# Heat and Mass Transfer Analysis of Nanofluid Flow Based onCu, Al2O3, and TiO2 over a Moving Rotating Plate and Impact of Various Nanoparticle Shapes

Authors: W. Abbas; M. M. Magdy
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The study of rotating nanofluid flows has a vital role in several applications such as in food processing, rotating machinery, cooling systems, and chemical fluid. The aims of the present work are to improve the thermophysical properties of convective flow and heat transfer for unsteady nanofluid past a moving rotating plate in the presence of ohmic, viscous dissipations, Brownian, and thermophoresis diffusion. The system is strained under the effect of strong magnetic field, and then the Hall current is considered. For this investigation, three different types of the nanoparticlesCu (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. The system governing equations are solved semianalytically using homotopy perturbation technique. In order to validate the present work, different comparisons are made under some special cases with previously published results and found an excellent agreement. It is observed that the shape of nanoparticles plays a substantial role to significantly determine the flow behaviour. Also, it can be found that the use of the cylindrical nanoparticle shape has better improvement for heat transfer rate compared with the other nanoparticle shapes.


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## Body

## 1. Introduction

In recent years, investigation of nanofluid flow has gained considerable interest by the researchers owing to increase in the implementations in different fields of technology, science, biomechanics, chemical, and nuclear industries. Nanofluid can be applied to engineering problems such as solar energy collection, heat exchangers, and engine cooling [1]. The term of nanofluids means the addition of small quantity of nanometer-sized particles nominally less than 100 nm into base fluids like oil, water, biofluids, ethylene, and lubricants. This term was first introduced by Choi [2]; he added small nanoparticles in a base fluid to increase the thermal conductivity. Eastmann et al. [3] discussed the thermal conductivity in ethylene glycol nanofluid-based copper nanoparticles. Since then, the growth of nanofluid mathematical modelling advances by demonstrating the characteristics of thermophoresis and Brownian diffusion. Chamkha et al. [4] studied the thermal radiation effect on nanofluid boundary layer flow past a vertical cone. Time-dependent viscosity effect on free convection heat transfer of magnetohydrodynamic nanofluid was investigated by Sheikholeslami et al. [5]. Hayat et al. [1] investigated the flow with heat and mass transfer characteristics in magnetohydrodynamic (MHD) squeezing flow between two surfaces. The effect of Joule heating and Hall current on nanofluid flow along a vertical cone with heat and mass transfer was investigated by Abbas and Sayed [6]. Mahanthesh et al. [7] analyzed the effect of the novel exponential space-dependent heat source on magnetohydrodynamic nanofluid flow past a stretchable rotating disk with cross-diffusion and the convective condition. Later, many researchers have been engaged in recent developments on nanofluids [8–11].Study of convective heat transfer and flow through magnetohydrodynamic nanofluid in the recent years has received significant attraction by many researchers for their useful applications in both engineering and sciences such as reactors, cooling or heating processes, and solar energy. Shehzad et al. [12] studied the convective heat transfer effects of nanofluid flow on a wavy channel with Brownian and thermophoresis diffusion effect. Heat transfer of a nanofluid over a stretching sheet with considering the effects of temperature jump, slip velocity, and radiation was investigated by Shen et al. [13]. Jahan et al. [14] reported the nanofluid flow with stability and regression analysis of a permeable stretching sheet over the heated surface. In other studies, the convective heat transfer through nanofluid bounded by a moving rotating frame is investigated due to the increase in engineering and technological process applications. Hamad and Pop [15] studied heat transfer of unsteady nanofluid flow for a moving oscillatory plate with constant heat source. Nanofluid flow through two plates with rotating system in presence of magnetic field was reported by Sheikholeslami and Ganji [16]. Hussain et al. [17] presented the analytical solution of unsteady flow past a moving rotating plate considering the effects of Hall current and chemical reaction. Many investigators studied the thermocapillary convective with heat transfer on the fluid/nanofluid flow field under different categories and conditions [18–21].The thermal radiation effect has attracted much attention for researchers for its enormous area of scientific and engineering applications such as electric power, food, solar cell panels, and medical industry [22]. Mahanthesh et al. [23] presented effects of nonlinear thermal radiation and suspended nanoparticles on mixed convection boundary layer nanofluid flow past a melting vertical surface. Kumar et al. [24] studied thermal analysis of bioconvection magnetonanofluid flow containing gyrotactic microorganisms with second-order velocity slip and convective conditions. Raza et al. [25] presented the combined effects of thermal radiation and magnetic field of molybdenum disulfide nanofluid on a channel with shape effects.Based on published literature, it is observed that the motion of nanofluid flow through a moving rotating plate has a significant potential role in several applications such as food processing, polymer processing, rotating machinery, cooling of metallic electronic component surfaces, and chemical fluid. Therefore, the objective of the present study is to investigate the effects of the nanoparticle shape, Brownian, and thermophoresis diffusion on the motion of MHD unsteady nanofluid flow through past a moving rotating plate in the presence of ohmic, viscous dissipations, and thermal radiation. Three different types of the nanoparticles are considered. These types are copper, aluminium oxide, and titanium dioxide with various shapes (spherical, cylindrical, and brick), and water is used as a base nanofluid. The problem is formulated and solved semianalytical using homotopy perturbation technique. Physical parameters are displayed and analyzed, and pertinent results are discussed in detail using graphs. Different comparisons are made under some special cases with previously published results which are conducted to validate the current results and prove efficiency of the methodology.


