AParametric Analysis of the Effect of HybridNanoparticles on the Flow Field and Homogeneous-Heterogeneous Reaction between Squeezing Plates

Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, 23200 KP, Pakistan Department of Pure and Applied Mathematics, "e University of Haripur, Haripur, KP, Pakistan Department of Mathematics, College of Science and Arts, Qassim University, Ar Rass, Saudi Arabia Department of Basic Sciences and Islamiat, University of Engineering and Technology, Peshawar, KP, Pakistan Department of Mathematics, University of Swabi, Swabi 23561, KP, Pakistan College of Industrial Engineering, King Khalid University, Abha 62529, Saudi Arabia Department of Mathematics, College of Sciences, University of Juba, Juba, South Sudan


Introduction
Pressure flows have many engineering, scientific, and technical applications in the industry such as lubrication system, moveable pistons, hydrodynamic engines, hydraulic lifts, scattering and formulation, chemical equipment processing, food processing, film damage, and frost damage syringes and nasogastric tubes. e initial study of squeezing flow was published by Stefan [1] who reported the lubrication method in his research. New doors were opened by Stephen's rewarding work for researchers on squeezing flows. Different researchers studied the flow of compression and followed him. From different research perspectives, this study has been pushed forward in recent years to compress the flow. Hayat et al. [2] have examined the squeezed flow of MHD fluid between two horizontal disks using homotopy analysis. e result of their investigation was found to be an increase in the velocity field of for augmenting values of micropolar parameter. Mustafa et al. [3] analyzed the fluid flow with magnetic effects upon thermal and mass transmission behavior of an incompressible viscous Casson fluid flow amid parallel plates. ey have noticed in this work that flow has been augmented with escalating values of squeezing parameter. e two-dimensional magnetized laminar constant Marangoni convection of the incompressible viscous fluid was explored by Mahanthesh et al. [4] by implementing the Runge-Kutta-Fehlberg technique. e authors have found that the boundary layer thickness and the increasing meridian convection have increased the fluid velocity within the flow area.
Ferroliquids are magnetic nanofluids suspended in nondirecting liquids such as water, hydrocarbons, and kerosene.
ese ferrofluids have various applications in medical science like cell partition, focusing of drug medication, and imaging of magnetic characters. e thermal and magnet functions of the ferrofluid flow were investigated by Neuringer [5]. Khan et al. [6] examined the influence of a homogeneous heat stream on flat-surface slip flows with heat transfer. ey evaluated three distinct ferrofluids with two distinct basic fluids (CoFe 2 O 4 , Fe 3 O 4 , and Mn − ZnFe 2 O 4 ) (water and kerosene). Rashad [7] examined the magnetic slip-flow function containing nonisothermal convection and radiation wedge kerosene based cobalt ferrofluid. Zaib et al. [8] investigated a mixed convective flow entropy of a vertical plate of magnetite ferrofluid. Ali et al. [9] recently discussed the magnetic dipole impact on micropolar fluid consisting of the EG and the water-based ferrofluids Fe and Fe 3 O 4 from a stretched sheet.
Hybrid nanoliquids, however, are deliberately captured by blending several different nanoparticles with better thermal and rheological characteristics. e introduction of hybrid ferroliquids is to increase heat transfer efficiency in fluid flow. It has several scientific applications such as dynamic sealing, naval sealing, dampening, and microfluidics. Suresh et al. [10] investigated the effect of dissipation on time-based flux comprising a rounded pipe of hybrid nanoliquid. ey have achieved a lower friction factor for nanolytes than for hybrid nanol. e pressurization decline in the volume percentage of the water-based Cu − TiO 2 hybrid nanolic was examined by Madhesh and Kalaiselvam [11]. Minea [12] revealed the association with the date of the temperature gradient of alumina hybrids and nanofluids.
e fluid flow characteristics of hybrid nanoliquids from water-based Ag-CuO were evaluated by Hayat and Nadeem [13]. Mebarek-Oudina [14] examined the thermal and hydrodynamic parameters of Titania nanoliquids that satisfy a cylinder annulus, the impact of annulus, Mahanthesh et al. [15] An exponential spatially dependent magneto slip heat source flow from an extendable rotation consisting of carbon nanofluids. Marzougui et al. [16] investigated the surface effects. Al 2 O 3 − CuO stability roughness and radiation nanolic hybrid through the widespread use of the model. Recently, Wakif et al. [17] examined the entropy examination through convective flow including nanoliquid by means of MHD with chamfers in a hole. e investigation of the attractive capacity has significant sales in MHD orientation; topography, astronomy, siphons, generators, medication, control of limit layer, and so on are many noticeable MHD applications. Alshomrani and Gul [18] inspected the slight film flow of water-based Al 2 O 3 and Cu nanofluid through an extended chamber under the effect of attractive capacity. e characteristics of magneto thermal transport, comprising a time dependent flux of liquid nanofluid thin film flow to a starched surface, were examined by Sandeep and Malvandi [19]. Sandeep [20] examined the characteristics of the hybrid nanolytic flux with various heat and drag forces. Ahmad and Nadeem [21] examined the magnetic effects of hybrid nanofluid with a heat sink/source on micropolitan fluid and achieved numerous findings for hybrid nanofluid and micropolar fluid. Hamrelaine et al. [22] examined the magnetic effect of Jeffery-Hamel flow between nonparallel permeable walls or permeable plates. e attractive impact on the radiative progression of the hybrid nanoliquid thin film with sporadic warmth sink/ source was examined by Anantha Kumar et al. [23]. Zaib et al. [24] got the comparability of various outcomes from magnetite ferroliquid passing on non-Newtonian blood stream with entropy age. Wakif et al. [25] assessed the impact of the magnetic field on progressions of Stokes' second issue with entropy generation. Recently, Kameswaran et al. [26] investigated homogeneous-heterogeneous reactions in nanofluid flow due to a microscopic stretch sheet. ey showed that the velocity profiles decrease with an increasing volume of the nanoparticles, while the liquid concentration is reversed by the volume of the nanoparticles for both Cu-water and Ag-water nanofluids.
e cited literature and similar other works show that no study is conducted to examine the combined effects of unstable flow between two squeezing plates in the presence of hybrid nanoparticles. erefore, using all the studies mentioned above, we analyzed the multifaceted and homogeneous chemical reaction effects on the flow between two compression plates in the presence of hybrid nanoparticles. Navier-Stokes equations, heat transfer, and homogeneous and multifaceted reactions are solved by the HAM and BVP4c. In this work, we analyzed, discussed, and obtained the effects of different parameters on velocity, temperature, concentration, skin friction coefficient, and Nusselt and Sherwood numbers through graphs and tables. Figure 1 shows a laminar, unsteady incompressible, and twodimensional and hybrid nanofluid flowing between horizontally parallel and squeezing plates with homogeneous and heterogeneous reactions. Hybrid nanoparticles comprise Fe 3 O 4 and Fe 3 O 4 + Co in the ethylene glycol with water as base fluid suspension. e plates are separated by a 2

