# Comparative Analysis of the Effect of Joule Heating and Slip Velocity on Unsteady Squeezing Nanofluid Flow

Authors: Hakeem Ullah; Hina Khan; Mehreen Fiza; Kashif Ullah; Saeed Islam; Seham M. Al-Mekhlafi
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

In this paper, we studied unsteady MHD nanofluid squeezing flow between two parallel plates considering the effect of Joule heating and thermal radiation. The governing equations in the form of partial differential equations (PDEs) are transformed into a system of ordinary differential equations (ODEs) with the help of similarity transformation. The obtained boundary value problem is solved analytically by optimal auxiliary function method (OAFM) and numerically by Runge–Kutta method of order 4 (RKMO4). The OAFM results are validated and compared to the results of RKMO4. The effects of physical parameters such as stretching parameterS, Prandtl number Pr, Eckert number Ec, magnetic number M, volume friction φ electric parameter E1, and porous parameter γ on the velocity, temperature, and concentration profiles are discussed with the help of plots. Also, the skin friction and Nusselt numbers effects are discussed with the help of tabular data. As the plates move apart, the Nusselt number and the skin friction coefficient decline and the Prandtl number decreases the temperature profile, whereas the stretching and Eckert number increases causing to increase the temperature field.


---

## Body

## 1. Introduction

A nanofluid is a fluid made up of nanoparticles, which are nanometer-sized particles having diameter less than 100 nm. The concept of nanofluid was given by Choi and Eastman [1]. These fluids are colloidal nanoparticle deferments in a base fluid such as metals, oxides, carbides, and carbon nanotubes are often used as nanoparticles in nanofluids and the base fluids contain water, ethylene glycol, oil, toluene, biofluids, and polymer solution. The nanoparticles are up to 5% of volume fraction in nanofluids. In recent years, many researchers have studied and reported nanofluid technology experimentally or numerically in the presence of heat transfer. The nanofluid have industrial and engineering applications such as electronic cooling devices, chemical factors, heat pumps, and heat exchangers [2–13]. Nanofluid have a variety of features that could make them beneficial in a variety of heat transfer applications. As the heating/cooling fluids have an important role in the energy efficient heat transfer materials. The heat and mass transfer is an important phenomenon in the nanofluid because of its industrial applications such as polymer formation, compression, power transmitting, lubricant system, and food processing. Stephen [14] introduced the idea of squeezing flow under lubrication. Domairry and Hatami studied the flow of squeezed nanofluid between two plates [15]. The unsteady flow of squeezing flow between two parallel plates is studied by Pourmehran et al. [16]. The unsteady squeezed flow is studied by Gupta and Ray [17]. This study has extended by Khan et al. [18] by considering the viscous dissipation properties. Magnetohydrodynamics (MHD) is the effect of magnetic field on the electrical conducting fluid, such as water and [19], which have been presented for the first time. This field have many applications in industry and engineering such as MHD sensors, MHD cooling reactors, and casting. The MHD and heat transfer analysis with thermal radiation of nanofluid is studied by Ibrahim and Shankar [20]. Malvandi and Ganji [21] studied the MHD and heat transfer of nanofluid. The impact of thermal radiation and slip on MHD nanofluid was studied by Haq et al. [22]. Govindaraju et al. [23] studied the entropy analysis of MHD nanofluid. Uddin et al. [24] studied the porous medium of MHD nanofluid flow on the horizontal plate. The stagnation point flow of MHD nanofluid is investigated by Hsaio [25]. The dissipation and chemical reaction analysis for MHD nanofluid is study by Kameswaran et al. [26]. Matin et al. [27] and Pal et al. [28] studied the dissipation analysis and heat transfer analysis over the stretching sheet. The analysis of porous medium on MHD nanofluid flow was studied by Zhang et al. [29]. Elshehbey and Ahmed [30] studied the Buongiorno nanofluid model. The thermal radiation and Ohmic dissipation effects on the MHD flow with heat transfer is studied by Olanreaju [31]. Ullah et al. studied the MHD nanofluid with thermal radiation [32], whereas Rashidi et al. [33] examined the MHD flow caused by heat generation. From the literature, it is shown that the MHD flow over stretching sheets with heat transfer, the effect of electric field, Ohmic dissipation joule, and thermal radiation have not been considered and very little consideration is devoted towards it in the viscous fluids. Having this view, the unsteady MHD nanofluid squeezing flow between two parallel plates considering the effect of Joule heating and thermal radiation is considered. The effect of electric and magnetic fields are considered in the momentum and energy equations, and thermal radiation and Ohmic dissipation are taken into description. The skin friction Nusselt number and Sherwood number are elaborated with the help of tables. The analytical and numerical techniques are used for the treatment of BVP. Normally the numerical techniques require the process of linearization and discretization, which may turn to divergent solutions in some cases. Recently Herisanu [34, 35] presented a new optimal technique OAFM that do not need the linearization/discretization and small parameters issues such as perturbation method. OAFM has a large convergence region, which control the convergence with the help of optimal constant. OAFM provides us the accurate solution at just the first iteration without using the complex mathematical algorithms, and even a low specified computer can run the algorithm easily, and also the procedure of OAFM is very easy in implementation and quick convergent as compared to the other semianalytical methods such as HAM and OHAM. Some recent development in this area can be seen in [37–41].The objective of this study is to find the analytical (OAFM) and numerical (RKMO4) solutions of unsteady MHD nanofluid squeezing flow considering the effect of Joule heating and thermal radiation. The OAFM results are validated and compared to numerical method results.