## 2. Mathematical Model and Formulation

### 2.1. Problem Description

Consider unsteady MHD nanofluid flow of ambient temperatureT∞ and concentration C∞ with heat and mass transfer through the porous medium past a moving flat plate in a rotating frame. The coordinate system x,y,z is selected in such a way that x−axis is taken along the plate, z−axis is perpendicular to it, and y−axis is normal to the xz plane. The physical model is described in Figure 1. Three different kinds of nanoparticles, Cu (copper), Al2O3 (aluminium oxide), and TiO2 (titanium dioxide), with water used as a base nanofluid are considered. The fluid and nanoparticles are assumed to have no slip between them, and both are in thermal equilibrium. Table 1 clearly lists the thermophysical properties of the base fluid and different nanoparticles [26, 27]. In this work, there are three shapes of nanoparticles which are spherical, cylindrical, and brick employed into account. Also, the model is assumed rotating about z−axis with constant angular velocity Ω. The fluid is permeated by applied time-dependent magnetic field which takes the form Bt=Ge−αt which is taken to be acting along the positive z-direction, where G and α are constants. It is assumed that the magnetic Reynolds number is to be very small so that the induced magnetic field can be neglected, and there is no applied external electric field. It is also assumed that the frequency of electron-atom collision to be relatively high so that the Hall effect cannot be neglected.Figure 1
Physical model of the problem.Table 1
Thermophysical properties [26, 27].
Thermophysical
Water
Copper
Aluminium oxide
Titanium dioxide
properties
H
2
O
Cu
Al
2
O
3
TiO
2
C
p
J
/
kgK
4179.00
385.00
765.00
686.00
ρ
kg
/
m
3
997.10
8933.00
3970.00
4250.00
K
W
/
mK
0.613
400.00
46.00
8.90
β
∗
×
10
−
5 (1/K)
21
1.67
0.63
1.05
σ
S
/
m
5.50
×
10
−
6
1163.10
×
10
7
131.70
×
10
7
2.38
×
10
6


### 2.2. Governing Equations

With above assumptions, the nanofluid flow is governed by conservations of continuity, momentum, energy, and concentration equations as(1)∇⋅V=0,∂V∂t+V⋅∇V−Ω×V=1ρnf∇μ⋅∇V+J×Bt+FT+FC+Fp,∂T∂t+V⋅∇T=1ρCpnf∇knf⋅∇T+J⋅Jσnf+DmkTρnfCs∇⋅∇CρsCpsDB∇T⋅∇C+DTT∞∇T⋅∇T+Q0T−T∞+φ−∇⋅qr,∂C∂t+V⋅∇C=∇⋅D∇C+DTT∞∇T+kcC−C∞,where ρnf is the nanofluid density, V denotes the velocity vector which has the following components: V=uz,t,vz,t,w0, μ is the nanofluid viscosity, FT is the thermal expansion term due to temperature difference, FC denotes the thermal expansion term due to concentration difference, FP is the porous medium term, Cpnf denotes the nanofluid specific heat at constant pressure, knf denotes the nanofluid thermal conductivity, σnf is the electric conductivity, Dm denotes the mass diffusivity, kT is the thermal diffusivity, Cs denotes the susceptibility of concentration, ρsCps denotes the nanoparticle heat capacity, and DB and DT denote the Brownian and thermophoresis coefficient, respectively. Q0 is the source constant, and φ is the viscous dissipation term. qr is the radiative heat flux, Tm is the mean temperature, and kc is the chemical reaction coefficient. J denotes the electric current density which is formulated by generalized Ohm’s law including Hall current as [28–30](2)J=σnfV×B−ξJ×B,where ξ is the Hall factor.Furthermore, radiative heat flux is formulated by Rosseland approximation as(3)qr=−4σ∗3α∗∂T4∂z.σ
∗ and α∗ denote the constant Stefan–Boltzman and mean absorption coefficient, respectively. Consider temperature differences are sufficiently small, and the term T4 can be expressed as a linear function into the Taylor series about T∞, and after neglecting higher-order terms, it yields(4)T4≅4T∞3T−3T∞4.Under the assumptions made above, the governing equations are as follows:(5)ρnf∂u∂t+w0∂u∂z+2Ωv=μnf∂2u∂z2−σnfB2t1+m2u+mv−μnfkpu+ρnfβTnf∗gT−T∞+ρnfβCnf∗gC−C∞,(6)ρnf∂v∂t+w0∂v∂z−2Ωu=μnf∂2v∂z2−σnfB2t1+m2v−mu−μnfkpv,(7)ρnfCPnf∂T∂t+w0∂T∂z=knf∂2T∂z2+μnf∂u∂z2+σnfB2t1+m2u2+v2+QT−T∞+DmkTρnfCs∂2C∂z2+ρsCPsDB∂C∂z∂T∂z+DTT∞∂T∂z2−16σ∗T∞33α∗∂2T∂z2,(8)∂C∂t+w0∂C∂z=D∂2C∂z2+kcC−C∞+DTT∞∂2T∂z2,where βTnf∗ denotes the nanofluid thermal expansion coefficient due to temperature difference, βCnf∗ denotes the nanofluid thermal expansion coefficient due to concentration difference, and m=σnfξB0 is the Hall parameter [31].The boundary conditions are(9)t≤0:u=0,w=0,T=T∞,C=C∞,z=0,t>0:u=U0,w=0,T=Tw,C=Cwz⟶∞,t>0:u⟶0,w⟶0,T⟶T∞,C⟶C∞.