Mathematical Formulation
Advances in Mathematical Physics with α as squeezed parameter. For α > 0, the plates are squeezing but when t � (1/α) and α < 0, the two plates are separated. e velocity field is often impacted by a uniform magnetic field B(t) � B 0 / ����� 1 − αt √ distributed along the y-axis. e upper and lower plates are held at steady temperature T u and T l , respectively. e proposed model of Chaudhary and Merkin [27] for homogeneous as well as heterogeneous reaction has been used in this study as described below. e homogeneous reaction for cubic autocatalyst surface is (1) e heterogeneous reaction upon catalyst surface is e quantities of the chemical sorts N1 and N2 are signified by a and b, respectively, while the initial conditions are denoted by k c . e reaction rate vanishes in exterior flow and beyond the boundary layer bottom, as shown by the equations above. e equations that governed the flow system are presented as [28,29].
Continuity equation is as follows: Navier-Stokes equation is as follows: Energy equation is as follows: Homogeneous and heterogeneous equations are as follows: In the above equations, T represents temperature, P represents pressure, ρ hnf represents effective density, (ρc p ) hnf represents effective heat power, and σ hnf represents electrical conductivity of nanofluid. e quantities (u, v) denote the fluid's nanofluid velocity component, T is the temperature, and D A and D B are the respective diffusion constants of the chemical sorts a and b. e permeability is provided by K * . k hnf nanofluid thermal conductivity, Q * heat generation, k hnf nanofluid thermal conductivity, k hnf nanofluid thermal conductivity, k hnf nanofluid thermal conduct.

