

12 Mathematical Problems in Engineering

u1(t) 1590J1

eminusαt

(1minus ϕ)52

MJ1 1+ eαt

1113872 1113873 + Ha2J1J4A1 1+ e

αt1113872 1113873 + GcJ1J3 1minus e

αt1113872 1113873 minus α(1minus s)GrJ1J2 1minus e

αt1113872 1113873 + R 1+ e

αt1113872 11138731113872 1113873

u2(t) 323J1

eminusαt

(1minus ϕ)52

MJ1 1+ eαt

1113872 1113873 + Ha2J1J4A1 1+ e

αt1113872 1113873 + GcJ1J3 1minus e

αt1113872 1113873 minus α(1+ s) minus GrJ1J2 1minus e

αt1113872 1113873 + R 1+ e

αt1113872 11138731113872 1113873

u3(t) 1590J1

(1minus ϕ)52

Ha2J1J4(1minus m) minus R minus αminus e

αt1113872 1113873

v1(t) 1590

(1minus ϕ)52

minuseminusαt

(R + αs) + Ha2J4J1

9015

eminusαt

+ m2

minus m3

+ A1eminusαt

1113874 11138751113874 1113875

v2(t) minus323

Ha2J1J4(1minus ϕ)

52+ R(1minus ϕ)

52+ Ha

2J1J4m(1minus ϕ)

52+323αe

minusαt(1minus ϕ)

52minus323

Ha2J1J4e

minusαt(1minus ϕ)

52

+323Reminusαt

(1minus ϕ)52

v3(t) minusMJ1(1minus ϕ)52

+ +Ha2J4m

3e

minusαt(1minus ϕ)

52323

minus 0125

θ1 minus32Pre

minusαt α+ J6Qr 1minus eαt( 1113857( 1113857

3J5J6+4Pre

minus2αt eαt minus 1( 1113857 seαt + DNBJ6 1minus eαt( 1113857( 1113857

J5J6minus256EcHa2J4Pre

minus2αt m2 + 1( 1113857 eαt + 1( 11138572

3J5+18

θ2 Pre

minus2αt eαt minus 1( 1113857 seαt + DNBJ6 1minus eαt( 1113857( 1113857

2J5J6

θ3 Pre

minusαt α+ J6Qr 1minus eαt( 1113857( 1113857

6J5J6

Γ1(t) Sce

minusαt αminus cc 1+ eαt( 1113857( 1113857

6

Γ2(t) 32Sce

minusαt αminus cc 1+ eαt( 1113857( 1113857

3minus 4Sce

minusαt(Sr + s) e

αtminus 11113872 1113873 minus 0125

Γ3(t) Sce

minusαt(Sr + s) eαt minus 1( 1113857

2

(34)

where A1 m3 + m2 + m + 1 and A2 minusm3 + m2 minus m

4 Solution Validation

To check the accuracy and efficiency of the methodology thecurrent results are verified by comparing with previouslypublished results in some special cases which are tabulated inTables 3ndash5 )e present results of different skin frictioncoefficient values Cf and Nusselt number NuRex for dif-ferent values of nanoparticle volume fraction ϕ and rotationparameter R are compared with those reported by Das [39]and Reddy et al [40] with both nanoparticles Cu and Al2O3which are illustrated in Tables 3 and 4 )e results obtainedcome with excellent agreement of previous studies

Furthermore the comparison for different values of theSherwood number minusShx against time t and chemical reac-tion cc with obtained results by Hussain et al [17] at ϕ 00is listed in Table 5 Also it can be seen that the presentobtained results are in an excellent agreement with the

previously published results )is proves the validity of thepresent work and shows how powerful the solution reachedby homotopy perturbation technique

5 Results and Discussion

)e influence of various physical parameters such asnanoparticle volume fraction ϕ shape of nanoparticles npHall current m Brownian NB thermophoretic NT andchemical reaction cc which are computed from the ap-proximate solution reported in the previous section is dis-played graphically and discussed )e graphical results areillustrated in Figures 2ndash11 )e numerical results are per-formed by taking values of some constant parameters as s

05 M 05 Pr 07 Ec 02 Gc 1 Gr 1 Sc 03

Du 05 G 5 and Sr 05Figures 2 and 3 display the effect of nanoparticle

volume fraction parameter ϕ on the fluid velocity u andtemperature profiles θ for either Cu minus water Al2O3 minus water

Mathematical Problems in Engineering 7

Table 3 Result comparison for Cf

Cf

ϕ RCuminuswater Al2O3minuswater

Present Das [39] Reddy et al [40] Present Das [39]

000 03 20355 20354 203542 20355 20354004 23158 23154 231535 05134 21831008 26111 26107 261068 21832 23384012 29120 29196 291953 25030 25020014 30792 30789 307891 10885 10872005 00 23790 23786 237862 22144 22143

05 24042 24040 240401 22333 2233315 25755 25755 257546 23669 2366625 28236 28235 28251 25697 25695

Table 4 Result comparison for NuxRex

NuxRex

ϕ RCuminuswater Al2O3minuswater

Present Das [39] Reddy et al [40] Present Das [39]

000 03 04607 04607 04607 04607 04607004 05135 05134 05134 05115 05114008 05695 05698 05698 05656 05655012 06304 06303 06303 06234 06233014 06622 06622 06622 06536 06537005 00 05272 05271 05271 05271 05271

05 mdash mdash mdash mdash mdash15 mdash mdash mdash mdash mdash25 05271 05271 05271 05272 05271

Table 5 Result comparison for minusShx

minusShx

ccdarr t⟶ 03 05 07 03 05 07Present Hussain et al [17]

02 0535254 0438632 0389005 054907 044765 0386382 0849652 0815633 0767225 085729 0820922 0781565 12252 116674 114632 124260 116967 1143922

ϕ = 0 007 01

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

Cu-waterAI2O3-waterTiO2-water

Figure 2 )e effect of volume fraction of nanoparticles on the velocity profile

8 Mathematical Problems in Engineering

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θϕ = 0 007 012

1

08

06

04

02

00 2 3 4 5 6 7 81

Cu-waterAI2O3-waterTiO2-water

z

Figure 3 )e effect of volume fraction nanoparticle on thetemperature profile

Cylindrical shape

Brick shape

Spherical shape

1

08

06

04

02

00 2 3 4 5 6 7 81

z

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θ

Cu-waterAI2O3-waterTiO2-water

Figure 4 )e effect of nanoparticle shape on the temperatureprofile

R = 1 2 4

Cu-waterTiO2-water

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

Figure 5 )e effect of rotation parameter on the velocity profile

m = 1 15 2

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 6 )e effect of Hall current on the velocity profile

NB = 02 05 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07m = 1γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 7 )e effect of Brownian motion parameter on the tem-perature profile

ϕ = 005NB = 07m = 1γc = 05Qs = 1Qr = 09R = 1

NT = 03 06 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 8 )e effect of thermophoresis parameter on the tem-perature profile

Mathematical Problems in Engineering 9

and TiO2 minus water nanofluid It is clear that increase innanoparticle volume fraction parameter ϕ decreases the fluidvelocity but the temperature profiles increase It is due to thefact that nanofluid density increases with increasing thevolume fraction of nanoparticles and it causes to slow downthe nanofluid flow velocity and increases the thickness of thethermal boundary layer which improves thermal conduc-tivity and increases surface heat transfer rate Also it isobserved that nanofluid Cu minus water heat transfer rate isgreater than Al2O3 and TiO2 rates So nanoparticles havinghigh thermal conductivity enhance heat transfer more thanthe particles having low thermal conductivity On thecontrary it is seen from Tables 2 and 3 that wall skin frictioncoefficient Cf and the Nusselt number NuRex increase withincreasing in nanoparticle volume fraction for both Cu minus

water and Al2O3 minus water)e influence of the nanoparticle shape on the tem-

perature distribution of nanofluid is studied through Fig-ure 4 )is figure indicates that the heat transfer rate ishigher for the cylindrical nanoparticle shape than thespherical and brick nanoparticle shapes for eitherCu minus water Al2O3 minus water and TiO2 minus water nanofluid)erefore the spherical nanoparticle shape tends to dragmore heat from the boundary layer because it has a greatersurface area while this effect is less evident for othernanoparticle shapes )us the use of the cylindrical nano-particle shape has better improvement in effective thermalconductivity when compared with other nanoparticle shapes(spherical and brick)

)e influence of rotation parameter R on the velocityprofiles against z is displayed in Figure 5 It observed that anincrease in rotation parameter leads to decreasing theboundary layer momentum so it decreases the values ofnanofluid velocity From Tables 2 and 3 it can be noted thatthe wall skin friction coefficient Cf increases with an in-crease in the rotation parameter but the Nusselt numberNuRex remains unchanged with both Cu minus water andAl2O3 minus water

Figure 6 presents the influence of Hall current parameterm on main velocity u It is clear that the velocity distributionof nanofluid reduces with an increase in Hall current pa-rameter)is happens naturally due to the fact that when themagnetic field is strengthened it enhances the opposingforce which is represented by the well-known Lorentz force

Figures 7 and 8 show the Brownian and thermophoresiseffects on the temperature profile θ It is seen that thetemperature distribution increases with increasing for theseparameters )is is due to the fact that the thermophoreticand Brownian parameters assist to improve the thermalboundary layer thickness Also it is observed that the heattransfer rate of Cu minus water nanofluid is greater thanTiO2 minus water

Figure 9 depicts the influence of chemical parameter cc

on the concentration profile Γ It is noticed that the increasein concentration profile is accompanied by a decrease in thechemical parameter On the contrary it is seen from Table 4that the Sherwood number minusShx increases due to increasingthe chemical parameter but the Sherwood number decreaseswhile increasing the time t )is implies that the chemical

Cu-waterTiO2-water

ϕ = 005NT = 07NB = 07m = 1

γc = 01 05 1

Qs = 1Qr = 09R = 1

0 2 3 4 5 6 7 81z

1

08

06

04

02

0

12

14

Γ

Figure 9 )e effect of chemical reaction parameter on the con-centration profile

Cu-waterTiO2-water

ϕ = 005

NB = 07

m = 1γc = 05

Qs = 1 2 3

Qr = 09R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

NT = 07

Figure 10 )e effect of heat source parameter on the temperatureprofile

ϕ = 005

NB = 07NT = 07

m = 1

γc = 05Qs = 1

Qr = 05 09 13

R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 11 )e effect of radiation parameter on the temperatureprofile

10 Mathematical Problems in Engineering

reaction has tendency to increase the mass transfer rate atthe surface plate and it is reduced with the progress of time

Figures 10 and 11 present the heat source parameter Qs

and thermal radiation parameter Qr effects on the tem-perature profiles It is clear from Figure 10 that with anincrease in the heat generation parameter the boundarylayer temperature decreases and as a consequence to de-crease the thermal boundary layer thickness In addition it isclear from Figure 11 that with an increase in the thermalradiation the temperature profiles of the nanofluid increaseIt is well known that the increase in the radiation parameterenhances the energy transport inside the fluid and as aconsequence to increase the heat transfer rate

