Heat transfer performance appraisal of hybrid and ternary nanofluid flows over a wedge considering viscous dissipation and quadratic thermal convection: Non-similar solution approach

Abstract

This study investigates the flow of ternary and hybrid nanofluids over a stretching wedge in a porous medium, incorporating quadratic thermal convection, Newtonian heating, and viscous dissipation. The hybrid nanofluid comprises magnesium and zinc oxides (MgO-ZnO), while the ternary nanofluid includes magnesium, zinc, and graphene oxides (MgO-ZnO-GO), with ethylene glycol as the base fluid. The model accounts for homogeneous-heterogeneous (HOM-HET) reactions and surface-catalyzed processes that enhance reaction rates by increasing molecular collision frequency. A non-similarity transformation is employed to convert the governing equations into a second-order truncated dimensionless form. Unlike similarity solutions, non-similar solutions offer enhanced applicability in fluid flow analysis. The bvp4c numerical method is used to compute velocity and temperature profiles versus varying physical parameters. Results indicate that the concentration of the ternary nanofluid decreases with increasing Falkner–Skan and surface-catalysis parameters. The hybrid nanofluid exhibits enhanced velocity and temperature profiles, while the ternary nanofluid achieves a higher heat transfer rate. Numerical results are validated, confirming the model’s reliability.

Introduction

Ternary nanofluids are fluids containing three types of nanoparticles suspended in a base liquid. These nanoparticles, often metals, metal oxides, or carbon nanotubes, alter the fluid’s properties, such as density, viscosity, and thermal conductivity. The third nanoparticle in the mix further changes these characteristics by influencing the size, shape, or surface chemistry of the particles, affecting their interaction with the fluid. The fluid’s rheological properties depend on shear rate, viscosity, and stress. Ternary and hybrid nanofluids are synthesized to optimize the thermal characteristics of conventional heat transfer liquids. These fluids have applications in the automotive industry, refrigeration and air conditioning, industrial processes, and biomedical applications. Kho et al. ¹ examined numerically the flow of the hybrid nanofluid over a permeable wedge considering the important effects of thermal radiation and the frictional heating. The interesting outcomes indicate that the rate of heat flux and hybrid nanofluid temperature increase with the thermal radiation parameter. The flow of the hybrid nanofluid over a 3D wedge influenced by nonlinear thermal radiation and a variable magnetic field with momentum and thermal slips is numerically analyzed by Rana et al. ² The salient results revealed that the velocity field is on the decline for the variable magnetic field and the slip condition. Yusuf et al. ³ discussed the flow of the hybrid nanofluid affected by chemical reactions and gyrotactic microorganisms in a porous medium over a wedge with melting heat impact. The problem is addressed by the Chebyshev spectral collocation method. The leading outcome disclosed that the fluid temperature is enhanced for the melting heat parameter. The ternary-hybrid nanoliquid flow caused by a double-stretched wedge surface with forced convection was described by Xiu et al. ⁴ The results showed that thermal transmission is quite sluggish in the scenario of the water flow. Next, Animasaun et al. ⁵ analyzed the unsteady ternary hybrid nanofluid flow considering static and moving wedges impacted by bioconvection and stretching velocity of the wall. The researchers concluded that for the stationary wedge, higher friction is achieved in comparison to the moving wedge. Berrehal et al. ⁶ computed the irreversibility analysis of the flow of hybrid nanofluid along a moving wedge with distinct shape effects. Yaseen et al. ⁷ obtained similar solutions of the bioconvective ternary-hybrid nanoliquid flow on three different geometries with modified Fourier law. The outcome revealed that the flow over the wedge exhibits the highest rate of mass transfer and gradient in microorganism density. In parallel with physics-based approaches, researchers have also adopted data-driven methods to analyze and optimize nanofluid behavior. For instance, Hai et al. ⁸ applied neural networks, gene expression programing, and multi-objective particle swarm optimization to predict and enhance the thermal performance of ternary hybrid nanofluids. Their findings emphasize the potential of computational intelligence techniques for performance prediction and optimization. However, such models often lack the physical insights provided by differential equation-based analyses.

Momentum and thermal boundary-layer theory^9–14 is widely used in engineering for heat transfer applications. It applies to fluid flow over solid surfaces, influencing various practical situations, such as friction in automobiles, cooling of metal or plastic sheets, steam and hot water pipes, and extruded wire production. A good number of studies may be found in the literature due to their immense importance.^15,16 Analyzing the boundary layer requires making the governing equations non-dimensional, which can be accomplished through either similarity or non-similarity methods. In similar flow phenomena, the dimensionless numbers remain constant along the flow direction. However, in non-similar flows, the fundamental flow properties change along the flow direction. In such cases, it becomes necessary to convert the PDEs into dimensionless PDEs by introducing two new independent dimensionless variables known as non-similarity and pseudo-similarity variables. Identifying a suitable variable that holds prevalent characteristics in a problem is more of an art than a science, and it necessitates a solid mathematical comprehension of the issue at hand. Due to its simplicity, many researchers^17–19 focused on studying similarity boundary-layer flows, leading to a substantial number of articles published on this topic. Non-similar boundary-layer flows, which occur naturally and find wide application in our everyday lives, are the focus of the investigation. Sparrow and Yu^20,21 noted that these flow patterns can arise as a result of dimensional velocity variations, transverse curvature, or mass transfer onto the sheet. In most physical situations, the similarity approach does not work for boundary layer equations. This is because the equations are nonlinear and difficult to solve mathematically. Numerical and analytical methodologies can be utilized to identify solutions for non-similar flows. Numerical approaches use computers to solve equations, whereas analytical methods seek accurate solutions to problems.

Gorla et al., ²² Duck et al., ²³ Sahu et al., ²⁴ Banu and Rees, ²⁵ and Chen ²⁶ include a list of research studies that employed numerical approaches to solve boundary layer equations. By discretizing the infinite domain into a finite set of points, boundary layer equation solutions may be computed using numerical techniques. However, the answers may contain inaccuracies or uncertainty as a result. This problem can be solved using analytical techniques by giving accurate answers to the equations in the infinite domain. Analytical techniques, however, can be challenging to utilize and might not be able to produce precise estimates for all variables. This is due to the dependence on perturbation techniques, which are frequently employed to develop analytical solutions, on small parameters. The perturbation approach may not be valid if these parameters are not small enough. Cimpean et al. ²⁷ investigated non-similar thermal boundary layer flow using both perturbation and analytical approaches. They applied the technique of local similarity, which was first introduced by Massoudi ²⁸ and Sparrow et al. ²⁰ By following this technique, it is presumed that the non-similar terms are considered to be very small and can be ignored by making $\zeta <<0$ mathematically. Here, PDEs will behave like ODEs. Nevertheless, as noted by Sparrow et al., ²⁰ the outcomes obtained here are of “uncertain accuracy.” Massoudi ²⁸ highlighted that this technique is true only for small estimates of non-similar parameter $\zeta .$ He used this technique over a model of a wedge and obtained two supplementary equations by taking the derivative of PDEs with respect to $\zeta .$ The numerical solution obtained for the resulting ODEs, that is, two original equations, two auxiliary equations, and heat equations simultaneously which is quite challenging. Khan et al. ²⁹ computed the non-similar solution with the chemical reactive and Joule heating of nanofluid flow along an exponentially elongated sheet. They concluded that entropy generation is positively impacted by an escalation in the magnetic parameter, and velocity is negatively influenced. More work on non-similar solutions can be found in Liu et al. ³⁰ and Khan et al. ³¹

