# Thermo-Magneto-Solutal Squeezing Flow of Nanofluid between Two Parallel Disks Embedded in a Porous Medium: Effects of Nanoparticle Geometry, Slip and Temperature Jump Conditions

Authors: M. G. Sobamowo; A. T. Akinshilo; A. A. Yinusa
Journal: Modelling and Simulation in Engineering
 (2018)

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## Abstract

The various applications of squeezing flow between two parallel surfaces such as those that are evident in manufacturing industries, polymer processing, compression, power transmission, lubricating system, food processing, and cooling amongst others call for further study on the effects of various parameters on the flow phenomena. In the present study, effects of nanoparticle geometry, slip, and temperature jump conditions on thermo-magneto-solutal squeezing flow of nanofluid between two parallel disks embedded in a porous medium are investigated, analyzed, and discussed. Similarity variables are used to transform the developed governing systems of nonlinear partial differential equations to systems of nonlinear ordinary differential equations. Homotopy perturbation method is used to solve the systems of the nonlinear ordinary differential equations. In order to verify the accuracy of the developed analytical solutions, the results of the homotopy perturbation method are compared with the results of the numerical method using the shooting method coupled with the fourth-order Runge–Kutta, and good agreements are established. Through the approximate analytical solutions, parametric studies are carried out to investigate the effects of nanoparticle size and shape, Brownian motion parameter, nanoparticle parameter, thermophoresis parameter, Hartmann number, Lewis number and pressure gradient parameters, slip, and temperature jump boundary conditions on thermo-solutal and hydromagnetic behavior of the nanofluid. This study will enhance and advance the understanding of nanofluidics such as energy conservation, friction reduction, and micromixing of biological samples.


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## Body

## 1. Introduction

The study of squeezing flow of fluid between two parallel surfaces has received considerable and appreciable attentions in the last few decades due to its various industrial and biological applications. These applications are evident in manufacturing industries, polymer processing, compression, power transmission, lubricating system, food processing, and cooling amongst others. In the past efforts to analyze fluid flow behavior between two parallel surfaces under the influences of various flow, fluid, and external properties, Mustafa et al. [1] presented an analysis of heat and mass transfer between parallel plates undergoing unsteady squeezing fluid flow while in the previous year, Hayat et al. [2] studied the squeezing flow behavior of a second grade fluid between parallel disk under the influence of presence of magnetic field. In an earlier work, Domairry and Aziz [3] applied the homotopy perturbation method to investigate the effects of suction and injection on magnetohydrodynamic squeezing flow of fluid between parallel disks. Also, Siddiqui et al. [4] investigated squeezing flow of viscous fluid between two parallel plates under the unsteady flow condition. In the same year, Rashidi et al. [5] examined the unsteady squeezing flow between parallel plates and presented different approximate analytical solutions. Three years later, Khan and Aziz [6] submitted a study on the flow of nanofluid between parallel plates due to natural convection. In a further work in the same year, the same authors [7] analyzed a double-diffusive natural convective boundary-layer fluid flow through porous media saturated with nanofluid while in an earlier work, Kuznetsov and Nield [8] studied the flow of nanofluid between two parallel plates considering the effect of the natural convective boundary layer. Hashimi et al. [9] presented stimulating exploration of analytical solutions for the study of squeezing flow of nanofluid. Most of the above reviews have been limited to the analysis of squeezing flow under no slip and no temperature jump boundary conditions. However, when flow system characteristic size is small or at a low flow pressure, the assumption of no slip boundary condition becomes insufficient and inadequate in predicting the flow behavior of the fluid under such scenario. Therefore, additional boundary conditions are required to adequately predict the low flow pressure or low system characteristics size. Moreover, it has been established that in many cases of fluid and flow problems, such as polymeric liquids, thin film problems, nanofluids, rarefied fluid problems, fluids containing concentrated suspensions, and flow on multiple interfaces, slip condition prevails at the boundary of the flow [10–21]. Such slip boundary condition was first initiated by Navier [22] upon which other researchers have built their analysis [10, 11]. Therefore, in recent years, the effects of slip effect on fluid flow have been considered by many researchers [12–21] due to its significance to most practical fluid flow situations. Furthermore, the effect of stretching sheet wall problem adopting natural convective boundary conditions was investigated by Yao et al. [23], Kandasamy et al. [24], and Makinde and Aziz [25]. More works on parametric studies of the flow process can be found in [26–34]. Additionally, analyses of magnetohydrodynamic flow in the porous medium have been presented in various works [35–50] while various studies of magneto-nanofluid flow have also been presented in the past works [51–68].The development of analytical solutions through some approximate analytical methods such as the Adomian decomposition method, homotopy analysis method, variation iteration method, and differential transformation method for the flow process has been semianalytic in procedures and applications. In the earlier efforts to develop total analytic methods for solving nonlinear equations, different perturbation methods such as regular, singular, and homotopy perturbation methods have been applied. However, among these methods, the traditional perturbation methods such as methods of regular and singular perturbations require the inclusion of a small parameter in the governing differential equations for the approximate analytical solution(s) to be realized for the governing differential equations. In the class of the newly developed approximate analytical methods for solving nonlinear equations, the homotopy perturbation method has been considered to be relatively simple with fewer requirements for mathematical rigour or skill. The homotopy perturbation method (HPM) gives highly accurate analytical solutions without the need for small perturbation parameters as required in traditional perturbations [69–73]. The results of HPM are completely reliable and physically realistic. Unlike the other approximate analytical methods, the HPM does not involve the search for a particular value that will satisfy the end boundary condition. Furthermore, to the best of our knowledge, there is no study where the thermal-diffusion and diffusion-thermal of magnetohydrodynamic squeezing unsteady flow of nanofluid between two parallel disks embedded in a porous medium under the influences of different nanoparticle geometries, slip and temperature jump conditions are analyzed. Therefore, in this work, the homotopy perturbation method is used to study the effects of nanoparticle geometry, slip, and temperature jump conditions on thermo-magneto-solutal squeezing flow of nanofluid between two parallel disks embedded in a porous medium. The obtained analytical solutions are used to investigate the effects of the Brownian motion parameter, thermophoresis parameter, Hartmann number, Lewis number and pressure gradient parameters, slip, and temperature jump boundary conditions on fluid behavior of the nanofluid.