### 2.3. Thermophysical Properties of Nanofluid

The nanofluid thermophysical properties were calculated using the following forms [6, 26]:Nanofluid effective dynamic viscosity, effective density, and heat capacity are given as(10)μnf=μf1−ϕ2.5,J1=ρfρnf=11−ϕ+ϕρs/ρf,J2=ρCpfρCpnf=11−ϕ+ϕρCps/ρCpf,whereϕ denotes the nanofluid volume fraction. Subscripts f, nf, and s denote the base fluid, nanofluid, and solid particle, respectively.Also, nanofluid effective electric conductivity and effective thermal expansion coefficient are given as(11)J3=σnfσf=1+3σs/σf−1ϕσs/σf+2−σs/σf−1ϕ,J4=ρβ∗nfρβ∗f=1−ϕ+ϕρβ∗sρβ∗f.Effective thermal conductivity and effect of nanoparticle shapes are considered by using the Hamilton and Crosser model as [26](12)J5=knfkf=ks+np−1kf−np−1ϕkf−ksks+np−1kf+ϕkf−ks,where np indicates the empirical nanoparticle shape factor which is given by(13)np=3Ψs.Here, Ψs is the sphericity of the nanoparticles which is defined as the ratio of the surface area of a sphere to the area of the particle with a volume equal to that of a particle and has value 1 np=3 for spherical-shaped nanoparticles and 0.5 np=6 for spherical-shaped nanoparticles. The characteristics of shape factor np for various nanoparticle shapes have been displayed in Table 2 [32].Table 2
Shape factor for various nanoparticle shapes [32].
Spherical
Cylindrical
Brick
Shapes of nanoparticles
Shape factornp
3
6
3.7Introducing the following nondimensional variables,(14)x^,y^,z^=x,y,zU0νf,t^=U02νft,u^=uU0,w^=wU0,θ=T−T∞Tw−T∞,Γ=C−C∞Cw−C∞.Governing equations (5)–(8) are reduced to nondimensional form after dropping the cap as(15)∂u∂t+s∂u∂z+Rv=J11−ϕ2.5∂2u∂z2−Ha2J1J31+m2u+mv+GrJ1J4θ+GcJ1J4Γ−MJ11−ϕ2.5u,(16)∂v∂t+s∂v∂z−Ru=J11−ϕ2.5∂2v∂z2−Ha2J1J31+m2v−mu−MJ11−ϕ2.5v,(17)∂θ∂t+s∂θ∂z=1PrJ5J2∂2θ∂z2+Ha2Ec1+m2J3J2u2+v2+QsJ2θ−QrJ2∂2θ∂z2+DuJ2∂2Γ∂z2+NBJ2∂θ∂z∂Γ∂z+NTJ2∂θ∂z2+Ec1−ϕ2.5J2∂u∂z2+∂v∂z2,(18)∂Γ∂t+s∂Γ∂z=1SC∂2Γ∂z2+γcΓ+Sr∂2θ∂z2,where R=2Ωνf/U02 is the rotation parameter, Ha2=μfB02σf/ρf2U02 is the Hartmann number squared, Gr=gνfβTf∗Tw−T∞/U03 is the Grashof number, Gc=gνfβCf∗Cw−C∞/U03 is the modified Grashof number, M=μf2/U02kρf2 is the porous medium parameter, Pr=CPfμf/kf is the Prandtl number, Ec=U02/CPfTw−T∞ is the Eckart number, Qs=Qνf/U02ρCpf is the heat source parameter, Qr=16σT∞3/3α∗νfρCpf is the radiation parameter, Du=DmkTCw−C∞/CsCPfνfTw−T∞ is the Dufour number, Brownian parameter NB=DBτCw−C∞/νf, NT=DTτTw−T∞/νfT∞ denotes thermophoresis parameter, nanoparticles and base fluid heat capacity ratio is τ=ρsCps/ρfCpf, Sc=νf/D is the Schmidt number, γc=kcνf/U02 is the chemical reaction parameter, and Sr=DTTw−T∞/T∞Cw−C∞ denotes the Soret number.The boundary conditions reduce to(19)t=0:u=0,w=0,θ⟶0,Γ⟶0,z=0,t>0:u=1,w=0,θ=1,Γ=1,z⟶∞,t>0:u⟶0,w⟶0,θ⟶0,Γ⟶0.