6 Conclusion

An analysis is done to solve the magnetohydro dynamicsunsteady nanofluid flow through the porous medium past amoving plate in a rotating frame with considering the heatand mass transfer )ree different types of the nanoparticlesnamely copper aluminium oxide and titanium dioxide andwater used as a base nanofluid are considered with variousshapes (spherical cylindrical and brick) )e important ef-fects of nanoparticle shape Brownian and thermophoresisdiffusion thermal radiation ohmic and viscous dissipationshave been included Homotopy perturbation method is ap-plied to solve the governing nonlinear partial differentialequations )e efficiency of the methodology has been vali-dated through comparison with the previously publishedresults and observed a very close agreement It is found that

)e main velocity component decreases with increaseof nanoparticle volume fraction On the contrarythermal properties increase with increase in nano-particle volume fraction)e enhancement of heat transfer rate and mechanicalproperty improvement is high in copper as comparedwith both aluminium oxide and titanium dioxide)ermal conductivity of cylindrical-shaped nanoparticlesis more efficient and possesses better improvement onheat transfer rate than the other shaped onesIncrease in the Brownian and thermophoresis pa-rameter is accompanied by an increase in the thermalboundary layer thicknessIncreasing Hall current and rotation parameters de-creases the main fluid velocity componentHPM has a close agreement and is recommended formany nonlinear problems

Data Availability

)e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] T Hayat R Sajjad A Alsaedi T Muhammad and R EllahildquoOn squeezed flow of couple stress nanofluid between twoparallel platesrdquo Results in Physics vol 7 pp 553ndash561 2017

[2] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition pp 99ndash105San Francisco CA USA 1995

[3] J A Eastman S U S Choi S Li W Yu and L J )ompsonldquoAnomalously increased effective thermal conductivities of eth-ylene glycol-based nanofluids containing copper nanoparticlesrdquoApplied Physics Letters vol 78 no 6 pp 718ndash720 2001

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

[5] M Sheikholeslami M M Rashidi T Hayat and D D GanjildquoFree convection of magnetic nanofluid considering MFDviscosity effectrdquo Journal of Molecular Liquids vol 218pp 393ndash399 2016

[6] W Abbas and E A Sayed ldquoHall current and joule heatingeffects on free convection flow of a nanofluid over a verticalcone in presence of thermal radiationrdquo ermal Sciencevol 21 pp 2603ndash2614 2017

[7] B Mahanthesh B Gireesha I Animasaun T Muhammadand N Shashikumar ldquoMHD flow of SWCNT and MWCNTnanoliquids past a rotating stretchable disk with thermal andexponential space dependent heat sourcerdquo Physica Scriptavol 94 no 8 Article ID 085214 2019

[8] I L Animasaun O K Koriko K S Adegbie et al ldquoCom-parative analysis between 36 nm and 47 nm alumina-waternanofluid flows in the presence of Hall effectrdquo Journal ofermal Analysis and Calorimetry vol 135 no 2 pp 873ndash886 2019

[9] I Ullah S Shafie I Khan and K L Hsiao ldquoBrownian dif-fusion and thermophoresis mechanisms in Casson fluid over amoving wedgerdquo Results in Physics vol 9 pp 183ndash194 2018

[10] P B S Kumar B J Gireesha R S R Gorla andB Mahanthesh ldquoMagnetohydrodynamic flow of Williamsonnanofluid due to an exponentially stretching surface in thepresence of thermal radiation and chemical reactionrdquo Journalof Nanofluids vol 6 no 2 pp 264ndash272 2017

[11] X Zhou Y Jiang X Li et al ldquoNumerical investigation of heattransfer enhancement and entropy generation of naturalconvection in a cavity containing nano liquid-metal fluidrdquoInternational Communications in Heat and Mass Transfervol 106 pp 46ndash54 2019

[12] N Shehzad A Zeeshan R Ellahi and K Vafai ldquoConvectiveheat transfer of nanofluid in a wavy channel Buongiornorsquosmathematical modelrdquo Journal of Molecular Liquids vol 222pp 446ndash455 2016

[13] B Shen L Zheng C Zhang and X Zhang ldquoBioconvectionheat transfer of a nanofluid over a stretching sheet with ve-locity slip and temperature jumprdquo ermal Science vol 21no 6 pp 2347ndash2356 2017

[14] S Jahan H Sakidin R Nazar and I Pop ldquoAnalysis of heattransfer in nanofluid past a convectively heated permeablestretchingshrinking sheet with regression and stability ana-lysesrdquo Results in Physics vol 10 pp 395ndash405 2018

[15] M A A Hamad and I Pop ldquoUnsteady MHD free convectionflow past a vertical permeable flat plate in a rotating frame ofreference with constant heat source in a nanofluidrdquo Heat andMass Transfer vol 47 no 12 pp 1517ndash1524 2011

Mathematical Problems in Engineering 11

[16] M Sheikholeslami and D D Ganji ldquo)ree dimensional heatand mass transfer in a rotating system using nanofluidrdquoPowder Technology vol 253 pp 789ndash796 2014

[17] S M Hussain J Jain G S Seth and M M Rashidi ldquoFreeconvective heat transfer with Hall effects heat absorption andchemical reaction over an accelerated moving plate in a ro-tating systemrdquo Journal of Magnetism and Magnetic Materialsvol 422 pp 112ndash123 2017

[18] X Zhou Y Jiang Y Hou and M Du ldquo)ermocapillaryconvection instability in annular two-layer system undervarious gravity levelsrdquo Microgravity Science and Technologyvol 31 no 5 pp 641ndash648 2019

[19] Y Jiang and X Zhou ldquoHeat transfer and entropy generationanalysis of nanofluids thermocapillary convection around abubble in a cavityrdquo International Communications in Heatand Mass Transfer vol 105 pp 37ndash45 2019

[20] Y Jiang X Zhou and Y Wang ldquoEffects of nanoparticleshapes on heat andmass transfer of nanofluid thermocapillaryconvection around a gas bubblerdquo Microgravity Science andTechnology pp 1ndash11 2019 In press

[21] Y Jiang X Zhou and Y Wang ldquoComprehensive heattransfer performance analysis of nanofluid mixed forced andthermocapillary convection around a gas bubble in mini-channelrdquo International Communications in Heat and MassTransfer vol 110 Article ID 104386 2020

[22] B Mahanthesh B J Gireesha R S R Gorla F M Abbasiand S A Shehzad ldquoNumerical solutions for magnetohy-drodynamic flow of nanofluid over a bidirectional non-linearstretching surface with prescribed surface heat flux bound-aryrdquo Journal of Magnetism and Magnetic Materials vol 417pp 189ndash196 2016

[23] B Mahanthesh B J Gireesha and I L Animasaun ldquoEx-ploration of non-linear thermal radiation and suspendednanoparticles effects on mixed convection boundary layerflow of nanoliquids on a melting vertical surfacerdquo Journal ofNanofluids vol 7 no 5 pp 833ndash843 2018

[24] P S Kumar B Mahanthesh B Gireesha and S ShehzadldquoQuadratic convective flow of radiated nano-jeffrey liquidsubject to multiple convective conditions and Cattaneo-Christov double diffusionrdquo Applied Mathematics and Me-chanics vol 39 no 9 pp 1311ndash1326 2018

[25] J Raza F Mebarek-Oudina and A Chamkha ldquoMagneto-hydrodynamic flow of molybdenum disulfide nanofluid in achannel with shape effectsrdquo Multidiscipline Modeling inMaterials and Structures vol 15 no 4 pp 737ndash757 2019

[26] T Hayat B Ahmed F M Abbasi and A Alsaedi ldquoHy-dromagnetic peristalsis of water based nanofluids with tem-perature dependent viscosity a comparative studyrdquo Journal ofMolecular Liquids vol 234 pp 324ndash329 2017

[27] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filledwith nanofluidsrdquo International Journal of Heat and FluidFlow vol 29 no 5 pp 1326ndash1336 2008

[28] M A El Kot and W Abbas ldquoNumerical technique of bloodflow through catheterized arteries with overlapping stenosisrdquoComputer Methods in Biomechanics and Biomedical Engi-neering vol 20 no 1 pp 45ndash58 2017

[29] H A Attia W Abbas A El-Din Abdin andM A M Abdeen ldquoEffects of ion slip and Hall current onunsteady Couette flow of a dusty fluid through porous mediawith heat transferrdquo High Temperature vol 53 no 6pp 891ndash898 2015

[30] H A Attia W Abbas and M A M Abdeen ldquoIon slip effecton unsteady Couette flow of a dusty fluid in the presence of

uniform suction and injection with heat transferrdquo Journal ofthe Brazilian Society of Mechanical Sciences and Engineeringvol 38 no 8 pp 2381ndash2391 2016

[31] H A Attia W Abbas M A M Abdeen and A A M SaidldquoHeat transfer between two parallel porous plates for Couetteflow under pressure gradient and Hall currentrdquo Sadhanavol 40 no 1 pp 183ndash197 2015

[32] M Sheikholeslami A Shafee M Ramzan and Z Li ldquoIn-vestigation of Lorentz forces and radiation impacts onnanofluid treatment in a porous semi annulus via Darcy lawrdquoJournal of Molecular Liquids vol 272 pp 8ndash14 2018

[33] P S Kumar B Gireesha B Mahanthesh and A J Chamkhaldquo)ermal analysis of nanofluid flow containing gyrotacticmicroorganisms in bioconvection and second-order slip withconvective conditionrdquo Journal of ermal Analysis andCalorimetry vol 136 no 5 pp 1947ndash1957 2019

[34] B Mahanthesh F Mabood B Gireesha and R Gorla ldquoEffectsof chemical reaction and partial slip on the three-dimensionalflow of a nanofluid impinging on an exponentially stretchingsurfacerdquo e European Physical Journal Plus vol 132 no 3p 113 2017

[35] B J Gireesha R S R Gorla and B Mahanthesh ldquoEffect ofsuspended nanoparticles on three-dimensional mhd flowheat and mass transfer of radiating eyring-powell fluid over astretching sheetrdquo Journal of Nanofluids vol 4 no 4pp 474ndash484 2015

[36] J-H He ldquoHomotopy perturbation techniquerdquo ComputerMethods in Applied Mechanics and Engineering vol 178no 3-4 pp 257ndash262 1999

[37] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo Interna-tional Journal of Non-Linear Mechanics vol 35 no 1pp 37ndash43 2000

[38] Y Wu and J-H He ldquoHomotopy perturbation method fornonlinear oscillators with coordinate-dependent massrdquo Re-sults in Physics vol 10 pp 270-271 2018

[39] K Das ldquoFlow and heat transfer characteristics of nanofluids ina rotating framerdquo Alexandria Engineering Journal vol 53no 3 pp 757ndash766 2014

[40] J V R Reddy V Sugunamma N Sandeep and C SulochanaldquoInfluence of chemical reaction radiation and rotation onMHD nanofluid flow past a permeable flat plate in porousmediumrdquo Journal of the Nigerian Mathematical Societyvol 35 no 1 pp 48ndash65 2016

12 Mathematical Problems in Engineering

Table 3 Result comparison for Cf

Cf

ϕ RCuminuswater Al2O3minuswater

Present Das [39] Reddy et al [40] Present Das [39]

000 03 20355 20354 203542 20355 20354004 23158 23154 231535 05134 21831008 26111 26107 261068 21832 23384012 29120 29196 291953 25030 25020014 30792 30789 307891 10885 10872005 00 23790 23786 237862 22144 22143

