Analysis of Heat Transfer of Mono and Hybrid Nanoﬂuid Flow between Two Parallel Plates in a Darcy Porous Medium with Thermal Radiation and Heat Generation/Absorption

: In the last two decades, academicians have concentrated on the nanoﬂuid squeezing ﬂow between parallel plates. The increasing energy demands and their applications have seen the focus shifted to the hybrid nanoﬂuid ﬂows, but so much is still left to be investigated. This analysis is executed to explore the symmetry of the MHD squeezing nanoﬂuid (MoS 2 /H 2 O) ﬂow and the hybrid nanoﬂuid (MoS 2 –SiO 2 /H 2 O–C 2 H 6 O 2 ) ﬂow between the parallel plates and their heat transport property. The heat transport phenomenon is analyzed with the magnetic ﬁeld, thermal radiation, heat source/sink, suction/injection effect, and porous medium. In the present model, the plate situated above is in the movement towards the lower plate, and the latter is stretching with a linear velocity. The prevailing PDEs depicting the modeled problem with the aforementioned effects are transformed via similarity transformations and solved via the “bvp4c” function, which is an inbuilt function in MATLAB software. The control of the factors on the ﬁelds of velocity and temperature, heat transfer rate, velocity boundary layer patterns, and streamlines is investigated. The solution proﬁles are visually shown and explained. Furthermore, the Nusselt number at the bottom plate is larger for the (MoS 2 –SiO 2 /H 2 O–C 2 H 6 O 2 ) hybrid nanoﬂuid than for the (MoS 2 /H 2 O) nanoﬂuid ﬂow. In the presence of suction/injection, the streamlines appear to be denser. In addition, the magnetic ﬁeld has a thinning consequence on the velocity boundary layer region. The results of this study apply to several thermal systems, engineering, and industrial processes, which utilize nanoﬂuid and hybrid nanoﬂuid for cooling and heating processes.