## 2. Basic Ideas of Optimal Auxiliary Functions Method [38, 39]

Assume that the nonlinear differential equation(1)LΘη+sη+ΝΘη=0,with the BCs(2)ΒΘη,dΘηdη=0.We write the solutions as follows:(3)Θη=Θ0η+Θ1η,Ei,i=1,2,3,……s.The initial approximate solutions is of the following form:(4)LΘ0η+sη=0,BΘ0η,dΘ0ηdη=0.And the first approximation solutions is as follows:(5)Θ1η,Ei+NΘ0η+Θ1η,Ei=0,BΘ1η,Ei,dΘ1η,Eidη=0.Also,(6)NΘ0η+Θ1η,Ei=NΘ0η+∑dΘ1kη,EikNkΘ0η.The approximated solution is obtained by using equation(3).The auxiliary functionsEis can be obtained by using the method of least square,(7)JEi,Ej=∫abR2m,λ,α,εdε,where(8)Rm,λ,α,ε=LΘEi,Ej+gε+NΘEi,Ej,i=1,2,…P,j=p+1,p+2…s.And(9)∂J∂E1=∂J∂E2=∂J∂E3=...∂J∂CP=∂J∂EP+1=...∂J∂Es=0.


## 3. Problem Formulation and Solution

We consider the unsteady two-dimensional flow of squeezed nanofluid between two parallel plates with heat and mass transfer with water as base fluid and nanoparticles as copper (Cu), silver (Ag), alumina (Al2O3), and titanium oxide (TiO2). A uniform magnetic field is applied vertically to the direction of flow and the plates. The separation of the plates is given as z=±l1−αt1/2=±ht, where l is the initial position (at time t  = 0). The flow is considered incompressible with no chemical reaction. The fluid is electrically conducting in the presence of applied magnetic field B¯=0,B0,0 and electric field E¯=0,0,−E0. The flow is due to squeezing. The electric and magnetic fields obeys the Ohms law “ J¯=σE¯+V¯×B¯,” where J¯ is the Joule current, σ is the electrical conduction, and E¯andV¯ are electric and velocity fields. The induced magnetic field and Hall current are ignored and the electric and magnetic field contribute in the momentum and thermal heat equation. The flow description can be seen in Figure 1.Figure 1
Schematic representation of the problem.The fundamental equations are as follows:(10)∂xu+∂yv=0,ρnf∂tu+u∂xu+v∂yu=−∂p∂x+μnf∂xxu+∂yyu−σnfE0B0−B02u,ρnf∂tv+u∂xv+v∂yv=−∂p∂y+μnf∂xxv+∂yyv,∂tT+u∂xT+v∂yT=knfρCpnf∂xxT+∂yyT+μnfρCpnf4∂xxu2+∂yu+∂xv2+σnfρuB0−E02,and∂tC+u∂xC+v∂yC=D∂xxC+∂yyC.With BCs,(11)v=dhdt,u=−L∂yu,C=Ch,T=Th,y=ht,v=∂yu=∂yT=0,C=C0,T=T0,y=0,and(12)μnf=μf1−ϕ2.5Brinkman,KnfKf=2Kf+Ks−2ϕKf−Ks2Kf+Ks+2ϕKf−KsGarnetts Maxwell,andσnf=1−ϕσf+ϕσs.Using the following similarity variable as [40].