Analytical and Numerical Analysis of the Squeezed Unsteady MHD Nanofluid Flow in the Presence of Thermal Radiation

In this study, the unsteady squeezing nanofluid flow between two plates with thermal radiation has been investigated. The governing equations of the flow model have been transformed to a set of nonlinear ordinary differential equations (ODEs) from a set of partial differential equations (PDEs) using a suitable similarity variable. The optimal auxiliary function method (OAFM) and Runge–Kutta method of order 4 (RK method of order 4) are used for the solution of the modeled problem. The variation of the squeezing number, Prandtl number, Eckert number, and thermal radiation has been presented. The magnetic field resists the flow velocity, and the Prandtl number resists the temperature distribution. The increase in volume fraction decreases the velocity profile whereas increases the temperature profile. The skin friction coefficient and the Nusselt number are inversely proportional to S. The effect of increasing values of Ec is to decrease the skin friction coefficient Cf and the heat transfer rate Nux . The increasing value of φ increases the skin friction coefficient and decreases the heat transfer rate.


Introduction
The nanofluid consists of the nanometer particle size of the fluid having less than 10^-9nm, such as copper, aluminum, silver, silicon, aluminum oxides, and graphite. The base fluids are water, oil, and ethylene glycol. Choi et al. [1] introduced the term nanofluid and heat transfer features of the fluids, such as thermal conductivity is enriched by the addition of nanoparticles into it [2,3]. The study of heat and mass transfer for squeezing unsteady viscous flow between two parallel plates has a wide range of physical applications, including lubrication systems, polymer processing, food processing, hydrodynamical machines, compression, and crop damage due to freezing, formation, and dispersion. Squeeze flow, also known as squeezing flow, squeezing film flow, or squeeze flow theory, is a flow in which a material is squeezed out between two parallel plates. Josef Stefan studied it in 1874 for the first time. There are several squeeze flow models that may be used to explain Newtonian and non-Newtonian fluids that are squeezed under various geometries and condi-tions. Squeeze flow is used in a variety of scientific and engineering areas, including welding engineering, and materials science, to name a few. Sheikholeslami et al. [4] used heat line analysis to simulate a two-phase simulation of nanofluid flow and heat transfer. Moreover, Sheikholeslami et al. [5] investigated the unsteady flow of a nanofluid squeezing between two parallel plates using the Adomian decomposition method (ADM). Also, the problem of squeezing flow between rotating disks has been studied by Hamza [6] and Bhattacharyya [7]. Magnetohydrodynamics (MHD) is the information of the magnetic assets of electrically conducting fluids. Plasmas, electrolytes, water, and liquid metals are examples of magneto fluids. Hannes Alfven [8] was the first who introduced the field of MHD. MHD has several applications in the field of industries and engineering such as plasma, crystal growth, MHD sensors, liquid-metal cooling of MHD casting, MHD power generation, and magnetic drug targeting. MHD depends on the strength of the magnetic field; the stronger the magnetic field, the greater is MHD effects and vice versa. MHD includes plasmas, molten metals, saltwater, and electrolytes. [9][10][11][12][13]. Seddiqui, Domairy, and Aziz et al. [14,15] explored two-dimensional MHD squeezing flow between parallel plates and parallel disc. Magnetic nanofluid is a one-of-a-kind material that combines the qualities of a liquid with a magnetic material [16]. Magneto-optical wavelength filters [17,18], optical modulators, [19], nonlinear optical materials [20], tunable optical fiber filters [21], optical gratings, and optical switches have all been discovered to use such fluids [22]. Changing the magnetic field can change a lot of the physical features of these fluids. They also served as an excellent model system for fundamental investigations. Manipulation of nanoparticles with carbon nanotubes has recently been shown to result in