Introduction
A high-heat-transfer rate is desired in many applications, including dehumidifiers, heat exchangers, evaporators, and power stations. As a result, concerned researchers aim to develop a variety of models and methods to increase the heat-transmission rate. Integrating nanoparticles into conventional fluids is a cutting-edge method for meeting the requirement for high-heat-transfer rates. When nanoparticles of one or more kinds are disseminated into a working fluid, the outcome is a nanofluid. Depending on the usage and its applications, nanoparticles of various materials are used, such as metal, oxides of metals, carbon nanotubes, carbides, etc. Mono nanofluids are formed when only one variety of nanoparticles are injected into the working fluid (Choi [1]). Many researchers (Choi et al. [2]) have investigated the characteristics of amalgamated and Polymer nanomaterials. polyester/titanium oxide (TiO 2 ), polymer/hydroxides, and polymer/carbon nanotubes (CNT).
The rate of heat transmission is one of the characteristics that differentiate between the nanofluids having one and more nanoparticles. With the aforementioned applications of hybrid nanofluids, some scholars have lately studied the flows of hybrid nanofluids. Of late, Ullah et al. [6] studied the aspects of entropy generation on a flow of hybrid nanofluid (SWCNT-MWCNT/kerosene oil) past a stretching sheet. Yaseen et al. [7] surveyed the influence of suction/injection on the hybrid nanofluid flow past a moving surface. Ullah et al. [8] published a study in which they analyzed the influence of the Hall current and the Lorentz force on ZnO-Ni/water hybrid nanofluid over a transient stretching and rotating disk. Garia et al. [9] surveyed the flow of nanofluid with MoS 2 and SiO 2 nanoparticles past a wedge and a cone and inferred the role of time lag during the transport of heat. A theoretical analysis of the influence of suction, injection, heat generation, and magnetohydrodynamic effects on a Williamson hybrid nanofluid (i.e., (MoS4-Cu)/water) over a stretching cylinder was done by Kavya et al. [10]. Raju et al. [11] inspected the heat transport property of two types of ternary hybrid nanofluids in an expanding or contracting porous channel. They considered ternary hybrid nanofluids with two different combinations of nanoparticles, graphene, carbon nanotubes, and aluminium oxide; and copper, silver, and copper oxide. Upadhya et al. [12] studied the entropy generation of an incompressible, steady Casson, micropolar, and hybrid nanofluid over a curved stretching sheet. They considered the hybrid nanofluid with silica SiO 2 and aluminium oxide Al 2 O 3 nanoparticles dispersed in water. They concluded that the micropolar fluid shows higher entropy generation compared to the Casson and hybrid nanofluid. Ullah et al. [13] investigated the thermal radiation and thermal slip parameter effects on the flow of hybrid nanoliquid past a stretchable rotating disk. They considered the hybrid nanoliquid with a combination of AA7072 and AA7075 nanoparticles and water.
Because of its uses in a variety of sectors, the squeezing flow induced by two parallel plates has grabbed the curiosity of many researchers in recent years. There are many applications but are not limited to the manufacturing and production fields, such as chocolate fillers, electric motors, hydraulic lifts, moving pistons, power transmission squeezed film, and polymer processing. In the aforementioned applications, the flow of fluid occurs when plates move towards or away from each other. Furthermore, the study of flow between the two parallel plates has received much importance due to their direct real-world applications in engineering sciences, such as viscometers, compression shaping, injection, the polymer and food industries, liquid-metal lubrication, and so on. Applications of squeezing problems can also be seen in tunnels of roads and railway lines, hydropower station caverns, water conveyance tunnels, underground sports halls, and oil and gas storage caverns. Singh et al. [14] surveyed the impact of velocity slip on the time-dependent flow of four different water-based nanofluids between two plates placed Symmetry 2022, 14, 1943 3 of 23 parallel to each other. Salehi et al. [15] scrutinized the heat-transport property of squeezing MHD hybrid nanofluid flow induced by the two infinite plates placed parallel to each other. They inferred from their findings that as the squeeze number grows so does the thermal profile. The influence of the amalgamation of two different nanoparticles, Cu and Al 2 O 3 , in the flow with water as a working fluid in the middle of two parallel plates was scrutinized by Khashi'ie et al. [16]. They discussed their model with the assumption that the flow was induced due to the movement of the upper plate and the deformation of the lower plate. Kapen et al. [17] conducted an analysis to scrutinize the consequence of injection on hybrid nanofluid flow (Cu-Al 2 O 3 /water) between two stationary parallel plates and performed the stability analysis of their solution. The two-dimensional squeezing unsteady MHD Casson fluid flow between two parallel plates with nonlinear radiation was scrutinized by Kumar et al. [18]. Shah et al. [19] conducted an analysis to scrutinize the effect of the Hall current and electric field on a flow of nanofluid with micropolar nature between two parallel and rotating plates. Li et al. [20] studied the axisymmetric transient squeezing flow of the Newtonian non-conducting fluid between the two circular horizontal plates in a porous medium.