Natural convection is driven by temperature-induced density differences, while forced convection relies on external forces like pumps or fans. Mixed convection combines both. Understanding mixed convection is key to studying non-uniform flows. Quadratic convection, prominent in water and molten metals with high thermal conductivity, significantly enhances heat transfer compared to linear convection. Lately, many researchers have focused on the role of thermal convection in numerous scenarios. Upreti et al. ³² studied the free convective flow of kerosene oil-based nanofluid with silver nanoparticles along a cone with frictional and Joule dissipations. They showed that when the Eckert number was raised, the momentum boundary layer diminished. Thriveni and Mahanthesh ³³ examined the flow of hybrid nanoliquid under the consequences of quadratic convection and thermal radiation with variable fluid properties and surface response technique. Quadratic convection results in a higher surface drag coefficient. The significance of quadratic thermal radiation and convection accompanied by dust particles in the nanofluid flow over a vertical plate is determined by Mahanthesh et al. ³⁴ The key observation revealed that heat transmission is superior in the case of nanoparticle inclusion into the base fluid, but reduces when dust particles are added to the liquid. Patil and Benawadi ³⁵ deliberated the numerical solution of the Williamson nanofluid flow affected by quadratic convection and multiple diffusions. It is observed that the fluid drag coefficient is higher for the quadratic convection parameter. This is followed by another publication by Patil and Kulkarni ³⁶ in which they discussed the Eyring-Powell nanofluid flow with quadratic nanofluid flow with multiple diffusions. Alsabery et al. ³⁷ numerically computed the thermal convection and entropy optimization of water-based nanofluid flow in an irregular cubical container with a solid cylinder. It is seen that mounting estimations of Rayleigh number diminish the Bejan number. The salient remark of the said exploration is that surface drag force boosts for large values of quadratic convection.

Chemical reactions are classified as heterogeneous or homogeneous based on the phases of the reactants and products. Heterogeneous reactions occur between different phases, like a liquid and a solid, and are influenced by factors like surface area and mass transfer. Homogeneous reactions occur within the same phase, such as between two gases or liquids, and typically proceed faster due to uniform reactant distribution. Both types are crucial in industrial and natural processes, and understanding their mechanisms is key for optimizing reactions. A simplified model for heterogeneous and homogeneous interactions in boundary layer flow was initially scrutinized by Chaudhary and Merkin.^38,39 According to Merkin, ⁴⁰ cubic autocatalysis in the fluid flow signifies the isothermal homogeneous reaction as:

$$C+2D\to 3D.$$

(1)

Moreover, a heterogeneous reaction over the catalytic surface is specified via

$$C\to D.$$

(2)

Here, $C$ and $D$ represent the chemical species along with concentrations $H_c$ and $H_d$ .

The reaction rate for a homogeneous reaction is illustrated as

$$\frac{\partial H_c}{\partial t}=\frac{\partial H_d}{\partial t}=-K^{*}{H}_c{\left(H_d\right)}^2.$$

(3)

At the fluid-solid contact, the rate of reaction is defined as:

$$D_c\frac{\partial H_c}{\partial n}=-D_d\frac{\partial H_d}{\partial n}=K_s{H}_c.$$

(4)

By employing the above theory in hybrid nanoliquid flow, Haq et al. ⁴¹ researched the influences of spongy media and homogeneous/heterogeneous reactions over the three varied geometries (cone, wedge, and plate). The observations proved that both heterogeneous and homogeneous reaction parameters diminish the concentration distribution. Further, in recent studies, scholars have studied homogeneous/heterogeneous reactions over distinct geometries.^42–44 In the current model, spongy media, and the surface of the wedge dwell with the same catalyst and no one has adopted this work for ternary nanoliquid flow over a wedge. As a result, the heterogeneous reaction happens on the surface of spongy media, which is classified as a surface-catalyzed reaction. The spongy media reaction rate is formulated by Hill and Root ⁴⁵ and is given by

$$R_p=-S{\mathrm{K}}_s{H}_c,$$

(5)

where the interfacial area of a porous medium is symbolized by $S.$

This article addresses the challenge of handling non-similar terms from similarity transformations, focusing on non-similar boundary-layer flows, which are crucial but less studied compared to similar flows. The research offers a non-similar solution for hybrid and ternary nanofluid flow near a wedge with quadratic convection and surface-catalyzed reactions. It includes effects like Newtonian heating, viscous dissipation, and homogeneous-heterogeneous reactions. Unlike past studies using similar transformations, this work employs a non-similar approach, solving it numerically, and presenting results through graphs and tables. The uniqueness of work in tabular form is presented in Table 1.

Table 1.

The originality of the current hybrid nanofluid flow model.

Authors	Hybrid nanofluid	Quadratic thermal convection	Viscous dissipation	Wedge	Non-similar solution
Kho et al. 1	Yes	No	Yes	Yes	No
Yusuf et al. 3	Yes	No	No	Yes	No
Xiu et al. 4	Yes	No	No	Yes	No
Present	Yes	Yes	Yes	Yes	Yes

In the proposed model, we aim to address the following questions:

• ➢ How does an increasing Falkner-Skan power law parameter influence velocity, temperature, concentration profiles, and boundary layer thickness, particularly under a positive pressure gradient?

• ➢ How do velocity ratio and porosity parameters influence the velocity distribution?

• ➢ How does an increasing surface-catalyzed parameter affect the concentration profile?

• ➢ How do increasing particle volume fraction values affect the heat transfer rate, wall drag coefficient, and surface roughness in ternary nanofluids compared to hybrid nanofluids, and how does nanoparticle inclusion influence thermal conductivity?

• ➢ How does the quadratic convection parameter influence wall heat transfer rate, surface drag, and vortex formation compared to linear convection, and what role does quadratic buoyancy force play in enhancing heat transfer and turbulence?

Mathematical analysis

The envisaged model is constructed based on the following assumptions:

• ➢ Steady ternary and hybrid nanofluid flows, based on ethylene glycol, passing over an impermeable wedge within a porous medium are considered.

