Optimization and sensitivity analysis of heat transfer rate of tri-hybrid nanofluid of stagnation point flow

Abstract

In manufacturing, especially in oil flow filtration, combustion systems, cooling turbines, and other areas, heat transfer performance through hybrid nanofluids (HNFs) is a key factor in achieving dominance of the final product. The present study deals with the movement of the fluid containing tri-hybrid nanoparticles on an excessively large stagnation point area on a smooth plate in a permeable medium. Additionally, the leading partial differential equations of the proposed model are converted to ordinary differential equations (ODEs) by incorporating similarity variables, and the fourth-order Runge–Kutta method is then used to solve these. To find the missing initial conditions of first-order ODEs, a shooting technique is also used. Furthermore, the consequences of heat transmission rate in the form of graphs and tables are explored. It is noticed that by enhancing the strength of the solid volume fraction the skin friction along the x-axis ( $f^{\mathrm{″}}(0)$ or $CFX$ ) increases as $\gamma \in (-2,10]$ and decreases for the values of $\gamma \in [-10,2)$ . But the converse of this behavior is true for $g^{\mathrm{″}}(0)=CFY$ . Moreover, the ternary fluid has taken the most significant effect on the Nusselt number, that is, 17.12978%, 8.43809%, and 19.20192% increment for Go, Ag and Cu type mono-nanofluid and 12.67309%, 18.13489%, and 13.65317% enhancement for HNF (Go-Ag, Go-Cu, Ag-Cu, respectively).

Introduction

Currently, academics are mostly focused on the issue of improving heat transmission in industrial processes. In the past, fluids such as water, oil, and ethylene glycol were commonly employed as cooling agents in many processes. However, it is worth noting that their heat transfer rates (HTRs) are quite modest. The examination of the flow of hybrid nanofluid (HNF) across an expanding surface holds significant applications in many industries such as paper and plastic production, wire coating, food manufacturing processes, rubber sheet manufacturing, and polymer sheet production. The nanofluid flow across a coiled sheet was examined by Gul et al. ¹ in a two-dimensional context. The study conducted by Elattar et al. ² examined the impact of hall current and chemical reaction on HNF flow caused by slender surfaces. Ali et al. ³ investigated the flow of 3D nanoliquid over a stretched sheet characterized by varying thermal conductivity. Allehiany et al. ⁴ described the flow of a magneto-micropolar fluid across a heated sheet. The findings of the study indicate that the Newtonian liquid exhibits a greater velocity in comparison to the micropolar liquid. In their study, Bilal et al. ⁵ recognized the phenomenon of MHD viscous Jeffrey thermal transfer flow across a permeable stretched sheet. Numerous recent studies on nanofluids and HNFs for different geometries have been published.^6–10 The utilization of nanofluids is observed in diverse domains, encompassing heat transfer apparatus, thermal control mechanisms for electronic gadgets, automotive radiators, refrigeration systems, and solar reservoirs. Therefore, the introduction of ternary HNF, a sophisticated kind of nanofluid, recognizes the need to achieve significant enhancements in the thermal conductivity of conventional fluids. This formulation aims to achieve a significantly high degree of heat conductivity. ¹¹ In theory, it is postulated that ternary HNF exhibits superior thermal characteristics when compared to both single and HNF. The technique of producing ternary HNFs, along with their stability and applications in heat transfer, was reviewed by Adun et al. ¹² In research done by Kashyap et al., ¹³ it was determined that the utilization of ternary HNF as a coolant does not provide advantageous outcomes, mostly owing to a marginal increase in the rate of heat transfer. Besides experimental studies, several numerical simulations examined the nature of the flow of HNFs of ternary systems under conditions of different geometrical and physical basis. Ahmed et al., ¹⁴ heat transfer exercise, was published through a flow conduit made up of a square. They developed a ternary nanofluid that was made of ZnO, Al2O3, and TiO2, mixed with water. The scientist confirmed the fact that the enhanced average and local heat transfer of the ternary nano-hybrid is fine to be used in an industrial context through the performed tests. The carried out by Algehyne et al. ¹⁵ revealed that various diffusion rates have either an increasing or decreasing effect on the thermal performance of a mixed nanofluid with three constituents. Sarfraz et al. ¹⁶ studied the energy conductivity for ternary HNF radiating from an asymmetric disc. In their study, Mahmood et al. ¹⁷ considered a ternary nano-hybrid fluid flow that undergoes stretching and shrinking along a curved surface. The inquiry has been conducted between the administrator dynamics and thermal characteristics. Maranna et al. ¹⁸ investigated the phenomena of ternary HNF conduction by wetting a shrinking sheet over the porous media taking into account the impact of radiation and Navier slip. The ternary nanofluid flow in a radiated channel was examined by Abbasi and Ashraf. ¹⁹ The binary/engine oil nanofluid was less productively potent pushing the ternary-hybrid/engine oil to stand out in the study. Ramana et al. ²⁰ investigated the behavior of a nanofluid that was ternary in a stretchable converging/diverging channel with the effect of both hot reservoir/cold sink and predefined porous media through their study. The researchers found out that the efficiency of energy diffusion in the diverging channel could be significantly improved by upping the non-dimensional (heat sink or source factor) and Eckert number. But for the diverging tube, the flow becomes significantly abrupt.