## 2. Model Development and Analytical Solutions

Consider an axisymmetrical flow of nanofluid through two parallel disks as shown in Figure1. The upper disk is moving towards the stationary lower disks under a uniform magnetic field strength applied perpendicular to disks as depicted in Figure 1. The fluid conducts electrical energy as it flows unsteadily under the influence of magnetic force field. It is assumed that the fluid structure is everywhere in thermodynamic equilibrium and the plate is maintained at constant temperature.Figure 1
The squeezing flow of nanofluid between the parallel circular plates.The details of the governing equation and nondimensional parameters have been described by Hashimi et al. [9] and Das et al. [20] which can be introduced under the stated assumptions in the previous studies as(1)∂u∂r+ur+∂w∂z=0,∂u∂t+u∂u∂r+w∂u∂z=−1ρnf⋅∂p∂r+vnf∂2u∂r2+∂2u∂z2+1r⋅∂u∂r−ur2−σnfB2tuρnf−νnfuKp,∂w∂t+u∂w∂r+w∂w∂z=−1ρnf⋅∂p∂z+vnf∂2w∂r2+∂2w∂z2+1r⋅∂w∂r−νnfwKp,∂T∂t+u∂T∂r+w∂T∂z=αnf∂2T∂r2+1r⋅∂T∂r+∂2T∂z2+τnfDB∂c∂r⋅∂T∂r+∂c∂z⋅∂T∂z+DTTm∂T∂r2+∂T∂z2,∂c∂t+u∂c∂r+w∂c∂z=DB,nf∂2c∂r2+1r⋅∂c∂r+∂2c∂z2+DTTm∂2T∂r2+1r⋅∂T∂r+∂2T∂z2,where(2)ρnf=1−ϕρf+ϕρs,μnf=μf1−ϕ2.5,σnf=σf1+3σs/σf−1ϕσs/σf+2ϕ−σs/σf−1ϕ,αnf=ρCpnfknf,ρCpnf=1−ϕρCpf+ϕρCps,knf=kfks+m−1kf−m−1kf−ksϕks+m−1kf+kf−ksϕ.The appropriate initial and boundary conditions given as(3)u=0,w=0,T=To,c=co,att=0,0≤z≤H,u=−β1∂u∂z,w=w01−αt,T=−γ1∂T∂z+Tw,c=cw,z=0,u=−β1∂u∂z,w=wh∂h∂t,T=γ1∂T∂z+Th,c=ch,atz=ht.Using the following dimensionless quantities and similarity transformations,(4)A=w0αH,β=β1H1−αt,γ=γ1H1−αt,Nt=τDBTw−Thv,Nb=τDBkw−khv,Le=vDB,Pr=vα,M=HB0σμ,Da=KH,C=αHμfvf∂p∂z,S=αH22ν,φ=c−chcw−ch,θ=T−ThTw−Th,w=αH21−αtfη,u=αr21−αtf′η,η=zH1−αt,τ=ρcpρcf,K1=1−ϕ+ϕρsρf,K2=1−ϕ+ϕρCpsρCpf,K3=ks+m−1kf−m−1kf−ksϕks+m−1kf+kf−ksϕ,B1=σs+m−1σf+m−1σs−σfϕσs+m−1σf−m−1σs−σfϕ.The dimensionless equations are given as(5)d4f∂η4−SK11−ϕ2.5ηd3fdη3+3d2fdη2+fdfdη−2dfdη⋅d3fdη3−M2B11−ϕ2.5d2fdη2−1DaB11−ϕ2.5d2fdη2=C,(6)d2θdη2+PrSK2K32fdθdη−ηdθdη+PrK31−ϕ2.5Nbdθdη⋅dφdη+Ntdθdη2=0,(7)d2φdη2+LeSK11−ϕ2.52f−ηdθdη+NtNb⋅d2φdη2=0.And the dimensionless boundary conditions are given as(8)f0=A,df0dη=−βd2f0dη2,f1=12,df1dη=βd2f1dη2,θ0=1−γdθdη,θ1=γdθdη,(9)φ0=0,φ1=0,where m in the above Hamilton Crosser’s model in (4) is the shape factor, and its numerical values for different shapes are given in Table 1.Table 1
The values of different shapes of nanoparticles [74, 75].
No.
Name
Shape
Shape factor (m)
1
Spherical
3.0
2
Platelet
5.7
3
Cylinder
4.8
4
Laminar
16.2
5
Brick
3.7The physical and thermal properties of the basefluid and nanoparticles are given in Tables2 and 3, respectively.Table 2
Physical and thermal properties of the basefluid [74, 75].
Base fluid
ρ (kg/m3)
C
p (J/kg K)
k (W/mK)
σ (Ω−1·m−1)
Pure water
997.1
4179
0.613
5.50
Ethylene glycol
1115
2430
0.253
1.07
Engine oil
884
1910
0.144
4.02
Kerosene
783
2010
0.145
4.01Table 3
Physical and thermal properties of nanoparticles [74, 75].
Nanoparticles
ρ (kg/m3)
C
p (J/kg K)
k (W/mK)
σ (Ω−1·m−1)
Copper (Cu)
8933
385
401
59.6
Aluminum oxide (Al2O3)
3970
765
40
16.7
SWCNTs
2600
42.5
6600
1.26
Silver (Ag)
10500
235.0
429
Titanium dioxide (TiO2)
4250
686.2
8.9538
Copper (II) oxide (CuO)
783
540
18It should be noted that the shape factorm=3/ψ, where ψ is the sphericity (the ratio of the surface area of the sphere and the surface area of the real particles with equal volumes). Sphericity of sphere, platelet, cylinder, laminar, and brick are 1.000, 0.526, 0.625, 0.185, and 0.811, respectively. Hamilton–Crosser’s model becomes Maxwell-Garnett’s model, when the shape factor of the nanoparticle is m=3. SWCNTs represent single-walled carbon nanotubes.


## 3. Method of Solution by the Homotopy Perturbation Method

The comparative advantages and the provision of acceptable analytical results with convenient convergence and stability [69–73] coupled with total analytic procedures of the homotopy perturbation method compel us to consider the method for solving the system of nonlinear differential equations in (5)–(7).


