Numerical Simulation of a Time-Dependent Electroviscous and Hybrid Nanofluid with Darcy-Forchheimer Effect between Squeezing Plates

In this article, the behavior of transient electroviscous fluid flow is investigated through squeezing plates containing hybrid nanoparticles. A hybrid nanofluid MoS2+Au/C2H6O2−H2O was formulated by dissolving the components of an inorganic substance such as molybdenum disulfide (MoS2) and gold (Au) in a base fluid of ethylene glycol/water. This hybrid non-liquid flow was modeled by various nonlinear mathematical fluid flow models and subsequently solved by numerical as well as analytical methods. For the numerical solution of nonlinear ODEs, a built-in function BVP4C was used in MATLAB, and the same problem was solved in MATHEMATICA by HAM. The result of the present problem related to the results obtained from the existing literature under certain conditions. The outcomes revealed that the concentration profiles were more sensitive to homogeneity diversity parameters. The simulation of the various physical parameters of the model indicated that the heat transfer through a mixture of hybrid nanofluids was greater than a simple nanofluid. In addition, the phenomenon of mixed convection was considered to improve the velocity of simple nanofluids and hybrid nanofluids, when both cases have low permeability. A rise in the volume fraction of the nanomaterials, Φ, was associated with an increase in the heat transfer rate. It was observed that the heat transfer rate of the hybrid nanofluids MoS2+Au/C2H6O2−H2O was higher than that of the single nanofluids MoS2/C2H6O2−H2O.


Introduction
Our survival in this modern age depends heavily on machines and technology. We are updating them in many ways for efficient use. A few years ago, a new technology was introduced to immerse nanosized materials into base liquids to improve the cooling capacity of certain machines. Nanofluids are used to improve the thermal conductivity and heat transfer rate of the base fluids. Typically, the fundamental liquids are triethylene glycol, oil, water, polymeric solutions, refrigerants, bioliquids, ethylene, and lubricants. The most commonly used nanoparticles are copper, gold, zirconium, aluminum, metal nitride AIN, Al 2 O 3 , diamond, carbon nanotubes, CuO metal carbides, and SiC. There are lots of qualities associative with nanofluids such as uniformity, long-lasting stability, of the stream. In a fully developed region, the skin friction and velocity do not show fluctuation in the streamwise direction. Many factors influence the physics of microchannel flow, such as the electric field, the electrical charge density, and the zeta potential of the microchannel. These factors can be included in the classical Navier-Stokes equation for a microchannel flow. Yang et al. [22] and Arulanandam and Li [23] numerically solved the fluid flow model in the microchannel. Their model used the Poisson-Boltzmann equation to describe the behavior of the electrical double layer potential; the Laplace equation explained the electrostatic field, and the modified version of the Navier-Stokes equations described the influence of the body forces that occur as a result of the interaction between zeta and the electrical potential. The numerical outcomes of their model were in line with their empirical observations. In existing and new biotechnologies application, it is important to understand the flow in micrometer-sized channels to improve new microfluidic instruments with the power to manipulate and transport liquid on a small scale [24]. In the last few years, much research has been conducted in the field of microfluidic and microscale instruments [25]. Surface tension and electrokinetic influence play a significant role in surface phenomena at micrometer scales [26,27].
Khan et al. [28][29][30] studied nanofluid flow in terms of mass and heat transfer in the presence of variable magnetic fields. In their proposed model, they analyzed the behavior of (Cu-H 2 O) nanofluid between two parallel discs under the influence of a variable magnetic field. Khan et al. [31][32][33][34] reviewed the Poisson-Boltzmann formulation, which comes from the assumption of thermodynamic equilibrium by imposing the condition that the distribution of the ions would not be affected by the movement of the fluid. However, while this is a reasonable assumption for the steady flow of electroosmotic liquids via linear microchannels, there are some important cases where the convective transport of ions has considerable influence. In these cases, the Nernst-Planck model should be used instead of the Poisson-Boltzmann model to generate an electric field in the region under consideration. Sajid et al. [35] examined the incompressible Prandle number micropolar fluid motion at a porous stretching sheet along with the impact of a temperaturedependent exponential heat source, viscous dissipation, higher-order chemical reaction, convective boundary conditions, and nonlinear thermal radiation. Jamshed et al. [36] examined the transient flow behavior of the Casson non-Newtonian nanofluid with respect to entropy generation and thermal transport. The influences of thermal solar transport and slip conditions for the Casson nanofluid flow were thoroughly investigated. To analyze nanofluids regarding flow behavior and thermal transport, slippery surface conditions were imposed under convective heat. The mathematical formulations for the Casson nanofluid flow and heat transfer were summarized through presumed boundary conditions. Waqas et al. [37] observed that the increasing need for modern technology to better enhance the heat transfer efficiency of thermal systems has made nanofluids more important in the last two decades. Keeping such applications in mind, the presence of the Carreau-Yasuda nanofluid moving microorganisms, thermal radiation, and the conditions of Robin's boundaries were examined.
Succeed in previous works [28,29], persuading to examine the joint effect of the electromagnetic force, inertia force, surface suction/injection, magnetic field, and ionized fluid in the squeezing flow of hybrid nanofluids between two parallel plates. We assumed that the lower plate was stretchable and permeable with a linear velocity. The mathematical model of (MoS 2 )/Au + base fluid ethylene glycol-water was formulated and then transformed into reduced differential equations via the similarity transformation. In the squeezing flow, the aforementioned combination of influences has not been studied before. After ionization, we studied the behaviour of the fluid as well. The PCM/BVP4C/HAM application was used to generate the results, which were validated based on the available numerical values from the previous works. This study is a novel and original consideration of an internal hybrid nanofluid flow. Numerical computation and graphical representation were completed for the Nussult number, velocity field, skin frictions, temperature, and ion distribution.