Heat generation/absorption and thermal radiation cannot be ignored at high-functioning temperatures. The awareness of the two aforementioned factors is a consideration in the design of appropriate tools since many technical activities occur at high temperatures. They also play an important part in many manufacturing units, such as glass manufacturing and construction of furnaces, and industrial-related applications, such as thrust systems in rockets, plasma mechanics, spaceships, nuclear plant ignition processes, solar radiations, and compressors of ships. The literature is rich in studies which deal with the effect of generation/absorption of heat or radiation. Ullah [21] analyzed the influence of heat source and thermal radiation on the flow of hybrid nanofluid. He considered the flow over the disk surface in a porous medium and hybrid nanofluid consisting of γ-ALOOH and AA7075 nanoparticles and gasoline oil as the working fluid. Henda et al. [22] looked upon the combined influence of radiation and heat sink/source on the third-grade bioconvective nanofluid flow induced by a stretched cylinder. Mishra and Kumar [23] analyzed the consequences of radiation on the heat-transmission performance of a nanofluid flow over a stretching sheet. Rawat and Kumar [24] published a study with an analysis of the heattransference property of nanofluid flow with copper nanoparticles past a stretching surface under the combined impact of heat sink/source and thermal radiation. Ullah et al. [25] studied the combined influence of a thermal-heat source and the exponential space heart source on the heat-transport rate of a nanofluid on an infinite and rotating disk. Sharma et al. [26] inspected the influence of thermal radiation on the graphene Maxwell nanofluid flow over a linearly stretched sheet. Ge-Jile et al. [27] scrutinized the heat-transport property of a nanofluid flow past a vertical cone in the presence of thermal radiation. They considered the nanofluid consisting of water as a working fluid with ferrous nanoparticles. Raju et al. [28] studied the cross-diffusion effects on the Carreau fluid flow over a wedge filled with gyro tactic microorganisms under the influence of thermal radiation. Ullah et al. [29] explored the influence of temperature-dependent viscosity and exponential heat source on the flow of a nanoliquid in a rotating system.
The flow passing through a porous media has a critical advantage in the diverse environmental and manufacturing settings, including the design of boilers, geophysics, and reactors that involve convective flow. Many academicians have investigated the uses and impact of the porous media on fluid flow because of the aforementioned applications. Nandeppanavar et al. [30] surveyed the heat-transport property of the time-independent flow of the Casson nanofluid past a stretching sheet in the porous medium formulated by Darcy's law. Shah et al. [31] surveyed the rotational flow of micropolar nanofluid with CNT nanoparticles in the several base fluids between the horizontal and parallel plates in a porous medium formulated by Darcy-Forchheimer's law. Shafiq et al. [32] surveyed the nanofluid flow with thermal slip and convective heating in a rotating structure embedded in a porous medium characterized by the Darcy-Forchheimer relation. Ahmad et al. [33] Symmetry 2022, 14,1943 4 of 23 published a comparative study describing the heat-transport property of mono nanofluid (Cu/water) and hybrid nanofluid (Al 2 O 3 -Cu/water) flow through a stretching surface in a porous medium. Mishra and Kumar [34] studied the impact of a porous medium on the heat-and mass-transmission property of a nanofluid flow past a wedge. Yaseen et al. [35] published a comparative study describing the heat-transport property of hybrid nanofluid (MoS 2 -SiO 2 /kerosene oil) and (MoS 2 /kerosene oil) nanofluid flow amid the two disks in a rotating state in a porous medium characterized by the Darcy-Forchheimer relation. Ullah et al. [36] investigated the significance of entropy generation in the flow of Ethylene glycol/water nanofluid in a rotating frame in a Darcy-Forchheimer porous medium. Hayat et al. [37] investigated the entropy optimization in the nonlinear mixed convective unsteady magnetohydrodynamic flow of nanomaterials in porous space. Li et al. [38] inspected the flow of MHD third-grade liquid through Darcy-Forchheimer's porous space with homogeneous-heterogeneous reactions. Ullah [39] performed the theoretical investigation of MHD nanofluid over a rotating and stretching disk in a Darcy-Forchheimer porous medium with zero-mass flux.
There has been a great amount of concern regarding the flow behavior and heattransmission characteristics of nanofluid squeezing flow amid two parallel plates in recent years. The growing interest is because of the practical and theoretical values of squeezing flow. Squeezing flow has substantial applications in chocolate fillers, electric motors, hydraulic lifts, moving pistons, power transmission squeezed film, and polymer processing. The additional benefits are seen when the model is exposed to radiation, or the system has to deal with the generation of heat while in operation. Considering the aforementioned applications, the current paper deals with the squeezing flow model of hybrid nanofluid/nanofluid between the two parallel plates. The literature still lacks a study that compares the heat-transmission property of the hybrid nanofluid and nanofluid. In