### 3.1. The Basic Idea of the Homotopy Perturbation Method

In order to establish the basic idea behind the homotopy perturbation method, consider a system of nonlinear differential equations given as(10)AU−fr=0,r∈Ω.With the boundary conditions(11)Bu,∂u∂η=0,r∈Γ,where A is a general differential operator, B is a boundary operator, fr is a known analytical function, and Γ is the boundary of the domain Ω.The operatorA can be divided into two parts, L and N, where L is a linear operator and N is a nonlinear operator. Equation (10) can be therefore rewritten as follows:(12)Lu+Nu−fr=0.By the homotopy technique, a homotopyUr,p:Ω×0,1→R can be constructed, which satisfies(13)HU,p=1−pLU−LUο+pAU−fr=0,p∈0,1,or(14)HU,p=LU−LUο+pLUο+pNU−fr=0.In the above (13) and (14), p∈0,1 is an embedding parameter and u0 is an initial approximation of equation of (10), which satisfies the boundary conditions.Also, from (13) and (14), one has(15)HU,0=LU−LU0=0.or(16)HU,0=AU−fr=0.The changing process ofp from zero to unity is just that of Ur,p from u0r to ur. This is referred to homotopy in topology. Using the embedding parameter p as a small parameter, the solutions of (13) and (14) can be assumed to be written as a power series in p as given in (17):(17)U=U0+pU1+p2U2+⋯.It should be pointed out that of all the values ofp are between 0 and 1, and p=1 produces the best result. Therefore, setting p=1, results in the approximation solution of (9):(18)u=limp→1U=U0+U1+U2+⋯.The basic idea expressed above is a combination of the homotopy and perturbation method. Hence, the method is called the homotopy perturbation method (HPM), which has eliminated the limitations of the traditional perturbation methods (regular and singular perturbation methods). On the other hand, this technique can have full advantages of the traditional perturbation techniques. The series (18) is convergent for most cases.


### 3.2. Application of the Homotopy Perturbation Method to the Present Problem

According to the homotopy perturbation method (HPM), one can construct an homotopy for (5)–(7) as(19)H1p,η=1−pd4f∂η4−C+pd4f∂η4−SK11−ϕ2.5ηd3fdη3+3d2fdη2+fdfdη−2dfdη⋅d3fdη3−M2B11−ϕ2.5d2fdη2−1DaB11−ϕ2.5d2fdη2−C=0,(20)H2p,η=1−pd2θdη2+pd2θdη2+PrSK2K32fdθdη−ηdθdη+PrK31−ϕ2.5Nbdθdη⋅dφdη+Ntdθdη2=0,(21)H3p,η=1−pd2φdη2+pd2φdη2+LeSK11−ϕ2.52f−ηdθdη+NtNb⋅d2φdη2=0.Taking power series of velocity, temperature, and concentration fields gives(22)f=f0+pf1+p2f2+⋯,(23)θ=θ0+pθ1+p2θ2+⋯,(24)φ=φ0+pφ1+p2φ2+⋯.Substituting (22) into (19) yields(25)p0:d4f0dη4,(26)p1:d4f1dη4−M2B11−ϕ2.5d2f0dη2−1DaB11−ϕ2.5d2f0dη2−C−SK11−ϕ2.53d2f0dη2+ηd3f0dη3+f0df0dη−2df0dη⋅d3f0dη3,(27)p2:d4f2dη4−d2f1dη2M2B11−ϕ2.5−1DaB11−ϕ2.5d2f1dη2−SK11−ϕ2.53d2f1dη2+d3f1dη3−df0dη2df1dη−f1−df1dη2df0dη−f0+2,⋮.The boundary conditions for (25)–(27) are(28)f00=A,f10=0,f20=0,df00dη=−βd2f00dη2,df10dη=−βd2f10dη2,df20dη=−βd2f20dη2,f01=12,f11=0,f21=0,df01dη=βd2f01dη2,df11dη=βd2f11dη2,df21dη=βd2f21dη2.Substituting (23) into (20) yields(29)p0:d2θ0dη2,(30)p1:d2θ1dη2−PrSK2K3ηdθ0dη−2dθ0dηf0+PrK31−ϕ2.5Nbdθ0dη⋅dφ0dη+Ntdθ0dη2,(31)p2:d2θ2dη2+PrSK2K32dθ0dηf1−dθ1dη+2dθ1dηf0+PrK31−ϕ2.5Nbdθ1dη⋅dφ0dη+Nbdθ0dη⋅dφ1dη+2Ntdθ0dη⋅dθ1dη.⋮.The boundary conditions for (29)–(31) are(32)θ00=1−γdθ0dη,θ10=−γdθ1dη,θ20=−γdθ2dη,θ01=γdθ0dη,θ11=γdθ1dη,θ21=γdθ2dη.Substituting (24) into (21) yields(33)p0:d2φ0dη2+NtNb⋅d2θ0dη2,(34)p1:d2φ1dη2+NtNb⋅d2φ1dη2−LeSK11−ϕ2.5ηdθ0dη−2dθ0dηf0,(35)p2:d2φ2dη2+LeSK11−ϕ2.52dθ0dηf1−ηdθ1dη+2dθ1dηf0+NtNb⋅d2θ2dη2.⋮.The boundary conditions for (33)–(35) are(36)φ00=0,φ10=0,φ20=0,φ01=0,φ11=0,φ21=0.On solving (25), applying the boundary condition (28) gives(37)f0=A−2A−1η36β−1+32A−1η212β−2−3β2A−1η6β−1.Also, on solving (29), applying the boundary condition (32) yields(38)θ0=η2γ−1+γ−12γ−1.And the solution of (33), by applying the boundary condition (36) is(39)φ0=0.On solving (26) and applying the boundary condition (28), one arrives at(40)f1=η4C24−SK11−ϕ2.516−A2SK11−ϕ2.58+3ASK11−ϕ2.516−SK11−ϕ2.52A−12486β−12+2A−11−ϕ2.512M2+1/DaB1+20SK1−10ASK1966β−1+C19η22+C20η36−ηC24+C12+C206−29SK11−ϕ2.5480−7A2SK11−ϕ2.560+43ASK11−ϕ2.5240−53SK11−ϕ2.52A−12100806β−12−8SK11−ϕ2.5−16ASK11−ϕ2.52A−133606β−1+57SK11−ϕ2.5−78ASK11−ϕ2.52A−114406β−12+2A−11−ϕ2.512M2+1/DaB1+20SK1−10ASK1966β−1−2A−11−ϕ2.524M2+1/DaB1+122SK1−64ASK14806β−1−57SK11−ϕ2.5−78ASK11−ϕ2.52A−114406β−1−η67SK11−ϕ2.52A−12966β−12+η717SK11−ϕ2.52A−123366β−12−2A−18SK11−ϕ2.5−16ASK11−ϕ2.533606β−1−η817SK11−ϕ2.52A−1211206β−12+η9SK11−ϕ2.52A−1210086β−12+η5SK11−ϕ2.5480+A2SK11−ϕ2.5120−ASK11−ϕ2.5120+5SK11−ϕ2.52A−12966β−12−2A−11−ϕ2.524M2+1/DaB1+122SK1−64ASK14806β−1.On solving (30), applying the boundary condition in (32) gives(41)θ1=PrSK2106K32A−1β−12γ−1η5−PrSK24K32A−16β−12γ−1η4+PrSK26K31−2γ−2A+4Aγ2γ−12+PrSK26K32A−16β−12γ−1η3−PrSK260K32A−16β−12γ−1η−PrNtK31−ϕ2.522γ−12−PrSK26K31−2γ−2A+4Aγ22γ−12.It can easily be shown that the solution of (34) after applying the boundary condition of (36) yields(42)φ1=PrSK2NbK3Ntγ1−2A56β−12γ−12−1−2Aη5106β−12γ−12+1−2A46β−12γ−12−γ1−2Aη426β−12γ−12−β1−2Aη36β−12γ−12+1−6β1−2A6β−12γ−12−1−2A−6β+12Aβη226β−12γ−12−Nt1−6βη22S6β−12γ−12+β1−2A6β−12γ−12+1−2A−6β+12Aβ26β−12γ−12+Nt1−6β2S6β−12γ−12−γ1−6β1−2A6β−12γ−12−γ1−2Aβ56β−12γ−12+γ1−2Aβη26β−12γ−12.In the same way,f2η,θ2η,andϕ2η are (27), (31), and (35) are solved using the boundary conditions in (28), (32), and (36), respectively. Although the resulting solutions and the other subsequent solutions are too long to be shown in this paper, they are included in the simulated results shown graphically in the results and discussion section.Substituting (37) and (40) into the power series (22) yields(43)fη=A−2A−1η36β−1+32A−1η212β−2−3β2A−1η6β−1+η4C24−SK11−ϕ2.516−A2SK11−ϕ2.58+3ASK11−ϕ2.516−SK11−ϕ2.52A−12486β−12+2A−11−ϕ2.512M2+1/DaB1+20SK1−10ASK1966β−1+C19η22+C20η36−ηC24+C12+C206−29SK11−ϕ2.5480−7A2SK11−ϕ2.560+43ASK11−ϕ2.5240−53SK11−ϕ2.52A−12100806β−12−8SK11−ϕ2.5−16ASK11−ϕ2.52A−133606β−1+57SK11−ϕ2.5−78ASK11−ϕ2.52A−114406β−12+2A−11−ϕ2.512M2+1/DaB1+20SK1−10ASK1966β−1−2A−11−ϕ2.524M2+1/DaB1+122SK1−64ASK14806β−1−57SK11−ϕ2.5−78ASK11−ϕ2.52A−114406β−1−η67SK11−ϕ2.52A−12966β−12+η717SK11−ϕ2.52A−123366β−12−2A−18SK11−ϕ2.5−16ASK11−ϕ2.533606β−1−η817SK11−ϕ2.52A−1211206β−12+η9SK11−ϕ2.52A−1210086β−12+η5SK11−ϕ2.5480+A2SK11−ϕ2.5120−ASK11−ϕ2.5120+5SK11−ϕ2.52A−12966β−12−2A−11−ϕ2.524M2+1/DaB1+122SK1−64ASK14806β−1.Also, substituting (38) and (41) into the power series (23) gives(44)θη=η2γ−1+γ−12γ−1+PrSK2106K32A−1β−12γ−1η5−PrSK24K32A−16β−12γ−1η4+PrSK26K31−2γ−2A+4Aγ2γ−12+PrSK26K32A−16β−12γ−1η3−PrSK260K32A−16β−12γ−1η−PrNtK31−ϕ2.522γ−12−PrSK23K3⋅1−2γ−2A+4Aγ2γ−12+⋯.Similarly, substituting (39) and (42) into the power series (24) yields(45)φη=PrSK2NbK3Ntγ1−2A56β−12γ−12−1−2Aη5106β−12γ−12+1−2A46β−12γ−12−γ1−2Aη426β−12γ−12−β1−2Aη36β−12γ−12+1−6β1−2A6β−12γ−12−1−2A−6β+12Aβη226β−12γ−12−Nt1−6βη22S6β−12γ−12+β1−2A6β−12γ−12+1−2A−6β+12Aβ26β−12γ−12+Nt1−6β2S6β−12γ−12−γ1−6β1−2A6β−12γ−12−γ1−2Aβ56β−12γ−12+γ1−2Aβη26β−12γ−12.