• ➢ The stretching velocity of a wedge is characterized as $U_W\left(X\right)=U_w{X}^m$ and velocity at the free stream is considered as $U_e\left(X\right)=U_{\infty }{X}^m.$

• ➢ For ternary and hybrid flows, $X-$ axis is assumed along the wedge whereas $Y-$ axis is in the outward direction to the surface of the wedge.

• ➢ The Falkner-Skan power law parameter is described as $m=\frac{\epsilon }{2-\epsilon }$ , where $\epsilon =\frac{\omega }{\pi },$ ⁴⁶ with $\left(\omega \right)$ signifies the wedge angle, and the Hartree pressure gradient is depicted by $\left(\epsilon \right).$

• ➢ The gravitational force $\left(g\right)$ acts downwards.

• ➢ The magnetic Reynolds number and electron-atom collision frequency are so small that the induced magnetic field and Hall current can be neglected.^47,48

• ➢ The non-linear Boussinesq approximation is employed, which says that quadratic density-temperature variation are represented as ${\beta }_o=\frac{1}{\rho }\left(\frac{\partial \rho }{\partial T}\right)$ and ${\beta }_1=\frac{1}{2\rho }\left(\frac{{\partial }^2\rho }{\partial T^2}\right)$ .

• ➢ Figure 1 is provided to illustrate the flow geometry.

Figure 1.

Flow geometry.

The vector form of governing equations is

$$\nabla ·\overrightarrow{S}=0,$$

(6)

$$(\overrightarrow{S}·\nabla )\overrightarrow{S}=-\frac{1}{\rho }\nabla P+\frac{1}{\rho }\nabla ·(\mu \nabla ·\overrightarrow{S})+\frac{1}{\rho }{\overrightarrow{J}}_c\times \overrightarrow{\beta }-\frac{\mu }{\overline{k}}\overrightarrow{S}+g\sin \left(\frac{\omega }{2}\right)[\frac{1}{\rho }\left(\frac{\partial \rho }{\partial T}\right)(T-T_{\infty })+\frac{1}{2\rho }\left(\frac{{\partial }^2\rho }{\partial T^2}\right){(T-T_{\infty })}^2],$$

(7)

$$(\overrightarrow{S}·\nabla )T=\alpha (\nabla ·(k\nabla T))+\left(\frac{\mu }{\left(\rho C_p\right)}\right)\mathrm{\Phi },$$

(8)

where, $\overrightarrow{S}$ illustrates the velocity field, ${\overrightarrow{J}}_c$ represents the current vector and $\mathrm{\Phi }$ is the viscous dissipation term.

The following set of governing equations using the Tiwari and Das model with boundary layer approximation is represented as^49,50:

$$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0,$$

(9)

$$\begin{array}{c}U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=U_e\left(X\right)\frac{\partial U_e\left(X\right)}{\partial X}+\frac{{\mu }_{\mathrm{THNF}}}{{\rho }_{\mathrm{THNF}}}\left(\frac{{\partial }^{2}U}{\partial Y^2}\right)-\frac{{\mu }_{\mathrm{THNF}}}{{\rho }_{\mathrm{THNF}}\overline{k}}(U-U_e\left(X\right))\hfill \\ -\frac{{\sigma }_{\mathrm{THNF}}{\beta }_o^2}{{\rho }_{\mathrm{THNF}}}(U-U_e\left(X\right))+\sin \left(\frac{\omega }{2}\right)[g{\left({\beta }_o\right)}_{\mathrm{THNF}}(T-T_{\infty })+g{\left({\beta }_1\right)}_{\mathrm{THNF}}{(T-T_{\infty })}^2],\hfill \end{array}$$

(10)

$$U\frac{\partial T}{\partial X}+V\frac{\partial T}{\partial Y}=\left(\frac{k_{\mathrm{THNF}}}{{\left(\rho C_p\right)}_{\mathrm{THNF}}}\right)\frac{{\partial }^{2}T}{\partial Y^2}+\left(\frac{{\mu }_{\mathrm{THNF}}}{{\left(\rho C_p\right)}_{\mathrm{THNF}}}\right){\left(\frac{\partial U}{\partial Y}\right)}^2,$$

(11)

$$U\frac{\partial H_c}{\partial X}+V\frac{\partial H_c}{\partial Y}=D_c\frac{{\partial }^2{H}_c}{\partial Y^2}-K^{*}{H}_c{\left(H_d\right)}^2-\mathrm{S}{\mathrm{K}}_s{H}_c,$$

(12)

$$U\frac{\partial H_d}{\partial X}+V\frac{\partial H_d}{\partial Y}=D_d\frac{{\partial }^2{H}_d}{\partial Y^2}+K^{*}{H}_c{\left(H_d\right)}^2+\mathrm{S}{\mathrm{K}}_s{H}_c.$$

(13)

With the below-mentioned boundary constraints ⁴⁹ :

$$\begin{array}{c}\multicolumn{1}{c}{U=U_w{X}^m,V=0,k_{\mathrm{THNF}}\left(\frac{\partial T}{\partial Y}\right)=-h_{s}T,D_c\frac{\partial H_c}{\partial Y}=K_s{H}_c,D_d\frac{\partial H_d}{\partial Y}=-K_s{H}_c,\mathrm{at}Y=0,}\\ \multicolumn{1}{c}{U\to U_{\infty }{X}^m,T\to T_{\infty },H_c\to H_o,H_d\to 0,\mathrm{as}Y\to \infty .}\end{array}$$

(14)

Non-similarity analysis (solution methodology)

The non-similarity transformations ⁴⁹ are defined below:

$$\begin{array}{c}\multicolumn{1}{c}{\eta =\sqrt{\frac{U_e\left(X\right)(m+1)}{2{\nu }_{F}X}}Y,\zeta =\frac{X}{L},U=U_e\left(X\right)\frac{\partial f}{\partial \eta },\theta (\zeta ,\eta )=\frac{T-T_{\infty }}{T_{\infty }},\Psi =\sqrt{\frac{U_e\left(X\right)2{\nu }_{F}X}{(m+1)}}f(\zeta ,\eta ),}\\ \multicolumn{1}{c}{h_c(\zeta ,\eta )=\frac{H_c}{H_o},h_d(\zeta ,\eta )=\frac{H_d}{H_o},v=-\sqrt{\frac{U_e\left(X\right)2{\nu }_F}{X(m+1)}}[\left(\frac{m+1}{2}\right)f(\zeta ,\eta )+\zeta \frac{\partial f}{\partial \zeta }+\left(\frac{m-1}{2}\right)\eta \frac{\partial f}{\partial \eta }].}\end{array}$$

(15)