Model Development
In this article, a two-dimensional unsteady hybrid nanofluid/nanofluid squeezing flow in the middle of two infinite parallel plates in a Darcy porous medium with a magnetic field B(t) = (1 − αt) −0.5 B 0 is considered ( Figure 1). In this model, the influence of thermal radiation and heat source/sink are also deliberated, and the buoyancy effect (gravitational force) is not taken into account. The upper plate and lower plate are . Moreover, the upper plate has velocity, (squeezing of fluid), and it is moving towards the lower plate. Note that the steady-state case of the model is recovered when α = 0. Furthermore, Furthermore, the current model deals with the fluid suction/injection, and v w = − V 0 1 − αt is the wall mass velocity; where V 0 > 0 for suction, V 0 < 0 for injection, and V 0 = 0 corresponds to an impermeable plate. In addition, the lower plate displaces with linear α and the lower plate is stretching/shrinking.
Moreover, the upper plate has velocity, , and it is moving towards the lower plate.
Note that the steady-state case of the model is recovered when 0 α = . Furthermore, the temperatures of the lower and upper plates are designated as T1 and T2, respectively. Furthermore, the current model deals with the fluid suction/injection, and the wall mass velocity; where 0 0 V > for suction, 0 0 V < for injection, and 0 0 V = corresponds to an impermeable plate. In addition, the lower plate displaces with linear veloc- and the lower plate is stretching/shrinking. In view of the aforementioned assumptions, the governing equations are as follows (see Refs. [16,40]): The current model considers the following boundary conditions: In view of the aforementioned assumptions, the governing equations are as follows (see Refs. [16,40]): The current model considers the following boundary conditions: In the aforementioned equation, (u, v) is the velocity in the (x, y) direction. Furthermore, the other symbol represents the following: T-temperature, ρ-density, µ-dynamic viscosity, C p -heat capacity, k-thermal conductivity, B(t)-magnetic field strength, φ * indicates porosity of the porous medium, k o -permeability of the porous medium, λ is the stretching/shrinking parameter,Q o -heat absorption/generation coefficient, and 'b' denotes the stretching/shrinking rate of the lower plate.
In the view of the Rosseland approximation, (q r ) is defined as (see Ref. [7]): where "σ * stands for the Stefan-Boltzmann constant, and K * hn f stands for mean absorption coefficient".
Under the assumption that the variation of temperature is low between the layers, the following relationship is used (see Ref. [7]): With simple computations using the Equations (6) and (7), Equation (3) is simplified as:  Table 1, we adopt the correlations by Devi and Devi [41], which are feasible and correct based on the experimental validation. These correlations are built based on the physical assumptions. The subscripts are used as follows: "f -base fluid, hnf -hybrid nanofluid, and nf -nanofluid". Table 1. Thermophysical properties of mono and hybrid nanofluid (see Devi and Devi [41]).