### 2.4. Engineering Interest Physical Quantities

For engineering interest, the local skin frictionCf, local Nusselt number Nux, and local Sherwood number Shx are defined as [33–35](20)Cf=τwρfU02,Nux=xqwkfTw−T∞,Shx=xqmDBTw−T∞,where τw, qw,andqm denote the wall shear stress and heat and mass fluxes, respectively, which are given by(21)τw=μnf∂u∂zz=0,qw=−knf∂T∂z+qrz=0,qm=−DB∂C∂zz=0.By substituting equation (14) into equations (20) and (21),(22)Nux=−knfkfRex1+Qrθ′0,where Rex=U0x/νf is the local Reynolds number.


## 2.1. Problem Description

Consider unsteady MHD nanofluid flow of ambient temperatureT∞ and concentration C∞ with heat and mass transfer through the porous medium past a moving flat plate in a rotating frame. The coordinate system x,y,z is selected in such a way that x−axis is taken along the plate, z−axis is perpendicular to it, and y−axis is normal to the xz plane. The physical model is described in Figure 1. Three different kinds of nanoparticles, Cu (copper), Al2O3 (aluminium oxide), and TiO2 (titanium dioxide), with water used as a base nanofluid are considered. The fluid and nanoparticles are assumed to have no slip between them, and both are in thermal equilibrium. Table 1 clearly lists the thermophysical properties of the base fluid and different nanoparticles [26, 27]. In this work, there are three shapes of nanoparticles which are spherical, cylindrical, and brick employed into account. Also, the model is assumed rotating about z−axis with constant angular velocity Ω. The fluid is permeated by applied time-dependent magnetic field which takes the form Bt=Ge−αt which is taken to be acting along the positive z-direction, where G and α are constants. It is assumed that the magnetic Reynolds number is to be very small so that the induced magnetic field can be neglected, and there is no applied external electric field. It is also assumed that the frequency of electron-atom collision to be relatively high so that the Hall effect cannot be neglected.Figure 1
Physical model of the problem.Table 1
Thermophysical properties [26, 27].
Thermophysical
Water
Copper
Aluminium oxide
Titanium dioxide
properties
H
2
O
Cu
Al
2
O
3
TiO
2
C
p
J
/
kgK
4179.00
385.00
765.00
686.00
ρ
kg
/
m
3
997.10
8933.00
3970.00
4250.00
K
W
/
mK
0.613
400.00
46.00
8.90
β
∗
×
10
−
5 (1/K)
21
1.67
0.63
1.05
σ
S
/
m
5.50
×
10
−
6
1163.10
×
10
7
131.70
×
10
7
2.38
×
10
6


## 2.2. Governing Equations

With above assumptions, the nanofluid flow is governed by conservations of continuity, momentum, energy, and concentration equations as(1)∇⋅V=0,∂V∂t+V⋅∇V−Ω×V=1ρnf∇μ⋅∇V+J×Bt+FT+FC+Fp,∂T∂t+V⋅∇T=1ρCpnf∇knf⋅∇T+J⋅Jσnf+DmkTρnfCs∇⋅∇CρsCpsDB∇T⋅∇C+DTT∞∇T⋅∇T+Q0T−T∞+φ−∇⋅qr,∂C∂t+V⋅∇C=∇⋅D∇C+DTT∞∇T+kcC−C∞,where ρnf is the nanofluid density, V denotes the velocity vector which has the following components: V=uz,t,vz,t,w0, μ is the nanofluid viscosity, FT is the thermal expansion term due to temperature difference, FC denotes the thermal expansion term due to concentration difference, FP is the porous medium term, Cpnf denotes the nanofluid specific heat at constant pressure, knf denotes the nanofluid thermal conductivity, σnf is the electric conductivity, Dm denotes the mass diffusivity, kT is the thermal diffusivity, Cs denotes the susceptibility of concentration, ρsCps denotes the nanoparticle heat capacity, and DB and DT denote the Brownian and thermophoresis coefficient, respectively. Q0 is the source constant, and φ is the viscous dissipation term. qr is the radiative heat flux, Tm is the mean temperature, and kc is the chemical reaction coefficient. J denotes the electric current density which is formulated by generalized Ohm’s law including Hall current as [28–30](2)J=σnfV×B−ξJ×B,where ξ is the Hall factor.Furthermore, radiative heat flux is formulated by Rosseland approximation as(3)qr=−4σ∗3α∗∂T4∂z.σ
∗ and α∗ denote the constant Stefan–Boltzman and mean absorption coefficient, respectively. Consider temperature differences are sufficiently small, and the term T4 can be expressed as a linear function into the Taylor series about T∞, and after neglecting higher-order terms, it yields(4)T4≅4T∞3T−3T∞4.Under the assumptions made above, the governing equations are as follows:(5)ρnf∂u∂t+w0∂u∂z+2Ωv=μnf∂2u∂z2−σnfB2t1+m2u+mv−μnfkpu+ρnfβTnf∗gT−T∞+ρnfβCnf∗gC−C∞,(6)ρnf∂v∂t+w0∂v∂z−2Ωu=μnf∂2v∂z2−σnfB2t1+m2v−mu−μnfkpv,(7)ρnfCPnf∂T∂t+w0∂T∂z=knf∂2T∂z2+μnf∂u∂z2+σnfB2t1+m2u2+v2+QT−T∞+DmkTρnfCs∂2C∂z2+ρsCPsDB∂C∂z∂T∂z+DTT∞∂T∂z2−16σ∗T∞33α∗∂2T∂z2,(8)∂C∂t+w0∂C∂z=D∂2C∂z2+kcC−C∞+DTT∞∂2T∂z2,where βTnf∗ denotes the nanofluid thermal expansion coefficient due to temperature difference, βCnf∗ denotes the nanofluid thermal expansion coefficient due to concentration difference, and m=σnfξB0 is the Hall parameter [31].The boundary conditions are(9)t≤0:u=0,w=0,T=T∞,C=C∞,z=0,t>0:u=U0,w=0,T=Tw,C=Cwz⟶∞,t>0:u⟶0,w⟶0,T⟶T∞,C⟶C∞.