## 3.1. The Basic Idea of the Homotopy Perturbation Method

In order to establish the basic idea behind the homotopy perturbation method, consider a system of nonlinear differential equations given as(10)AU−fr=0,r∈Ω.With the boundary conditions(11)Bu,∂u∂η=0,r∈Γ,where A is a general differential operator, B is a boundary operator, fr is a known analytical function, and Γ is the boundary of the domain Ω.The operatorA can be divided into two parts, L and N, where L is a linear operator and N is a nonlinear operator. Equation (10) can be therefore rewritten as follows:(12)Lu+Nu−fr=0.By the homotopy technique, a homotopyUr,p:Ω×0,1→R can be constructed, which satisfies(13)HU,p=1−pLU−LUο+pAU−fr=0,p∈0,1,or(14)HU,p=LU−LUο+pLUο+pNU−fr=0.In the above (13) and (14), p∈0,1 is an embedding parameter and u0 is an initial approximation of equation of (10), which satisfies the boundary conditions.Also, from (13) and (14), one has(15)HU,0=LU−LU0=0.or(16)HU,0=AU−fr=0.The changing process ofp from zero to unity is just that of Ur,p from u0r to ur. This is referred to homotopy in topology. Using the embedding parameter p as a small parameter, the solutions of (13) and (14) can be assumed to be written as a power series in p as given in (17):(17)U=U0+pU1+p2U2+⋯.It should be pointed out that of all the values ofp are between 0 and 1, and p=1 produces the best result. Therefore, setting p=1, results in the approximation solution of (9):(18)u=limp→1U=U0+U1+U2+⋯.The basic idea expressed above is a combination of the homotopy and perturbation method. Hence, the method is called the homotopy perturbation method (HPM), which has eliminated the limitations of the traditional perturbation methods (regular and singular perturbation methods). On the other hand, this technique can have full advantages of the traditional perturbation techniques. The series (18) is convergent for most cases.