Stagnation point flow is a fluid flow in a specific form when a fluid, such as gas or liquid, is forced to bypass a solid object around the perimeter, leading to the velocity of the liquid being equal to zero at a designated spot we call the stagnation point. There can be several reasons why this kind of fluid flow can be important, like aircraft and missiles aerodynamic profile, combustion, and thermal transfer. The aerodynamics which is involved in airplane designing and weapons predicting high-speed projectiles is explained by way of the static-point flow. Just like that, through proper designing of the aircraft fuselage and wing features, engineers can therefore reduce stagnation points and effectively improve lift. Certainly, this is the essence of attaining an exceptional propulsion system and enhanced maneuverability. Static point flow is used in combustion when the rapidity and accuracy of combustion through incoming air need to be established. Engineers obtain both fuel efficiency and low emissions by being skillful in this by infusing fuel and oxygen in a position near the stagnation point, giving rise to a consistent and steady flame. Static-point flow denotes a popular technique operating in the domain of heat transfer used in designing high-temperature heat sinks, air-counters, etc., (e.g., gas turbines and rocket engines). Machine engineers can dissipate the heat from the surface of the equipment effectively while also guiding a fluid to the stagnation point thus improving hydrodynamics of the flow. This prevents damage and improves performance. Static points occur when objects in the flow area experience excessive pressure and essential fluid velocity, causing the fluid to completely halt. Studying fluid flow around stagnation sites considerably enhances the design of industrial and scientific processes. In recent times, there has been a growing recognition of the ability to confine molecules or other particles inside fluid flows at locations of stagnation. The achievement of the stagnation point movement has been seen in several sectors, particularly in the domains of biology, chemistry, and medicine,^21–25 after the isolation and targeting of particles or cells. The notion of a two-dimensional stagnation points motion issue approaching a solid wall was suggested by Hiemenz in 1911. ²⁶ Ariel delivered a concise and accurate reply to a query that had elicited a momentary hesitation. The notion of two-dimensional stagnation point movement across elongating and compressing plates was established by Mahapatra and Gupta,^27,28 as well as Wang. ²⁹ The complexity of flow generation arises from the accumulation of specific stagnation point flow and the reduction of the elongating surface, both of which have an impact on the velocity of boundary layer motion, the heat transmission ratio, and the mass transmission ratio. Chamkha et al. ³⁰ investigated a linearly stratified stagnation flow (Hiemenz flow) of coupled heat and mass transfer by mixed convection in the presence of internal heat generation or absorption effects and an externally applied magnetic field. The plate surface has a power law variation in both wall temperature and concentration and is embedded in a uniform Darcian porous medium. It is permeable to allow for potential fluid wall suction or blowing. The effects of the Lorentz force and the convective heating boundary on second-grade nanofluid flow in conjunction with a Riga pattern was examined by Gangadhar et al. ³¹ Gorla and Chamkha ³² explored the effect of natural convection past a horizontal plate in a porous medium saturated in a nanofluid. In their respective studies, Bhattacharyya et al., ³³ Zaimi, and Ishak ³⁴ examined the influence of expeditions on the occurrence of stagnation point motion problems. Both excursion and thermal declines were assessed by Aman et al. ³⁵ and Fauzi et al. ³⁶ to extend/decline wall stagnation point motion. The primary emphasis of the findings was on the distributions of velocity, temperature, skin friction, and local Nusselt number. The investigation and examination focused on the movement of fluids containing nanoparticles at stationary locations have made significant contributions to the comprehension of the thermal characteristics that influence fluid dynamics. Mustafa et al. ³⁷ investigated the movement of nanofluids along a stagnation point in the direction of an extended surface. Controverting this, Bachok et al. ³⁸ and Kameswaran et al. ³⁹ looked into the fluid moving over the two-dimensional stretching and shrinking surface at a stationary point. From the study by Mansur et al., ⁴⁰ the magnetohydrodynamic motion developing at a stagnation point was evaluated during the fluid flow in a channel that has either expansion or contraction under nanoparticles contained material. The research results showed that the cases of the relaxing scenario and the escalating situation were solved by the special approach, whereas the common approach was applied in the case of the diminishing scenario. The slip boundary on the stagnation point flow problem in nanofluids has been tackled because the scholars attempted to replace the older phenomenon called fluid confinement which does not work in nano and micro settings. Mohamed et al. ⁴¹ proceeded to introduce a nanofluid that was moved, at a stagnation point on a stretched surface, never ceasing to flow. It raises a contradiction with the fact that Mahatha et al. ⁴² explored the fluid motion of a nanofluid stagnation on a stretching surface taking into account the velocity slip and at the same time heat radiation. The nanofluid simulation described above employed Buongiorno's ⁴³ methodology, and the findings demonstrated a strong correlation between the decrease in heat transmission rate at the surface and the influence of Brownian movement and thermophoresis components. Nanofluids, including a combination of base fluid and nanoparticles, are intelligent fluids that have garnered significant attention in several industrial domains. These applications have been extensively studied in theoretical research, encompassing their potential use in renewable energy generation within chemical processes, ⁴⁴ as well as their potential in stimulating oil and gas reserves. ⁴⁵

Motivation of the study

The investigators’ proficiency and ambition enabled them to successfully publish their findings on the heat transmission of stagnation point flow, which was generated through extensive research and the application of numerical methods. The motivation behind this research work can be attributed to the following factors:

In 2020, Weidman ⁴⁶ successfully explored an advanced group of irregular stagnation point motions considering the shear-to-strain ratio and permeable media. An approximate method was employed to determine leading nonlinear ordinary differential equations (ODEs).