Boundary Conditions.
e squeezing flow under consideration has the undermentioned conditions at boundaries: Use the transformations [2] Upper Plate y-axis

Advances in Mathematical Physics
Implementing equations (7a) and (7b), we have the following system of equations: and the boundary conditions are reduced to Here, S � (αl 2 /2v f ) is the squeezed Reynolds number, P r � (μc p /K) is the Prandtl number, Ha � lB 0 is the heterogeneous reaction strength, and δ � (D A /D B ) is the ratio of the diffusion coefficients. Here, it is considered that A and B diffusion coefficients of chemical species are of comparable size. e other hypothesis is that D A and D B are equivalent, so δ � (D A /D B ) � 1, and G(η) + H(η) � 1 [30].

Coefficients of Interest.
e local Nusselt number (Nu), Sherwood number (Sh), and skin friction coefficient (Cf) are some of the coefficients of interest in engineering.
By using A i for the dimensionless constant, the result will be as follows: e total volume fraction of nanoparticles is represented by ϕ. e volume fractions of the discrete nanoparticles are represented by the symbols ϕ1 and ϕ2; the density of first, second, and base fluids of nanoparticles is ρ 1 , ρ 2 , and ρ f ; and the electrical conductivity of 1st, 2nd, and base fluids of nanoparticles is σ 1 , σ 2 and σ f .

Approximate Analytical Solution
To solve system of equations (8)- (11), the analytic method HAM is used. Due to HAM, the functions f(9), θ(9), H(9), and G(9) can be stated by a set of base functions 9 c , c ≥ 0 as where a ξ , b ξ , c ξ , and d ξ are the constant coefficients to be determined. Initial approximations are chosen as follows:

Advances in Mathematical Physics
e auxiliary operators are chosen as with the following properties: where ξ 1 , ξ 2 , ξ 3 , ξ 4 , ξ 5 , ξ 6 , ξ 7 , ξ 8 , ξ 9 , and ξ 10 are arbitrary constants. e zeroth order deformation problems can be obtained as e nonlinear operators of (17)- (20) are defined as where ϖ is an embedding parameter; Z f , Z θ , Z H , and Z G are the nonzero auxiliary parameter; and N f , N θ , N H , and N G are the nonlinear parameters.
For ϖ � 1, we have Differentiating the deformation equations (25)-(28) Ψ − times with respect to ϖ and putting ϖ � 0, we have subject to the boundary conditions where and

Optimizing the Convergence of Control Parameter
It is important to note that the series solutions (46)-(49) include Z f , Z θ , Z H , and Z G nonzero auxiliary parameters that define the convergence area as well as rate of the homotopy series solutions. e residual error to obtain the maximum values of Z f , Z θ , Z H , and Z G was used as Due to Liao where ϵ t Ψ is the total squared residual error.

Analysis of Error
An error analysis is conducted to ensure a minimum residual error for the efficiency of the analysis. HAM and BVP4c solve the problem analytically and numerically. e analysis is conducted using an approximation order of the 40th order. For this analysis, the validity of HAM techniques is also evaluated using the Mathematica software BVPh 2.0 for maximum residual error 10 −40 . e results are compared with the numerical solution of BVP4c using Matlab for the authentication and consistency of the HAM solution. e reliability of the two methods for various concerning physical parameters is investigated by error analysis in Figure 2 and Tables 1-12. Figure 2 shows that up to the 16th transition series, the maximum average residual errors of f ″ (η), −θ ′ (η), −H ′ (η), and −G ′ (η) are almost gradually decreased. e cumulative residual error for the various approximation orders of fixed P r � −0.5, S � −0.5, L � 0.01, Table 1. Table 2 shows various orders of approximation with distinct average squared residual 00509. e comparison of the analytical and numerical values by HAM and BVP4c is shown in Table 3 for various values of η and fixed values of other parameters, σ1 � 0.74 × 10 6 , σ2 � 1.602 × 10 7 , and σ3 � 0.00509.