(13)u=αx21−αtf′η,θ=T−T0Th−T0,ϕ=C−C0Ch−C0,η=yl1−αt1/2.With the help of similarity variable obtain in the above codes,(14)f″″−SA11−φ2.5f′f″+3f″+ηf‴−ff‴−−M2E1−f′=0,−M2E1−f′,and(15)A1=1−φ↔+φ↔ρSρf,A2=1−φ↔+φ↔ρCpsρCpf,andA3=knfkf.Here, A1, A2, and A3 are dimensionless constants,S=al2/2vf is the squeeze number, Pr=μfpCpf/pfkf is the Prandtl number, Sc=vf/Dnf is Schmidt number, and δ=L/l1−αt1/2 is the velocity slip parameter,Ec=pf/pCpfαη/21−αt2 is the Eckert number,M2=σB02/ρα is magnetic number, and E1=2E01−αt/B0 is the electrical number.The BCs are as follows:(16)θ′0=0,f″0=0,f0=0,ϕ0=0,f1=1,f′1=−δf″1,θ1=1,andϕ1=1.The physical quantities of interest are the skin-friction coefficientCf, the Nusselt number Nux, and the Sherwood number Shx, defined as follows:(17)Nux=IqwkfTh−T0,Cf=Twρnfv2w,Shx=lmwDCh−C0,wheremw=−Dnf∂yCy=ht,andTw=μnf∂yuy=ht.Using the dimensionless form as follows:(18)−A3θ′1=Nux,Sh=−ϕ1,andCf=f″1A11−φ2.5,we obtained(19)fiv−SA11−φ2.5f′f″+3f″+ηf‴ff‴−M2E1−f′=0,θ′′+PrSA2A3θ′f−ηθ′+EcPrA31−φ2.5f″24ϕ2+f′2+M2Prf′−E12=0,andϕ′′+ScSfϕ′−ScSηϕ′=0.Boundary conditions are as follows:(20)θ′0=0,f0=0,f′0=0,ϕ0=0,f1=1,f′1=−δf″1,θ1=1,andϕ1=1.The linear and nonlinear operators are given as follows:(21)Lfη=fivη,Lθη=θ″η,ϕLθη=ϕ″η,Nfη=SA11−φ2.5f′f″+3f″′+ηf″′−ff″′−M2E1−f′=0,Nθη=PrSA2A3θ′f−ηθ′+EcPrA31−φ2.54ϕ2f′2+f″2+M2Prf′−E12=0,andNϕη=ScSfϕ′−ScSηϕ′=0.We have,(22)θ′0=0,f′′0=0,f0=0,ϕ0=0,whenx=0,f1=1,f′1=−δf′1,θ1=1,ϕ1=1,whenx=1,ϕ″η=0,andϕ1=1.The initial solutions are as follows:(23)fη=123η−η3,θη=η2,andϕη=η.Also,(24)Nfη=−SA11−ϕ2.5−3η−9−3x3−x22+33x−x32,Nθη=PrSA2A33x2−x4−2ηx+PrEcA31−φ2.5−5x2921−x2+12x4,andNϕη=ScS3x−x32−η=0.The first approximation, we have the following equation:(25)fivη+D1η,η2,Ei−SA11−ϕ2.5−3η−9x+3−3x22−3xM−3x−x32−3+3E1x=0,θ″η+D3η,η2,EpPrSA2A33x2−x4−2ηx+PrEcA31−ϕ2.5−5x2921−x2+12x4+D4η,η2,Em=0ϕ″η+D5η,η2,ErScS3x−x32−η+D6η,η2,En=0.,The OAF can be chosen freely as follows:(26)D1f0η,Ei=−E1+E2η,D2f0η,Ej=−E3+E4ηe−η−E5+E6η+E7η2e−2η,D3f0η,EP=0,D4f0η,Em=−E8+E9ηe−η−E10+E11η+E12η2e−2η,D5f0η,Er=−E13+E14ηe−η−E15+E16η+E17η2e−2ηD6f0η,En=−E18+E19ηe−η−E20+E21η+E22η2e−2η.We get,(27)fivη+D1η,η2,Ei−SA11−ϕ2.5−3η−9x+3−3x22−3x−3x−x32−3+3ME1x=0,θ″η+D3η,η2,EpPrSA2A33x2−x4−2ηx+PrEcA31−ϕ2.5−5x2921−x2+12x4+D4η,η2,Em=0,andϕ″η+D5η,η2,ErScS3x−x32−η+D6η,η2,En=0.The final results is furnished by using the convergence control constants Es.