By using equation (15) for equations (9)–(14), we derived the subsequent dimensionless system:

$$\begin{array}{c}\left(\frac{{\mu }_{\mathrm{THNF}}}{{\rho }_{\mathrm{THNF}}}\right)\left(\frac{m+1}{2}\right)\frac{{\partial }^{3}f}{\partial {\eta }^3}+\left(\frac{m+1}{2}\right)f\frac{{\partial }^{2}f}{\partial {\eta }^2}-m{\left(\frac{\partial f}{\partial \eta }\right)}^2+m-\zeta (\frac{\partial f}{\partial \eta }\frac{{\partial }^{2}f}{\partial \eta \partial \zeta }-\frac{\partial f}{\partial \zeta }\frac{{\partial }^{2}f}{\partial {\eta }^2})-\hfill \\ \left(\frac{{\mu }_{\mathrm{THNF}}}{{\rho }_{\mathrm{THNF}}}\right)r_1{\zeta }^{1-m}(\frac{\partial f}{\partial \eta }-1)+{\zeta }^{2m-1}{G}_r\sin \left(\frac{\omega }{2}\right)\left[{\left({\beta }_o\right)}_{\mathrm{THNF}}(\theta +\frac{{\left({\beta }_1\right)}_{\mathrm{THNF}}}{{\left({\beta }_o\right)}_{\mathrm{THNF}}}{G}_R{\theta }^2)\right]\hfill \\ -\left(\frac{{\sigma }_{\mathrm{THNF}}}{{\rho }_{\mathrm{THNF}}}\right)M{\zeta }^{1-m}(\frac{\partial f}{\partial \eta }-1)=0,\hfill \end{array}$$

(16)

$$\begin{array}{c}\left(\frac{k_{\mathrm{THNF}}}{{\left(\rho C_p\right)}_{\mathrm{THNF}}}\right)\left(\frac{1}{\mathrm{Pr}}\right)\frac{{\partial }^2\theta }{\partial {\eta }^2}+\left(\frac{m+1}{2}\right)f\frac{\partial \theta }{\partial \eta }-\zeta (\frac{\partial f}{\partial \eta }\frac{\partial \theta }{\partial \zeta }-\frac{\partial f}{\partial \zeta }\frac{\partial \theta }{\partial \eta })+\hfill \\ \left(\frac{{\mu }_{\mathrm{THNF}}}{{\left(\rho C_p\right)}_{\mathrm{THNF}}}\right)\left(\frac{m+1}{2}\right){\zeta }^{2m}\mathrm{Ec}{\left(\frac{{\partial }^{2}f}{\partial {\eta }^2}\right)}^2=0,\hfill \end{array}$$

(17)

$$\begin{array}{c}\left(\frac{m+1}{2}\right)\left(\frac{1}{\mathrm{Sc}}\right)\frac{{\partial }^2{h}_c}{\partial {\eta }^2}+\left(\frac{m+1}{2}\right)f\frac{\partial h_c}{\partial \eta }-\zeta (\frac{\partial f}{\partial \eta }\frac{\partial h_c}{\partial \zeta }-\frac{\partial f}{\partial \zeta }\frac{\partial h_c}{\partial \eta })\hfill \\ -k_{\mathrm{HO}}{\zeta }^{1-m}\left(h_c{h}_d^2\right)-k_{\mathrm{sv}}{\zeta }^{1-m}{h}_c=0,\hfill \end{array}$$

(18)

$$\begin{array}{c}\left(\frac{m+1}{2}\right)\left(\frac{\Upsilon }{\mathrm{Sc}}\right)\frac{{\partial }^2{h}_d}{\partial {\eta }^2}+\left(\frac{m+1}{2}\right)f\frac{\partial h_d}{\partial \eta }-\zeta (\frac{\partial f}{\partial \eta }\frac{\partial h_d}{\partial \zeta }-\frac{\partial f}{\partial \zeta }\frac{\partial h_d}{\partial \eta })\hfill \\ +k_{\mathrm{HO}}{\zeta }^{1-m}\left(h_c{h}_d^2\right)+k_{\mathrm{sv}}{\zeta }^{1-m}{h}_c=0,\hfill \end{array}$$

(19)

with boundary constraints

$$\frac{\partial f}{\partial \eta }(\zeta ,0)=\delta ,\left(\frac{m+1}{2}\right)f(\zeta ,0)+\zeta \frac{\partial f}{\partial \zeta }(\zeta ,0)=0,\left(\frac{k_{\mathrm{THNF}}}{k_F}\right)\frac{\partial \theta }{\partial \eta }=-\sqrt{\frac{2}{m+1}}{\zeta }^{\frac{1-m}{2}}{B}_i(1+\theta (\zeta ,0)),$$

$$\frac{\partial h_c}{\partial \eta }=\sqrt{\frac{2}{m+1}}{\zeta }^{\frac{1-m}{2}}{\mathrm{k}}_{\mathrm{HE}}{h}_c,\Upsilon \frac{\partial h_d}{\partial \eta }=-\sqrt{\frac{2}{m+1}}{\zeta }^{\frac{1-m}{2}}{\mathrm{k}}_{\mathrm{HE}}{h}_c,$$

$$\frac{\partial f}{\partial \eta }(\zeta ,\infty )\to 1,\theta (\zeta ,\infty )\to 0,h_c(\zeta ,\infty )\to 1,h_d(\zeta ,\infty )\to 0.$$

(20)

Further, considering the fact that both species diffuse in the same medium, ³⁸ implies $D_c=D_d,$ and $\Upsilon =1,$ we get

$$\begin{array}{c}\left(\frac{m+1}{2}\right)\left(\frac{1}{\mathrm{Sc}}\right)\frac{{\partial }^2{h}_c}{\partial {\eta }^2}+\left(\frac{m+1}{2}\right)f\frac{\partial h_c}{\partial \eta }-\zeta (\frac{\partial f}{\partial \eta }\frac{\partial h_c}{\partial \zeta }-\frac{\partial f}{\partial \zeta }\frac{\partial h_c}{\partial \eta })\hfill \\ -k_{\mathrm{HO}}{\zeta }^{1-m}{h}_c{(1-h_c)}^2-k_{\mathrm{sv}}{\zeta }^{1-m}{h}_c=0,\hfill \end{array}$$

(21)

with modified wall constraints

$$\frac{\partial h_c}{\partial \eta }=\sqrt{\frac{2}{m+1}}{\zeta }^{\frac{1-m}{2}}{k}_{\mathrm{HE}}{h}_c,h_c(\zeta ,\infty )\to 1.$$

(22)

First level of truncation

Upon the first truncation, ²⁵ the expressions with $\zeta \frac{\partial (.)}{\partial \zeta }$ are neglected in the equations (16)–(20).