Results and Discussion
For various values of parameters S, Ha, P r , Q, R, Sc, K1, K2, and ϕ, the system of nonlinear equations (8)- (11), resulting in the boundary conditions equation (12), is numerically Advances in Mathematical Physics 7 resolved using HAM and BVP4c numerical routines. e local skin friction coefficient as well as the local rates of heat and mass transfer at the surface of the squeezing plates, both of which are extremely important in terms of physical properties, is also computed. Because there are so many physical parameters in the current problem, a wide range of results can be obtained. e distributions of temperature, velocity, pressure, and mass transfer are obtained by solving and Ha � 0.1.   Figure 3 in the case of both SNF and HNF. In reality, for larger S, the upper plate slides downward, which puts more stress on nanoparticles, and consequently velocity components f(η), f ′ (η) are amplified. Effect of S on θ(η) is demonstrated in Figure 3. For bigger S, the top plate shift downwards, and interatomic collision nanoparticles increase; hence, the temperature increases. e influence on concentration profiles H(η) and G(η) of the parameter S can be observed in Figure 3 in both SNF and HNF. e concentration profile H(η) decreases and G(η) increases. It has been discovered that as S increases, the homogeneous chemical reaction increases, resulting in a decrease in viscosity. However, the H(η) indicates the reverse of the G(η) above can seen in Figure 3. e impact of Hartmann number Ha show in Figure 3 on f(η) against the similarity variable η for the phenomenon of both SNF and HNF are decreases. When the HNF flow is applied, the fluid velocity is reduced by moving down the horizontal axis and the SNF flow       velocity is increased for continued positive changes in Ha values while the same behavior is displayed as in the figure, for both situations. It is also found that the maximum velocity of HNF is greater than that of SNF. e impact on the temperature profile of the Prandtl number P r is seen in Figure 4. It is examined to show θ(η) in the case of SNF and the inclining performance for high values of P r in the case of HNF. Essentially, larger P r values enhance the boundary layer thickness that stimulates the nanoparticle's cooling effect due to thermal diffusion ratio. As contrast to in the HNF nanoparticles are tightly packed SNF. e impact on the temperature profile of Q (heat generation parameter) is seen in Figure 4. For rising values of heat parameter Q, T u > T l indicates a further transfer of heat from the surface into the fluid, thus raising the fluid temperature for both SNF and HNF. e growing temperature behavior in SNF and HNF for the larger value of R is shown in Figure 4. However, it is commonly recognized that the heat transfer phenomena of their radiation process emit the energy via fluid particles, so that more heat is created during flow. us, the thermal boundary layer with more R is defined as a development. Figure 5 indicates a rise in the Schmidt number Sc for SNF and HNF concentration profile of H(η) and G(η) and, as a result, reduction in the thickness of concentration boundary layer. e reactant concentration is observed to be increasing at a quicker rate when the diffusion coefficient of species is reduced; i.e., higher Sc values lead to a more rapid increase of the flow field concentration. e impact of homogeneous parameter k1 and a heterogeneous parameter k2 on the concentration profile H(η) can be seen in Figure 6. Increase in the concentration profiles H(η) it is observed that due to increase in k1. According to this an increase in the homogeneous chemical reaction parameter which decreases viscosity. e concentration profile increases as the thickness of the boundary layer decreases for lower strength values of homogeneous reaction parameter. It is also proved that concentration profile is lower for the situation of HNF when compared with SNF. In both situations of SNF and HNF, the concentration boundary layer of the reactants decreases. However, they coincide for smaller values of η which physically means that the homogeneous and heterogeneous reactions have no effect on the concentration of the reactants. e strong conduct of heterogeneous reaction parameters K2 on the distribution is studied in Figure 6. e distribution of the concentration is increasing towards the surface of the plate and decreasing away from the surface with smaller values of K2. When compared to HNF, the concentration distribution is smaller (SNF). It is noticed in Figure 6 that the behavior of concentration profile G(η) and H(η) is opposite. is shows the effects of the homogeneous chemical reaction parameter k1 and the heterogeneous chemical reaction parameter k2 on