## 4. Results and Discussion

### 4.1. Graphical Discussion

In this section, the results are discussed in detail with the help of graphs. Figure2 shows the effect of the magnetic and electric fields on the velocity profile. As the magnetic and electric fields increases, the velocity profile decreases. Since the magnetic and electric field oppose the electrically conducting fluid and as a result the velocity of the fluid reduces. Figure 3 shows the effect of stretching parameter s on the velocity field. Growth in the stretching parameter causes the velocity profile to rise. Since the stretching parameter is increased, it assists the flow, and hence the velocity of the fluid is increased. The effect of the volume friction on the velocity profile can be seen in Figure 4. The volume friction reduces the velocity profile and act as opposing force to the flow. The effect of the Prandtl number and volume friction numbers on the temperature profile are observed in Figures 5 and 6. The temperature profile is reduced when increasing the Prandtl and volume friction numbers. Since the Prandtl/volume friction number increase kinetic energy of the particle and the elastic collision of the particles reduces the temperature profile. The effect of stretching parameter and Eckert number are depicted in Figures 7 and 8. The same behavior of both the parameter is observed for the temperature profile as it increases the temperature profile. Also the effect of the stretching parameter and Schmidt number on the concentration profile can be seen in Figures 9–11. By increasing the stretching and Schmidt numbers the concentration profile increases. Also, the effects of magnetic and electric fields on the temperature profiles are given in Figures 12 and 13. The temperature profile decreases by increasing the magnetic field. It is due to the fact that increasing the magnetic field increases the elastic collision of the nanoparticles, which reduces the temperature profile. We obtain the opposite effect of the electric field on the temperature profile as compared to the magnetic effect on the temperature profile.Figure 2
M versusfη.Figure 3
S versusfη.Figure 4
φ versus fη.Figure 5
Pr versusθη.Figure 6
φ versus θη.Figure 7
s versusθη.Figure 8
Ec versusθη.Figure 9
S versusθη.Figure 10
Sc versusθη.Figure 11
E1 versusfη.Figure 12
M versusθη.Figure 13
E1 versusθη.