## 3.2. Application of the Homotopy Perturbation Method to the Present Problem

According to the homotopy perturbation method (HPM), one can construct an homotopy for (5)–(7) as(19)H1p,η=1−pd4f∂η4−C+pd4f∂η4−SK11−ϕ2.5ηd3fdη3+3d2fdη2+fdfdη−2dfdη⋅d3fdη3−M2B11−ϕ2.5d2fdη2−1DaB11−ϕ2.5d2fdη2−C=0,(20)H2p,η=1−pd2θdη2+pd2θdη2+PrSK2K32fdθdη−ηdθdη+PrK31−ϕ2.5Nbdθdη⋅dφdη+Ntdθdη2=0,(21)H3p,η=1−pd2φdη2+pd2φdη2+LeSK11−ϕ2.52f−ηdθdη+NtNb⋅d2φdη2=0.Taking power series of velocity, temperature, and concentration fields gives(22)f=f0+pf1+p2f2+⋯,(23)θ=θ0+pθ1+p2θ2+⋯,(24)φ=φ0+pφ1+p2φ2+⋯.Substituting (22) into (19) yields(25)p0:d4f0dη4,(26)p1:d4f1dη4−M2B11−ϕ2.5d2f0dη2−1DaB11−ϕ2.5d2f0dη2−C−SK11−ϕ2.53d2f0dη2+ηd3f0dη3+f0df0dη−2df0dη⋅d3f0dη3,(27)p2:d4f2dη4−d2f1dη2M2B11−ϕ2.5−1DaB11−ϕ2.5d2f1dη2−SK11−ϕ2.53d2f1dη2+d3f1dη3−df0dη2df1dη−f1−df1dη2df0dη−f0+2,⋮.The boundary conditions for (25)–(27) are(28)f00=A,f10=0,f20=0,df00dη=−βd2f00dη2,df10dη=−βd2f10dη2,df20dη=−βd2f20dη2,f01=12,f11=0,f21=0,df01dη=βd2f01dη2,df11dη=βd2f11dη2,df21dη=βd2f21dη2.Substituting (23) into (20) yields(29)p0:d2θ0dη2,(30)p1:d2θ1dη2−PrSK2K3ηdθ0dη−2dθ0dηf0+PrK31−ϕ2.5Nbdθ0dη⋅dφ0dη+Ntdθ0dη2,(31)p2:d2θ2dη2+PrSK2K32dθ0dηf1−dθ1dη+2dθ1dηf0+PrK31−ϕ2.5Nbdθ1dη⋅dφ0dη+Nbdθ0dη⋅dφ1dη+2Ntdθ0dη⋅dθ1dη.⋮.The boundary conditions for (29)–(31) are(32)θ00=1−γdθ0dη,θ10=−γdθ1dη,θ20=−γdθ2dη,θ01=γdθ0dη,θ11=γdθ1dη,θ21=γdθ2dη.Substituting (24) into (21) yields(33)p0:d2φ0dη2+NtNb⋅d2θ0dη2,(34)p1:d2φ1dη2+NtNb⋅d2φ1dη2−LeSK11−ϕ2.5ηdθ0dη−2dθ0dηf0,(35)p2:d2φ2dη2+LeSK11−ϕ2.52dθ0dηf1−ηdθ1dη+2dθ1dηf0+NtNb⋅d2θ2dη2.⋮.The boundary conditions for (33)–(35) are(36)φ00=0,φ10=0,φ20=0,φ01=0,φ11=0,φ21=0.On solving (25), applying the boundary condition (28) gives(37)f0=A−2A−1η36β−1+32A−1η212β−2−3β2A−1η6β−1.Also, on solving (29), applying the boundary condition (32) yields(38)θ0=η2γ−1+γ−12γ−1.And the solution of (33), by applying the boundary condition (36) is(39)φ0=0.On solving (26) and applying the boundary condition (28), one arrives at(40)f1=η4C24−SK11−ϕ2.516−A2SK11−ϕ2.58+3ASK11−ϕ2.516−SK11−ϕ2.52A−12486β−12+2A−11−ϕ2.512M2+1/DaB1+20SK1−10ASK1966β−1+C19η22+C20η36−ηC24+C12+C206−29SK11−ϕ2.5480−7A2SK11−ϕ2.560+43ASK11−ϕ2.5240−53SK11−ϕ2.52A−12100806β−12−8SK11−ϕ2.5−16ASK11−ϕ2.52A−133606β−1+57SK11−ϕ2.5−78ASK11−ϕ2.52A−114406β−12+2A−11−ϕ2.512M2+1/DaB1+20SK1−10ASK1966β−1−2A−11−ϕ2.524M2+1/DaB1+122SK1−64ASK14806β−1−57SK11−ϕ2.5−78ASK11−ϕ2.52A−114406β−1−η67SK11−ϕ2.52A−12966β−12+η717SK11−ϕ2.52A−123366β−12−2A−18SK11−ϕ2.5−16ASK11−ϕ2.533606β−1−η817SK11−ϕ2.52A−1211206β−12+η9SK11−ϕ2.52A−1210086β−12+η5SK11−ϕ2.5480+A2SK11−ϕ2.5120−ASK11−ϕ2.5120+5SK11−ϕ2.52A−12966β−12−2A−11−ϕ2.524M2+1/DaB1+122SK1−64ASK14806β−1.On solving (30), applying the boundary condition in (32) gives(41)θ1=PrSK2106K32A−1β−12γ−1η5−PrSK24K32A−16β−12γ−1η4+PrSK26K31−2γ−2A+4Aγ2γ−12+PrSK26K32A−16β−12γ−1η3−PrSK260K32A−16β−12γ−1η−PrNtK31−ϕ2.522γ−12−PrSK26K31−2γ−2A+4Aγ22γ−12.It can easily be shown that the solution of (34) after applying the boundary condition of (36) yields(42)φ1=PrSK2NbK3Ntγ1−2A56β−12γ−12−1−2Aη5106β−12γ−12+1−2A46β−12γ−12−γ1−2Aη426β−12γ−12−β1−2Aη36β−12γ−12+1−6β1−2A6β−12γ−12−1−2A−6β+12Aβη226β−12γ−12−Nt1−6βη22S6β−12γ−12+β1−2A6β−12γ−12+1−2A−6β+12Aβ26β−12γ−12+Nt1−6β2S6β−12γ−12−γ1−6β1−2A6β−12γ−12−γ1−2Aβ56β−12γ−12+γ1−2Aβη26β−12γ−12.In the same way,f2η,θ2η,andϕ2η are (27), (31), and (35) are solved using the boundary conditions in (28), (32), and (36), respectively. Although the resulting solutions and the other subsequent solutions are too long to be shown in this paper, they are included in the simulated results shown graphically in the results and discussion section.Substituting (37) and (40) into the power series (22) yields(43)fη=A−2A−1η36β−1+32A−1η212β−2−3β2A−1η6β−1+η4C24−SK11−ϕ2.516−A2SK11−ϕ2.58+3ASK11−ϕ2.516−SK11−ϕ2.52A−12486β−12+2A−11−ϕ2.512M2+1/DaB1+20SK1−10ASK1966β−1+C19η22+C20η36−ηC24+C12+C206−29SK11−ϕ2.5480−7A2SK11−ϕ2.560+43ASK11−ϕ2.5240−53SK11−ϕ2.52A−12100806β−12−8SK11−ϕ2.5−16ASK11−ϕ2.52A−133606β−1+57SK11−ϕ2.5−78ASK11−ϕ2.52A−114406β−12+2A−11−ϕ2.512M2+1/DaB1+20SK1−10ASK1966β−1−2A−11−ϕ2.524M2+1/DaB1+122SK1−64ASK14806β−1−57SK11−ϕ2.5−78ASK11−ϕ2.52A−114406β−1−η67SK11−ϕ2.52A−12966β−12+η717SK11−ϕ2.52A−123366β−12−2A−18SK11−ϕ2.5−16ASK11−ϕ2.533606β−1−η817SK11−ϕ2.52A−1211206β−12+η9SK11−ϕ2.52A−1210086β−12+η5SK11−ϕ2.5480+A2SK11−ϕ2.5120−ASK11−ϕ2.5120+5SK11−ϕ2.52A−12966β−12−2A−11−ϕ2.524M2+1/DaB1+122SK1−64ASK14806β−1.Also, substituting (38) and (41) into the power series (23) gives(44)θη=η2γ−1+γ−12γ−1+PrSK2106K32A−1β−12γ−1η5−PrSK24K32A−16β−12γ−1η4+PrSK26K31−2γ−2A+4Aγ2γ−12+PrSK26K32A−16β−12γ−1η3−PrSK260K32A−16β−12γ−1η−PrNtK31−ϕ2.522γ−12−PrSK23K3⋅1−2γ−2A+4Aγ2γ−12+⋯.Similarly, substituting (39) and (42) into the power series (24) yields(45)φη=PrSK2NbK3Ntγ1−2A56β−12γ−12−1−2Aη5106β−12γ−12+1−2A46β−12γ−12−γ1−2Aη426β−12γ−12−β1−2Aη36β−12γ−12+1−6β1−2A6β−12γ−12−1−2A−6β+12Aβη226β−12γ−12−Nt1−6βη22S6β−12γ−12+β1−2A6β−12γ−12+1−2A−6β+12Aβ26β−12γ−12+Nt1−6β2S6β−12γ−12−γ1−6β1−2A6β−12γ−12−γ1−2Aβ56β−12γ−12+γ1−2Aβη26β−12γ−12.