HAM results
Numerical results    the concentration profiles H(η) and G(η). It has been observed that increasing k1 causes the concentration profiles H(η) and G(η) to increase. is is due to the fact that as the homogeneous chemical reaction parameter rises, the viscosity declines. However, the heterogeneous parameter k2 yields the opposite result, as shown in Figure 6. is is because as k2 increases, diffusion decreases, and the concentration of less diffused particles grows. e impacts of nanoparticles volume fraction ϕ on the velocity profiles f(η), θ(η), H(η), and G(η) in case of SNF and HNF are shown in Figure 7. It can be clearly seen that the volume fraction parameter ϕ is increased with the increase in the  e influence on concentration profiles f(η), θ(η), H(η), and G(η) of the parameter S can be observed in Figure 8 in nanoparticles. All the concentration profiles are increased while G(η) decreases, it is observed that due to increase in S. Figure 9 indicates that the f(η) and H(η) concentration profiles have increased with rising values of Schmidt number Sc, while θ(η) and G(η) decrease. Figure 10 displays the effects of the homogeneous and heterogeneous chemical reaction parameters k1 and k2 on the concentration profiles f(η), H(η), and G(η). It has been observed that k1 increases and k2 decreases on the concentration profile f(η); both decrease on the concentration profile H(η); and both increase on G(η). e purpose of Tables 5-12 is to test the impact of various physical parameters numerically. As can be seen from the tables, all of the results are in good settlement with the BVP4c and HAM results. It is observed that effects of skin friction coefficient, velocity, temperature, and Nusselt and Sherwood numbers, both homogeneous as well as the heterogeneous parameters cause increment in the mass transfer rate. A decrease in the skin friction coefficient, as well as the heat and mass transfer rate, is caused by an increment in the internal heat generation parameter. A similar set of results has been observed when the squeezing parameter S is increased. As the squeezing parameter S is enhanced, the friction factor reduces and the local Nusselt and Sherwood numbers increase. e skin friction coefficient showed a downward trend, showing that the fluid was being drawn by the floor. Tables 3-12 It is acknowledged from Table 5 that perhaps the skin friction factor f ′ (η) tends to increase the squeezing number S; the disruptive effect can be seen for the Hartmann number Ha and the volume fraction ϕ of nanoparticles and hybrid nanoparticles. From Table 6, it has been noticed that the squeezing number S has declining influence on heat transfer efficiency, but Prandtl number P r , Q (heat generation parameter), radiation R, and volume fraction ϕ of nanoparticles and hybrid nanoparticles are greatly influencing the heat transfer efficiency. Table 7

Conclusion
is investigation gives a mathematical answer for dissecting the impacts of flow between two squeezing plates with a homogeneous and heterogeneous reaction in the presence of hybrid nanoparticles. e impact of dimensional overseeing boundaries on velocity, temperature, profiles with skin friction, and local Nusselt and Sherwood numbers is examined with the assistance of graphs and tables. e effects of the current examination are listed below: (i) e skin friction coefficient augmenting for growing values of solid volume fraction ϕ. (ii) e increasing values of the squeezing parameter S reduced the friction factor and local Nusselt numbers, but from Table 7 Sherwood numbers are decreasing and increasing on the −H(0) and −G(0). (iii) e Prandtl number, heat generation, and radiation parameters appear to increase in the local Nusselt numbers, as shown in Tables 9, 11, and 12.
(iv) Homogeneous-heterogeneous parameters assist in observing the flow's utility profiles. (v) e local Nusselt and friction factor in fluid viscosity are similar as Sc increases and the homogeneous parameter decreases while heterogeneous parameter increases, as shown in Table 13. Upper plate's temperature T l : Lower plate's temperature P: Pressure ρ hnf : Effective density (ρc p ) hnf : Effective heat capacity σ hnf : Electrical conductivity of nanofluid u, v: Nanofluid velocity D A , D B : Diffusion coefficients of the chemical species δ: Ratio of the diffusion coefficients K * : Permeability k hnf : Nanofluid thermal conductivity

Data Availability
No data were used in this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.