Properties Nanofluid Hybrid Nanofluid
Dynamic viscosity µ n f Density

Transforming the Governing Equations Using Similarity Transformation
The succeeding similarity variables are employed for transforming the equations described in Section 2.1 (see Ref. [16]): where T 0 is the reference temperature for hybrid nanofluid flow. Using Equation (9), Equations (2), (4), and (8) are altered as: where sents the temperature-ratio parameter. Furthermore, "λ = 0 implies the lower plate is static, λ < 0 implies shrinking, and λ > 0 implies the stretching of the lower plate". Moreover,

Nusselt Numbers
The Nusselt numbers that represent the heat transfer rates, (Nu x1 ) and (Nu x2 ) plates are defined as: at the lower plate : and at the upper plate : where q w + q r is the heat flux in the presence of thermal radiation, which is defined as: Symmetry 2022, 14, 1943 8 of 23 Substituting the values in Equation (13) from the Equations (9) and (14), the dimensionless forms of the Nusselt numbers are:

Numerical Method
The dimensionless Equations (10)-(12) are numerically solved utilizing "bvp4c solver" in the MATLAB software. The "bvp4c solver" is very efficient for the boundary value problems [42] and several researchers (see References [7,43]) have utilized it to analyze their flow models. The "bvp4c" solver is used through the following syntax: "sol = bvp4c (@OdeBVP, @OdeBC, solinit, options)" For solving, the following substitutions are made: (y 1 , y 2 , y 3 , y 4 , y 5 , The MATLAB syntax written to solve the system is as follows: and the initial conditions associated with the above system are as follows: y 0 (2) = λ, y 0 (1) = S, y 0 (5) = δ, y 1 (2) = 0, y 1 (1) = Sq 2 , y 1 (5) = 1 Authors have utilized the aforementioned procedure to numerically solve the system represented by Equations (17) and (18). The missing values, f (1), f (0),θ (1) and θ (0)", are required to initiate the procedure of finding the numerical solution. The step size to carry out the computations was set as 0.01 and η = 1. When the boundary conditions given in Equation (18) were fulfilled, the process of determining the numerical solution was terminated. Authors have validated the present results by presenting a comparison with the results of the published study of Hayat et al. [44] and Khashi'ie et al. [16] (see Table 3). A good agreement between the results is seen. Hence, this validates the results of this study.