$$\begin{array}{c}\left(\frac{{\mu }_{\mathrm{THNF}}}{{\rho }_{\mathrm{THNF}}}\right)\left(\frac{m+1}{2}\right)\frac{{\partial }^{3}f}{\partial {\eta }^3}+\left(\frac{m+1}{2}\right)f\frac{{\partial }^{2}f}{\partial {\eta }^2}-m{\left(\frac{\partial f}{\partial \eta }\right)}^2+m-\left(\frac{{\mu }_{\mathrm{THNF}}}{{\rho }_{\mathrm{THNF}}}\right)r_1{\zeta }^{1-m}(\frac{\partial f}{\partial \eta }-1)+\hfill \\ {\zeta }^{2m-1}{G}_r\sin \left(\frac{\omega }{2}\right)\left[{\left({\beta }_o\right)}_{\mathrm{THNF}}(\theta +\frac{{\left({\beta }_1\right)}_{\mathrm{THNF}}}{{\left({\beta }_o\right)}_{\mathrm{THNF}}}{G}_R{\theta }^2)\right]-\left(\frac{{\sigma }_{\mathrm{THNF}}}{{\rho }_{\mathrm{THNF}}}\right)M{\zeta }^{1-m}(\frac{\partial f}{\partial \eta }-1)=0,\hfill \end{array}$$

(23)

$$\left(\frac{k_{\mathrm{THNF}}}{{\left(\rho C_p\right)}_{\mathrm{THNF}}}\right)\left(\frac{1}{\mathrm{Pr}}\right)\frac{{\partial }^2\theta }{\partial {\eta }^2}+\left(\frac{m+1}{2}\right)f\frac{\partial \theta }{\partial \eta }+\left(\frac{{\mu }_{\mathrm{THNF}}}{{\left(\rho C_p\right)}_{\mathrm{THNF}}}\right)\left(\frac{m+1}{2}\right){\zeta }^{2m}\mathrm{Ec}{\left(\frac{{\partial }^{2}f}{\partial {\eta }^2}\right)}^2=0,$$

(24)

$$\left(\frac{m+1}{2}\right)\left(\frac{1}{\mathrm{Sc}}\right)\frac{{\partial }^2{h}_c}{\partial {\eta }^2}+\left(\frac{m+1}{2}\right)f\frac{\partial h_c}{\partial \eta }-k_{\mathrm{HO}}{\zeta }^{1-m}{h}_c{(1-h_c)}^2-k_{\mathrm{sv}}{\zeta }^{1-m}{h}_c=0,$$

(25)

with the converted boundary conditions as given below:

$$\frac{\partial f}{\partial \eta }(\zeta ,0)=\delta ,\left(\frac{m+1}{2}\right)f(\zeta ,0)=0,\left(\frac{k_{\mathrm{THNF}}}{k_F}\right)\frac{\partial \theta }{\partial \eta }=-\sqrt{\frac{2}{m+1}}{\zeta }^{\frac{1-m}{2}}{B}_i(1+\theta (\zeta ,0)),$$

$$\frac{\partial h_c}{\partial \eta }=\sqrt{\frac{2}{m+1}}{\zeta }^{\frac{1-m}{2}}{k}_{\mathrm{HE}}{h}_c,h_c(\zeta ,\infty )\to 1,\frac{\partial f}{\partial \eta }(\zeta ,\infty )\to 1,\theta (\zeta ,\infty )\to 0.$$

(26)

Second level of truncation

Following the subsequent underlying relations during the truncation process ²¹ :

$$\frac{\partial f}{\partial \zeta }=F,\frac{\partial \theta }{\partial \zeta }=\mathrm{\Theta },\frac{\partial h_c}{\partial \zeta }=R,$$

(27)

we get the subsequent system of equations:

$$\begin{array}{c}\left(\frac{{\mu }_{\mathrm{THNF}}}{{\rho }_{\mathrm{THNF}}}\right)\left(\frac{m+1}{2}\right)\frac{{\partial }^{3}f}{\partial {\eta }^3}+\left(\frac{m+1}{2}\right)f\frac{{\partial }^{2}f}{\partial {\eta }^2}-m{\left(\frac{\partial f}{\partial \eta }\right)}^2+m-\zeta (\frac{\partial f}{\partial \eta }\frac{\partial F}{\partial \eta }-F\frac{{\partial }^{2}f}{\partial {\eta }^2})-\hfill \\ \left(\frac{{\mu }_{\mathrm{THNF}}}{{\rho }_{\mathrm{THNF}}}\right)r_1{\zeta }^{1-m}(\frac{\partial f}{\partial \eta }-1)+{\zeta }^{2m-1}{G}_r\sin \left(\frac{\omega }{2}\right)\left[{\left({\beta }_o\right)}_{\mathrm{THNF}}(\theta +\frac{{\left({\beta }_1\right)}_{\mathrm{THNF}}}{{\left({\beta }_o\right)}_{\mathrm{THNF}}}{G}_R{\theta }^2)\right]=0,\hfill \end{array}$$

(28)

$$\begin{array}{c}\left(\frac{k_{\mathrm{THNF}}}{{\left(\rho C_p\right)}_{\mathrm{THNF}}}\right)\left(\frac{1}{\mathrm{Pr}}\right)\frac{{\partial }^2\theta }{\partial {\eta }^2}+\left(\frac{m+1}{2}\right)f\frac{\partial \theta }{\partial \eta }-\zeta (\frac{\partial f}{\partial \eta }\mathrm{\Theta }-F\frac{\partial \theta }{\partial \eta })+\left(\frac{{\mu }_{\mathrm{THNF}}}{{\left(\rho C_p\right)}_{\mathrm{THNF}}}\right)\left(\frac{m+1}{2}\right)\hfill \\ {\zeta }^{2m}\mathrm{Ec}{\left(\frac{{\partial }^{2}f}{\partial {\eta }^2}\right)}^2=0,\hfill \end{array}$$

(29)

$$\begin{array}{c}\left(\frac{m+1}{2}\right)\left(\frac{1}{\mathrm{Sc}}\right)\frac{{\partial }^2{h}_c}{\partial {\eta }^2}+\left(\frac{m+1}{2}\right)f\frac{\partial h_c}{\partial \eta }-\zeta (\frac{\partial f}{\partial \eta }R-F\frac{\partial h_c}{\partial \eta })-k_{\mathrm{HO}}{\zeta }^{1-m}{h}_c{(1-h_c)}^2\hfill \\ -k_{\mathrm{sv}}{\zeta }^{1-m}{h}_c=0,\hfill \end{array}$$