increased thermal conductivity. The thermal characteristics of nanofluids are the parameters that are critical to the performance of nanofluids. Thermal conductivity, specific heat, viscosity, and heat transfer coefficient are the four variables. Solar collector thermal performance is mostly determined by how thermal characteristics behave under various operating situations. Temperature, environmental conditions, type of base fluid, particle size and form of the nanoparticle, and volume concentration are all operating parameters. Taking into account all of these factors, choosing the right nanofluid is critical for optimum performance. Based on this idea, scientist studies many important flow models for various physical aspects [23][24][25][26]. Explicit Runge-Kutta techniques have become popular for wave simulations due to their great accuracy and low memory requirements [27]. The traditional fourth-order Runge-Kutta technique requires three memory places per dependent variable [28]; however, low-storage approaches may be constructed that only require two memory sites per dependent variable [29]. A third-order Runge-Kutta technique can readily accomplish this characteristic, while a fourth-order approach requires an additional stage [30]. Because the evaluation of the derivative function is the fundamental cost of integration, and each level necessitates a function evaluation, the new stage implies a large rise in cost. Some of the recent development in the related field can be seen in [26,[31][32][33][34][35][36][37][38][39][40][41][42]. In this paper, we propose the OAFM [43,44] for the squeezed unsteady MHD nanofluid flow in the presence of thermal radiation. The validity of OAFM is based on the auxiliary function which optimally controls the convergence of the solution. The efficiency of OAFM is proved in comparison to the numerical solutions obtained by Runge-Kutta method of order 4. In the present work, we show how solutions to the modeled problem can be obtained using OAFM, without the need for complex and complicated calculations with low specification of computer with high accuracy. Moreover, OAFM presented here is less computational work and simple in applications at the first iteration. Up to now, the squeezed unsteady MHD nanofluid flow in the presence of thermal radiation has not been studied. The purpose of this study is to provide an analytical solution of the squeezed unsteady MHD nanofluid flow in the presence of thermal radiation by using the newly developed method OAFM. In the above-mentioned problem, analytical and numerical methods are used for the solution of the problem. The numerical methods required the linearization and discretiza-tion techniques and huge computer memory with operating time; we show how solutions to the boundary value problem can be obtained using OAFM, without the need for complex and involved mathematical algorithms, and at a relatively low computing memory and easy approach with high accuracy at the first iteration.
2. Basic Mathematical Theory of OAFM [43,44] Let us look at the OAFM for the differential equation where L, N denotes the linear and nonlinear operators; s denotes the source function, f ðκÞ, and is an unknown function at this stage; the initial/boundary conditions are Because finding an accurate solution to severely nonlinear equations is extremely difficult, the proposed approximation is as follows: Using Equation (3) in Equation (1), we have where E k , k = 1, 2 ⋯ s are control convergence parameters to be determined. The initial approximation is determined as The first approximation is obtained as The nonlinear term is expressed as The last term in Equation (7) seems difficult to solve, so to avoid this difficulty and to fast the convergence of the 2 Journal of Nanomaterials solution. Equation (6) can be written as

Convergence of the Method.
In order to obtain the convergent solution, we calculate the optimal constants also known as control convergence constant by method of least squares: These optimal constants are resubmitted into original equation to get the series solution.
where I is equation domain.
The unknown constants are established as The mth order approximate solution can be obtained by these constants so obtained.