Mahapatra and Sidui ⁴⁷ investigated the non-Newtonian viscoelastic fluid of disproportionate Homan stagnation point motion over a rigid slice. An approximate method is utilized to derive numerical outcomes of coupled ODEs. They extended the paper of Weidman. ⁴⁶

The flow over a vertical surface that is extending or contracting while experiencing second-order velocity slip and heat vault is explored by Soid et al. ⁴⁸ Partial differential equations (PDEs) are converted into regular (indistinguishability) differential equations employing a modification of variables. The confine value problems solver bvp4c in Matlab software is used to numerically resolve these equations and associated confine situations. For these comparison equations, consequences from interpretative values of boundless variable, dual (first and second) outcomes are determined.

Objectives of the study

Based on the literature cited above, the leading purpose of the present investigation is as follows:

Design a mathematical model of stagnation point motion of ternary-hybrid nanofluid (THNF) in a permeable medium.

To determine calculation outcomes of leading nonlinear similarity equations with the aid of a built-in MATLAB routine called bvp4c.

Optimize the HTR of THNF by supervised machine learning approaches such as response surface methodology (RSM).

Mathematical formulation of the problem

The following are the flow assumptions of the problem (depicted in Figure 1):

The flow field is modeled near a non-axisymmetric stagnation point on a flat plate in a porous medium.

The porous media has a permeability parameter K, and fluid flow follows Darcy's law without considering Forchheimer's inertial effects.

It is assumed that the flow in the porous medium is incompressible, fully developed, and steady.

The potential flow velocities are stated as $u_{e*}(x,y)=$ $(a+b)x$ , $v_{e*}(x,y)=(a-b)y$ and $w_{e*}(z)=-2az$ , corresponding to the Cartesian coordinates $(x,y,z)$ .

The surface temperature of the flat plate is supposed to change linearly with position, as $T_w=T_{\mathrm{\infty }}+T_{\circ }x$ , where $T_{\mathrm{\infty }}$ and $T_{\circ }$ are for ambient and characteristic temperature and the surface temperature.

The thermophysical properties of the fluid and nanoparticles are assumed to be constant and assessed using standard property correlations.

Figure 1.

Physical sketch of the problem.

The leading PDEs of said problem are ⁴⁹ as follows:

$${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+{\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }y}+{\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }z}=0}}}$$

(1)

$$u{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}+w{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }z}=u_{e*}{\displaystyle \frac{d{u}_{e*}}{dx}+{\displaystyle \frac{{\mu }_{Thnf}}{{\rho }_{Thnf}}{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}-{\displaystyle \frac{1}{K*}{\displaystyle \frac{{\mu }_{Thnf}}{{\rho }_{Thnf}}(u-u_{e*})}}}}}}}}$$

(2)

$$u{\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }x}+v{\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }y}+w{\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }z}=u_{e*}{\displaystyle \frac{d{v}_{e*}}{dy}+{\displaystyle \frac{{\mu }_{Thnf}}{{\rho }_{Thnf}}{\displaystyle \frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{z}^2}-{\displaystyle \frac{1}{K*}{\displaystyle \frac{{\mu }_{Thnf}}{{\rho }_{Thnf}}(v-v_{e*})}}}}}}}}$$

(3)

$$u{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }y}+w{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }z}={\displaystyle \frac{k_{Thnf}}{{(\rho C_p)}_{Thnf}}{\displaystyle \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{z}^2}}}}}}$$

(4)

Concern to confines conditions:

$$\begin{array}{c}u=v=w=0,T=T_w(x)atz=0\\ u\to u_{e*},v\to v_{e*},w\to w_{e*}(z),T\to T_{\mathrm{\infty }}asz\to \mathrm{\infty }\end{array}\}$$

(5)

Here, $K*,T,\rho C_p,\mu ,k,$ and $\rho$ are presented for porous media permeability, temperature, heat capacity, dynamic consistency, diffusivity, and consistency of fluid, respectively.

Equations (1)–(4) are higher of PDEs and it is difficult to solve numerically. Therefore, we will introduce similarity variables that transform these PDEs into ODEs. So, the appropriate similarity variables are as follows:

$$\begin{array}{rl}u=& (a+b)x{f}^{\mathrm{\prime }}(\eta ),v=(a-b)y{g}^{\mathrm{\prime }}(\eta ),\\ w=& -\sqrt{a{\nu }_f}[(a+b)f(\eta )+(a-b)g(\eta )],\\ \theta (\eta )=& {\displaystyle \frac{T-T_{\mathrm{\infty }}}{T_w-T_{\mathrm{\infty }}},\eta =z\sqrt{{\displaystyle \frac{a}{{\nu }_f}}}}\end{array}$$

(6)

By inserting equation (6) into equations (1)–(4) we obtain succeeding ODEs:

$${\displaystyle \frac{{\mathring{\mathrm{A}}}_1}{{\mathring{\mathrm{A}}}_2}{\displaystyle \frac{d^{3}f}{d{\eta }^3}+(1+\gamma )\left(f{\displaystyle \frac{d^{2}f}{d{\eta }^2}+1-{{\displaystyle \frac{df}{d\eta }}}^2}\right)+(1-\gamma )g{\displaystyle \frac{d^{2}f}{d{\eta }^2}-{\displaystyle \frac{{\mathring{\mathrm{A}}}_1}{{\mathring{\mathrm{A}}}_2}\sigma \left({\displaystyle \frac{df}{d\eta }-1}\right)=0}}}}$$