## 2.3. Thermophysical Properties of Nanofluid

The nanofluid thermophysical properties were calculated using the following forms [6, 26]:Nanofluid effective dynamic viscosity, effective density, and heat capacity are given as(10)μnf=μf1−ϕ2.5,J1=ρfρnf=11−ϕ+ϕρs/ρf,J2=ρCpfρCpnf=11−ϕ+ϕρCps/ρCpf,whereϕ denotes the nanofluid volume fraction. Subscripts f, nf, and s denote the base fluid, nanofluid, and solid particle, respectively.Also, nanofluid effective electric conductivity and effective thermal expansion coefficient are given as(11)J3=σnfσf=1+3σs/σf−1ϕσs/σf+2−σs/σf−1ϕ,J4=ρβ∗nfρβ∗f=1−ϕ+ϕρβ∗sρβ∗f.Effective thermal conductivity and effect of nanoparticle shapes are considered by using the Hamilton and Crosser model as [26](12)J5=knfkf=ks+np−1kf−np−1ϕkf−ksks+np−1kf+ϕkf−ks,where np indicates the empirical nanoparticle shape factor which is given by(13)np=3Ψs.Here, Ψs is the sphericity of the nanoparticles which is defined as the ratio of the surface area of a sphere to the area of the particle with a volume equal to that of a particle and has value 1 np=3 for spherical-shaped nanoparticles and 0.5 np=6 for spherical-shaped nanoparticles. The characteristics of shape factor np for various nanoparticle shapes have been displayed in Table 2 [32].Table 2
Shape factor for various nanoparticle shapes [32].
Spherical
Cylindrical
Brick
Shapes of nanoparticles
Shape factornp
3
6
3.7Introducing the following nondimensional variables,(14)x^,y^,z^=x,y,zU0νf,t^=U02νft,u^=uU0,w^=wU0,θ=T−T∞Tw−T∞,Γ=C−C∞Cw−C∞.Governing equations (5)–(8) are reduced to nondimensional form after dropping the cap as(15)∂u∂t+s∂u∂z+Rv=J11−ϕ2.5∂2u∂z2−Ha2J1J31+m2u+mv+GrJ1J4θ+GcJ1J4Γ−MJ11−ϕ2.5u,(16)∂v∂t+s∂v∂z−Ru=J11−ϕ2.5∂2v∂z2−Ha2J1J31+m2v−mu−MJ11−ϕ2.5v,(17)∂θ∂t+s∂θ∂z=1PrJ5J2∂2θ∂z2+Ha2Ec1+m2J3J2u2+v2+QsJ2θ−QrJ2∂2θ∂z2+DuJ2∂2Γ∂z2+NBJ2∂θ∂z∂Γ∂z+NTJ2∂θ∂z2+Ec1−ϕ2.5J2∂u∂z2+∂v∂z2,(18)∂Γ∂t+s∂Γ∂z=1SC∂2Γ∂z2+γcΓ+Sr∂2θ∂z2,where R=2Ωνf/U02 is the rotation parameter, Ha2=μfB02σf/ρf2U02 is the Hartmann number squared, Gr=gνfβTf∗Tw−T∞/U03 is the Grashof number, Gc=gνfβCf∗Cw−C∞/U03 is the modified Grashof number, M=μf2/U02kρf2 is the porous medium parameter, Pr=CPfμf/kf is the Prandtl number, Ec=U02/CPfTw−T∞ is the Eckart number, Qs=Qνf/U02ρCpf is the heat source parameter, Qr=16σT∞3/3α∗νfρCpf is the radiation parameter, Du=DmkTCw−C∞/CsCPfνfTw−T∞ is the Dufour number, Brownian parameter NB=DBτCw−C∞/νf, NT=DTτTw−T∞/νfT∞ denotes thermophoresis parameter, nanoparticles and base fluid heat capacity ratio is τ=ρsCps/ρfCpf, Sc=νf/D is the Schmidt number, γc=kcνf/U02 is the chemical reaction parameter, and Sr=DTTw−T∞/T∞Cw−C∞ denotes the Soret number.The boundary conditions reduce to(19)t=0:u=0,w=0,θ⟶0,Γ⟶0,z=0,t>0:u=1,w=0,θ=1,Γ=1,z⟶∞,t>0:u⟶0,w⟶0,θ⟶0,Γ⟶0.