05 24042 24040 240401 22333 2233315 25755 25755 257546 23669 2366625 28236 28235 28251 25697 25695

Table 4 Result comparison for NuxRex

NuxRex

ϕ RCuminuswater Al2O3minuswater

Present Das [39] Reddy et al [40] Present Das [39]

000 03 04607 04607 04607 04607 04607004 05135 05134 05134 05115 05114008 05695 05698 05698 05656 05655012 06304 06303 06303 06234 06233014 06622 06622 06622 06536 06537005 00 05272 05271 05271 05271 05271

05 mdash mdash mdash mdash mdash15 mdash mdash mdash mdash mdash25 05271 05271 05271 05272 05271

Table 5 Result comparison for minusShx

minusShx

ccdarr t⟶ 03 05 07 03 05 07Present Hussain et al [17]

02 0535254 0438632 0389005 054907 044765 0386382 0849652 0815633 0767225 085729 0820922 0781565 12252 116674 114632 124260 116967 1143922

ϕ = 0 007 01

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

Cu-waterAI2O3-waterTiO2-water

Figure 2 )e effect of volume fraction of nanoparticles on the velocity profile

8 Mathematical Problems in Engineering

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θϕ = 0 007 012

1

08

06

04

02

00 2 3 4 5 6 7 81

Cu-waterAI2O3-waterTiO2-water

z

Figure 3 )e effect of volume fraction nanoparticle on thetemperature profile

Cylindrical shape

Brick shape

Spherical shape

1

08

06

04

02

00 2 3 4 5 6 7 81

z

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θ

Cu-waterAI2O3-waterTiO2-water

Figure 4 )e effect of nanoparticle shape on the temperatureprofile

R = 1 2 4

Cu-waterTiO2-water

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

Figure 5 )e effect of rotation parameter on the velocity profile

m = 1 15 2

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 6 )e effect of Hall current on the velocity profile

NB = 02 05 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07m = 1γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 7 )e effect of Brownian motion parameter on the tem-perature profile

ϕ = 005NB = 07m = 1γc = 05Qs = 1Qr = 09R = 1

NT = 03 06 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 8 )e effect of thermophoresis parameter on the tem-perature profile

Mathematical Problems in Engineering 9

and TiO2 minus water nanofluid It is clear that increase innanoparticle volume fraction parameter ϕ decreases the fluidvelocity but the temperature profiles increase It is due to thefact that nanofluid density increases with increasing thevolume fraction of nanoparticles and it causes to slow downthe nanofluid flow velocity and increases the thickness of thethermal boundary layer which improves thermal conduc-tivity and increases surface heat transfer rate Also it isobserved that nanofluid Cu minus water heat transfer rate isgreater than Al2O3 and TiO2 rates So nanoparticles havinghigh thermal conductivity enhance heat transfer more thanthe particles having low thermal conductivity On thecontrary it is seen from Tables 2 and 3 that wall skin frictioncoefficient Cf and the Nusselt number NuRex increase withincreasing in nanoparticle volume fraction for both Cu minus

water and Al2O3 minus water)e influence of the nanoparticle shape on the tem-

perature distribution of nanofluid is studied through Fig-ure 4 )is figure indicates that the heat transfer rate ishigher for the cylindrical nanoparticle shape than thespherical and brick nanoparticle shapes for eitherCu minus water Al2O3 minus water and TiO2 minus water nanofluid)erefore the spherical nanoparticle shape tends to dragmore heat from the boundary layer because it has a greatersurface area while this effect is less evident for othernanoparticle shapes )us the use of the cylindrical nano-particle shape has better improvement in effective thermalconductivity when compared with other nanoparticle shapes(spherical and brick)

)e influence of rotation parameter R on the velocityprofiles against z is displayed in Figure 5 It observed that anincrease in rotation parameter leads to decreasing theboundary layer momentum so it decreases the values ofnanofluid velocity From Tables 2 and 3 it can be noted thatthe wall skin friction coefficient Cf increases with an in-crease in the rotation parameter but the Nusselt numberNuRex remains unchanged with both Cu minus water andAl2O3 minus water

Figure 6 presents the influence of Hall current parameterm on main velocity u It is clear that the velocity distributionof nanofluid reduces with an increase in Hall current pa-rameter)is happens naturally due to the fact that when themagnetic field is strengthened it enhances the opposingforce which is represented by the well-known Lorentz force

Figures 7 and 8 show the Brownian and thermophoresiseffects on the temperature profile θ It is seen that thetemperature distribution increases with increasing for theseparameters )is is due to the fact that the thermophoreticand Brownian parameters assist to improve the thermalboundary layer thickness Also it is observed that the heattransfer rate of Cu minus water nanofluid is greater thanTiO2 minus water

Figure 9 depicts the influence of chemical parameter cc

on the concentration profile Γ It is noticed that the increasein concentration profile is accompanied by a decrease in thechemical parameter On the contrary it is seen from Table 4that the Sherwood number minusShx increases due to increasingthe chemical parameter but the Sherwood number decreaseswhile increasing the time t )is implies that the chemical

Cu-waterTiO2-water

ϕ = 005NT = 07NB = 07m = 1

γc = 01 05 1

Qs = 1Qr = 09R = 1

0 2 3 4 5 6 7 81z

1

08

06

04

02

0

12

14

Γ

Figure 9 )e effect of chemical reaction parameter on the con-centration profile

Cu-waterTiO2-water

ϕ = 005

NB = 07

m = 1γc = 05

Qs = 1 2 3

Qr = 09R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

NT = 07

Figure 10 )e effect of heat source parameter on the temperatureprofile

ϕ = 005

NB = 07NT = 07

m = 1

γc = 05Qs = 1

Qr = 05 09 13

R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 11 )e effect of radiation parameter on the temperatureprofile

10 Mathematical Problems in Engineering

reaction has tendency to increase the mass transfer rate atthe surface plate and it is reduced with the progress of time

Figures 10 and 11 present the heat source parameter Qs

and thermal radiation parameter Qr effects on the tem-perature profiles It is clear from Figure 10 that with anincrease in the heat generation parameter the boundarylayer temperature decreases and as a consequence to de-crease the thermal boundary layer thickness In addition it isclear from Figure 11 that with an increase in the thermalradiation the temperature profiles of the nanofluid increaseIt is well known that the increase in the radiation parameterenhances the energy transport inside the fluid and as aconsequence to increase the heat transfer rate

6 Conclusion

An analysis is done to solve the magnetohydro dynamicsunsteady nanofluid flow through the porous medium past amoving plate in a rotating frame with considering the heatand mass transfer )ree different types of the nanoparticlesnamely copper aluminium oxide and titanium dioxide andwater used as a base nanofluid are considered with variousshapes (spherical cylindrical and brick) )e important ef-fects of nanoparticle shape Brownian and thermophoresisdiffusion thermal radiation ohmic and viscous dissipationshave been included Homotopy perturbation method is ap-plied to solve the governing nonlinear partial differentialequations )e efficiency of the methodology has been vali-dated through comparison with the previously publishedresults and observed a very close agreement It is found that

)e main velocity component decreases with increaseof nanoparticle volume fraction On the contrarythermal properties increase with increase in nano-particle volume fraction)e enhancement of heat transfer rate and mechanicalproperty improvement is high in copper as comparedwith both aluminium oxide and titanium dioxide)ermal conductivity of cylindrical-shaped nanoparticlesis more efficient and possesses better improvement onheat transfer rate than the other shaped onesIncrease in the Brownian and thermophoresis pa-rameter is accompanied by an increase in the thermalboundary layer thicknessIncreasing Hall current and rotation parameters de-creases the main fluid velocity componentHPM has a close agreement and is recommended formany nonlinear problems

Data Availability

)e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] T Hayat R Sajjad A Alsaedi T Muhammad and R EllahildquoOn squeezed flow of couple stress nanofluid between twoparallel platesrdquo Results in Physics vol 7 pp 553ndash561 2017

[2] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition pp 99ndash105San Francisco CA USA 1995

[3] J A Eastman S U S Choi S Li W Yu and L J )ompsonldquoAnomalously increased effective thermal conductivities of eth-ylene glycol-based nanofluids containing copper nanoparticlesrdquoApplied Physics Letters vol 78 no 6 pp 718ndash720 2001

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

[5] M Sheikholeslami M M Rashidi T Hayat and D D GanjildquoFree convection of magnetic nanofluid considering MFDviscosity effectrdquo Journal of Molecular Liquids vol 218pp 393ndash399 2016

[6] W Abbas and E A Sayed ldquoHall current and joule heatingeffects on free convection flow of a nanofluid over a verticalcone in presence of thermal radiationrdquo ermal Sciencevol 21 pp 2603ndash2614 2017

[7] B Mahanthesh B Gireesha I Animasaun T Muhammadand N Shashikumar ldquoMHD flow of SWCNT and MWCNTnanoliquids past a rotating stretchable disk with thermal andexponential space dependent heat sourcerdquo Physica Scriptavol 94 no 8 Article ID 085214 2019

[8] I L Animasaun O K Koriko K S Adegbie et al ldquoCom-parative analysis between 36 nm and 47 nm alumina-waternanofluid flows in the presence of Hall effectrdquo Journal ofermal Analysis and Calorimetry vol 135 no 2 pp 873ndash886 2019

[9] I Ullah S Shafie I Khan and K L Hsiao ldquoBrownian dif-fusion and thermophoresis mechanisms in Casson fluid over amoving wedgerdquo Results in Physics vol 9 pp 183ndash194 2018

[10] P B S Kumar B J Gireesha R S R Gorla andB Mahanthesh ldquoMagnetohydrodynamic flow of Williamsonnanofluid due to an exponentially stretching surface in thepresence of thermal radiation and chemical reactionrdquo Journalof Nanofluids vol 6 no 2 pp 264ndash272 2017

[11] X Zhou Y Jiang X Li et al ldquoNumerical investigation of heattransfer enhancement and entropy generation of naturalconvection in a cavity containing nano liquid-metal fluidrdquoInternational Communications in Heat and Mass Transfervol 106 pp 46ndash54 2019

[12] N Shehzad A Zeeshan R Ellahi and K Vafai ldquoConvectiveheat transfer of nanofluid in a wavy channel Buongiornorsquosmathematical modelrdquo Journal of Molecular Liquids vol 222pp 446ndash455 2016

[13] B Shen L Zheng C Zhang and X Zhang ldquoBioconvectionheat transfer of a nanofluid over a stretching sheet with ve-locity slip and temperature jumprdquo ermal Science vol 21no 6 pp 2347ndash2356 2017

[14] S Jahan H Sakidin R Nazar and I Pop ldquoAnalysis of heattransfer in nanofluid past a convectively heated permeablestretchingshrinking sheet with regression and stability ana-lysesrdquo Results in Physics vol 10 pp 395ndash405 2018

[15] M A A Hamad and I Pop ldquoUnsteady MHD free convectionflow past a vertical permeable flat plate in a rotating frame ofreference with constant heat source in a nanofluidrdquo Heat andMass Transfer vol 47 no 12 pp 1517ndash1524 2011