## 4. Results and Discussion

In order to verify the accuracy of the homotopy perturbation method, the developed nonlinear equations are also solved using the shooting method coupled with the Runge–Kutta method. Table4 shows the comparison of the results of the numerical method (NM) and the homotopy perturbation method. From the table, it could be inferred that the results of the present work agree with the results of the numerical method using the shooting method with the Runge-Kutta method.Table 4
Comparison of values of the temperature jump parameter (γ) when A=2,S=1,Nt=Nb=0.2,ϕ=0,β=0.05,M=0.5,Le=Pr=1.0,andC=0.
γ
Dimensionless temperature
NM
Present work
0.00
1.0000
1.0000
0.10
1.2231
1.2232
0.15
1.4053
1.4053
0.20
1.6214
1.6215
0.25
1.7312
1.7311
0.30
1.8656
1.8656
0.35
2.2013
2.2011
0.40
3.1052
3.1052In order to further establish the accuracy of the solution of the HPM, the results of the present study (in the absence of the slip parameter, i.e., no-slip condition) are further compared with the results of the numerical method using the shooting method with the six-order Runge–Kutta method. The values of the local Nusselt numberNuL and local Sherwood number ShL have been calculated for various values of Nb and Nt. An excellent agreement found between the two set of results is as shown in Table 5. Therefore, the use of the homotopy perturbation method for the analysis of the double-diffusive models is justified.Table 5
Comparison of the values of the local Nusselt and local Sherwood number for various values ofNb and Nt with β=ϕ=γ=0.
N
b
N
t
N
u
L
=
−
θ
′
1
S
h
L
=
−
φ
′
1
NM [20]
HAM [9]
Present method
NM [20]
HAM [9]
Present method
0.1
0.1
0.5263
0.5263
0.5263
0.8660
0.8660
0.8660
0.5
0.6343
0.6343
0.6343
0.5301
0.5300
0.5300
1.0
0.7864
0.7864
0.7864
0.4860
0.4860
0.4860
1.5
0.9557
0.9557
0.9556
0.4699
0.4699
0.4698
1.5
0.5
1.1768
1.1768
1.1768
0.4018
0.4018
0.4018
1.0
1.4858
1.4858
1.4857
0.1262
0.1262
0.1261
1.5
1.8231
1.8231
1.8230
0.3908
0.3908
0.3908
2.0
2.1792
2.1792
2.1792
1.1680
1.1678
1.1678In order to further establish the accuracy of the solution of HPM, the results of the present study (in the absence of the slip parameter) are further compared with the results of the numerical method using the shooting method with the Runge–Kutta method as presented in the literature [20]. The values of the local Nusselt number NuL and local Sherwood number ShL have been calculated for various values of Nb and Nt. An excellent agreement is found between the two set of results as shown in Table 5. Also, the results that agree with the previous studies using the homotopy analysis method have been presented in the literature [9]. Therefore, the use of the homotopy perturbation method for the analysis of the thermal-diffusion and diffusion-thermal of magnetohydrodynamic squeezing unsteady flow of nanofluid between two parallel disks embedded in a porous medium under the influences of slip and temperature jump conditions is justified.Figure2 shows the effects of nanoparticle volume fraction on the nanofluid dynamic viscosity ratio. It is inferred from the results that nanofluid dynamic viscosity ratio decreases as the nanoparticle volume fraction increases. The variation of nanoparticle volume fraction with thermal conductivities ratio of copper (II) oxide-water nanofluid with different particle shapes is shown in Figure 3. It is depicted in Figure 3 that the thermal conductivity of nanofluid varies linearly and increases with increase in nanoparticle volume fraction. Irrespective of the nanoparticle volume fraction, it is established that the nanofluid with spherical shape nanoparticles has the lowest thermal conductivity ratio while the nanofluid with lamina shape nanoparticles has the highest thermal conductivity ratio. This also shows that the thermal conductivity ratio of the nanofluid is directly proportional to the nanoparticles shape factor. The suspensions of nanoparticles with a high shape factor or low sphericity in a basefluid gives higher thermal conductivity ratio to the fluid as a nanofluid than when nanoparticles with a low shape factor or high sphericity is suspended in the same fluid.Figure 2
Variation of nanofluid dynamic viscosity ratio.Figure 3
Effects of nanoparticle shape on thermal conductivity with nanoparticle volume fraction conductivity ratio of nanofluid.The influence of nanoparticle volume fraction on the velocity of the nanofluid during the squeezing flow is depicted in Figure4. It is established from the results that as the nanoparticle volume fraction increases, the velocity of the nanofluid increases between 0≤η≤0.42 (not accurately determined) because the nanoparticle volume fraction increases the skin friction coefficient of the nanofluid. However, the trend of this effect is reversed when η>0.42.Figure 4
Effect of nanoparticle volume fraction on the velocity profile whenA=2,S=1,Nt=Nb=0.2,γ=0.1,β=0.05,M=0.5,Le=Pr=1.0,andC=−1.Effect of the slip parameter on the velocity profile of the flow process is illustrated in Figure5. It is shown in the figure that the radial velocity component increases with an increase in the slip parameter near the lower disk, that is, η>0 and η>0.5 (not accurately determined). A reverse case is recorded as the flow approaches the upper disk, that is, η>0.5 and η<1 (not accurately determined). The trend in the graph and the behavior of the fluid as depicted in the figure can be physically explained that as the slip parameter increases, there is a corresponding decrease in shear stress which consequently increases the radial velocity component near the stationary lower disk while a reverse trend occurs as the nanofluid moves and flow approaches the upper disk. Figure 6 shows the effect of the increasing Hartmann parameter (M) on the velocity profile of the nanofluid flow between the parallel disk. It is observed as presented in the figure that, at increasing values of M, the velocity decreases slightly near the lower disk and as the upper disk is approached the velocity increases slightly due to the increase in the boundary layer thickness caused by the Lorentz or magnetic force field; that is, for the electrically conducting fluid in the presence of magnetic field, there is a Lorentz force which slows down the motion of the fluid in the boundary layer region. Moreover, during the squeezing flow (when the disks are moving towards each other), the situation together with the Lorentz force creates adverse pressure gradient in the flow. Whenever such forces act over a long time, there might be a point separation and back flow occurs. However, when the disks are moving apart, the reason for the behavior is quite different as there exist a vacant space and the nanofluid in that region goes with high velocity so as not to violate the mass conservation. That is, since the mass flow rate is kept conservative, decrease in the fluid velocity near the wall region will be compensated by increasing the fluid velocity near the central region.Figure 5
Effect of the slip parameter on the velocity profile whenA=2,S=1,Nt=Nb=0.2,γ=0.1,M=0.5,Le=Pr=1.0,andC=−1.Figure 6
Effect of the Hartmann parameter on the velocity profile whenA=2,S=1,Nt=Nb=0.2,γ=0.1,β=0.05,M=0.5,Le=Pr=1.0,andC=−1.As squeeze parameter (S) increases which is demonstrated in Figure 7, the radial velocity component increases. The effect is maximum at the lower disk and minimum at the upper disk. Figure 8 depicts the effect of increasing the pressure term (C) on the velocity of the fluid flow; it is shown that with, increasing C, a very slight increase in the velocity component is observed.Figure 7