(30)

with transformed boundary constraints

$$\frac{\partial f}{\partial \eta }(\zeta ,0)=\delta ,\left(\frac{m+1}{2}\right)f(\zeta ,0)+\zeta F(\zeta ,0)=0,\left(\frac{k_{\mathrm{THNF}}}{k_F}\right)\frac{\partial \theta }{\partial \eta }=-\sqrt{\frac{2}{m+1}}{\zeta }^{\frac{1-m}{2}}{B}_i(1+\theta (\zeta ,0)),$$

$$\frac{\partial h_c}{\partial \eta }=\sqrt{\frac{2}{m+1}}{\zeta }^{\frac{1-m}{2}}{k}_{\mathrm{HE}}{h}_c,h_c(\zeta ,0)\to 1,\frac{\partial f}{\partial \eta }(\zeta ,\infty )\to 1,\theta (\zeta ,\infty )\to 0.$$

(31)

Now by taking the derivative of equations (27)–(30), we get

$$\begin{array}{c}\left(\frac{{\mu }_{\mathrm{THNF}}}{{\rho }_{\mathrm{THNF}}}\right)\left(\frac{m+1}{2}\right)\frac{{\partial }^{3}F}{\partial {\eta }^3}+\left(\frac{m+1}{2}\right)[F\frac{{\partial }^{2}f}{\partial {\eta }^2}+f\frac{{\partial }^{2}F}{\partial {\eta }^2}]-2m\left(\frac{\partial f}{\partial \eta }\right)\left(\frac{\partial F}{\partial \eta }\right)-\left(\frac{\partial f}{\partial \eta }\frac{\partial F}{\partial \eta }\right)-\hfill \\ \left(F\frac{{\partial }^{2}f}{\partial {\eta }^2}\right)-\zeta [{\left(\frac{\partial F}{\partial \eta }\right)}^2+F\frac{{\partial }^{2}F}{\partial {\eta }^2}]-\left(\frac{{\mu }_{\mathrm{THNF}}}{{\rho }_{\mathrm{THNF}}}\right)r_1[(1-m){\zeta }^{-m}(\frac{\partial f}{\partial \eta }-1)+{\zeta }^{1-m}\left(\frac{\partial F}{\partial \eta }\right)]+\hfill \\ +G_r\sin \left(\frac{\omega }{2}\right)\left(\begin{array}{c}\left[(2m-1){\zeta }^{2m}{\left({\beta }_o\right)}_{\mathrm{THNF}}(\theta +\frac{{\left({\beta }_1\right)}_{\mathrm{THNF}}}{{\left({\beta }_o\right)}_{\mathrm{THNF}}}{G}_R{\theta }^2)\right]+\hfill \\ \multicolumn{1}{c}{{\zeta }^{2m-1}{\left({\beta }_o\right)}_{\mathrm{THNF}}(\mathrm{\Theta }+2\frac{{\left({\beta }_1\right)}_{\mathrm{THNF}}}{{\left({\beta }_o\right)}_{\mathrm{THNF}}}{G}_R\theta \mathrm{\Theta })}\end{array}\right)=0,\hfill \end{array}$$

(32)

$$\begin{array}{c}\left(\frac{k_{\mathrm{THNF}}}{{\left(\rho C_p\right)}_{\mathrm{THNF}}}\right)\left(\frac{1}{\mathrm{Pr}}\right)\left(\frac{m+1}{2}\right)\frac{{\partial }^2\mathrm{\Theta }}{\partial {\eta }^2}+\left(\frac{m+1}{2}\right)(F\frac{\partial \theta }{\partial \eta }+f\frac{\partial \mathrm{\Theta }}{\partial \eta })-(\frac{\partial f}{\partial \eta }\mathrm{\Theta }-F\frac{\partial \theta }{\partial \eta })\hfill \\ -\zeta (\frac{\partial F}{\partial \eta }\mathrm{\Theta }-F\frac{\partial \mathrm{\Theta }}{\partial \eta })+\left(\frac{{\mu }_{\mathrm{THNF}}}{{\left(\rho C_p\right)}_{\mathrm{THNF}}}\right)\mathrm{Ec}(m+1)\left[\begin{array}{c}\left(m\right){\zeta }^{2m-1}{\left(\frac{{\partial }^{2}f}{\partial {\eta }^2}\right)}^2+\hfill \\ \multicolumn{1}{c}{{\zeta }^{2m}\left(\frac{{\partial }^{2}f}{\partial {\eta }^2}\right)\left(\frac{{\partial }^{2}F}{\partial {\eta }^2}\right)}\end{array}\right]=0,\hfill \end{array}$$

(33)

$$\begin{array}{c}\left(\frac{1}{\mathrm{Sc}}\right)\left(\frac{m+1}{2}\right)\frac{{\partial }^{2}R}{\partial {\eta }^2}+\left(\frac{m+1}{2}\right)(F\frac{\partial h_c}{\partial \eta }+f\frac{\partial R}{\partial \eta })-(\frac{\partial f}{\partial \eta }R-F\frac{\partial h_c}{\partial \eta })-\hfill \\ \zeta (R\frac{\partial F}{\partial \eta }-F\frac{\partial R}{\partial \eta })-k_{\mathrm{HO}}(1-h_c)\left[\begin{array}{c}(1-m){\zeta }^{-m}{h}_c(1-h_c)+\hfill \\ \multicolumn{1}{c}{{\zeta }^{1-m}(R(1-h_c)-2h_{c}R)}\end{array}\right]\hfill \\ -k_{\mathrm{sv}}[\left({\zeta }^{1-m}\right)R+(1-m){\zeta }^{-m}{h}_c]=0,\hfill \end{array}$$

(34)

with the boundary constraints given as follows:

$$\begin{array}{c}\frac{\partial F}{\partial \eta }(\zeta ,0)=0,F(\zeta ,0)=0,\frac{\partial F}{\partial \eta }(\zeta ,\infty )\to 0,F(\zeta ,\infty )\to 0,\mathrm{\Theta }(\zeta ,\infty )\to 0,\hfill \\ \left(\frac{k_{\mathrm{THNF}}}{k_F}\right)\frac{\partial \mathrm{\Theta }}{\partial \eta }(\zeta ,0)=-B_i\sqrt{\frac{2}{m+1}}[\left(\frac{1-m}{2}\right){\zeta }^{-\left(\frac{m+1}{2}\right)}(1+\theta (\zeta ,0))-{\zeta }^{\left(\frac{1-m}{2}\right)}\mathrm{\Theta }(\zeta ,0)],\hfill \\ \frac{\partial R}{\partial \eta }=k_{\mathrm{HE}}\sqrt{\frac{2}{m+1}}[{\zeta }^{\left(\frac{1-m}{2}\right)}R(\zeta ,0)+\left(\frac{1-m}{2}\right){\zeta }^{-\left(\frac{m+1}{2}\right)}{h}_c(\zeta ,0)],R(\zeta ,\infty )\to 0.\hfill \end{array}$$

(35)