Mathematical Problems in Engineering 11

[16] M Sheikholeslami and D D Ganji ldquo)ree dimensional heatand mass transfer in a rotating system using nanofluidrdquoPowder Technology vol 253 pp 789ndash796 2014

[17] S M Hussain J Jain G S Seth and M M Rashidi ldquoFreeconvective heat transfer with Hall effects heat absorption andchemical reaction over an accelerated moving plate in a ro-tating systemrdquo Journal of Magnetism and Magnetic Materialsvol 422 pp 112ndash123 2017

[18] X Zhou Y Jiang Y Hou and M Du ldquo)ermocapillaryconvection instability in annular two-layer system undervarious gravity levelsrdquo Microgravity Science and Technologyvol 31 no 5 pp 641ndash648 2019

[19] Y Jiang and X Zhou ldquoHeat transfer and entropy generationanalysis of nanofluids thermocapillary convection around abubble in a cavityrdquo International Communications in Heatand Mass Transfer vol 105 pp 37ndash45 2019

[20] Y Jiang X Zhou and Y Wang ldquoEffects of nanoparticleshapes on heat andmass transfer of nanofluid thermocapillaryconvection around a gas bubblerdquo Microgravity Science andTechnology pp 1ndash11 2019 In press

[21] Y Jiang X Zhou and Y Wang ldquoComprehensive heattransfer performance analysis of nanofluid mixed forced andthermocapillary convection around a gas bubble in mini-channelrdquo International Communications in Heat and MassTransfer vol 110 Article ID 104386 2020

[22] B Mahanthesh B J Gireesha R S R Gorla F M Abbasiand S A Shehzad ldquoNumerical solutions for magnetohy-drodynamic flow of nanofluid over a bidirectional non-linearstretching surface with prescribed surface heat flux bound-aryrdquo Journal of Magnetism and Magnetic Materials vol 417pp 189ndash196 2016

[23] B Mahanthesh B J Gireesha and I L Animasaun ldquoEx-ploration of non-linear thermal radiation and suspendednanoparticles effects on mixed convection boundary layerflow of nanoliquids on a melting vertical surfacerdquo Journal ofNanofluids vol 7 no 5 pp 833ndash843 2018

[24] P S Kumar B Mahanthesh B Gireesha and S ShehzadldquoQuadratic convective flow of radiated nano-jeffrey liquidsubject to multiple convective conditions and Cattaneo-Christov double diffusionrdquo Applied Mathematics and Me-chanics vol 39 no 9 pp 1311ndash1326 2018

[25] J Raza F Mebarek-Oudina and A Chamkha ldquoMagneto-hydrodynamic flow of molybdenum disulfide nanofluid in achannel with shape effectsrdquo Multidiscipline Modeling inMaterials and Structures vol 15 no 4 pp 737ndash757 2019

[26] T Hayat B Ahmed F M Abbasi and A Alsaedi ldquoHy-dromagnetic peristalsis of water based nanofluids with tem-perature dependent viscosity a comparative studyrdquo Journal ofMolecular Liquids vol 234 pp 324ndash329 2017

[27] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filledwith nanofluidsrdquo International Journal of Heat and FluidFlow vol 29 no 5 pp 1326ndash1336 2008

[28] M A El Kot and W Abbas ldquoNumerical technique of bloodflow through catheterized arteries with overlapping stenosisrdquoComputer Methods in Biomechanics and Biomedical Engi-neering vol 20 no 1 pp 45ndash58 2017

[29] H A Attia W Abbas A El-Din Abdin andM A M Abdeen ldquoEffects of ion slip and Hall current onunsteady Couette flow of a dusty fluid through porous mediawith heat transferrdquo High Temperature vol 53 no 6pp 891ndash898 2015

[30] H A Attia W Abbas and M A M Abdeen ldquoIon slip effecton unsteady Couette flow of a dusty fluid in the presence of

uniform suction and injection with heat transferrdquo Journal ofthe Brazilian Society of Mechanical Sciences and Engineeringvol 38 no 8 pp 2381ndash2391 2016

[31] H A Attia W Abbas M A M Abdeen and A A M SaidldquoHeat transfer between two parallel porous plates for Couetteflow under pressure gradient and Hall currentrdquo Sadhanavol 40 no 1 pp 183ndash197 2015

[32] M Sheikholeslami A Shafee M Ramzan and Z Li ldquoIn-vestigation of Lorentz forces and radiation impacts onnanofluid treatment in a porous semi annulus via Darcy lawrdquoJournal of Molecular Liquids vol 272 pp 8ndash14 2018

[33] P S Kumar B Gireesha B Mahanthesh and A J Chamkhaldquo)ermal analysis of nanofluid flow containing gyrotacticmicroorganisms in bioconvection and second-order slip withconvective conditionrdquo Journal of ermal Analysis andCalorimetry vol 136 no 5 pp 1947ndash1957 2019

[34] B Mahanthesh F Mabood B Gireesha and R Gorla ldquoEffectsof chemical reaction and partial slip on the three-dimensionalflow of a nanofluid impinging on an exponentially stretchingsurfacerdquo e European Physical Journal Plus vol 132 no 3p 113 2017

[35] B J Gireesha R S R Gorla and B Mahanthesh ldquoEffect ofsuspended nanoparticles on three-dimensional mhd flowheat and mass transfer of radiating eyring-powell fluid over astretching sheetrdquo Journal of Nanofluids vol 4 no 4pp 474ndash484 2015

[36] J-H He ldquoHomotopy perturbation techniquerdquo ComputerMethods in Applied Mechanics and Engineering vol 178no 3-4 pp 257ndash262 1999

[37] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo Interna-tional Journal of Non-Linear Mechanics vol 35 no 1pp 37ndash43 2000

[38] Y Wu and J-H He ldquoHomotopy perturbation method fornonlinear oscillators with coordinate-dependent massrdquo Re-sults in Physics vol 10 pp 270-271 2018

[39] K Das ldquoFlow and heat transfer characteristics of nanofluids ina rotating framerdquo Alexandria Engineering Journal vol 53no 3 pp 757ndash766 2014

[40] J V R Reddy V Sugunamma N Sandeep and C SulochanaldquoInfluence of chemical reaction radiation and rotation onMHD nanofluid flow past a permeable flat plate in porousmediumrdquo Journal of the Nigerian Mathematical Societyvol 35 no 1 pp 48ndash65 2016

12 Mathematical Problems in Engineering

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θϕ = 0 007 012

1

08

06

04

02

00 2 3 4 5 6 7 81

Cu-waterAI2O3-waterTiO2-water

z

Figure 3 )e effect of volume fraction nanoparticle on thetemperature profile

Cylindrical shape

Brick shape

Spherical shape

1

08

06

04

02

00 2 3 4 5 6 7 81

z

R = 1NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

θ

Cu-waterAI2O3-waterTiO2-water

Figure 4 )e effect of nanoparticle shape on the temperatureprofile

R = 1 2 4

Cu-waterTiO2-water

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09m = 1

Figure 5 )e effect of rotation parameter on the velocity profile

m = 1 15 2

1

08

06

04

02

0

u

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07NB = 07γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 6 )e effect of Hall current on the velocity profile

NB = 02 05 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

ϕ = 005NT = 07m = 1γc = 05Qs = 1Qr = 09R = 1

Cu-waterTiO2-water

Figure 7 )e effect of Brownian motion parameter on the tem-perature profile

ϕ = 005NB = 07m = 1γc = 05Qs = 1Qr = 09R = 1

NT = 03 06 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 8 )e effect of thermophoresis parameter on the tem-perature profile

Mathematical Problems in Engineering 9

and TiO2 minus water nanofluid It is clear that increase innanoparticle volume fraction parameter ϕ decreases the fluidvelocity but the temperature profiles increase It is due to thefact that nanofluid density increases with increasing thevolume fraction of nanoparticles and it causes to slow downthe nanofluid flow velocity and increases the thickness of thethermal boundary layer which improves thermal conduc-tivity and increases surface heat transfer rate Also it isobserved that nanofluid Cu minus water heat transfer rate isgreater than Al2O3 and TiO2 rates So nanoparticles havinghigh thermal conductivity enhance heat transfer more thanthe particles having low thermal conductivity On thecontrary it is seen from Tables 2 and 3 that wall skin frictioncoefficient Cf and the Nusselt number NuRex increase withincreasing in nanoparticle volume fraction for both Cu minus

water and Al2O3 minus water)e influence of the nanoparticle shape on the tem-

perature distribution of nanofluid is studied through Fig-ure 4 )is figure indicates that the heat transfer rate ishigher for the cylindrical nanoparticle shape than thespherical and brick nanoparticle shapes for eitherCu minus water Al2O3 minus water and TiO2 minus water nanofluid)erefore the spherical nanoparticle shape tends to dragmore heat from the boundary layer because it has a greatersurface area while this effect is less evident for othernanoparticle shapes )us the use of the cylindrical nano-particle shape has better improvement in effective thermalconductivity when compared with other nanoparticle shapes(spherical and brick)

)e influence of rotation parameter R on the velocityprofiles against z is displayed in Figure 5 It observed that anincrease in rotation parameter leads to decreasing theboundary layer momentum so it decreases the values ofnanofluid velocity From Tables 2 and 3 it can be noted thatthe wall skin friction coefficient Cf increases with an in-crease in the rotation parameter but the Nusselt numberNuRex remains unchanged with both Cu minus water andAl2O3 minus water

Figure 6 presents the influence of Hall current parameterm on main velocity u It is clear that the velocity distributionof nanofluid reduces with an increase in Hall current pa-rameter)is happens naturally due to the fact that when themagnetic field is strengthened it enhances the opposingforce which is represented by the well-known Lorentz force

Figures 7 and 8 show the Brownian and thermophoresiseffects on the temperature profile θ It is seen that thetemperature distribution increases with increasing for theseparameters )is is due to the fact that the thermophoreticand Brownian parameters assist to improve the thermalboundary layer thickness Also it is observed that the heattransfer rate of Cu minus water nanofluid is greater thanTiO2 minus water

Figure 9 depicts the influence of chemical parameter cc

on the concentration profile Γ It is noticed that the increasein concentration profile is accompanied by a decrease in thechemical parameter On the contrary it is seen from Table 4that the Sherwood number minusShx increases due to increasingthe chemical parameter but the Sherwood number decreaseswhile increasing the time t )is implies that the chemical

Cu-waterTiO2-water

ϕ = 005NT = 07NB = 07m = 1

γc = 01 05 1

Qs = 1Qr = 09R = 1

0 2 3 4 5 6 7 81z

1

08

06

04

02

0

12

14

Γ

Figure 9 )e effect of chemical reaction parameter on the con-centration profile

Cu-waterTiO2-water

ϕ = 005

NB = 07

m = 1γc = 05

Qs = 1 2 3

Qr = 09R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

NT = 07

Figure 10 )e effect of heat source parameter on the temperatureprofile

ϕ = 005

NB = 07NT = 07

m = 1

γc = 05Qs = 1

Qr = 05 09 13

R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 11 )e effect of radiation parameter on the temperatureprofile

10 Mathematical Problems in Engineering

reaction has tendency to increase the mass transfer rate atthe surface plate and it is reduced with the progress of time