Effect of the squeezing parameter on the velocity profile whenA=2,S=1,β=0.05,Nt=Nb=0.2,γ=0.1,M=0.5,Le=Pr=1.0,andC=−1.Figure 8
Effect of pressure gradient on velocity profile whenA=2,S=1,Nt=Nb=0.2,γ=0.1,M=0.5,Le=Pr=1.0,andβ=0.05.The effect of the temperature jump parameter (γ) on the temperature profile is shown in Figure 9. It is observed that as the temperature jump parameter increases, temperature distribution increases towards the lower disk where it decreases towards the upper disk. In the absence of slip, that is, γ=0, it is observed that temperature distribution equals to unity at the lower plate and zero at the upper plate. Influence of the thermophoresis parameter (Nt) is demonstrated in Figure 10 which depicts that, with increasing values of Nt, the temperature distribution increases and is maximum at the lower disk but falls rapidly towards the upper disk. The effect of the squeeze parameter (S) on temperature distribution is displayed in Figure 11. The figure illustrates that, at increasing values S, the temperature distribution at the lower disk reduces while the temperature distribution at the upper disk increases which can be physically explained as increase in S that leads to a corresponding decrease in kinematic viscosity and vice versa. It is obvious from Figure 12 that the increasing pressure term (C) has no significant effect on temperature distribution.Figure 9
Effect of the jump parameter on the temperature profile whenA=2,S=1,Nt=Nb=0.2,β=0.05,M=0.5,Le=Pr=1.0,andC=−1.Figure 10
Effect of the thermophoresis parameter on the temperature profile whenA=2,S=1,γ=0.1,Nb=0.2,β=0.05,M=0.5,Le=Pr=1.0,andC=−1.Figure 11
Effect of the squeeze parameter on the temperature profile whenA=2,γ=0.1,Nt=Nb=0.2,β=0.05,M=0.5,Le=Pr=1.0,andC=1.Figure 12
Effect of the pressure gradient on the temperature profile whenA=2,S=1,Nt=Nb=0.2,β=0.05,M=0.5,Le=Pr=1.0,andγ=0.1.Figure13 displays the influence of nanoparticle volume fraction on the temperature profile of the nanofluid during the squeezing flow. The figure shows that the temperature increases as the nanoparticle volume fraction increases in the nanofluid. The reason behind this behavior is that when the two disks move towards each other (squeezing flow) and the nanoparticle volume fraction increases, there is more collision between the nanoparticles and the particles with the boundary surface of the disk. Consequently, the resulting friction gives rise to increases in temperature within the nanofluid near the boundary region. It should be pointed out that an opposite to the trend witnessed during the squeezing flow is established when the disks are moving apart.Figure 13
Effect of nanoparticle volume fraction on the temperature profile whenA=2,S=1,Nt=Nb=0.2,β=0.05,M=0.5,Le=Pr=1.0,γ=0.1,andC=−1.The effect of the nanoparticle shape factor on the temperature profile of the nanofluid during the squeezing flow is shown in Figure14. The different shaped nanoparticles considered are sphere, brick, cylinder, and lamina. It is shown from the figure that the temperature increases as the nanoparticle shape factor increases. This is because thermal conductivity of the nanofluid increases with increase in the nanoparticle shape factor. Regardless of the nanoparticle volume fraction, it is established that the nanofluid with spherical shape nanoparticles has the lowest thermal conductivity ratio while the nanofluid with lamina shape nanoparticles has the highest thermal conductivity ratio. The suspensions of nanoparticles with a high shape factor or low sphericity in a basefluid gives higher thermal conductivity ratio to the fluid as a nanofluid than when nanoparticles with a low shape factor or high sphericity is suspended in the same fluid. Therefore, as the nanoparticle volume fraction increases due to the increased shape factor, there are increased interactions and collisions between the nanoparticles and the particles with the boundary surface of the disk, and as a consequence, more friction is created that gives rise to increase in temperature within the nanofluid near the boundary region.Figure 14
Effect of the nanoparticle shape factor on the temperature profile whenA=2,S=1,Nt=Nb=0.2,β=0.05,M=0.5,Le=Pr=3.0,γ=0.1,andC=−1.Influence of the Lewis number (Le) on the concentration profile of the nanofluid flow process is depicted in Figure 15. It is illustrated that increasing Le evokes a corresponding increase in concentration distribution at the region closer to the lower disk, though it decreases rapidly as it moves towards the upper disk. Thermophoresis parameter (Nt) effect on the concentration profile of the flow of nanofluid is observed in Figure 16, at increasing values of Nt the concentration profile increases significantly but towards the upper disk there is a steady reduction. Figure 17 depicts the effect of the Brownian motion parameter (Nb) on the concentration profile of the nanofluid flow. As observed and presented in the figure, it is established that increasing value of Nb concentration profile decreases significantly, which falls rapidly as it approaches the upper disk at suction. Also, when the squeeze parameter (S) is increased, the effect on the concentration profile of the nanofluid flow process is shown in Figure 18. From the figure, it is shown that the concentration distribution increases significantly near the wall close to the lower disk but falls rapidly as the upper disk is approached during suction. Figures 19 and 20 display the influence of nanoparticle volume fraction and nanoparticle shape factor on the concentration profile of the nanofluid, respectively. It can be seen in the figure that a maximum temperature occurs between 0.2<η<0.3 for all the nanoparticle volume fraction and nanoparticle shape factor considered. The figure shows that the concentration of the nanofluid increases as the nanoparticle volume fraction and nanoparticle shape factor increase. The increase of nanoparticle volume fraction or the use of nanoparticle with a high shape factor in a basefluid increases the surface area for chemical reaction. Therefore, as the nanoparticle volume fraction or nanoparticle shape factor increases, there are increased chemical interactions between the nanoparticles and the particles with the boundary surface of the disk, and as a consequence, more species are formed which increase the concentration of the species within the nanofluid.Figure 15
Effect of the Lewis number on the concentration profile whenA=2,S=1,Nt=Nb=0.2,β=0.05,M=0.5,γ=0.1,Pr=1.0,andC=−1.Figure 16
Effect of the thermophoresis parameter on the concentration profile whenA=2,S=1,γ=0.1,Nb=0.2,β=0.05,M=0.5,Le=Pr=1.0,andC=−1.Figure 17
Effect of the Brownian motion parameter on the concentration profile whenA=2,S=1,Nt=0.2,γ=0.1,β=0.05,M=0.5,Le=Pr=1.0,andC=−1.Figure 18
Effect of the squeeze parameter on the concentration profile whenA=2,Nt=Nb=0.2,γ=0.1,β=0.05,M=0.5,Le=Pr=1.0,andC=−1.Figure 19
Effect of nanoparticle volume fraction on the concentration profile whenA=2,S=1,Nt=Nb=0.2,β=0.05,M=0.5,Le=Pr=1.0,γ=0.1,andC=−1.Figure 20
Effect of the nanoparticle shape factor on the concentration profile whenA=2,S=1,Nt=Nb=0.2,β=0.05,M=0.5,Le=Pr=1.0,γ=0.1,andC=−1.