(7)

$$\begin{array}{r}{\displaystyle \frac{{\mathring{\mathrm{A}}}_1}{{\mathring{\mathrm{A}}}_2}{\displaystyle \frac{d^{3}g}{d{\eta }^3}+(1+\gamma )\left(g{\displaystyle \frac{d^{2}g}{d{\eta }^2}+1-{{\displaystyle \frac{dg}{d\eta }}}^2}\right)}}\\ +(1-\gamma )f{\displaystyle \frac{d^{2}g}{d{\eta }^2}-{\displaystyle \frac{{\mathring{\mathrm{A}}}_1}{{\mathring{\mathrm{A}}}_2}\sigma \left({\displaystyle \frac{dg}{d\eta }-1}\right)=0}}\end{array}$$

(8)

$${\displaystyle \frac{1}{Pr}{\displaystyle \frac{{\mathring{\mathrm{A}}}_3}{{\mathring{\mathrm{A}}}_4}{\displaystyle \frac{d^2\theta }{d{\eta }^2}+(1+\gamma )\left(f{\displaystyle \frac{d\theta }{d\eta }-{\displaystyle \frac{df}{d\eta }\theta }}\right)+(1-\gamma )g{\displaystyle \frac{d\theta }{d\eta }=0}}}}$$

(9)

Concern to confine conditions:

$$\begin{array}{r}\begin{array}{c}f(\eta =0)=0,{\displaystyle \frac{df}{d\eta }{|}_{\eta =0}=0,g(\eta =0)=0,}\\ {\displaystyle \frac{dg}{d\eta }{|}_{\eta =0}=0,\theta (\eta =0)=1}\\ {\displaystyle \frac{df}{d\eta }{|}_{\eta \to \mathrm{\infty }}\to 1,{\displaystyle \frac{dg}{d\eta }{|}_{\eta \to \mathrm{\infty }}\to 1,\theta (\eta ){|}_{\eta \to \mathrm{\infty }}=0}}\end{array}\}\end{array}$$

(10)

where,

$${\mathring{\mathrm{A}}}_1={\displaystyle \frac{{\mu }_{Thnf}}{{\mu }_f},{\mathring{\mathrm{A}}}_2={\displaystyle \frac{{\rho }_{Thnf}}{{\rho }_f},{\mathring{\mathrm{A}}}_3={\displaystyle \frac{k_{Thnf}}{k_f},{\mathring{\mathrm{A}}}_4={\displaystyle \frac{{(\rho C_p)}_{Thnf}}{{(\rho C_p)}_f},}}}}$$

are thermos-physical parameters of Tri-hybrid nanofluid and $\gamma =\frac{b}{a},\sigma =\frac{{\nu }_f}{aK*}$ are shear-to-strain rate and porous medium parameters, respectively. Besides this, from equations (7) and (8), we came to know that symmetries can be observed if the following axioms hold:

$$\begin{array}{c}i)f(\eta ,-\gamma )=g(\eta ,\gamma )\\ ii)g(\eta ,-\gamma )=f(\eta ,\gamma )\end{array}\}$$

(11)

Table 1 is for correlations of tri-hybrid nanofluid and Table 2 is for thermos-physical properties of tri-hybrid nanofluids.

Table 1.

Tri-hybrid nanofluid correlation. ⁵⁰

Properties	Correlations
Dynamic viscosity	${\mu }_{Thnf}=\frac{{\mu }_f}{{(1-{\phi }_1)}^{2.5}{(1-{\phi }_2)}^{2.5}{(1-{\phi }_3)}^{2.5}}$
Density	${\rho }_{Thnf}=(1-{\phi }_1)\{(1-{\phi }_2)[(1-{\phi }_3){\rho }_f+{\rho }_3{\phi }_3]+{\rho }_2{\phi }_2\}+{\rho }_1{\phi }_1$
Thermal conductivity	$\frac{k_{Thnf}}{k_{hnf}}=\frac{k_1+2k_{hnf}-2{\phi }_1(k_{hnf}-k_1)}{k_1+2k_{hnf}+{\phi }_1(k_{hnf}-k_1)}$ $\frac{k_{hnf}}{k_{nf}}=\frac{k_2+2k_{nf}-2{\phi }_2(k_{nf}-k_2)}{k_2+2k_{nf}+{\phi }_2(k_{nf}-k_2)}$ $\frac{k_{nf}}{k_f}=\frac{k_3+2k_f-2{\phi }_3(k_f-k_3)}{k_3+2k_f+{\phi }_3(k_f-k_3)}$
Heat capacity	$\frac{{(\rho C_p)}_{Thnf}}{{(\rho C_p)}_f}=\frac{{(\rho C_p)}_1{\phi }_1}{{(\rho C_p)}_f}+(1-{\phi }_1)\left[(1-{\phi }_2)\left\{(1-{\phi }_3)+\frac{{(\rho C_p)}_3{\phi }_3}{{(\rho C_p)}_f}\right\}+\frac{{(\rho C_p)}_2{\phi }_2}{{(\rho C_p)}_f}\right]$

Table 2.