Figures 10 and 11 present the heat source parameter Qs

and thermal radiation parameter Qr effects on the tem-perature profiles It is clear from Figure 10 that with anincrease in the heat generation parameter the boundarylayer temperature decreases and as a consequence to de-crease the thermal boundary layer thickness In addition it isclear from Figure 11 that with an increase in the thermalradiation the temperature profiles of the nanofluid increaseIt is well known that the increase in the radiation parameterenhances the energy transport inside the fluid and as aconsequence to increase the heat transfer rate

6 Conclusion

An analysis is done to solve the magnetohydro dynamicsunsteady nanofluid flow through the porous medium past amoving plate in a rotating frame with considering the heatand mass transfer )ree different types of the nanoparticlesnamely copper aluminium oxide and titanium dioxide andwater used as a base nanofluid are considered with variousshapes (spherical cylindrical and brick) )e important ef-fects of nanoparticle shape Brownian and thermophoresisdiffusion thermal radiation ohmic and viscous dissipationshave been included Homotopy perturbation method is ap-plied to solve the governing nonlinear partial differentialequations )e efficiency of the methodology has been vali-dated through comparison with the previously publishedresults and observed a very close agreement It is found that

)e main velocity component decreases with increaseof nanoparticle volume fraction On the contrarythermal properties increase with increase in nano-particle volume fraction)e enhancement of heat transfer rate and mechanicalproperty improvement is high in copper as comparedwith both aluminium oxide and titanium dioxide)ermal conductivity of cylindrical-shaped nanoparticlesis more efficient and possesses better improvement onheat transfer rate than the other shaped onesIncrease in the Brownian and thermophoresis pa-rameter is accompanied by an increase in the thermalboundary layer thicknessIncreasing Hall current and rotation parameters de-creases the main fluid velocity componentHPM has a close agreement and is recommended formany nonlinear problems

Data Availability

)e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] T Hayat R Sajjad A Alsaedi T Muhammad and R EllahildquoOn squeezed flow of couple stress nanofluid between twoparallel platesrdquo Results in Physics vol 7 pp 553ndash561 2017

[2] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition pp 99ndash105San Francisco CA USA 1995

[3] J A Eastman S U S Choi S Li W Yu and L J )ompsonldquoAnomalously increased effective thermal conductivities of eth-ylene glycol-based nanofluids containing copper nanoparticlesrdquoApplied Physics Letters vol 78 no 6 pp 718ndash720 2001

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

[5] M Sheikholeslami M M Rashidi T Hayat and D D GanjildquoFree convection of magnetic nanofluid considering MFDviscosity effectrdquo Journal of Molecular Liquids vol 218pp 393ndash399 2016

[6] W Abbas and E A Sayed ldquoHall current and joule heatingeffects on free convection flow of a nanofluid over a verticalcone in presence of thermal radiationrdquo ermal Sciencevol 21 pp 2603ndash2614 2017

[7] B Mahanthesh B Gireesha I Animasaun T Muhammadand N Shashikumar ldquoMHD flow of SWCNT and MWCNTnanoliquids past a rotating stretchable disk with thermal andexponential space dependent heat sourcerdquo Physica Scriptavol 94 no 8 Article ID 085214 2019

[8] I L Animasaun O K Koriko K S Adegbie et al ldquoCom-parative analysis between 36 nm and 47 nm alumina-waternanofluid flows in the presence of Hall effectrdquo Journal ofermal Analysis and Calorimetry vol 135 no 2 pp 873ndash886 2019

[9] I Ullah S Shafie I Khan and K L Hsiao ldquoBrownian dif-fusion and thermophoresis mechanisms in Casson fluid over amoving wedgerdquo Results in Physics vol 9 pp 183ndash194 2018

[10] P B S Kumar B J Gireesha R S R Gorla andB Mahanthesh ldquoMagnetohydrodynamic flow of Williamsonnanofluid due to an exponentially stretching surface in thepresence of thermal radiation and chemical reactionrdquo Journalof Nanofluids vol 6 no 2 pp 264ndash272 2017

[11] X Zhou Y Jiang X Li et al ldquoNumerical investigation of heattransfer enhancement and entropy generation of naturalconvection in a cavity containing nano liquid-metal fluidrdquoInternational Communications in Heat and Mass Transfervol 106 pp 46ndash54 2019

[12] N Shehzad A Zeeshan R Ellahi and K Vafai ldquoConvectiveheat transfer of nanofluid in a wavy channel Buongiornorsquosmathematical modelrdquo Journal of Molecular Liquids vol 222pp 446ndash455 2016

[13] B Shen L Zheng C Zhang and X Zhang ldquoBioconvectionheat transfer of a nanofluid over a stretching sheet with ve-locity slip and temperature jumprdquo ermal Science vol 21no 6 pp 2347ndash2356 2017

[14] S Jahan H Sakidin R Nazar and I Pop ldquoAnalysis of heattransfer in nanofluid past a convectively heated permeablestretchingshrinking sheet with regression and stability ana-lysesrdquo Results in Physics vol 10 pp 395ndash405 2018

[15] M A A Hamad and I Pop ldquoUnsteady MHD free convectionflow past a vertical permeable flat plate in a rotating frame ofreference with constant heat source in a nanofluidrdquo Heat andMass Transfer vol 47 no 12 pp 1517ndash1524 2011

Mathematical Problems in Engineering 11

[16] M Sheikholeslami and D D Ganji ldquo)ree dimensional heatand mass transfer in a rotating system using nanofluidrdquoPowder Technology vol 253 pp 789ndash796 2014

[17] S M Hussain J Jain G S Seth and M M Rashidi ldquoFreeconvective heat transfer with Hall effects heat absorption andchemical reaction over an accelerated moving plate in a ro-tating systemrdquo Journal of Magnetism and Magnetic Materialsvol 422 pp 112ndash123 2017

[18] X Zhou Y Jiang Y Hou and M Du ldquo)ermocapillaryconvection instability in annular two-layer system undervarious gravity levelsrdquo Microgravity Science and Technologyvol 31 no 5 pp 641ndash648 2019

[19] Y Jiang and X Zhou ldquoHeat transfer and entropy generationanalysis of nanofluids thermocapillary convection around abubble in a cavityrdquo International Communications in Heatand Mass Transfer vol 105 pp 37ndash45 2019

[20] Y Jiang X Zhou and Y Wang ldquoEffects of nanoparticleshapes on heat andmass transfer of nanofluid thermocapillaryconvection around a gas bubblerdquo Microgravity Science andTechnology pp 1ndash11 2019 In press

[21] Y Jiang X Zhou and Y Wang ldquoComprehensive heattransfer performance analysis of nanofluid mixed forced andthermocapillary convection around a gas bubble in mini-channelrdquo International Communications in Heat and MassTransfer vol 110 Article ID 104386 2020

[22] B Mahanthesh B J Gireesha R S R Gorla F M Abbasiand S A Shehzad ldquoNumerical solutions for magnetohy-drodynamic flow of nanofluid over a bidirectional non-linearstretching surface with prescribed surface heat flux bound-aryrdquo Journal of Magnetism and Magnetic Materials vol 417pp 189ndash196 2016

[23] B Mahanthesh B J Gireesha and I L Animasaun ldquoEx-ploration of non-linear thermal radiation and suspendednanoparticles effects on mixed convection boundary layerflow of nanoliquids on a melting vertical surfacerdquo Journal ofNanofluids vol 7 no 5 pp 833ndash843 2018

[24] P S Kumar B Mahanthesh B Gireesha and S ShehzadldquoQuadratic convective flow of radiated nano-jeffrey liquidsubject to multiple convective conditions and Cattaneo-Christov double diffusionrdquo Applied Mathematics and Me-chanics vol 39 no 9 pp 1311ndash1326 2018

[25] J Raza F Mebarek-Oudina and A Chamkha ldquoMagneto-hydrodynamic flow of molybdenum disulfide nanofluid in achannel with shape effectsrdquo Multidiscipline Modeling inMaterials and Structures vol 15 no 4 pp 737ndash757 2019

[26] T Hayat B Ahmed F M Abbasi and A Alsaedi ldquoHy-dromagnetic peristalsis of water based nanofluids with tem-perature dependent viscosity a comparative studyrdquo Journal ofMolecular Liquids vol 234 pp 324ndash329 2017

[27] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filledwith nanofluidsrdquo International Journal of Heat and FluidFlow vol 29 no 5 pp 1326ndash1336 2008

[28] M A El Kot and W Abbas ldquoNumerical technique of bloodflow through catheterized arteries with overlapping stenosisrdquoComputer Methods in Biomechanics and Biomedical Engi-neering vol 20 no 1 pp 45ndash58 2017

[29] H A Attia W Abbas A El-Din Abdin andM A M Abdeen ldquoEffects of ion slip and Hall current onunsteady Couette flow of a dusty fluid through porous mediawith heat transferrdquo High Temperature vol 53 no 6pp 891ndash898 2015

[30] H A Attia W Abbas and M A M Abdeen ldquoIon slip effecton unsteady Couette flow of a dusty fluid in the presence of

uniform suction and injection with heat transferrdquo Journal ofthe Brazilian Society of Mechanical Sciences and Engineeringvol 38 no 8 pp 2381ndash2391 2016

[31] H A Attia W Abbas M A M Abdeen and A A M SaidldquoHeat transfer between two parallel porous plates for Couetteflow under pressure gradient and Hall currentrdquo Sadhanavol 40 no 1 pp 183ndash197 2015

[32] M Sheikholeslami A Shafee M Ramzan and Z Li ldquoIn-vestigation of Lorentz forces and radiation impacts onnanofluid treatment in a porous semi annulus via Darcy lawrdquoJournal of Molecular Liquids vol 272 pp 8ndash14 2018

[33] P S Kumar B Gireesha B Mahanthesh and A J Chamkhaldquo)ermal analysis of nanofluid flow containing gyrotacticmicroorganisms in bioconvection and second-order slip withconvective conditionrdquo Journal of ermal Analysis andCalorimetry vol 136 no 5 pp 1947ndash1957 2019

[34] B Mahanthesh F Mabood B Gireesha and R Gorla ldquoEffectsof chemical reaction and partial slip on the three-dimensionalflow of a nanofluid impinging on an exponentially stretchingsurfacerdquo e European Physical Journal Plus vol 132 no 3p 113 2017

[35] B J Gireesha R S R Gorla and B Mahanthesh ldquoEffect ofsuspended nanoparticles on three-dimensional mhd flowheat and mass transfer of radiating eyring-powell fluid over astretching sheetrdquo Journal of Nanofluids vol 4 no 4pp 474ndash484 2015

[36] J-H He ldquoHomotopy perturbation techniquerdquo ComputerMethods in Applied Mechanics and Engineering vol 178no 3-4 pp 257ndash262 1999

[37] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo Interna-tional Journal of Non-Linear Mechanics vol 35 no 1pp 37ndash43 2000

[38] Y Wu and J-H He ldquoHomotopy perturbation method fornonlinear oscillators with coordinate-dependent massrdquo Re-sults in Physics vol 10 pp 270-271 2018

[39] K Das ldquoFlow and heat transfer characteristics of nanofluids ina rotating framerdquo Alexandria Engineering Journal vol 53no 3 pp 757ndash766 2014