The dimensionless parameters corresponding to the aforementioned set of equations are provided below:

$$\begin{array}{c}\multicolumn{1}{c}{G_r=\frac{T_{\infty }g{\left({\beta }_o\right)}_F}{U_{\infty }^2{L}^{2m-1}},G_R=\frac{T_{\infty }{\left({\beta }_1\right)}_F}{{\left({\beta }_o\right)}_F},r_1=\frac{{\mu }_F{L}^{1-m}}{{\rho }_F{U}_{\infty }\overline{k}},\mathrm{Ec}=\frac{{\left(U_{\infty }{L}^m\right)}^2}{{\left(C_p\right)}_F{T}_{\infty }},\mathrm{Pr}=\frac{{\left(\rho C_p\right)}_F{\nu }_F}{k_F},k_{\mathrm{HO}}=\frac{K^{*}{H}_o^2{L}^{1-m}}{U_{\infty }},}\\ \multicolumn{1}{c}{k_{\mathrm{sv}}=K_s{K}_{\mathrm{HE}},K_s=\frac{S{D}_c{L}^{\frac{1-m}{2}}}{\sqrt{U_{\infty }{v}_F}},K_{\mathrm{HE}}=\frac{K_s\sqrt{v_F}{L}^{\frac{1-m}{2}}}{D_c\sqrt{U_{\infty }}},\Upsilon =\frac{D_c}{D_d},B_i=\frac{h_s}{k_F}\sqrt{\frac{L^{1-m}{\nu }_F}{U_{\infty }}},\delta =\frac{U_w}{U_{\infty }},\mathrm{Sc}=\frac{{\nu }_F}{D_c},\mathrm{Re}=\frac{U_{\infty }{L}^{m+1}}{{\nu }_F}.}\end{array}$$

(36)

Quantities of physical interest

The surface drag coefficient $C_f\sqrt{\mathrm{Re}}$ and heat transfer rate $N_u/\sqrt{\mathrm{Re}}$ are computed as follows:

$$C_f=\frac{{\tau }_s}{{\rho }_F{U}_e^2\left(X\right)},N_u=\frac{X{q}_s}{k_F(T-T_{\infty })},$$

(37)

with

$${\tau }_s={\mu }_{\mathrm{THNF}}\left(\frac{\partial U}{\partial Y}\right),q_s=-k_{\mathrm{THNF}}\left(\frac{\partial T}{\partial Y}\right).$$

(38)

The dimensionless forms of the above equations are:

$$C_f\sqrt{\mathrm{Re}}=\frac{{\mu }_{\mathrm{THNF}}}{{\mu }_F}\sqrt{\frac{m+1}{2}}{\zeta }^{-\left(\frac{m+1}{2}\right)}\frac{{\partial }^{2}f}{\partial {\eta }^2},N_u/\sqrt{\mathrm{Re}}=\left(\frac{k_{\mathrm{THNF}}}{k_F}\right)B_i[1+\frac{1}{\theta \left(0\right)}].$$

(39)

Thermal and physical features of ternary-hybrid nanofluid flow

These features are as follows^51–53:

• (i) Viscosity and density:

$$\begin{array}{c}\multicolumn{1}{c}{{\mu }_{\mathrm{HNF}}={\mu }_F/{(1-{\phi }_{s1})}^{2.5}{(1-{\phi }_{s2})}^{2.5},}\\ \multicolumn{1}{c}{{\mu }_{\mathrm{THNF}}={\mu }_F/{(1-{\phi }_{s1})}^{2.5}{(1-{\phi }_{s2})}^{2.5}{(1-{\phi }_{s3})}^{2.5},}\\ \multicolumn{1}{c}{\frac{{\rho }_{\mathrm{HNF}}}{{\rho }_F}=(1-{\phi }_{s2})((1-{\phi }_{s1})+{\phi }_{s1}\frac{{\rho }_{s1}}{{\rho }_F})+{\phi }_{s2}\frac{{\rho }_{s2}}{{\rho }_F},}\end{array}$$

$$\frac{{\rho }_{\mathrm{THNF}}}{{\rho }_F}=(1-{\phi }_{s3})\left((1-{\phi }_{s2})[(1-{\phi }_{s1})+{\phi }_{s1}\frac{{\rho }_{s1}}{{\rho }_F}]\right)+{\phi }_{s2}\frac{{\rho }_{s2}}{{\rho }_F}+{\phi }_{s3}\frac{{\rho }_{s3}}{{\rho }_F}.$$

(40)

• (ii) Specific heat capacity:

$$\frac{{\left(\rho C_p\right)}_{\mathrm{HNF}}}{{\left(\rho C_p\right)}_F}=(1-{\phi }_{s2})((1-{\phi }_{s1})+{\phi }_{s1}\frac{{\left(\rho C_p\right)}_{s1}}{{\left(\rho C_p\right)}_F})+{\phi }_{s2}\frac{{\left(\rho C_p\right)}_{s2}}{{\left(\rho C_p\right)}_F},$$

$$\frac{{\left(\rho C_p\right)}_{\mathrm{THNF}}}{{\left(\rho C_p\right)}_F}=(1-{\phi }_{s3})\left((1-{\phi }_{s2})[(1-{\phi }_{s1})+{\phi }_{s1}\frac{{\left(\rho C_p\right)}_{s1}}{{\left(\rho C_p\right)}_F}]\right)+{\phi }_{s2}\frac{{\left(\rho C_p\right)}_{s2}}{{\left(\rho C_p\right)}_F}+{\phi }_{s3}\frac{{\left(\rho C_p\right)}_{s3}}{{\left(\rho C_p\right)}_F},$$

(41)

• (iii) Thermal expansion coefficients:

$$\frac{{\left(\rho {\beta }_o\right)}_{\mathrm{HNF}}}{{\left(\rho {\beta }_o\right)}_F}=(1-{\phi }_{s2})((1-{\phi }_{s1})+{\phi }_{s1}\frac{{\left(\rho {\beta }_o\right)}_{s1}}{{\left(\rho {\beta }_o\right)}_F})+{\phi }_{s2}\frac{{\left(\rho {\beta }_o\right)}_{s2}}{{\left(\rho {\beta }_o\right)}_F},$$

$$\frac{{\left(\rho {\beta }_o\right)}_{\mathrm{THNF}}}{{\left(\rho {\beta }_o\right)}_F}=(1-{\phi }_{s3})\left((1-{\phi }_{s2})[(1-{\phi }_{s1})+{\phi }_{s1}\frac{{\left(\rho {\beta }_o\right)}_{s1}}{{\left(\rho {\beta }_o\right)}_F}]\right)+{\phi }_{s2}\frac{{\left(\rho {\beta }_o\right)}_{s2}}{{\left(\rho {\beta }_o\right)}_F}+{\phi }_{s3}\frac{{\left(\rho {\beta }_o\right)}_{s3}}{{\left(\rho {\beta }_o\right)}_F},$$