Velocity Profile
This subsection has been prepared to examine the characteristics of involved parameters on the velocity distributions. Figures 2-8 display the deviation in fluid velocity for varied values of involved parameters. Figure 2 exhibits the behavior of velocity when the hybrid nanofluid/nanofluid flow between the parallel plates when the upper plate is moving. It is observed that velocity at/near the lower plate is affected by the stretching of the lower plate. Figure 3 displays the deviation in velocity for varied values of the magnetic parameter (M). Near the lower plate, the velocity falls with an increment in the value of the magnetic parameter. However, a transition point exists near η ∼ 0.5. After this transition point, the velocity shows contrary behavior. The interface of a strong magnetic field represents remarkable decay of the flow of fluid. As M is assuming the high values, the Lorentz forces come into play, which reduces the liquid flow. The existence of a magnetic field challenges the flowing status and finally decelerates the radial velocity. Thus, magnetohydrodynamics is a procedure that controls fluid motion. Figure 4 represents the alterations in the velocity as a function of the Darcy number (Da). Near the lower plate, velocity surges with an increment in the Darcy number. However, a transition point exists near η ∼ 0.5. After this transition point, the velocity shows contrary behavior. Darcy number (Da) represents the "relative effect of the permeability of the medium versus its cross-sectional area", and permeability evaluates the surface capacity to flow fluid through its membrane. The increasing permeability near the lower plate restricts the motion of the fluid. As an outcome, when the Darcy number (Da) is increased, the velocity profiles fall near the upper plate due to more resistance to the flow. parameter (M). Near the lower plate, the velocity falls with an increment in the value of the magnetic parameter. However, a transition point exists near 0.5 η . After this transition point, the velocity shows contrary behavior. The interface of a strong magnetic field represents remarkable decay of the flow of fluid. As M is assuming the high values, the Lorentz forces come into play, which reduces the liquid flow. The existence of a magnetic field challenges the flowing status and finally decelerates the radial velocity. Thus, magnetohydrodynamics is a procedure that controls fluid motion.   the magnetic parameter. However, a transition point exists near 0.5 η . After this transition point, the velocity shows contrary behavior. The interface of a strong magnetic field represents remarkable decay of the flow of fluid. As M is assuming the high values, the Lorentz forces come into play, which reduces the liquid flow. The existence of a magnetic field challenges the flowing status and finally decelerates the radial velocity. Thus, magnetohydrodynamics is a procedure that controls fluid motion.   After this transition point, the velocity shows contrary behavior. Darcy number (Da) represents the "relative effect of the permeability of the medium versus its cross-sectional area", and permeability evaluates the surface capacity to flow fluid through its membrane. The increasing permeability near the lower plate restricts the motion of the fluid. As an outcome, when the Darcy number (Da) is increased, the velocity profiles fall near the upper plate due to more resistance to the flow.           Impressions of the shrinking/stretching parameter (λ) on the non-dimensional velocity are showcased in Figure 5. It is seen that the velocity, f (η), shows dual behavior with (λ). There exists a transition point near η ∼ 0.3. The velocity, f (η), upsurges in the locality of the lower disk, but after the transition point, the velocity, f (η), decreases. This result indicates that the stretching of the lower plate enhances the velocity in the neighborhood of the lower plate. However, as the upper plate is in movement towards the direction of the lower plate, the behavior of the velocity with an increment in the parameter (λ) is reversed. Figure 6 depicts the alterations in the velocity profile as a function of the squeezing parameter (Sq). The squeezing effect is instigated from the upper plate due to its movement towards the lower plate. It is seen that as the squeezing parameter (Sq) upsurges, the velocity of the fluid also increases. Figure 7 reveals the velocity profile with variation in the suction/injection parameter (S). The figure displays that the velocity rises with the application of injection in comparison to suction. Suction/injection is often used as a tool for avoiding the boundary layer separation, and in the present model, the increase in velocity is witnessed when the injection is applied, as seen in the figure. Hence, in the present model, the injection is more effective to defer the separation of the boundary layer.
In the Figures 2-7