[40] J V R Reddy V Sugunamma N Sandeep and C SulochanaldquoInfluence of chemical reaction radiation and rotation onMHD nanofluid flow past a permeable flat plate in porousmediumrdquo Journal of the Nigerian Mathematical Societyvol 35 no 1 pp 48ndash65 2016

12 Mathematical Problems in Engineering

and TiO2 minus water nanofluid It is clear that increase innanoparticle volume fraction parameter ϕ decreases the fluidvelocity but the temperature profiles increase It is due to thefact that nanofluid density increases with increasing thevolume fraction of nanoparticles and it causes to slow downthe nanofluid flow velocity and increases the thickness of thethermal boundary layer which improves thermal conduc-tivity and increases surface heat transfer rate Also it isobserved that nanofluid Cu minus water heat transfer rate isgreater than Al2O3 and TiO2 rates So nanoparticles havinghigh thermal conductivity enhance heat transfer more thanthe particles having low thermal conductivity On thecontrary it is seen from Tables 2 and 3 that wall skin frictioncoefficient Cf and the Nusselt number NuRex increase withincreasing in nanoparticle volume fraction for both Cu minus

water and Al2O3 minus water)e influence of the nanoparticle shape on the tem-

perature distribution of nanofluid is studied through Fig-ure 4 )is figure indicates that the heat transfer rate ishigher for the cylindrical nanoparticle shape than thespherical and brick nanoparticle shapes for eitherCu minus water Al2O3 minus water and TiO2 minus water nanofluid)erefore the spherical nanoparticle shape tends to dragmore heat from the boundary layer because it has a greatersurface area while this effect is less evident for othernanoparticle shapes )us the use of the cylindrical nano-particle shape has better improvement in effective thermalconductivity when compared with other nanoparticle shapes(spherical and brick)

)e influence of rotation parameter R on the velocityprofiles against z is displayed in Figure 5 It observed that anincrease in rotation parameter leads to decreasing theboundary layer momentum so it decreases the values ofnanofluid velocity From Tables 2 and 3 it can be noted thatthe wall skin friction coefficient Cf increases with an in-crease in the rotation parameter but the Nusselt numberNuRex remains unchanged with both Cu minus water andAl2O3 minus water

Figure 6 presents the influence of Hall current parameterm on main velocity u It is clear that the velocity distributionof nanofluid reduces with an increase in Hall current pa-rameter)is happens naturally due to the fact that when themagnetic field is strengthened it enhances the opposingforce which is represented by the well-known Lorentz force

Figures 7 and 8 show the Brownian and thermophoresiseffects on the temperature profile θ It is seen that thetemperature distribution increases with increasing for theseparameters )is is due to the fact that the thermophoreticand Brownian parameters assist to improve the thermalboundary layer thickness Also it is observed that the heattransfer rate of Cu minus water nanofluid is greater thanTiO2 minus water

Figure 9 depicts the influence of chemical parameter cc

on the concentration profile Γ It is noticed that the increasein concentration profile is accompanied by a decrease in thechemical parameter On the contrary it is seen from Table 4that the Sherwood number minusShx increases due to increasingthe chemical parameter but the Sherwood number decreaseswhile increasing the time t )is implies that the chemical

Cu-waterTiO2-water

ϕ = 005NT = 07NB = 07m = 1

γc = 01 05 1

Qs = 1Qr = 09R = 1

0 2 3 4 5 6 7 81z

1

08

06

04

02

0

12

14

Γ

Figure 9 )e effect of chemical reaction parameter on the con-centration profile

Cu-waterTiO2-water

ϕ = 005

NB = 07

m = 1γc = 05

Qs = 1 2 3

Qr = 09R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

NT = 07

Figure 10 )e effect of heat source parameter on the temperatureprofile

ϕ = 005

NB = 07NT = 07

m = 1

γc = 05Qs = 1

Qr = 05 09 13

R = 1

1

08

06

04

02

0

θ

0 2 3 4 5 6 7 81z

Cu-waterTiO2-water

Figure 11 )e effect of radiation parameter on the temperatureprofile

10 Mathematical Problems in Engineering

reaction has tendency to increase the mass transfer rate atthe surface plate and it is reduced with the progress of time

Figures 10 and 11 present the heat source parameter Qs

and thermal radiation parameter Qr effects on the tem-perature profiles It is clear from Figure 10 that with anincrease in the heat generation parameter the boundarylayer temperature decreases and as a consequence to de-crease the thermal boundary layer thickness In addition it isclear from Figure 11 that with an increase in the thermalradiation the temperature profiles of the nanofluid increaseIt is well known that the increase in the radiation parameterenhances the energy transport inside the fluid and as aconsequence to increase the heat transfer rate

6 Conclusion

An analysis is done to solve the magnetohydro dynamicsunsteady nanofluid flow through the porous medium past amoving plate in a rotating frame with considering the heatand mass transfer )ree different types of the nanoparticlesnamely copper aluminium oxide and titanium dioxide andwater used as a base nanofluid are considered with variousshapes (spherical cylindrical and brick) )e important ef-fects of nanoparticle shape Brownian and thermophoresisdiffusion thermal radiation ohmic and viscous dissipationshave been included Homotopy perturbation method is ap-plied to solve the governing nonlinear partial differentialequations )e efficiency of the methodology has been vali-dated through comparison with the previously publishedresults and observed a very close agreement It is found that

)e main velocity component decreases with increaseof nanoparticle volume fraction On the contrarythermal properties increase with increase in nano-particle volume fraction)e enhancement of heat transfer rate and mechanicalproperty improvement is high in copper as comparedwith both aluminium oxide and titanium dioxide)ermal conductivity of cylindrical-shaped nanoparticlesis more efficient and possesses better improvement onheat transfer rate than the other shaped onesIncrease in the Brownian and thermophoresis pa-rameter is accompanied by an increase in the thermalboundary layer thicknessIncreasing Hall current and rotation parameters de-creases the main fluid velocity componentHPM has a close agreement and is recommended formany nonlinear problems

Data Availability

)e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] T Hayat R Sajjad A Alsaedi T Muhammad and R EllahildquoOn squeezed flow of couple stress nanofluid between twoparallel platesrdquo Results in Physics vol 7 pp 553ndash561 2017

[2] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition pp 99ndash105San Francisco CA USA 1995

[3] J A Eastman S U S Choi S Li W Yu and L J )ompsonldquoAnomalously increased effective thermal conductivities of eth-ylene glycol-based nanofluids containing copper nanoparticlesrdquoApplied Physics Letters vol 78 no 6 pp 718ndash720 2001

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

[5] M Sheikholeslami M M Rashidi T Hayat and D D GanjildquoFree convection of magnetic nanofluid considering MFDviscosity effectrdquo Journal of Molecular Liquids vol 218pp 393ndash399 2016

[6] W Abbas and E A Sayed ldquoHall current and joule heatingeffects on free convection flow of a nanofluid over a verticalcone in presence of thermal radiationrdquo ermal Sciencevol 21 pp 2603ndash2614 2017

[7] B Mahanthesh B Gireesha I Animasaun T Muhammadand N Shashikumar ldquoMHD flow of SWCNT and MWCNTnanoliquids past a rotating stretchable disk with thermal andexponential space dependent heat sourcerdquo Physica Scriptavol 94 no 8 Article ID 085214 2019

[8] I L Animasaun O K Koriko K S Adegbie et al ldquoCom-parative analysis between 36 nm and 47 nm alumina-waternanofluid flows in the presence of Hall effectrdquo Journal ofermal Analysis and Calorimetry vol 135 no 2 pp 873ndash886 2019

[9] I Ullah S Shafie I Khan and K L Hsiao ldquoBrownian dif-fusion and thermophoresis mechanisms in Casson fluid over amoving wedgerdquo Results in Physics vol 9 pp 183ndash194 2018

[10] P B S Kumar B J Gireesha R S R Gorla andB Mahanthesh ldquoMagnetohydrodynamic flow of Williamsonnanofluid due to an exponentially stretching surface in thepresence of thermal radiation and chemical reactionrdquo Journalof Nanofluids vol 6 no 2 pp 264ndash272 2017

[11] X Zhou Y Jiang X Li et al ldquoNumerical investigation of heattransfer enhancement and entropy generation of naturalconvection in a cavity containing nano liquid-metal fluidrdquoInternational Communications in Heat and Mass Transfervol 106 pp 46ndash54 2019

[12] N Shehzad A Zeeshan R Ellahi and K Vafai ldquoConvectiveheat transfer of nanofluid in a wavy channel Buongiornorsquosmathematical modelrdquo Journal of Molecular Liquids vol 222pp 446ndash455 2016

[13] B Shen L Zheng C Zhang and X Zhang ldquoBioconvectionheat transfer of a nanofluid over a stretching sheet with ve-locity slip and temperature jumprdquo ermal Science vol 21no 6 pp 2347ndash2356 2017

[14] S Jahan H Sakidin R Nazar and I Pop ldquoAnalysis of heattransfer in nanofluid past a convectively heated permeablestretchingshrinking sheet with regression and stability ana-lysesrdquo Results in Physics vol 10 pp 395ndash405 2018

[15] M A A Hamad and I Pop ldquoUnsteady MHD free convectionflow past a vertical permeable flat plate in a rotating frame ofreference with constant heat source in a nanofluidrdquo Heat andMass Transfer vol 47 no 12 pp 1517ndash1524 2011

Mathematical Problems in Engineering 11

[16] M Sheikholeslami and D D Ganji ldquo)ree dimensional heatand mass transfer in a rotating system using nanofluidrdquoPowder Technology vol 253 pp 789ndash796 2014

[17] S M Hussain J Jain G S Seth and M M Rashidi ldquoFreeconvective heat transfer with Hall effects heat absorption andchemical reaction over an accelerated moving plate in a ro-tating systemrdquo Journal of Magnetism and Magnetic Materialsvol 422 pp 112ndash123 2017

[18] X Zhou Y Jiang Y Hou and M Du ldquo)ermocapillaryconvection instability in annular two-layer system undervarious gravity levelsrdquo Microgravity Science and Technologyvol 31 no 5 pp 641ndash648 2019

[19] Y Jiang and X Zhou ldquoHeat transfer and entropy generationanalysis of nanofluids thermocapillary convection around abubble in a cavityrdquo International Communications in Heatand Mass Transfer vol 105 pp 37ndash45 2019

[20] Y Jiang X Zhou and Y Wang ldquoEffects of nanoparticleshapes on heat andmass transfer of nanofluid thermocapillaryconvection around a gas bubblerdquo Microgravity Science andTechnology pp 1ndash11 2019 In press

[21] Y Jiang X Zhou and Y Wang ldquoComprehensive heattransfer performance analysis of nanofluid mixed forced andthermocapillary convection around a gas bubble in mini-channelrdquo International Communications in Heat and MassTransfer vol 110 Article ID 104386 2020

[22] B Mahanthesh B J Gireesha R S R Gorla F M Abbasiand S A Shehzad ldquoNumerical solutions for magnetohy-drodynamic flow of nanofluid over a bidirectional non-linearstretching surface with prescribed surface heat flux bound-aryrdquo Journal of Magnetism and Magnetic Materials vol 417pp 189ndash196 2016

