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 parameter , Prandtl number , Eckert number Ec, magnetic number , volume friction electric parameter , and porous parameter 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.
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 equationwith the BCs
We write the solutions as follows:
The initial approximate solutions is of the following form:
And the first approximation solutions is as follows:
Also,
The approximated solution is obtained by using equation(3).
The auxiliary functions Eis can be obtained by using the method of least square,where
And
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 , where 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 and electric field . The flow is due to squeezing. The electric and magnetic fields obeys the Ohms law “ ,” where is the Joule current, is the electrical conduction, and 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.
The fundamental equations are as follows:
With BCs,
and
Using the following similarity variable as [40].
With the help of similarity variable obtain in the above codes,
and
Here, A1, A2, and A3 are dimensionless constants, is the squeeze number, is the Prandtl number, is Schmidt number, and is the velocity slip parameter, is the Eckert number, is magnetic number, and is the electrical number.
The BCs are as follows:
The physical quantities of interest are the skin-friction coefficient , the Nusselt number , and the Sherwood number , defined as follows:
Using the dimensionless form as follows:
we obtained
Boundary conditions are as follows:
The linear and nonlinear operators are given as follows:
We have,
The initial solutions are as follows:
Also,
The first approximation, we have the following equation:
The OAF can be chosen freely as follows:
We get,
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. Figure 2 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.
4.2. Tables Discussion
The influence of S on the skin-friction coefficient , Nusselt number , and the Sherwood number are tabulated in Table 1. By increasing S, the and decreases while increases. The influence of Ec on , , and are shown in Table 2. By increasing Sc, the , , and decreases. Also the influence of M is tabulated for various values of M on , , and in Table 3. From this table it is clear that by increasing M, reduces while and increases. The effect of nanoparticle volume fraction on , , and is presented in Table 4. By increasing , increases while and decreases. Again the present method is validated by comparing the results as given in Table 5.
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, the and decreases while Sh increases(ii)By increasing Sc, the , , and Sh decreases.(iii)By increasing M, reduces while and Sh increases(iv)By increasing , increases while 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
Data Availability
All data are available in the paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.