## 2.4. Engineering Interest Physical Quantities

For engineering interest, the local skin frictionCf, local Nusselt number Nux, and local Sherwood number Shx are defined as [33–35](20)Cf=τwρfU02,Nux=xqwkfTw−T∞,Shx=xqmDBTw−T∞,where τw, qw,andqm denote the wall shear stress and heat and mass fluxes, respectively, which are given by(21)τw=μnf∂u∂zz=0,qw=−knf∂T∂z+qrz=0,qm=−DB∂C∂zz=0.By substituting equation (14) into equations (20) and (21),(22)Nux=−knfkfRex1+Qrθ′0,where Rex=U0x/νf is the local Reynolds number.


## 3. Methodology

Here, the homotopy perturbation method that was first developed by He [36, 37] and improved by Wu and He [38] to solve governing equations (15)–(18) subject to boundary condition (19) is used. To illustrate the fundamental ideas of this method, we first consider the time-dependent differential equation as follows:(23)ϒζλ,t−fλ,t=0,λ∈ℜ,where ϒ is the operator, ζλ,t denotes the unknown function, fλ,t is a known function, λ are spatial variables, t denote temporal independent variables, and ℜ is the domain.The operatorϒ can be decomposed into the linear part Lλ,t, and Nλ,t is the remaining part. Therefore, equation (23) can be rewritten as(24)Lλ,t+Nλ,t−fλ,t=0,λ∈ℜ.Homotopy traditional technique constructs a homotopy functionΨλ,t;p as follows:(25)HΨλ,t;p=1−pLΨλ,t;p−Lf0λ,t+pϒΨλ,t;p−fλ,t=0,where p∈0,1 is an embedding parameter and f0λ,t is an initial guess approximation of equation (23) which satisfies boundary and initial condition.Consequently, the solution of equation (23) is expressed as(26)Ψλ,t;p=∑i=0nΨiλ,tpi.Ideal approximation solution can be obtained whenp=1 as(27)ζλ,t=∑i=0nζiλ,t.According to equation (24), the appropriate initial arbitrary guess approximation and linear operators for each equation are proposed as(28)u0=y1+e−t,v0=y1+e−t,θ0=y1−e−t,Γ0=y1−e−t.Lu=∂2u∂z2,Lv=∂2v∂z2,Lθ=∂2θ∂z2,LΓ=∂2Γ∂z2.Now, HPM constructed equations which satisfy equation (25) to governing nonlinear differential equations (15)–(18) as follows:(29)Hu;p=pJ11−ϕ2.5Lu+1−p−∂u∂t−s∂u∂z−Rv−Ha2J1J31+m2u+mv+GrJ1J4θ+GcJ1J4Γ−MJ11−ϕ2.5u=0,(30)Hv;p=pJ11−ϕ2.5Lv+1−p−∂v∂t−s∂v∂z−Ru−Ha2J1J31+m2v−mu−MJ11−ϕ2.5v=0,(31)Hθ;p=p1PrJ5J2−QrJ2Lθ+1−p∂θ∂t+s∂θ∂z+Ha2Ec1+m2J3J2u2+v2+QsJ2θ+DuJ2∂2Γ∂z2+NBJ2∂θ∂z∂Γ∂z+NTJ2∂θ∂z2+Ec1−ϕ2.5J2∂u∂z2+∂v∂z2=0,(32)HΓ;p=p1SCLΓ+1−p−∂Γ∂t−s∂Γ∂z+γcΓ+Sr∂2θ∂z2=0.And, assuming the solution of equations (29)–(32), it can be written in the following forms:(33)uz,t=u1tz+u2tz2+u3tz3+⋯,vz,t=v1tz+v2tz2+v3tz3+⋯,θz,t=θ1tz+θ2tz2+θ3tz3+⋯,Γz,t=Γ1tz+Γ2tz2+Γ3tz3+⋯.Substituting equation (26) into equations (27)–(31) after some simplifications and rearranging depending on powers of p-terms and solved using conditions (19), the following approximations can be obtained:(34)u1t=1590J1e−αt1−ϕ5/2MJ11+eαt+Ha2J1J4A11+eαt+GcJ1J31−eαt−α1−sGrJ1J21−eαt+R1+eαt,u2t=323J1e−αt1−ϕ5/2MJ11+eαt+Ha2J1J4A11+eαt+GcJ1J31−eαt−α1+s−GrJ1J21−eαt+R1+eαt,u3t=1590J11−ϕ5/2Ha2J1J41−m−R−α−eαt,v1t=15901−ϕ5/2−e−αtR+αs+Ha2J4J19015e−αt+m2−m3+A1e−αt,v2t=−323Ha2J1J41−ϕ5/2+R1−ϕ5/2+Ha2J1J4m1−ϕ5/2+323αe−αt1−ϕ5/2−323Ha2J1J4e−αt1−ϕ5/2+323Re−αt1−ϕ5/2,v3t=−MJ11−ϕ5/2++Ha2J4m3e−αt1−ϕ5/2323−0.125,θ1=−32Pre−αtα+J6Qr1−eαt3J5J6+4Pre−2αteαt−1seαt+DNBJ61−eαtJ5J6−256EcHa2J4Pre−2αtm2+1eαt+123J5+18,θ2=Pre−2αteαt−1seαt+DNBJ61−eαt2J5J6,θ3=Pre−αtα+J6Qr1−eαt6J5J6,Γ1t=Sce−αtα−γc1+eαt6,Γ2t=32Sce−αtα−γc1+eαt3−4Sce−αtSr+seαt−1−0.125,Γ3t=Sce−αtSr+seαt−12,where A1=m3+m2+m+1andA2=−m3+m2−m.