[23] B Mahanthesh B J Gireesha and I L Animasaun ldquoEx-ploration of non-linear thermal radiation and suspendednanoparticles effects on mixed convection boundary layerflow of nanoliquids on a melting vertical surfacerdquo Journal ofNanofluids vol 7 no 5 pp 833ndash843 2018

[24] P S Kumar B Mahanthesh B Gireesha and S ShehzadldquoQuadratic convective flow of radiated nano-jeffrey liquidsubject to multiple convective conditions and Cattaneo-Christov double diffusionrdquo Applied Mathematics and Me-chanics vol 39 no 9 pp 1311ndash1326 2018

[25] J Raza F Mebarek-Oudina and A Chamkha ldquoMagneto-hydrodynamic flow of molybdenum disulfide nanofluid in achannel with shape effectsrdquo Multidiscipline Modeling inMaterials and Structures vol 15 no 4 pp 737ndash757 2019

[26] T Hayat B Ahmed F M Abbasi and A Alsaedi ldquoHy-dromagnetic peristalsis of water based nanofluids with tem-perature dependent viscosity a comparative studyrdquo Journal ofMolecular Liquids vol 234 pp 324ndash329 2017

[27] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filledwith nanofluidsrdquo International Journal of Heat and FluidFlow vol 29 no 5 pp 1326ndash1336 2008

[28] M A El Kot and W Abbas ldquoNumerical technique of bloodflow through catheterized arteries with overlapping stenosisrdquoComputer Methods in Biomechanics and Biomedical Engi-neering vol 20 no 1 pp 45ndash58 2017

[29] H A Attia W Abbas A El-Din Abdin andM A M Abdeen ldquoEffects of ion slip and Hall current onunsteady Couette flow of a dusty fluid through porous mediawith heat transferrdquo High Temperature vol 53 no 6pp 891ndash898 2015

[30] H A Attia W Abbas and M A M Abdeen ldquoIon slip effecton unsteady Couette flow of a dusty fluid in the presence of

uniform suction and injection with heat transferrdquo Journal ofthe Brazilian Society of Mechanical Sciences and Engineeringvol 38 no 8 pp 2381ndash2391 2016

[31] H A Attia W Abbas M A M Abdeen and A A M SaidldquoHeat transfer between two parallel porous plates for Couetteflow under pressure gradient and Hall currentrdquo Sadhanavol 40 no 1 pp 183ndash197 2015

[32] M Sheikholeslami A Shafee M Ramzan and Z Li ldquoIn-vestigation of Lorentz forces and radiation impacts onnanofluid treatment in a porous semi annulus via Darcy lawrdquoJournal of Molecular Liquids vol 272 pp 8ndash14 2018

[33] P S Kumar B Gireesha B Mahanthesh and A J Chamkhaldquo)ermal analysis of nanofluid flow containing gyrotacticmicroorganisms in bioconvection and second-order slip withconvective conditionrdquo Journal of ermal Analysis andCalorimetry vol 136 no 5 pp 1947ndash1957 2019

[34] B Mahanthesh F Mabood B Gireesha and R Gorla ldquoEffectsof chemical reaction and partial slip on the three-dimensionalflow of a nanofluid impinging on an exponentially stretchingsurfacerdquo e European Physical Journal Plus vol 132 no 3p 113 2017

[35] B J Gireesha R S R Gorla and B Mahanthesh ldquoEffect ofsuspended nanoparticles on three-dimensional mhd flowheat and mass transfer of radiating eyring-powell fluid over astretching sheetrdquo Journal of Nanofluids vol 4 no 4pp 474ndash484 2015

[36] J-H He ldquoHomotopy perturbation techniquerdquo ComputerMethods in Applied Mechanics and Engineering vol 178no 3-4 pp 257ndash262 1999

[37] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo Interna-tional Journal of Non-Linear Mechanics vol 35 no 1pp 37ndash43 2000

[38] Y Wu and J-H He ldquoHomotopy perturbation method fornonlinear oscillators with coordinate-dependent massrdquo Re-sults in Physics vol 10 pp 270-271 2018

[39] K Das ldquoFlow and heat transfer characteristics of nanofluids ina rotating framerdquo Alexandria Engineering Journal vol 53no 3 pp 757ndash766 2014

[40] J V R Reddy V Sugunamma N Sandeep and C SulochanaldquoInfluence of chemical reaction radiation and rotation onMHD nanofluid flow past a permeable flat plate in porousmediumrdquo Journal of the Nigerian Mathematical Societyvol 35 no 1 pp 48ndash65 2016

12 Mathematical Problems in Engineering

reaction has tendency to increase the mass transfer rate atthe surface plate and it is reduced with the progress of time

Figures 10 and 11 present the heat source parameter Qs

and thermal radiation parameter Qr effects on the tem-perature profiles It is clear from Figure 10 that with anincrease in the heat generation parameter the boundarylayer temperature decreases and as a consequence to de-crease the thermal boundary layer thickness In addition it isclear from Figure 11 that with an increase in the thermalradiation the temperature profiles of the nanofluid increaseIt is well known that the increase in the radiation parameterenhances the energy transport inside the fluid and as aconsequence to increase the heat transfer rate

6 Conclusion

An analysis is done to solve the magnetohydro dynamicsunsteady nanofluid flow through the porous medium past amoving plate in a rotating frame with considering the heatand mass transfer )ree different types of the nanoparticlesnamely copper aluminium oxide and titanium dioxide andwater used as a base nanofluid are considered with variousshapes (spherical cylindrical and brick) )e important ef-fects of nanoparticle shape Brownian and thermophoresisdiffusion thermal radiation ohmic and viscous dissipationshave been included Homotopy perturbation method is ap-plied to solve the governing nonlinear partial differentialequations )e efficiency of the methodology has been vali-dated through comparison with the previously publishedresults and observed a very close agreement It is found that

)e main velocity component decreases with increaseof nanoparticle volume fraction On the contrarythermal properties increase with increase in nano-particle volume fraction)e enhancement of heat transfer rate and mechanicalproperty improvement is high in copper as comparedwith both aluminium oxide and titanium dioxide)ermal conductivity of cylindrical-shaped nanoparticlesis more efficient and possesses better improvement onheat transfer rate than the other shaped onesIncrease in the Brownian and thermophoresis pa-rameter is accompanied by an increase in the thermalboundary layer thicknessIncreasing Hall current and rotation parameters de-creases the main fluid velocity componentHPM has a close agreement and is recommended formany nonlinear problems

Data Availability

)e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] T Hayat R Sajjad A Alsaedi T Muhammad and R EllahildquoOn squeezed flow of couple stress nanofluid between twoparallel platesrdquo Results in Physics vol 7 pp 553ndash561 2017

[2] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition pp 99ndash105San Francisco CA USA 1995

[3] J A Eastman S U S Choi S Li W Yu and L J )ompsonldquoAnomalously increased effective thermal conductivities of eth-ylene glycol-based nanofluids containing copper nanoparticlesrdquoApplied Physics Letters vol 78 no 6 pp 718ndash720 2001

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

[5] M Sheikholeslami M M Rashidi T Hayat and D D GanjildquoFree convection of magnetic nanofluid considering MFDviscosity effectrdquo Journal of Molecular Liquids vol 218pp 393ndash399 2016

[6] W Abbas and E A Sayed ldquoHall current and joule heatingeffects on free convection flow of a nanofluid over a verticalcone in presence of thermal radiationrdquo ermal Sciencevol 21 pp 2603ndash2614 2017

[7] B Mahanthesh B Gireesha I Animasaun T Muhammadand N Shashikumar ldquoMHD flow of SWCNT and MWCNTnanoliquids past a rotating stretchable disk with thermal andexponential space dependent heat sourcerdquo Physica Scriptavol 94 no 8 Article ID 085214 2019

[8] I L Animasaun O K Koriko K S Adegbie et al ldquoCom-parative analysis between 36 nm and 47 nm alumina-waternanofluid flows in the presence of Hall effectrdquo Journal ofermal Analysis and Calorimetry vol 135 no 2 pp 873ndash886 2019

[9] I Ullah S Shafie I Khan and K L Hsiao ldquoBrownian dif-fusion and thermophoresis mechanisms in Casson fluid over amoving wedgerdquo Results in Physics vol 9 pp 183ndash194 2018

[10] P B S Kumar B J Gireesha R S R Gorla andB Mahanthesh ldquoMagnetohydrodynamic flow of Williamsonnanofluid due to an exponentially stretching surface in thepresence of thermal radiation and chemical reactionrdquo Journalof Nanofluids vol 6 no 2 pp 264ndash272 2017

[11] X Zhou Y Jiang X Li et al ldquoNumerical investigation of heattransfer enhancement and entropy generation of naturalconvection in a cavity containing nano liquid-metal fluidrdquoInternational Communications in Heat and Mass Transfervol 106 pp 46ndash54 2019

[12] N Shehzad A Zeeshan R Ellahi and K Vafai ldquoConvectiveheat transfer of nanofluid in a wavy channel Buongiornorsquosmathematical modelrdquo Journal of Molecular Liquids vol 222pp 446ndash455 2016

[13] B Shen L Zheng C Zhang and X Zhang ldquoBioconvectionheat transfer of a nanofluid over a stretching sheet with ve-locity slip and temperature jumprdquo ermal Science vol 21no 6 pp 2347ndash2356 2017

[14] S Jahan H Sakidin R Nazar and I Pop ldquoAnalysis of heattransfer in nanofluid past a convectively heated permeablestretchingshrinking sheet with regression and stability ana-lysesrdquo Results in Physics vol 10 pp 395ndash405 2018

[15] M A A Hamad and I Pop ldquoUnsteady MHD free convectionflow past a vertical permeable flat plate in a rotating frame ofreference with constant heat source in a nanofluidrdquo Heat andMass Transfer vol 47 no 12 pp 1517ndash1524 2011

Mathematical Problems in Engineering 11

[16] M Sheikholeslami and D D Ganji ldquo)ree dimensional heatand mass transfer in a rotating system using nanofluidrdquoPowder Technology vol 253 pp 789ndash796 2014

[17] S M Hussain J Jain G S Seth and M M Rashidi ldquoFreeconvective heat transfer with Hall effects heat absorption andchemical reaction over an accelerated moving plate in a ro-tating systemrdquo Journal of Magnetism and Magnetic Materialsvol 422 pp 112ndash123 2017

[18] X Zhou Y Jiang Y Hou and M Du ldquo)ermocapillaryconvection instability in annular two-layer system undervarious gravity levelsrdquo Microgravity Science and Technologyvol 31 no 5 pp 641ndash648 2019

[19] Y Jiang and X Zhou ldquoHeat transfer and entropy generationanalysis of nanofluids thermocapillary convection around abubble in a cavityrdquo International Communications in Heatand Mass Transfer vol 105 pp 37ndash45 2019

[20] Y Jiang X Zhou and Y Wang ldquoEffects of nanoparticleshapes on heat andmass transfer of nanofluid thermocapillaryconvection around a gas bubblerdquo Microgravity Science andTechnology pp 1ndash11 2019 In press

[21] Y Jiang X Zhou and Y Wang ldquoComprehensive heattransfer performance analysis of nanofluid mixed forced andthermocapillary convection around a gas bubble in mini-channelrdquo International Communications in Heat and MassTransfer vol 110 Article ID 104386 2020