Thermophysical properties. ⁵⁰

Physical properties	$\mathit{A}\mathit{g}$	$\mathit{G}\mathit{o}$	$\mathit{C}\mathit{u}$	$\mathit{K}\mathit{e}\mathit{r}\mathit{o}\mathit{s}\mathit{e}\mathit{n}\mathit{e}\mathit{o}\mathit{i}\mathit{l}$
$\rho \left(\frac{kg}{m^3}\right)$	10’500	1800	632	783
$C_p\left(\frac{J}{KgK}\right)$	235	717	531.8	2’090
$k\left(\frac{W}{mK}\right)$	429	5’000	765	0.145

Physical parameters of possession

The skin friction coefficient along x and y directions is $CFX$ and $CFY$ and is defined as follows:

$$Cfx={\displaystyle \frac{{\mu }_{Thnf}}{{\rho }_f{u}_{e*}^2}{\left({\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }z}}\right)}_{z=0},Cfy={\displaystyle \frac{{\mu }_{Thnf}}{{\rho }_f{v}_{e*}^2}{\left({\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }z}}\right)}_{z=0},Nux=-{\displaystyle \frac{x{k}_{Thnf}}{k_f(T_w-T_{\mathrm{\infty }})}{\left({\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }z}}\right)}_{z=0}}}}$$

Hence,

$$\begin{array}{c}CFX=\sqrt{1+\gamma }\left(\sqrt{{\displaystyle \frac{u_{e*}x}{{\nu }_f}}}\right)Cfx={\mathring{\mathrm{A}}}_1{\displaystyle \frac{d^{2}f}{d{\eta }^2}{|}_{\eta =0}}\\ CFY=\sqrt{1-\gamma }\left(\sqrt{{\displaystyle \frac{v_{e*}y}{{\nu }_f}}}\right)Cfy={\mathring{\mathrm{A}}}_1{\displaystyle \frac{d^{2}g}{d{\eta }^2}{|}_{\eta =0}}\\ NUX={\displaystyle \frac{\sqrt{1+\gamma }}{\sqrt{\left({\displaystyle \frac{u_{e*}x}{{\nu }_f}}\right)}}Nux=-{\mathring{\mathrm{A}}}_3{\displaystyle \frac{d\theta }{d\eta }{|}_{\eta =0}}}\end{array}\}$$

(12)

Research methodology

The numerical approach known as fourth-order Runge–Kutta is utilized to solve equations (7)–(9) under the influence of boundary conditions (11). Equations (7–9) represent higher order ODEs, therefore necessitating their conversion into a system of first-order ODEs. However, it is worth noting that three starting conditions were absent, namely $\frac{d^{2}f}{d{\eta }^2}{|}_{\eta =0},\frac{d^{2}g}{d{\eta }^2}{|}_{\eta =0},\frac{d\theta }{d\eta }{|}_{\eta =0}$ . The shooting approach was utilized to determine the missing beginning conditions. After identifying the missing requirements, the solution is calculated and has been determined to satisfy the boundary conditions (11). We computed the graphical and numerical results of skin friction $(CFX,CFY)$ and Nusselt number $(NUX)$ for numerous values of $\gamma ,\sigma ,$ and solid volume fraction $\phi$ . As mentioned above, one of our main objectives is to optimize the Nusselt number for variations of these handling parameters. We employed RSM. We used second-order imitation that transforms the response area:

$$Y=Y({\alpha }_{\circ },{\alpha }_{ii},{\alpha }_{ij},x_i,x_{i,j},ϵ)={\alpha }_{\circ }+\sum _{i=1}^m{\alpha }_i{x}_i+\sum _{i=1}^m{\alpha }_{ii}{x}_{ii}^2+\sum _{i=1}^{m-1}\sum _{j=i+1}^m{\alpha }_{ij}{x}_i{x}_j+ϵ-$$

(13)

Box and Wilson ⁵¹ establish Central Composite Designs (CCD) for appropriate second-order imitation. Moreover, CCD in m variables essential 2 m factorial runs along 2 m auxiliary amalgamations named significant points beside the reconcile axes of classified variable standards and slightly single center point. The essential points need to resolve coefficients of the second-order imitation.

$$Y=\widehat{{\alpha }_{\circ }}+x^{\mathrm{\prime }}b+x^{\mathrm{\prime }}Bx$$

(14)

$$x=\left[\begin{array}{c}{x}_1\\ x_2\\ .\\ .\\ .\\ x_m\end{array}\right],b=\left[\hat{\begin{array}{c}{\alpha }_1\\ \widehat{\begin{array}{c}{\alpha }_2\\ .\\ .\\ .\\ \hat{{\alpha }_m}\end{array}}\end{array}}\right],B=\left[\begin{array}{c}\widehat{{\alpha }_{11}}\\ \widehat{\begin{array}{c}{\alpha }_{21}\\ .\\ .\\ .\\ \hat{{\alpha }_{m1}}\end{array}}\end{array}\begin{array}{c}\widehat{{\alpha }_{12}}\\ \widehat{\begin{array}{c}{\alpha }_{22}\\ .\\ .\\ .\\ \hat{{\alpha }_{m2}}\end{array}}\end{array}\begin{array}{c}\widehat{{\alpha }_{1m}}\\ \widehat{\begin{array}{c}{\alpha }_{2m}\\ .\\ .\\ .\\ \hat{{\alpha }_{mm}}\end{array}}\end{array}\right]$$

is used to estimate the stationary points involved in equation (13). Once, the stationary points are estimated then called by:

$$x_s=-{\displaystyle \frac{1}{2}{B}^{-1}b}$$

After that, we processed it as the advanced interior point of the principal parameter $P_i$ for finishing range $[l_i^{final},u_i^{final}]$ . The upcoming means is to estimate the direction of elevated ascent or descent of Y and then the course of investigations is organized across the computed path in advance of no further decline in consequence is noticeable.