## 4. Solution Validation

To check the accuracy and efficiency of the methodology, the current results are verified by comparing with previously published results in some special cases which are tabulated in Tables3–5. The present results of different skin friction coefficient values Cf and Nusselt number Nu/Rex for different values of nanoparticle volume fraction ϕ and rotation parameter R are compared with those reported by Das [39] and Reddy et al. [40] with both nanoparticles Cu and Al2O3 which are illustrated in Tables 3 and 4. The results obtained come with excellent agreement of previous studies.Table 3
Result comparison forCf.
C
f
ϕ
R
Cu
−water
Al
2
O
3
−water
Present
Das [39]
Reddy et al. [40]
Present
Das [39]
0.00
0.3
2.0355
2.0354
2.03542
2.0355
2.0354
0.04
2.3158
2.3154
2.31535
0.5134
2.1831
0.08
2.6111
2.6107
2.61068
2.1832
2.3384
0.12
2.9120
2.9196
2.91953
2.5030
2.5020
0.14
3.0792
3.0789
3.07891
1.0885
1.0872
0.05
0.0
2.3790
2.3786
2.37862
2.2144
2.2143
0.5
2.4042
2.4040
2.40401
2.2333
2.2333
1.5
2.5755
2.5755
2.57546
2.3669
2.3666
2.5
2.8236
2.8235
2.8251
2.5697
2.5695Table 4
Result comparison forNux/Rex.
Nu
x
/
Re
x
ϕ
R
Cu
−water
Al
2
O
3
−water
Present
Das [39]
Reddy et al. [40]
Present
Das [39]
0.00
0.3
0.4607
0.4607
0.4607
0.4607
0.4607
0.04
0.5135
0.5134
0.5134
0.5115
0.5114
0.08
0.5695
0.5698
0.5698
0.5656
0.5655
0.12
0.6304
0.6303
0.6303
0.6234
0.6233
0.14
0.6622
0.6622
0.6622
0.6536
0.6537
0.05
0.0
0.5272
0.5271
0.5271
0.5271
0.5271
0.5
—
—
—
—
—
1.5
—
—
—
—
—
2.5
0.5271
0.5271
0.5271
0.5272
0.5271Table 5
Result comparison for−Shx.
−
Sh
x
γ
c
↓
t
⟶
0.3
0.5
0.7
0.3
0.5
0.7
Present
Hussain et al. [17]
0.2
0.535254
0.438632
0.389005
0.54907
0.44765
0.38638
2
0.849652
0.815633
0.767225
0.85729
0.820922
0.78156
5
1.2252
1.16674
1.14632
1.24260
1.16967
1.143922Furthermore, the comparison for different values of the Sherwood number−Shx against time t and chemical reaction γc with obtained results by Hussain et al. [17] at ϕ=0.0 is listed in Table 5. Also, it can be seen that the present obtained results are in an excellent agreement with the previously published results. This proves the validity of the present work and shows how powerful the solution reached by homotopy perturbation technique.