### 4.2. Tables Discussion

The influence of S on the skin-friction coefficientCf, Nusselt number Nux, and the Sherwood number Sh are tabulated in Table 1. By increasing S, the Cf and Nux decreases while Sh increases. The influence of Ec on Cf, Nux, and Sh are shown in Table 2. By increasing Sc, the Cf, Nux, and Sh decreases. Also the influence of M is tabulated for various values of M on Cf, Nux, and Sh in Table 3. From this table it is clear that by increasing M, Cf reduces while Nux and Sh increases. The effect of nanoparticle volume fraction φ on Cf, Nux, and Sh is presented in Table 4. By increasing φ, Cf increases while Nux and Sh decreases. Again the present method is validated by comparing the results as given in Table 5.Table 1
Impact of S onCf, Nux, and Sh.
sCfNuxShx0.1−2.545260.25486−2.0658790.2−1.342950.187977−1.06942540.3−2.094190.104693−1.00000000.4−3.0546890.054896−0.9875630.5−4.0214580.0012458−0.568963Table 2
Impact of Ec onCf, Nux, and Sh.
EcCfNuxShx0.1−2.548620.025486−0.568960.2−2.521450.087562−0.568750.3−2.478560.082542−0.589520.4−2.365480.081546−0.487563Table 3
Impact ofM on Cf, Nux, and Sh.
MCfNuxShx1−2.8547850.035689−0.4875962−2.885460.087546−0.4875933−2.985630.087562−0.4882364−2.8452630.045785−0.489596Table 4
Impact ofφ on Cf, Nux, and Sh.
φCfNuxShx0.10−1.021540.054256−0.632560.20−1.0214850.054120−0.6585230.30−1.0214960.051243−0.6587450.40−1.214580.050852−0.532110Table 5
Comparison of OAFM and RKMO4 results.
ηfηθηfηθη00.0000000000.0000000000.0000000000.0000000000.10.141358791.141358780.141358630.141358780.20.280666390.280666380.280666690.280666380.30.415781370.415781310.415781360.415781360.40.544379790.544379750.544379890.544379780.50.663858370.663858350.663858520.663858360.60.771231320.771231310.771231310.771231310.70.863018530.863018530.863018510.863018520.80.935123640.935123650.935123620.935123620.90.982700440.982700450.982700410.982700421.01.000000001.000000001.000000001.00000000


## 4.1. Graphical Discussion

In this section, the results are discussed in detail with the help of graphs. Figure2 shows the effect of the magnetic and electric fields on the velocity profile. As the magnetic and electric fields increases, the velocity profile decreases. Since the magnetic and electric field oppose the electrically conducting fluid and as a result the velocity of the fluid reduces. Figure 3 shows the effect of stretching parameter s on the velocity field. Growth in the stretching parameter causes the velocity profile to rise. Since the stretching parameter is increased, it assists the flow, and hence the velocity of the fluid is increased. The effect of the volume friction on the velocity profile can be seen in Figure 4. The volume friction reduces the velocity profile and act as opposing force to the flow. The effect of the Prandtl number and volume friction numbers on the temperature profile are observed in Figures 5 and 6. The temperature profile is reduced when increasing the Prandtl and volume friction numbers. Since the Prandtl/volume friction number increase kinetic energy of the particle and the elastic collision of the particles reduces the temperature profile. The effect of stretching parameter and Eckert number are depicted in Figures 7 and 8. The same behavior of both the parameter is observed for the temperature profile as it increases the temperature profile. Also the effect of the stretching parameter and Schmidt number on the concentration profile can be seen in Figures 9–11. By increasing the stretching and Schmidt numbers the concentration profile increases. Also, the effects of magnetic and electric fields on the temperature profiles are given in Figures 12 and 13. The temperature profile decreases by increasing the magnetic field. It is due to the fact that increasing the magnetic field increases the elastic collision of the nanoparticles, which reduces the temperature profile. We obtain the opposite effect of the electric field on the temperature profile as compared to the magnetic effect on the temperature profile.Figure 2
M versusfη.Figure 3
S versusfη.Figure 4
φ versus fη.Figure 5
Pr versusθη.Figure 6
φ versus θη.Figure 7
s versusθη.Figure 8
Ec versusθη.Figure 9
S versusθη.Figure 10
Sc versusθη.Figure 11
E1 versusfη.Figure 12
M versusθη.Figure 13
E1 versusθη.