Results and discussions

The incorporated nonlinear ODEs were numerically resolved using the MATLAB built-in procedure bvp4c for a wide range of unknown parameters. We used $\eta =4$ to ensure that the velocity and heat profiles match the broad confines criteria as η approaches infinity. Because the prevailing issue may have several neglected inceptive conditions, the bvp4c solution requires a guess for solicited equations (7–9). Furthermore, the estimate must relate specific concerns and demonstrate the solution's performance. Table 3 shows the values of $f^{\mathrm{″}}(0)$ and the preceding results when ${\varphi }_{Thnf}=0$ (regular fluid) and $\sigma =0$ for an axisymmetric situation $(\gamma =0)$ . It should be noted that the current values of $f^{\mathrm{″}}(0)$ correspond to previously publicized data. Figure 2 depicts the influence of nanoparticle concentrations ranging from 1 to 3%. It should be intimated that as ${\phi }_{Thnf}$ rises values of skin friction along the x-axis ( $f^{\mathrm{″}}(0)$ or CFX) grow as $\gamma \in (-2,10]$ and decreases for the values of $\gamma \in [-10,2)$ . However, $g^{\mathrm{″}}(0)=CFY$ exhibits the opposite behavior. We discovered that the wall had negligible friction in both x and y directions at $\gamma ={\gamma }_{\circ }=\mp 2.5489$ for all nanoparticle concentrations. Furthermore, $f^{\mathrm{″}}(0)=g^{\mathrm{″}}(0)$ for axisymmetric $(\gamma =0)$ . Figure 3 depicts the accomplishment of nanoparticle preoccupation with the Nusselt number (NUX).

Figure 2.

Effect of solid volume fraction on $f^{\mathrm{″}}(0),g^{\mathrm{″}}(0)$ .

Figure 3.

Effect of solid volume fraction on NUX.

Table 3.

Validation of the numerical results for ${\phi }_{Thnf}=0,\sigma =0,\gamma =0$ .

Wani 49	Soid et al. 48	Present result ${\mathit{f}}^{\mathrm{″}}(0)$
1.311938	1.311938	1.3119375

The warm transmission rate increases with increasing immersion of NPs for $\gamma \in [-10,10]$ . The Nusselt number has a global maximum value at $\gamma =10$ and a minimum as $\gamma \to -1$ . The crucial values for the Nusselt number are $\gamma =-1,10$ , which correspond to the lowest and highest values. Figure 4 illustrates the impact of the porous medium parameter $\sigma$ on CFX and CFY at various $\gamma$ concentrations, with $Pr=6.2$ and $\phi =1\text{\%}.$ The skin friction coefficient has a direct relationship with the porous medium parameter $\sigma$ . The Nusselt number for $\sigma \in [-10,-1.152]$ shows an opposite trend.

Figure 4.

Effect of $\sigma$ on CFX and CFY.

Figure 5 compares the Nusselt numbers for three different types of fluids: nanofluids, hybrid fluids, and ternary-hybrid nanofluids. Figure 6 shows that, in the case of mono-nanofluid, the rate of warm exchange enhances as the volume ratios of Ag and Go are increased. Go-Ag and Ag-Cu for HNF, though. This is caused by Ag and Go's strong thermal conductivity; this is common knowledge. Commencing from the study, it is noted that the ternary fluid containing hybrid nanoparticles has taken the most significant effect on Nusselt number, that is, 17.12978%, 8.43809%, and 19.20192% increment for Go, Ag, and Cu type mono-nanofluid is seen. Similarly, 12.67309%, 18.13489%, and 13.65317% enhancement for HNF (Go-Ag, Go-Cu, Ag-Cu, respectively) are seen. Therefore, we concluded that the ternary nanofluid model can be more efficient in increasing the warm transmission rate related to mono and hybrid-type nanofluid imitation.

Figure 5.

Effect of $\sigma$ on NUX.

Figure 6.

Nusselt number NUX for various combinations of nanoparticles.

Result and discussion of response surface methodology

Examined the response of three factors of γ, σ, and φ on Nux. The adequate approach is defined and sensitivity analysis of essential factors to perform Nux (reduced Nusselt number). Here we work out on Nux. Table 4 represents the parameters of their level for Nusselt number and Table 5 represents the coded values and response values for experimental design.

Table 4.

Parameters with their levels for NUX.

		Levels
Parameters	Symbols	−1	0	1
$\sigma$	A	0	1	2
$\gamma$	B	−5	0	5
$\phi$	C	0.01	0.02	0.03

Table 5.

Experimental design and responses.

	Coded values
Runs	A	B	C	Response (NUX)
1	1	−1	1	3.26311682
2	−1	−1	−1	4.0986692
3	1	−1	−1	1.99308453
4	1	1	−1	3.13689202
5	0	0	0	1.99308453
6	0	0	0	1.88551497
7	1	1	1	1.99308453
8	−1	1	1	1.99308453
9	0	0	0	1.95289106
10	0	−1	0	1.99308453
11	−1	−1	1	2.71897024
12	−1	0	0	4.02299527
13	0	0	−1	2.9940403
14	0	0	0	4.00582394
15	0	0	1	4.19448971
16	1	0	0	4.17585535
17	0	0	0	2.83766466
18	0	1	0	1.99308453
19	−1	1	−1	2.07151137
20	0	0	0	2.03332191

Statistical analysis

In this agreement, the statistical analysis executed 20 runs for Nux. Tables 6 and 7 shows the coded coefficient and statistical analysis influence. This model is appropriate for calculating the value of Nux because the better value of R² for Nux i.e., 99.56% which attained by accomplishment technique and statistical analysis of the imitation. Further, the R²-Adj amount for Nux is 99.16% which is less than R² but imitation appropriate investigational data adequately (See Table 8).