Formulation of the Problem
The flow and heat transfer of two-dimensional unsteady squeezing nanofluid via the lateral plates is observed in this study, as shown in Figure 1.
At any nondimensional time t, the distance between the two plates is given as The two plates are squeezed until they touch each other at 1/α, whereas at α < 0, the two plates are separated. Here, α is a constant, l is the initial position (at t = 0), and z is the axial coordinate, which is obviously zero from the flow zone, with the flow model evaluated along the x and y axes. The nondimensional time throughout the flow is represented by the variable t. The heat source and viscous dissipation effect as a result of friction caused by fluid flow shear are both protected. As a result, this behavior occurs when the Eckert number is very high. Meanwhile, the copper material was enclosed in a nanofluid. A uniform magnet is used perpendicular to the flow. The following are the governing equations: ∂T ∂t The velocities in the x and y directions are represented by u and v, respectively. While p, T, T∞, f , and k nf are the effective density, dynamic viscosity, heat capacity, and thermal conductivity of the nanofluid; ρnf , μnf , (ρCp), and k nf are the effective density, dynamic viscosity, heat capacity, and thermal conductivity of the nanofluid, respectively.
The radiative heat flux in Equation (4) is given by the Rosseland formula as The Stefan-Boltzmann constant and the mean absorption number, respectively, are σ * and k * . We assume that the temperature variation among the flow is greatly constrained, and that the expression T 4 may be regarded a linear function of temperature, based on various research, As a result, T 4 is enlarged by disregarding the higher-order terms and utilizing Taylor series expansion about T.
Substituting Equations (21) and (22) into Equation (15), we obtain To begin, introduce the following quantities: where N denotes heat/thermal radiation as defined by Equation (10), and A 2 and A 3 denote dimensionless constants as defined by Equations (25) and (26)

Journal of Nanomaterials
Here, S is the squeezing integer; P r and Ec are the Prandtl and Eckert numbers, respectively.
The following quantities are categorically used for practical interest as defined Equation (24) provides the following result: The linear and nonlinear operators of Equations (11) and (12) are

Journal of Nanomaterials
From Equation (5), we have has solution as Based on Equation (23), we get  In the first approximation based on Equations (8), (32), (33), and (24), we get with boundary conditions The OAF can be chosen freely as We obtained the first approximate solution And its solution is given as by putting the values of the optimal constants obtained from the method of least square.

Numerical Method of Solution
The differential Equations (25) and (26) along with the side condition (28) have been solved using the fourth-order Runge-Kutta method (FORKM) along with the shooting technique. The nonlinear Equations (25) and (26) of fourth and second order are reduced to a set of six first-order simultaneous equations as follows: The boundary condition now became

Graphical Discussion.
The main focus of the study is to develop a mathematical model of an unstable nanofluid flow squeezed between parallel plates, as illustrated in Figure 1. Figure 2 shows the effect of A 1 on the velocity profile f . An increase in A 1 causes to increase the velocity profile f . Figures 3 and 4 demonstrate the influence of the squeeze number on the velocity profile. The motion of the plates represented by squeezing flow is indicated by the squeeze number S. When S > 0, the plates are moving separately, but when S < 0, the plates are moving collectively. Positive and negative squeezing values have distinct effects on the velocity profile. The velocity rises when the absolute value of the squeeze number is 0.5, but drops when it is >0.5. An increase in the stretching parameter causes to increase the velocity profile. The strecthing parameter assists the flow velocity. Figure 5 shows the effect of A 2 on the temperature profile θðηÞ. The temperature profile increases by increasing A 2 . Also, the effect of A 3 on the temperature profile θðηÞ is given in Figure 6. By increasing the values of A 3 , a reverse effect as compared to A 2 has been observed. The influence of the Prandtl number P r on temperature distributions is seen in Figure 7. With a large number of P r , the temperature distribution obviously decreases, whereas with a small number of P r , it grows. Fluids with a low Prandtl number have a higher thermal diffusivity than fluids with a high Prandtl number. A high P r causes the thermal boundary layer to 10 Journal of Nanomaterials decrease as a result. The impact is much more noticeable for a small number of P r because the thermal boundary layer thickness is so high. The impact of Eckert number and Prandtl number on the temperature profile is shown in Figure 8. It is observed that the temperature rises substantially with rising E c . The effect of s and δ on the temperature profile is given in Figures 9 and 10. Again, an increase in s and δcauses to decrease the temperature profile. The effect of the thermal radiation N on the temperature profile is given in Figure 11. The increase in thermal radiations causes to increase the temperature profile, since the thermal radiations increase the kinetic energy of the particles and their collision is caused to rise the temperature profile. The effect of magnetic field on the velocity profile is presented in Figure 12.
The increase in magnetic field M reduces the velocity profile, since the magnetic field is applied perpendicular to the flow, and hence the conducting fluid particles feel the opposite force of magnetic field and hence reduce the velocity profile. The effect of volume fraction of nanofluid on the velocity and temperature profiles is given in Figures 13 and 14, respectively. The increase in volume fraction ϕ decreases the velocity profile whereas increases the temperature profile.

