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

In this paper, we studied unsteady MHD nanouid squeezing ow between two parallel plates considering the eect of Joule heating and thermal radiation.e governing equations in the form of partial dierential equations (PDEs) are transformed into a system of ordinary dierential equations (ODEs) with the help of similarity transformation.e obtained boundary value problem is solved analytically by optimal auxiliary function method (OAFM) and numerically by Runge–Kutta method of order 4 (RKMO4). e OAFM results are validated and compared to the results of RKMO4. e eects of physical parameters such as stretching parameter S, Prandtl number Pr, Eckert number Ec, magnetic numberM, volume friction φ electric parameter E1, and porous parameter c on the velocity, temperature, and concentration pro les are discussed with the help of plots. Also, the skin friction and Nusselt numbers eects are discussed with the help of tabular data. As the plates move apart, the Nusselt number and the skin friction coecient decline and the Prandtl number decreases the temperature pro le, whereas the stretching and Eckert number increases causing to increase the temperature eld.


Introduction
A nano uid is a uid made up of nanoparticles, which are nanometer-sized particles having diameter less than 100 nm. e concept of nano uid was given by Choi and Eastman [1]. ese uids are colloidal nanoparticle deferments in a base uid such as metals, oxides, carbides, and carbon nanotubes are often used as nanoparticles in nano uids and the base uids contain water, ethylene glycol, oil, toluene, bio uids, and polymer solution. e nanoparticles are up to 5% of volume fraction in nano uids. In recent years, many researchers have studied and reported nano uid technology experimentally or numerically in the presence of heat transfer. e nano uid have industrial and engineering applications such as electronic cooling devices, chemical factors, heat pumps, and heat exchangers [2][3][4][5][6][7][8][9][10][11][12][13]. Nano uid have a variety of features that could make them bene cial in a variety of heat transfer applications. As the heating/cooling uids have an important role in the energy e cient heat transfer materials. e heat and mass transfer is an important phenomenon in the nano uid because of its industrial applications such as polymer formation, compression, power transmitting, lubricant system, and food processing. Stephen [14] introduced the idea of squeezing ow under lubrication. Domairry and Hatami studied the ow of squeezed nano uid between two plates [15]. e unsteady ow of squeezing ow between two parallel plates is studied by Pourmehran et al. [16]. e unsteady squeezed ow is studied by Gupta and Ray [17].
is study has extended by Khan et al. [18] by considering the viscous dissipation properties. Magnetohydrodynamics (MHD) is the e ect of magnetic eld on the electrical conducting uid, such as water and [19], which have been presented for the rst time. is eld have many applications in industry and engineering such as MHD sensors, MHD cooling reactors, and casting. e 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. e 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. e stagnation point flow of MHD nanofluid is investigated by Hsaio [25]. e 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. e analysis of porous medium on MHD nanofluid flow was studied by Zhang et al. [29]. Elshehbey and Ahmed [30] studied the Buongiorno nanofluid model. e 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. e effect of electric and magnetic fields are considered in the momentum and energy equations, and thermal radiation and Ohmic dissipation are taken into description. e skin friction Nusselt number and Sherwood number are elaborated with the help of tables. e 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][38][39][40][41]. e 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. e OAFM results are validated and compared to numerical method results.

Basic Ideas of Optimal Auxiliary Functions
Method [38,39] Assume that the nonlinear differential equation with the BCs We write the solutions as follows: e initial approximate solutions is of the following form: And the first approximation solutions is as follows: Also, e approximated solution is obtained by using equation (3). e auxiliary functions Eis can be obtained by using the method of least square, where And

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 (Al 2 O 3 ), and titanium oxide (TiO 2 ). A uniform magnetic field is applied vertically to the direction of flow and the plates. e separation of the plates is given as where l is the initial position (at time t = 0). e flow is considered incompressible with no chemical reaction. e fluid is electrically conducting in the presence of applied magnetic field B � (0, B 0 , 0) and electric field E � (0, 0, −E 0 ). e flow is due to squeezing. e electric and magnetic elds obeys the Ohms law " J σ(E + V × B)," where J is the Joule current, σ is the electrical conduction, and E and V are electric and velocity elds. e induced magnetic eld and Hall current are ignored and the electric and magnetic eld contribute in the momentum and thermal heat equation. e ow description can be seen in Figure 1.
e fundamental equations are as follows: and Using the following similarity variable as [40].
e BCs are as follows: e physical quantities of interest are the skin-friction coefficient Cf, the Nusselt number Nu x , and the Sherwood number Sh x , defined as follows: where m w � −D nf z y C y�h(t) , and Using the dimensionless form as follows: (1), and we obtained Boundary conditions are as follows: e linear and nonlinear operators are given as follows: We have, e initial solutions are as follows: Also, e rst approximation, we have the following equation: Mathematical Problems in Engineering e OAF can be chosen freely as follows: We get, e nal results is furnished by using the convergence control constants Es.

Graphical Discussion.
In this section, the results are discussed in detail with the help of graphs. Figure 2 shows the e ect of the magnetic and electric elds on the velocity pro le. As the magnetic and electric elds increases, the velocity pro le decreases. Since the magnetic and electric eld oppose the electrically conducting uid and as a result the velocity of the uid reduces. Figure 3 shows the e ect of stretching parameter s on the velocity eld. Growth in the stretching parameter causes the velocity pro le to rise. Since the stretching parameter is increased, it assists the ow, and hence the velocity of the uid is increased. e e ect of the volume friction on the velocity pro le can be seen in Figure 4. e volume friction reduces the velocity pro le and act as opposing force to the ow. e e ect of the Prandtl number and volume friction numbers on the temperature pro le are observed in Figures 5 and 6. e temperature pro le 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 pro le. e e ect of stretching parameter and Eckert number are depicted in Figures 7 and 8. e same behavior of both the parameter is observed for the temperature pro le as it increases the temperature pro le. Also the e ect of the stretching parameter and Schmidt number on the concentration pro le can be seen in Figures 9-11. By increasing the stretching and Schmidt numbers the concentration pro le increases. Also, the e ects of magnetic and electric elds on the temperature pro les are given in Figures 12 and 13. e temperature pro le decreases by increasing the magnetic eld. It is due to the fact that increasing the magnetic eld increases the elastic collision of the nanoparticles, which reduces the temperature pro le. We obtain the opposite e ect of the electric eld on the temperature pro le as compared to the magnetic e ect on the temperature pro le.

Tables Discussion.
e in uence of S on the skinfriction coe cient Cf, Nusselt number Nux, and the Sherwood number Sh are tabulated in Table 1  Mathematical Problems in Engineering 7 increases. e e ect 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.

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

Data Availability
All data are available in the paper.

Conflicts of Interest
e authors declare that they have no con icts of interest.