## 4.2. Tables Discussion

The influence of S on the skin-friction coefficientCf, Nusselt number Nux, and the Sherwood number Sh are tabulated in Table 1. By increasing S, the Cf and Nux decreases while Sh increases. The influence of Ec on Cf, Nux, and Sh are shown in Table 2. By increasing Sc, the Cf, Nux, and Sh decreases. Also the influence of M is tabulated for various values of M on Cf, Nux, and Sh in Table 3. From this table it is clear that by increasing M, Cf reduces while Nux and Sh increases. The effect of nanoparticle volume fraction φ on Cf, Nux, and Sh is presented in Table 4. By increasing φ, Cf increases while Nux and Sh decreases. Again the present method is validated by comparing the results as given in Table 5.Table 1
Impact of S onCf, Nux, and Sh.
sCfNuxShx0.1−2.545260.25486−2.0658790.2−1.342950.187977−1.06942540.3−2.094190.104693−1.00000000.4−3.0546890.054896−0.9875630.5−4.0214580.0012458−0.568963Table 2
Impact of Ec onCf, Nux, and Sh.
EcCfNuxShx0.1−2.548620.025486−0.568960.2−2.521450.087562−0.568750.3−2.478560.082542−0.589520.4−2.365480.081546−0.487563Table 3
Impact ofM on Cf, Nux, and Sh.
MCfNuxShx1−2.8547850.035689−0.4875962−2.885460.087546−0.4875933−2.985630.087562−0.4882364−2.8452630.045785−0.489596Table 4
Impact ofφ on Cf, Nux, and Sh.
φCfNuxShx0.10−1.021540.054256−0.632560.20−1.0214850.054120−0.6585230.30−1.0214960.051243−0.6587450.40−1.214580.050852−0.532110Table 5
Comparison of OAFM and RKMO4 results.
ηfηθηfηθη00.0000000000.0000000000.0000000000.0000000000.10.141358791.141358780.141358630.141358780.20.280666390.280666380.280666690.280666380.30.415781370.415781310.415781360.415781360.40.544379790.544379750.544379890.544379780.50.663858370.663858350.663858520.663858360.60.771231320.771231310.771231310.771231310.70.863018530.863018530.863018510.863018520.80.935123640.935123650.935123620.935123620.90.982700440.982700450.982700410.982700421.01.000000001.000000001.000000001.00000000


## 5. Conclusion

The OAFM results are identical to the results obtain from RKMO4 results. OAFM provide us a convenient way to control the convergence in the large flexible region with the help of optimal constants. OAFM contain less computational work and can be easily handle by a low specification computer. The OAFM provides us the first iteration results, which in comparison is proving that the method is simply applicable and provide us the accurate solution even at first iteration.From the above discussion the following points is of importance.(i)
By increasing S, theCf and Nux decreases while Sh increases(ii)
By increasing Sc, theCf, Nux, and Sh decreases.(iii)
By increasingM, Cf reduces while Nux and Sh increases(iv)
By increasingφ, Cf increases while Nux and Sh decreases(v)
Schmidt number increases the concentration profile(vi)
The electrical current resists the flow whereas the stretching parameter assists the flow velocity(vii)
The Prandtl number decreases the temperature profile whereas the stretching and Eckert number increases causes to increase the temperature field(viii)
The stretching and Schmidt numbers increase causes to decrease the concentration profile(ix)
The mention techniques can be applied in future for more complex physical models



---

*Source: 8452862-2022-08-16.xml*