Tables' Discussion.
The results of OAFM are validated in comparison with the results obtained from the Runge-Kutta method of order 4 along with absolute errors as given in Tables 1 and 2. The OAFM results obtained at just one iteration provide us a simple way to control the convergence and nearly identical to the results obtained from RKM fourth-order method. The effects of the squeeze number S on the skin friction coefficient, Cf , and the Nusselt number, Nu x , are given in Table 3. From Table 3, it is obvious that the skin friction coefficient and the Nusselt number are inversely proportional to S.       Table 6. From this table, it is concluded that the increasing value of φ increases the skin friction coefficient and decreases the heat transfer rate.

Conclusions
In this study, a new analytical method is suggested for the solution of the model problem. We obtain the first-order series solution for the governing equations of the model problem and achieved the first-order solution with high accuracy. For the accuracy and validity of OAFM, results are compared with numerical method. For comparison, it is concluded that the OAFM is very accurate and simple in application. OAFM is very easy in applicable to high nonlinear initial and boundary value problems even if the nonlinear initial/boundary value problem does not contain the small parameter. In comparison with other analytical methods, OAFM is very easy in applicability and provides us good results of more complex nonlinear initial/boundary value problems. OAFM contains the optimal auxiliary constants through which we can control the convergence as OAFM contains the auxiliary functions D 1 , D 2 , D 3 , E 4 in which the optimal constants E m , E n , E r , E p and the control convergence parameters exist to play an important role to get the convergent solution which is obtained rigorously. The computational work in OAFM is less when compared to other methods, and even a low specification computer can do the computational work easily. The less computational work and rapid convergent solution at just the first iteration enable us to implement this efficient method in our future work for more complex models arising from real-world problems. The numerical method required maximum space and time as compared to OAFM is the short method and is very rapidly convergent. Numerical methods required to have large computational work and required the latest computer for computational work.
Based on the results and discussion, some points are presented here: (i) The motion of the plates represented by squeezing flow is indicated by the squeeze number S. When S > 0, the plates are moving separately, but when S < 0, the plates are moving collectively. Positive and negative squeezing values have distinct effects on the velocity profile. The velocity rises when the absolute value of the squeeze number is 0.5, but drops when it is >0.5 (ii) With a large number of P r , the temperature distribution obviously decreases, whereas with a small number of P r , it grows. Fluids with a low Prandtl number have a higher thermal diffusivity than fluids with a high Prandtl number. A high P r causes the thermal boundary layer to decrease as a result. The impact is much more noticeable for a small number of P r because the thermal boundary layer thickness is so high (iii) The increase in magnetic field M reduces the velocity profile, since the magnetic field is applied perpendicular to the flow, and hence, the conducting fluid particles feel the opposite force of magnetic field and hence reduce the velocity profile (iv) The increase in volume fraction ϕ decreases the velocity profile whereas increases the temperature profile Distance between the plates J w : Mass flux k: Thermal conductivity (Wm −1 K −1 ) Kr: Rotation parameter k: The boundary parameter M: Magnetic parameter m: Hall parameter n e : Number density of electron Nb: Brownian motion Nt: Thermophoretic parameter Νu: Nusselt number O: Origen P: Fluid pressure ðPaÞ Pr: Prandtl number Q w : Heat flux ðWm −2 Þ