[22] B Mahanthesh B J Gireesha R S R Gorla F M Abbasiand S A Shehzad ldquoNumerical solutions for magnetohy-drodynamic flow of nanofluid over a bidirectional non-linearstretching surface with prescribed surface heat flux bound-aryrdquo Journal of Magnetism and Magnetic Materials vol 417pp 189ndash196 2016

[23] B Mahanthesh B J Gireesha and I L Animasaun ldquoEx-ploration of non-linear thermal radiation and suspendednanoparticles effects on mixed convection boundary layerflow of nanoliquids on a melting vertical surfacerdquo Journal ofNanofluids vol 7 no 5 pp 833ndash843 2018

[24] P S Kumar B Mahanthesh B Gireesha and S ShehzadldquoQuadratic convective flow of radiated nano-jeffrey liquidsubject to multiple convective conditions and Cattaneo-Christov double diffusionrdquo Applied Mathematics and Me-chanics vol 39 no 9 pp 1311ndash1326 2018

[25] J Raza F Mebarek-Oudina and A Chamkha ldquoMagneto-hydrodynamic flow of molybdenum disulfide nanofluid in achannel with shape effectsrdquo Multidiscipline Modeling inMaterials and Structures vol 15 no 4 pp 737ndash757 2019

[26] T Hayat B Ahmed F M Abbasi and A Alsaedi ldquoHy-dromagnetic peristalsis of water based nanofluids with tem-perature dependent viscosity a comparative studyrdquo Journal ofMolecular Liquids vol 234 pp 324ndash329 2017

[27] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filledwith nanofluidsrdquo International Journal of Heat and FluidFlow vol 29 no 5 pp 1326ndash1336 2008

[28] M A El Kot and W Abbas ldquoNumerical technique of bloodflow through catheterized arteries with overlapping stenosisrdquoComputer Methods in Biomechanics and Biomedical Engi-neering vol 20 no 1 pp 45ndash58 2017

[29] H A Attia W Abbas A El-Din Abdin andM A M Abdeen ldquoEffects of ion slip and Hall current onunsteady Couette flow of a dusty fluid through porous mediawith heat transferrdquo High Temperature vol 53 no 6pp 891ndash898 2015

[30] H A Attia W Abbas and M A M Abdeen ldquoIon slip effecton unsteady Couette flow of a dusty fluid in the presence of

uniform suction and injection with heat transferrdquo Journal ofthe Brazilian Society of Mechanical Sciences and Engineeringvol 38 no 8 pp 2381ndash2391 2016

[31] H A Attia W Abbas M A M Abdeen and A A M SaidldquoHeat transfer between two parallel porous plates for Couetteflow under pressure gradient and Hall currentrdquo Sadhanavol 40 no 1 pp 183ndash197 2015

[32] M Sheikholeslami A Shafee M Ramzan and Z Li ldquoIn-vestigation of Lorentz forces and radiation impacts onnanofluid treatment in a porous semi annulus via Darcy lawrdquoJournal of Molecular Liquids vol 272 pp 8ndash14 2018

[33] P S Kumar B Gireesha B Mahanthesh and A J Chamkhaldquo)ermal analysis of nanofluid flow containing gyrotacticmicroorganisms in bioconvection and second-order slip withconvective conditionrdquo Journal of ermal Analysis andCalorimetry vol 136 no 5 pp 1947ndash1957 2019

[34] B Mahanthesh F Mabood B Gireesha and R Gorla ldquoEffectsof chemical reaction and partial slip on the three-dimensionalflow of a nanofluid impinging on an exponentially stretchingsurfacerdquo e European Physical Journal Plus vol 132 no 3p 113 2017

[35] B J Gireesha R S R Gorla and B Mahanthesh ldquoEffect ofsuspended nanoparticles on three-dimensional mhd flowheat and mass transfer of radiating eyring-powell fluid over astretching sheetrdquo Journal of Nanofluids vol 4 no 4pp 474ndash484 2015

[36] J-H He ldquoHomotopy perturbation techniquerdquo ComputerMethods in Applied Mechanics and Engineering vol 178no 3-4 pp 257ndash262 1999

[37] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo Interna-tional Journal of Non-Linear Mechanics vol 35 no 1pp 37ndash43 2000

[38] Y Wu and J-H He ldquoHomotopy perturbation method fornonlinear oscillators with coordinate-dependent massrdquo Re-sults in Physics vol 10 pp 270-271 2018

[39] K Das ldquoFlow and heat transfer characteristics of nanofluids ina rotating framerdquo Alexandria Engineering Journal vol 53no 3 pp 757ndash766 2014

[40] J V R Reddy V Sugunamma N Sandeep and C SulochanaldquoInfluence of chemical reaction radiation and rotation onMHD nanofluid flow past a permeable flat plate in porousmediumrdquo Journal of the Nigerian Mathematical Societyvol 35 no 1 pp 48ndash65 2016

12 Mathematical Problems in Engineering

[16] M Sheikholeslami and D D Ganji ldquo)ree dimensional heatand mass transfer in a rotating system using nanofluidrdquoPowder Technology vol 253 pp 789ndash796 2014

[17] S M Hussain J Jain G S Seth and M M Rashidi ldquoFreeconvective heat transfer with Hall effects heat absorption andchemical reaction over an accelerated moving plate in a ro-tating systemrdquo Journal of Magnetism and Magnetic Materialsvol 422 pp 112ndash123 2017

[18] X Zhou Y Jiang Y Hou and M Du ldquo)ermocapillaryconvection instability in annular two-layer system undervarious gravity levelsrdquo Microgravity Science and Technologyvol 31 no 5 pp 641ndash648 2019

[19] Y Jiang and X Zhou ldquoHeat transfer and entropy generationanalysis of nanofluids thermocapillary convection around abubble in a cavityrdquo International Communications in Heatand Mass Transfer vol 105 pp 37ndash45 2019

[20] Y Jiang X Zhou and Y Wang ldquoEffects of nanoparticleshapes on heat andmass transfer of nanofluid thermocapillaryconvection around a gas bubblerdquo Microgravity Science andTechnology pp 1ndash11 2019 In press

[21] Y Jiang X Zhou and Y Wang ldquoComprehensive heattransfer performance analysis of nanofluid mixed forced andthermocapillary convection around a gas bubble in mini-channelrdquo International Communications in Heat and MassTransfer vol 110 Article ID 104386 2020

[22] B Mahanthesh B J Gireesha R S R Gorla F M Abbasiand S A Shehzad ldquoNumerical solutions for magnetohy-drodynamic flow of nanofluid over a bidirectional non-linearstretching surface with prescribed surface heat flux bound-aryrdquo Journal of Magnetism and Magnetic Materials vol 417pp 189ndash196 2016

[23] B Mahanthesh B J Gireesha and I L Animasaun ldquoEx-ploration of non-linear thermal radiation and suspendednanoparticles effects on mixed convection boundary layerflow of nanoliquids on a melting vertical surfacerdquo Journal ofNanofluids vol 7 no 5 pp 833ndash843 2018

[24] P S Kumar B Mahanthesh B Gireesha and S ShehzadldquoQuadratic convective flow of radiated nano-jeffrey liquidsubject to multiple convective conditions and Cattaneo-Christov double diffusionrdquo Applied Mathematics and Me-chanics vol 39 no 9 pp 1311ndash1326 2018

[25] J Raza F Mebarek-Oudina and A Chamkha ldquoMagneto-hydrodynamic flow of molybdenum disulfide nanofluid in achannel with shape effectsrdquo Multidiscipline Modeling inMaterials and Structures vol 15 no 4 pp 737ndash757 2019

[26] T Hayat B Ahmed F M Abbasi and A Alsaedi ldquoHy-dromagnetic peristalsis of water based nanofluids with tem-perature dependent viscosity a comparative studyrdquo Journal ofMolecular Liquids vol 234 pp 324ndash329 2017

[27] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filledwith nanofluidsrdquo International Journal of Heat and FluidFlow vol 29 no 5 pp 1326ndash1336 2008

[28] M A El Kot and W Abbas ldquoNumerical technique of bloodflow through catheterized arteries with overlapping stenosisrdquoComputer Methods in Biomechanics and Biomedical Engi-neering vol 20 no 1 pp 45ndash58 2017

[29] H A Attia W Abbas A El-Din Abdin andM A M Abdeen ldquoEffects of ion slip and Hall current onunsteady Couette flow of a dusty fluid through porous mediawith heat transferrdquo High Temperature vol 53 no 6pp 891ndash898 2015

[30] H A Attia W Abbas and M A M Abdeen ldquoIon slip effecton unsteady Couette flow of a dusty fluid in the presence of

uniform suction and injection with heat transferrdquo Journal ofthe Brazilian Society of Mechanical Sciences and Engineeringvol 38 no 8 pp 2381ndash2391 2016

[31] H A Attia W Abbas M A M Abdeen and A A M SaidldquoHeat transfer between two parallel porous plates for Couetteflow under pressure gradient and Hall currentrdquo Sadhanavol 40 no 1 pp 183ndash197 2015

[32] M Sheikholeslami A Shafee M Ramzan and Z Li ldquoIn-vestigation of Lorentz forces and radiation impacts onnanofluid treatment in a porous semi annulus via Darcy lawrdquoJournal of Molecular Liquids vol 272 pp 8ndash14 2018

[33] P S Kumar B Gireesha B Mahanthesh and A J Chamkhaldquo)ermal analysis of nanofluid flow containing gyrotacticmicroorganisms in bioconvection and second-order slip withconvective conditionrdquo Journal of ermal Analysis andCalorimetry vol 136 no 5 pp 1947ndash1957 2019

[34] B Mahanthesh F Mabood B Gireesha and R Gorla ldquoEffectsof chemical reaction and partial slip on the three-dimensionalflow of a nanofluid impinging on an exponentially stretchingsurfacerdquo e European Physical Journal Plus vol 132 no 3p 113 2017

[35] B J Gireesha R S R Gorla and B Mahanthesh ldquoEffect ofsuspended nanoparticles on three-dimensional mhd flowheat and mass transfer of radiating eyring-powell fluid over astretching sheetrdquo Journal of Nanofluids vol 4 no 4pp 474ndash484 2015

[36] J-H He ldquoHomotopy perturbation techniquerdquo ComputerMethods in Applied Mechanics and Engineering vol 178no 3-4 pp 257ndash262 1999

[37] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo Interna-tional Journal of Non-Linear Mechanics vol 35 no 1pp 37ndash43 2000

[38] Y Wu and J-H He ldquoHomotopy perturbation method fornonlinear oscillators with coordinate-dependent massrdquo Re-sults in Physics vol 10 pp 270-271 2018

[39] K Das ldquoFlow and heat transfer characteristics of nanofluids ina rotating framerdquo Alexandria Engineering Journal vol 53no 3 pp 757ndash766 2014

[40] J V R Reddy V Sugunamma N Sandeep and C SulochanaldquoInfluence of chemical reaction radiation and rotation onMHD nanofluid flow past a permeable flat plate in porousmediumrdquo Journal of the Nigerian Mathematical Societyvol 35 no 1 pp 48ndash65 2016

12 Mathematical Problems in Engineering