Formulation
Molybdenum disulfide (MoS 2 ) and gold (Au) in water and ethylene glycol formation was considered to flow between two infinite parallel plates, as shown in Figure 1 from the lower plate. Therefore, the porous lower plate was included in the physical illustration for the possible fluid suction/injection with the wall mass velocity denoted as V w = − V 0 1−αt ; V 0 > 0 for suction, V 0 < 0 for injection, and V 0 = 0 corresponds to an impermeable plate. A further assumption was that the lower and upper plates were maintained at fixed temperatures T1 and T2, respectively. Furthermore, the lower plate was stretchable with the linear velocity u w = − bx 1−αt ; t < 1 α , while the inclusion of the time-dependent magnetic field was formulated as H = B 0 1−αt . The fundamental governing equations are as follows [28][29][30]: Continuity equation: ∂u ∂x Momentum equations with electroviscous and uniform magnetic effect [28,29]: Poisson Equation [38,39]: Nernst-Planck Equations [38,39]: Energy Equation [28]: Mass Transfer Equation [29]: where U = ∂v ∂x − ∂u ∂y are the associate conditions at the lower and upper plates [28]. Here, (ρC p ) hn f is the heat capacity of the hybrid nanofluid, T is the temperature, C is the concentration, n + and n − are positive and negative ions, K * is the permeability of porous space, Fr = C * b √ K * is the non-uniform inertia coefficient of the porous medium [40], C * b is the drag coefficient, B 0 is the strength of the magnetic field, W is the electrical potential of ions, B = ρk 2 T 2 ε 0 ε 2z 2 e 2 µ 2 is fixed at a constant temperature, Q 0 is the heat generation, K 2 = 2z 2 e 2 n 0 ε 0 εk b T is the inverse Debye constant, ρ hn f is the fluid density of the hybrid nanofluid, σ hn f is electrical conductivity of hybrid nanofluid, and µ hn f is the kinematic viscosity of the hybrid nanofluid. Nanofluids are defined as [28]: where κ hn f is the thermal conductivity of the hybrid nanofluid, κ b f is the thermal conductivity of the base fluid, and Φ 1 and Φ 2 are the volume fractions of hybrid nanoparticles.
The boundary conditions of the proposed model are as follows [28]: The following similarity transformations [28] are considered for reducing the PDE (1)-(7) to an ODE system, Therefore, Equation (1) is satisfied and the remaining Equations (2)-(7) transform into the following form where is the local porosity parameter, λ > 0 refers to the stretching lower plate, λ = 0 denotes the fixed/static lower plate, Fr is the Forchheimer number [40], Sc = is the heat source/sink substrate, and the Brownian motion is . T 0 and C 0 are the ambient temperature and nanoparticle concentration, T 2 and C 2 are any reference temperature and concentration chosen unequal to T 0 and C 0 [28,40], with constants δ = T 1 −T 0 T 2 −T 0 and ω = C 1 −C 0 C 2 −C 0 . The boundary conditions in their reduced form are as follows, The required physical parameters are the Nusselt number and the skin friction coefficient at the upper plate and lower plate, which can be defined as, and where