Temperature Profile
This subsection has been prepared to examine the characteristics of the involved parameters on the temperature distributions. Figures 8-16 display the variation in temperature, θ(η), for diverse values of involved parameters. Figure 8 displays the deviation in temperature, θ(η), when the hybrid nanofluid/nanofluid flow between the parallel plates and when the upper plate is in movement. Figure 9 displays the deviation in temperature, θ(η), for diverse values of the magnetic parameter (M). It is perceived that the enhancement in the magnetic field decreases the temperature in both cases. The magnetic field instigates a Lorentz force that opposes the movement, which in opportunity gives a decrease in the thermal diffusion. The decreasing velocity near the lower plate is due to the application of a stronger magnetic field and causes the temperature to fall. Figure 10 represents the alterations in the temperature as a function of the Darcy number (Da). Darcy number (Da) represents the "relative effect of the permeability of the medium versus its cross-sectional area". The outcome implies that temperature rises with an upsurge in the Darcy number (Da). The flow experiences resistance while moving through the porous medium; hence, the temperature increases.
Impressions of the stretching/shrinking parameter (λ) on the temperature are showcased in Figure 11. In the present model, "λ = 0 implies the lower plate is static, λ < 0 implies shrinking, and λ > 0 implies the stretching of the lower plate". It is seen that the velocity, f (η), rises with the rising values of (λ). The results imply that the stretching of the lower plate enhances the temperature of the flow. Furthermore, as the extent of the shrinking of the lower plate increases, the temperature falls. Figure 12 exhibits the deviation in temperature, θ(η), for diverse values of the "volume fraction" (i.e., ϕ = ϕ 1 = ϕ 2 ). The plot directed that the rising value of volume-fraction parameters (i.e., ϕ = ϕ 1 = ϕ 2 ) raised the temperature. "Dispersion of supplementary nanoparticles supplements the capacity of hybrid nanofluid to transmit heat (i.e., thermal conductivity)". Thus, the temperature rises with the increasing value of ϕ. decrease in the thermal diffusion. The decreasing velocity near the lower plate is due to the application of a stronger magnetic field and causes the temperature to fall. Figure 10 represents the alterations in the temperature as a function of the Darcy number (Da). Darcy number (Da) represents the "relative effect of the permeability of the medium versus its cross-sectional area". The outcome implies that temperature rises with an upsurge in the Darcy number (Da). The flow experiences resistance while moving through the porous medium; hence, the temperature increases.    Figure 13 depicts the alterations in the temperature with varying squeezing parameters (Sq). The figure clearly shows that when the parameter (Sq) is increased, the temperature decreases. This implies that the movement of the upper plate towards the lower    Figure 13 depicts the alterations in the temperature with varying squeezing parameters (Sq). The figure clearly shows that when the parameter (Sq) is increased, the temperature decreases. This implies that the movement of the upper plate towards the lower  Figure 13 depicts the alterations in the temperature with varying squeezing parameters (Sq). The figure clearly shows that when the parameter (Sq) is increased, the temperature decreases. This implies that the movement of the upper plate towards the lower plates restricts the thermal diffusion, and hence, the temperature decreases. Figure 14 reveals the inclining upshots of temperature when the "heat generation/absorption parameter" (Q) rises. The higher estimations of a parameter (Q) cause temperature to grow. The positive values of the parameter (Q) imply the generation of heat in the system, and increasing values correspond to more amounts of heat being generated. Hence, the temperature rises with higher estimations of the heat-generation parameter (Q). Figure 15 demonstrates the variation trend of non-dimensional temperature, θ(η), due to fluctuating values of the suction/injection parameter (S). The rise in θ(η) is observed when parameter S is increased. Temperature is found to be higher for the suction value in comparison to the injection value. Suction corresponds to sucking out the layers separated from the boundary layer. The fluid layers gain momentum with the implementation of suction, and hence, the temperature increases. Figure 16 displays the variation in the θ(η) for diverse values of the radiation parameter (R d ). It is witnessed that the θ(η) rises with the increment in the radiation parameter (R d ). An increase in thermal radiation leads to a decline in the coefficient of heat absorption, which elevates the fluid temperature. Thus, the increased quantity of heat transmitted in the area as a result of enhanced radiation raises the temperature θ(η).
In temperature rises with higher estimations of the heat-generation parameter ( ) Q . Figure   15 demonstrates the variation trend of non-dimensional temperature, ( ) θ η , due to fluctuating values of the suction/injection parameter (S). The rise in ( ) θ η is observed when parameter S is increased. Temperature is found to be higher for the suction value in comparison to the injection value. Suction corresponds to sucking out the layers separated from the boundary layer. The fluid layers gain momentum with the implementation of suction, and hence, the temperature increases. Figure 16 displays the variation in the ( ) θ η for diverse values of the radiation parameter (Rd). It is witnessed that the ( ) θ η rises with the increment in the radiation parameter (Rd). An increase in thermal radiation leads to a decline in the coefficient of heat absorption, which elevates the fluid temperature. Thus, the increased quantity of heat transmitted in the area as a result of enhanced radiation raises the temperature ( ) θ η .            Figure 18 displays the velocity boundary layer patterns when the flow is under the influence of a magnetic field (M = 14) and in the absence of a magnetic field (M = 0). From these aforementioned figures, we can do a comparatively analyze and visualize that the magnetic field has a thinning consequence on the velocity boundary layer region. The reason for this thinning consequence with the magnetic field is the generation of the Lo-  The tangent at any point in the streamline gives the direction of the fluid velocity at that point. It is clearly seen from Figure 17a,c that the streamlines become denser in the presence of suction or injection in comparison to the absence of both suction and injection (S = 0). These results further confirm the role of suction and injection in delaying the separation of the boundary layer during the flow. Figure 18 displays the velocity boundary layer patterns when the flow is under the influence of a magnetic field (M = 14) and in the absence of a magnetic field (M = 0). From these aforementioned figures, we can do a comparatively analyze and visualize that the magnetic field has a thinning consequence on the velocity boundary layer region. The reason for this thinning consequence with the magnetic field is the generation of the Lorentz force due to the magnetic field, which opposes the movement of fluid.        Table 4 displays the behavior of the Nusselt numbers, Nu * x1 and Nu * x2 , at the lower and upper plate for different values of involved parameters. The Nusselt numbers represent the heat transference rates at the lower and upper plates. It is observed that the Darcy number (Da) has a positive correlation with the Nusselt number, Nu * x2 , at the upper plate. However, contrary behavior is seen with the Nusselt number, Nu * x1 , at the lower plate. Furthermore, the Nusselt number, Nu * x1 , at the lower plate is positively correlated with the "magnetic parameter (M) and squeezing parameter (Sq)". However, contrary behavior is seen with the Nusselt number, Nu * x2 , at the upper plate. In addition, the rising values of the "heat-generation/absorption parameter, stretching/shrinking parameter (λ), and suction/injection parameter (S)" acts to enhance the Nusselt number, Nu * x2 , at the upper plate. The rising values of "volume fraction" (ϕ 1 , ϕ 2 ) and thermal radiation parameter (R d ) cause the Nusselt numbers at both the plates (Nu * x1 and Nu * x2 ) to fall. In the present flow model, the Nusselt number at the lower plate is higher for MoS 2 -SiO 2 /H 2 O-C 2 H 6 O 2 (hybrid nanofluid) flow in comparison to MoS 2 /H 2 O (nanofluid) flow. However, contrary behavior is seen at the upper plate.

Conclusions
This study investigates the "unsteady MHD two-dimensional squeezing flow of the MoS 2 -SiO 2 /H 2 O-C 2 H 6 O 2 (hybrid nanofluid) and MoS 2 /H 2 O (nanofluid) between two parallel plates". Modeling of the flow is done with the assumptions of "heat-generation/absorption effect, porous medium, radiation, and injection/suction effect". The principal equations are handled by the "bvp4c" function of the MATLAB software.
Some vital conclusions of the present analysis are: • The injection effect and the shrinking of the lower plate aid the hybrid nanofluid flow.

•
The thermal radiation parameter and heat sink/source parameter have a positive correlation with the thermal field.

•
The hybrid nanofluid flow has a higher Nusselt number at the lower plate than the nanofluid.

•
The streamlines become denser under the influence of suction and injection effects.

•
The presence of a magnetic field has a thinning consequence on the velocity boundary layer region.

•
The results of this study apply to several thermal systems, engineering, and industrial processes, which utilize nanofluid and hybrid nanofluid for cooling and heating processes.
Derivation of the Flow Problem in Appendix A.