## 5. Results and Discussion

The influence of various physical parameters such as nanoparticle volume fractionϕ, shape of nanoparticles np, Hall current m, Brownian NB, thermophoretic NT, and chemical reaction γc which are computed from the approximate solution reported in the previous section is displayed graphically and discussed. The graphical results are illustrated in Figures 2–11. The numerical results are performed by taking values of some constant parameters as s=0.5,M=0.5,Pr=0.7,Ec=0.2,Gc=1,Gr=1,Sc=0.3,Du=0.5,G=5,andSr=0.5.Figure 2
The effect of volume fraction of nanoparticles on the velocity profile.Figure 3
The effect of volume fraction nanoparticle on the temperature profile.Figure 4
The effect of nanoparticle shape on the temperature profile.Figure 5
The effect of rotation parameter on the velocity profile.Figure 6
The effect of Hall current on the velocity profile.Figure 7
The effect of Brownian motion parameter on the temperature profile.Figure 8
The effect of thermophoresis parameter on the temperature profile.Figure 9
The effect of chemical reaction parameter on the concentration profile.Figure 10
The effect of heat source parameter on the temperature profile.Figure 11
The effect of radiation parameter on the temperature profile.Figures2 and 3 display the effect of nanoparticle volume fraction parameter ϕ on the fluid velocity u and temperature profiles θ for either Cu−water, Al2O3−water, and TiO2−water nanofluid. It is clear that increase in nanoparticle volume fraction parameter ϕ decreases the fluid velocity, but the temperature profiles increase. It is due to the fact that nanofluid density increases with increasing the volume fraction of nanoparticles, and it causes to slow down the nanofluid flow velocity and increases the thickness of the thermal boundary layer which improves thermal conductivity and increases surface heat transfer rate. Also, it is observed that nanofluid Cu−water heat transfer rate is greater than Al2O3 and TiO2 rates. So, nanoparticles having high thermal conductivity enhance heat transfer more than the particles having low thermal conductivity. On the contrary, it is seen from Tables 2 and 3 that wall skin friction coefficient Cf and the Nusselt number Nu/Rex increase with increasing in nanoparticle volume fraction for both Cu−water and Al2O3−water.The influence of the nanoparticle shape on the temperature distribution of nanofluid is studied through Figure4. This figure indicates that the heat transfer rate is higher for the cylindrical nanoparticle shape than the spherical and brick nanoparticle shapes for either Cu−water, Al2O3−water, and TiO2−water nanofluid. Therefore, the spherical nanoparticle shape tends to drag more heat from the boundary layer because it has a greater surface area, while this effect is less evident for other nanoparticle shapes. Thus, the use of the cylindrical nanoparticle shape has better improvement in effective thermal conductivity when compared with other nanoparticle shapes (spherical and brick).The influence of rotation parameterR on the velocity profiles against z is displayed in Figure 5. It observed that an increase in rotation parameter leads to decreasing the boundary layer momentum, so it decreases the values of nanofluid velocity. From Tables 2 and 3, it can be noted that the wall skin friction coefficient Cf increases with an increase in the rotation parameter, but the Nusselt number Nu/Rex remains unchanged with both Cu−water and Al2O3−water.Figure6 presents the influence of Hall current parameter m on main velocity u. It is clear that the velocity distribution of nanofluid reduces with an increase in Hall current parameter. This happens naturally due to the fact that when the magnetic field is strengthened, it enhances the opposing force which is represented by the well-known Lorentz force.Figures7 and 8 show the Brownian and thermophoresis effects on the temperature profile θ. It is seen that the temperature distribution increases with increasing for these parameters. This is due to the fact that the thermophoretic and Brownian parameters assist to improve the thermal boundary layer thickness. Also, it is observed that the heat transfer rate of Cu−water nanofluid is greater than TiO2−water.Figure9 depicts the influence of chemical parameter γc on the concentration profile Γ. It is noticed that the increase in concentration profile is accompanied by a decrease in the chemical parameter. On the contrary, it is seen from Table 4 that the Sherwood number −Shx increases due to increasing the chemical parameter, but the Sherwood number decreases while increasing the time t. This implies that the chemical reaction has tendency to increase the mass transfer rate at the surface plate, and it is reduced with the progress of time.Figures10 and 11 present the heat source parameter Qs and thermal radiation parameter Qr effects on the temperature profiles. It is clear from Figure 10 that, with an increase in the heat generation parameter, the boundary layer temperature decreases and as a consequence to decrease the thermal boundary layer thickness. In addition, it is clear from Figure 11 that, with an increase in the thermal radiation, the temperature profiles of the nanofluid increase. It is well known that the increase in the radiation parameter enhances the energy transport inside the fluid and as a consequence to increase the heat transfer rate.


## 6. Conclusion

An analysis is done to solve the magnetohydro dynamics unsteady nanofluid flow through the porous medium past a moving plate in a rotating frame with considering the heat and mass transfer. Three different types of the nanoparticles, namely, copper, aluminium oxide, and titanium dioxide, and water used as a base nanofluid are considered with various shapes (spherical, cylindrical, and brick). The important effects of nanoparticle shape, Brownian and thermophoresis diffusion, thermal radiation, ohmic and viscous dissipations have been included. Homotopy perturbation method is applied to solve the governing nonlinear partial differential equations. The efficiency of the methodology has been validated through comparison with the previously published results and observed a very close agreement. It is found thatThe main velocity component decreases with increase of nanoparticle volume fraction. On the contrary, thermal properties increase with increase in nanoparticle volume fractionThe enhancement of heat transfer rate and mechanical property improvement is high in copper as compared with both aluminium oxide and titanium dioxideThermal conductivity of cylindrical-shaped nanoparticles is more efficient and possesses better improvement on heat transfer rate than the other shaped onesIncrease in the Brownian and thermophoresis parameter is accompanied by an increase in the thermal boundary layer thicknessIncreasing Hall current and rotation parameters decreases the main fluid velocity componentHPM has a close agreement and is recommended for many nonlinear problems



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*Source: 9606382-2020-05-07.xml*