Numerical Solution by PCM
This section presents the numerical solution of the proposed mathematical model for the best selection of the continuation parameters through the algorithm of a wellknown numerical scheme (PCM). The PCM algorithm is used to solve the nonlinear ODEs in (11)-(16) through the boundary condition given in Equation (17): •

First order of ODE
We use the variables below to transform the PDEs given in (11)-(16) into the first order of ODEs.
By putting these new variables into Equations (11)- (16), we obtain the following transformed equations as below.
and the boundary conditions become • Introducing parameter q gives ODEs in the q-parameter group To obtain ODEs in the q-parameter group, we need the q-parameter in Equations (21)-(26); therefore, • Differentiation by q reaches the following system with respect to the sensitivities of the parameter-q Differentiating the Equations (28)-(33) w.r.t by q where h 1 is the coefficient matrix, e 1 is the remainder, and where y 1 and v 1 are the vector functions. By resolving the two types of Cauchy problems for each component, the system of the ODEs are satisfied automatically.
and we have the boundary conditions. •

Use of Numerical Solution
An absolute scheme has been considered for the resolution of the problem v i+1 • Take the corresponding coefficients As mentioned, the boundaries are commonly used for P i , where 1 ≤ i ≤ 14; for the solution of the ODEs, we used the equation d 2 = 0, and its corresponding matrix representation as given below.