## 5. Conclusion

In this study, effects of nanoparticle geometry, slip, and temperature jump conditions on thermo-magneto-solutal squeezing flow of nanofluid between two parallel disks embedded in a porous medium have been investigated analytically using the homotopy perturbation method. Also, the influences of various flow parameters such as thermophoresis, the Brownian motion, and the Lewis number and pressure gradient on flow, heat, and mass transfer of the process were investigated. It is established in the study that the addition of nanoparticles to the basefluid enhances its thermal conductivity. Based on the study, the following remarks were made:(i)
Irrespective of the nanoparticle volume fraction, the thermal conductivity ratio of the nanofluid is directly proportional to the nanoparticles shape factor. This fact presents overall effects on the velocity, temperature, and concentration of the nanofluid during the flow process.(ii)
The radial velocity component increases with an increase in the slip parameter near the lower disk. A reverse case is recorded as the flow approaches the upper disk.(iii)
Increasing the values of the magnetic field parameter, the velocity decreases slightly near the lower disk and increases towards the upper disk.(iv)
As the squeeze number increases, the radial velocity component increases and the temperature distribution at the lower disk decreases, while the temperature at the upper disk increases and the concentration distribution increases significantly near the wall close to the lower disk but falls rapidly as the upper disk is approached during suction.(v)
Increasing the pressure gradient term, a very slight increase in the velocity component was observed while the pressure gradient term has no significant effect on temperature distribution.(vi)
The temperature and concentration of the nanofluid increase as the nanoparticle volume fraction and nanoparticle shape factor of the nanofluid increase.(vii)
As the temperature jump parameter increases, temperature distribution increases towards the lower disk where it decreases towards the upper disk.(viii)
Increasing values of the thermophoresis parameter, the temperature distribution increases and is maximum at the lower disk but falls rapidly towards the upper disk, while the concentration profile increases significantly, but towards the upper disk there is a steady reduction.(ix)
Also, increasing the Lewis number evokes a corresponding increase in concentration distribution at the region closer to the lower disk.(x)
Increasing the value of the Brownian number, the concentration profile decreases significantly, which falls rapidly as it approaches the upper disk at suction.Important significance of the study includes energy conservation, friction reduction, and micromixing of biological samples.



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*Source: 7364634-2018-06-03.xml*