Table 6.

Coded coefficients.

Term	Coef	SE coef	T-value	p-value	VIF
Constant	1.9912	0.0283	70.39	0.000
γ	0.5547	0.0260	21.32	0.000	1.00
σ	−0.0693	0.0260	−2.66	0.024	1.00
φ	0.0667	0.0260	2.56	0.028	1.00
γ*γ	1.5579	0.0496	31.40	0.000	1.82
σ*σ	−0.0100	0.0496	−0.20	0.845	1.82
φ*φ	0.0046	0.0496	0.09	0.928	1.82
γ*σ	0.1009	0.0291	3.47	0.006	1.00
γ*φ	0.0121	0.0291	0.42	0.687	1.00
σ*φ	−0.0011	0.0291	−0.04	0.970	1.00

Table 7.

Analysis of variance.

Source	DF	Adj SS	Adj MS	F-value	p-value
Model	9	15.3377	1.70419	251.70	0.000
Linear	3	3.1696	1.05654	156.04	0.000
Γ	1	3.0771	3.07709	454.47	0.000
Σ	1	0.0481	0.04805	7.10	0.024
Φ	1	0.0445	0.04447	6.57	0.028
Square	3	12.0854	4.02846	594.98	0.000
γ*γ	1	6.6742	6.67420	985.74	0.000
σ*σ	1	0.0003	0.00027	0.04	0.845
φ*φ	1	0.0001	0.00006	0.01	0.928
Two-way interaction	3	0.0827	0.02757	4.07	0.040
γ*σ	1	0.0815	0.08152	12.04	0.006
γ*φ	1	0.0012	0.00117	0.17	0.687
σ*φ	1	0.0000	0.00001	0.00	0.970
Error	10	0.0677	0.00677
Lack-of-fit	5	0.0677	0.01354	*	*
Pure error	5	0.0000	0.00000
Total	19	15.4054

Table 8.

Model summary.

S	R-sq	R-sq(adj)	R-sq(pred)
0.0822847	99.56%	99.16%	96.59%

Analysis of variance and model estimation

To achieve an estimated and regression equation, the experimental model is performed under altered experimental settings. The residual plot is acquired by analysis of variance and commencing data into systematic software. The residual plot is given in Figure 7. In the residual plot, difference in the observed y-value (from the scattered plot) and predicted y-value is known as residual. According to the figure, normal probability plot is in positive correlation as all points near approximate straight lines which represents normality. In residual Histogram halves do not appear as a mirror image so is skew symmetrical distribution. Observed and fitted values display a good correlation if compared to the residual diagram and fitted value. The greatest residual was observed to be in the proximity of 0.15 for Nux. By applying RSM, we get equation1 which is a common relationship between their effective factors.

Figure 7.

Residual plot.

From Table 7, it is observed that the p-value of $\sigma {}^2,\phi {}^2,\gamma .\phi ,\sigma .\phi >0.05$ therefore $\sigma {}^2,\phi {}^2,\gamma .\phi ,\sigma .\phi$ has an insignificant impact on Response function Nux.

$$\mathrm{NUX}=1.933+0.0859\gamma -0.047\sigma +4.9\phi +0.06232{\gamma }^2-0.0100\sigma {}^2+46\phi {}^2+0.02019\gamma .\sigma +0.242\gamma .\phi -0.11\sigma .\phi$$

(15)

From Table 7, it is noticed that p-value of $\sigma {}^2,\phi {}^2,\gamma .\phi ,\sigma .\phi >0.05$ therefore $\sigma {}^2,\phi {}^2,\gamma .\phi ,\sigma .\phi$ has an insignificant impact on the Response function Nux. So, after removing the insignificant factors which have a p-value greater than 0.05 the regression equation for Nux is reduced into the following equation:

$$\mathrm{NUX}=1.933+0.0859\gamma -0.047\sigma +4.9\phi +0.06232{\gamma }^2+46\phi {}^2+0.02019\gamma .\sigma$$

(16)

Figure 8 shows the combined effect of solid volume fraction $\phi$ and porous parameter $\sigma$ on the Nusselt number. From this figure, it is noticed that a higher value of solid volume fraction is more effective for any value of the porous parameter on the Nusselt number. On the other hand, the minimum value of the Nusselt number can be obtained by setting a low range of solid volume fraction. So, it is concluded that the Nusselt number can be optimized by setting the higher range and lower range of the values of solid volume fraction. The effect of solid volume fraction and shear-to-strain rate is depicted in Figures 9 and 10. It is worth noticing that the Nusselt number can be minimized by setting $\gamma =0$ and maximized for $\gamma =10,\phi =0.2$ . The parabolic nature of the graphs can be seen for the combined effect of porous parameter $\sigma$ and shear-to-strain rate $\gamma$ . For axisymmetric case $\gamma =0$ , the Nusselt number is minimum for any value of porous parameter. However, the Nusselt number can be maximized by setting $\gamma =10,\sigma =2$ .