Results and Discussion
In this section, the effects of different involved flow parameters on the velocity and electric field components are discussed graphically. The flow of hybrid nanofluids, variable electric fields, and heat/mass transfer phenomenon between two long parallel plates were observed. The system of highly nonlinear ODEs in Equations (11)-(16) with boundary conditions given in Equation (17) was solved analytically as well as numerically through HAM (Mathematica), PCM (Matlab), and BVP4C (Matlab). The local heat and mass transfer rate and the local skin friction coefficient on the surface of the squeezing plates were estimated, which was necessary in terms of the physical properties. The ongoing problem has a variety of physical parameters that plays a key role in the flow of fluid. The velocity profile, pressure distribution, mass, and heat transfers were calculated by solving the system of the ODEs in Equations (11)- (16), and their numerical results are graphically shown in Figures 2-11 for various physical parameter values. In this section, the effects of the different flow parameters of the nonlinear ODEs in Equations (11)-(16) following the boundary conditions in Equation (17) are given. The statistics in Table 1 provide complete information about the thermophysical properties of the nanomaterials (Molybdenum disulfide (MoS 2 ), Gold(Au)) and Ethylene-glycol with water (C 2 H 6 O 2 − H 2 O). It is necessary to mention that our proposed model produces good results when compared with the results of the models available in the current literature, and the comparison of the numerical results with (Najiyah et al. [28]) is shown in Tables 2 and 3 for the upper and lower plates. Table 4 illustrates that the numerical and analytical outcomes of three important flow parameters, skin friction, and Nusselt number, which were obtained through three different numerical and analytical schemes (BVP4c, PCM, and HAM). The impacts of these fluid flow parameters are illustrated quantitatively through various tables and graphs for the velocity components f (η), f (η), electrical potential P(η), electric field components G(η), H(η), temperature profile θ(η), and mass transfer. This section deals to analyze the influence of the involved parameters. It is necessary to note that the positive values of Sq reflect the top plate apart from the bottom plate whereas the negative values signify that the top plate attracted towards stationary bottom plate. The low or high value of Sq can be thought of as slow or rapid vertical velocity at the top plate, or the increment in the distance between the two plates, respectively.       Table 4. Comparison of the numerical and analytical results of three methods PCM, BVP4C, and HAM for the skin friction and Nusselt number, with various physical parameters and Φ 1 = Φ 2 = 0.1.   Figure 4b. In essence, a rise in Fr causes fluids to become more resilient, leading to decreased f (η). Physically, boosting the amplitude of Fr reduces the interior nanofluid velocity. But after the middle area, the effect upon velocity distribution for rises of Fr. There is a reduction in velocity distribution, but the velocity distribution of the fluid flow increases when η > 0.5. An increment in Sq leads to a reduction in the concentration profiles. Therefore, H(η) and G(η) increase and become a parabolic shape in Figure 5. Figure 6 illustrates that the impact of H(η) and G(η) profile enhance due to the augmenting value of Schmidt number Sc for MoS 2 + Au/C 2 H 6 O 2 − H 2 O and Au/C 2 H 6 O 2 − H 2 O and so, the width of the concentration boundary layer is reduced. The reactant concentration rises rapidly as the diffusion coefficient of the species reduces, i.e., larger values of Sc cause a rapid increment in the concentration of the flow field, However, with increasing Schmidt number Sc values, the profile of H(η) reduces. The impact of the injection parameter S at θ(η) is explained for the higher absolute value of S in Figure 7a, which reflects that the upper plate moving down led to a decrease in the atomic collision in the nanocomponents and, consequently, a decline in the temperature and concentration profiles due to the injection parameter S, as shown in Figure 7b. The impact of Q on φ(η) and θ(η) can be seen in Figure 8 for both MoS 2 + Au/C 2 H 6 O 2 − H 2 O and Au/C 2 H 6 O 2 − H 2 O. It has been concluded that the influence of Q (heat generation parameter) at the concentration profile θ(η) and φ(η) having an augmenting behaviour of velocity and temperature for MoS 2 + Au/C 2 H 6 O 2 − H 2 O and Au/C 2 H 6 O 2 − H 2 O. The augmenting values of Q (heat generation parameter) reflect T2 > Tl. i.e., which signify more transfer of heat from surface to the liquid and consequently the temperature of the fluid is raised for both MoS 2 + Au/C 2 H 6 O 2 − H 2 O and Au/C 2 H 6 O 2 − H 2 O. We observed an increment in the temperature profile with an increment in Q. Accordingly, there is an increase in uniform chemical reactions, which reduces viscosity. As can be seen in Figure 8b, the two properties of the fluid flow vary in opposite directions. In addition, reverse behavior is also seen for θ(η) in Figure 8a. The outcome is based on the assumption that heat generation would rise the nuclear energy in the fluid. Figure 9a,b depicts the effect of thermophoresis parameters and Brownian motion Nt and Nb on the concentration and temperature profile. The temperature distribution has been shown to increase monotonically from lower plate to the upper plate. It is worth noticing that the two key parameters Nt and Nb emphasis the thermophoresis and Brownian motion effects respectively. Figure 10 shows how the volume percentage of nanomaterials Φ affects the concentration profile φ(η) in both the (SNF) and (SNF) scenarios (HNF). It is clear that increasing the volume fraction of the nano-components Φ increases the velocity profile for both (HNF) and (SNF). The increase in the volume fraction of nanocomponents is owing to an indirect relationship between dynamic viscosity and the volume fraction of nanofluids, as demonstrated by the physical phenomena. As a result, the viscosity of the conventional fluid decreases in comparison to a rise in Φ, and the fluid flow increases. The effect of Le on φ is depicted in Figure 11. With an increase in Le, the volume percentage of nanoparticles increases, as does the thickness of the boundary layer. As a result, the concentration profile increases.

Concluding Remarks
The behaviour of a squeezing flow of hybrid nanoparticles under the influence of an electric field is explained in this article. BVP4C, PCM, and HAM were used to solve the suggested model of nanofluid flow numerically and analytically. The model's solution was used to investigate the effects of physical parameters on the flow field, heat and mass profiles. The numerical outcomes of the proposed model were also validated using the results from the literature, and a good correlation was established. The following are the closing statements, which are based on the findings:

•
The velocity profile shows gradually opposite behavior as a result of the augmenting value of the suction/injection parameter S.

•
The increment in nanomaterial volume fraction reflects the same rising influence at the velocity profile for the value of M and Sq. Informed Consent Statement: Not applicable.

Data Availability Statement:
The data is available on reasonable request from the corresponding author.