Figure 8.

Combined effect of solid volume fraction and porous parameter on Nusselt number.

Figure 9.

Combined effect of solid volume fraction and shear-to-strain rate on Nusselt number.

Figure 10.

Combined effect of porous parameter and shear-to-strain rate on Nusselt number.

Sensitivity analysis

Sensitivity analysis is a principal consequential mechanism for model estimation. It is a technique for predicting the consequences of an interpretation of contingency proceed to be definite and correspond to the prevailing anticipations. Additionally, certain values of individualistic parameters (factors) regard an explicit reliant parameter (response) in a defined set of assumptions. Sensitivity analysis mentors us in ascertaining critical governing supervise points in research study and also substantiating and certifying the model. Additionally, it can be designed to examine how a diversity in the inputs of a model accomplishes the model outputs. Furthermore, we study how the change in warm transmission is attained by permeable parameters and others. Moreover, the sensitivity analysis imitation contains N self-reliant parameters $x=\{x_1,x_2,\dots ,x_n\}$ and reliant parameter $y=f(x_1,x_2,\dots ,x_n)=f(x)$ . Also, we obtain the partial derivative of the reliant variable (response) w.r.t self-reliant variables (factors) individually. $S_i=\frac{\mathrm{\partial }f(x)}{\mathrm{\partial }{x}_i}$ Here, S = sensitivity and x = self-reliant parameters (factors). The consequences of the imitation are gained by utilizing the values of self-reliant variables at low (−1) moderate (0) and high levels (+1).^52–56

The sensitivity of the imitation prevailed by utilizing the regression equation (16). The partial derivatives of the response function (NUX) w.r.t factors ( $\gamma$ , σ, $\phi$ ) define the measurement of sensitivity which are as follows:

$${\displaystyle \frac{\mathrm{\partial }NUX}{\mathrm{\partial }\gamma }=0.0859+0.12464\gamma +0.02019\sigma }$$

(17)

$${\displaystyle \frac{\mathrm{\partial }NUX}{\mathrm{\partial }\sigma }=-0.047+0.02019\gamma }$$

(18)

$${\displaystyle \frac{\mathrm{\partial }NUX}{\mathrm{\partial }\phi }=4.9+92\phi }$$

(19)

Table 9 displays the sensitivity of the response function (NUX) for different degrees of $\gamma$ , σ, and $\phi$ . The positive value of sensitivity indicates progress in the response function through the improvement of input components. Nevertheless, the negative number indicates a decrease in the response function due to an increase in input variables. The sensitivity of the response function is demonstrated by the bar charts depicted in Figure 11. The bars exhibit a high level of sensitivity to the response function (NUX) due to the non-negative nature of the factor variable. Furthermore, the bars exhibit a decrease in sensitivity to the response function concerning the factor variable, resulting in a negative outcome.

Figure 11.

Sensitivity analysis of NUX with σ=0 level and all combinations of other factors.

Table 9.

Sensitivity analysis of NUX with $\phi =0$ level, and all $\gamma ,\sigma$ combinations.

$\sigma$	$\gamma$	Sensitivity
		$\frac{\mathrm{\partial }NUX}{\mathrm{\partial }\gamma }$	$\frac{\mathrm{\partial }NUX}{\mathrm{\partial }\sigma }$	$\frac{\mathrm{\partial }NUX}{\mathrm{\partial }\phi }$
−1	−1	−1.1605	−0.2489	6.7400
	0	0.0859	−0.0470	6.7400
	1	1.3323	0.1549	6.7400
0	−1	−1.1403	−0.2489	6.7400
	0	0.1061	−0.0470	6.7400
	1	1.3525	0.1549	6.7400
1	−1	−1.1201	−0.2489	6.7400
	0	0.1263	−0.0470	6.7400
	1	1.3727	0.1549	6.7400

Conclusions

This work examined the area of non-axisymmetric stagnation of a flat plate in a porous medium using tri-hybrid nanoparticles. An appropriate similarity transformation was employed to convert the controlling PDEs into a set of nonlinear ODEs, which were subsequently solved numerically. The primary findings of the research are enumerated as follows:

The wall experiences zero friction in both x and y directions at $\gamma ={\gamma }_{\circ }=\mp 2.5489$ for each level of nanoparticle concentration.

HTR increases as immersion of nanoparticles increases for $\gamma \in [-10,10]$ . The highest and lowest values of the Nusselt number are obtained at $\gamma =10$ and $\gamma =-1$ , respectively.

The rate of warm exchange increases as the volume ratios of Ag and Go are increased in the case of mono-nanofluid. However, for HNF, the rate of heat transfer enhancement is seen for Go-Ag and Ag-Cu.

The ternary nanofluid containing hybrid nanoparticles has the most significant effect on Nusselt number, with a 17.12978%, 8.43809%, and 19.20192% increment for Go, Ag, and Cu type mono-nanofluid, respectively. Similarly, 12.67309%, 18.13489%, and 13.65317% enhancement for HNF (Go-Ag, Go-Cu, Ag-Cu, respectively) is seen.

Nusselt number can be minimized by setting $\gamma =0$ , and maximized for $\gamma =10,\phi =0.2$ when the effect of solid volume fraction and shear-to-strain rate is considered.

Nusselt number can be maximized by setting $\gamma =10,\sigma =2$ when the combined effect of porous parameter $\sigma$ and shear-to-strain rate $\gamma$ is considered.