$$\frac{{\left(\rho {\beta }_1\right)}_{\mathrm{HNF}}}{{\left(\rho {\beta }_1\right)}_F}=(1-{\phi }_{s2})((1-{\phi }_{s1})+{\phi }_{s1}\frac{{\left(\rho {\beta }_1\right)}_{s1}}{{\left(\rho {\beta }_1\right)}_F})+{\phi }_{s2}\frac{{\left(\rho {\beta }_1\right)}_{s2}}{{\left(\rho {\beta }_1\right)}_F},$$

$$\frac{{\left(\rho {\beta }_1\right)}_{\mathrm{THNF}}}{{\left(\rho {\beta }_1\right)}_F}=(1-{\phi }_{s3})\left((1-{\phi }_{s2})[(1-{\phi }_{s1})+{\phi }_{s1}\frac{{\left(\rho {\beta }_1\right)}_{s1}}{{\left(\rho {\beta }_1\right)}_F}]\right)+{\phi }_{s2}\frac{{\left(\rho {\beta }_1\right)}_{s2}}{{\left(\rho {\beta }_1\right)}_F}+{\phi }_{s3}\frac{{\left(\rho {\beta }_1\right)}_{s3}}{{\left(\rho {\beta }_1\right)}_F},$$

(42)

• (iv) Thermal conductivity:

$$\begin{array}{c}\frac{k_{\mathrm{THNF}}}{k_{\mathrm{HNF}}}=\frac{k_{s3}+2k_{\mathrm{HNF}}-2{\phi }_{s3}(k_{\mathrm{HNF}}-k_{s3})}{k_{s3}+2k_{\mathrm{HNF}}+{\phi }_{s3}(k_{\mathrm{HNF}}-k_{s3})},\hfill \\ \frac{k_{\mathrm{HNF}}}{k_{\mathrm{NF}}}=\frac{k_{s2}+2k_{\mathrm{NF}}-2{\phi }_{s2}(k_{\mathrm{NF}}-k_{s2})}{k_{s2}+2k_{\mathrm{NF}}+{\phi }_{s2}(k_{\mathrm{NF}}-k_{s2})},\hfill \\ \frac{k_{\mathrm{NF}}}{k_F}=\frac{k_{s1}+2k_F-2{\phi }_{s1}(k_F-k_{s1})}{k_{s1}+2k_F+{\phi }_{s1}(k_F-k_{s1})},\hfill \end{array}$$

(43)

The thermophysical characteristics of the aforementioned traits are shown in Table 2.

Table 2.

Thermal and physical attributes are listed as follows.^54–57

Base liquid/nanoparticles	$\left(\rho \right)$	$\left(k\right)$	$\left(C_p\right)$	$\left({\beta }_{o,1}\right)$
Ethylene glycol	1115	0.253	2430	$11\times {10}^5$
MgO (Magnesium oxide)	3580	45	955	$3.36\times {10}^{-5}$
ZnO (Zinc oxide)	5600	147	514	$3.02\times {10}^{-5}$
GO (Graphene oxide)	1800	5000	717	$28.4\times {10}^{-5}$

Numerical computation

To investigate the findings of the velocity, temperature, and concentration distributions, a numerical solution is built in MATLAB using the bvp4c algorithm. In this approach, a finite difference scheme is used, which is a method of collocation with the order four. Numerical outcomes obtained for the current problem are restricted to a tolerance of ${10}^{-6}$ . In this method, the decision of ${\eta }_{\infty }=6$ , guarantees that every numerical solution is asymptotic. Using the factors shown below, the system of PDEs is converted into first-order ODEs and equations (26)–(33) become

$$\begin{array}{c}{y}_1\to f,y_2\to f\prime ,y_3\to f″,y{y}_1\to f″\prime ,\hfill \\ y{y}_1=\left(\frac{B}{A}\right)\left(\frac{2}{m+1}\right)\times \left[\begin{array}{c}m{\left(y_2\right)}^2-\left(\frac{m+1}{2}\right)y_1{y}_3-m+\zeta (y_2{y}_5-y_4{y}_3)\hfill \\ \multicolumn{1}{c}{+A{r}_1{\zeta }^{1-m}(y_2-1)-C{\zeta }^{1-2m}{G}_r\sin \left(\frac{\omega }{2}\right)\times }\\ \multicolumn{1}{c}{(y_7+\left(\frac{D}{C}\right)G_R{\left(y_7\right)}^2)}\end{array}\right],\hfill \end{array}$$

$$\begin{array}{c}{y}_4\to F,y_5\to F\prime ,y_6\to F″,y{y}_2\to F″\prime ,\hfill \\ y{y}_2=\left(\frac{B}{A}\right)\left(\frac{2}{m+1}\right)\times \left[\begin{array}{c}-\left(\frac{m+1}{2}\right)(y_4{y}_3+y_1{y}_6)+(1+2m)y_2{y}_5-y_4{y}_3\hfill \\ \multicolumn{1}{c}{+\zeta ({\left(y_5\right)}^2-y_4{y}_6)+A{r}_1((1-m){\zeta }^{-m}(y_2-1)+{\zeta }^{1-m}{y}_5)}\\ \multicolumn{1}{c}{-G_r\sin \left(\frac{\omega }{2}\right)C\left(\begin{array}{c}(2m-1){\zeta }^{2m}(y_7+\left(\frac{D}{C}\right)G_R{\left(y_7\right)}^2)\hfill \\ \multicolumn{1}{c}{+{\zeta }^{1-2m}(y_9+2\left(\frac{D}{C}\right)G_R{y}_7{y}_9)}\end{array}\right)}\end{array}\right],\hfill \end{array}$$

$$\begin{array}{c}{y}_7\to \theta ,y_8\to \theta \prime ,y{y}_3\to \theta ″,\hfill \\ y{y}_3=\mathrm{Pr}\left(\frac{E}{F}\right)\left(\frac{2}{m+1}\right)\times \left[\begin{array}{c}-\left(\frac{m+1}{2}\right)y_1{y}_8+\zeta (y_2{y}_9-y_4{y}_8)-\left(\frac{A}{E}\right)\left(\frac{m+1}{2}\right)\hfill \\ \multicolumn{1}{c}{\times \mathrm{Ec}{\zeta }^{2m}{\left(y_3\right)}^2}\end{array}\right],\hfill \end{array}$$

$$\begin{array}{c}{y}_9\to \mathrm{\Theta },y_{10}\to \mathrm{\Theta }\prime ,y{y}_4\to \mathrm{\Theta }″,\hfill \\ y{y}_4=\mathrm{Pr}\left(\frac{E}{F}\right)\left(\frac{2}{m+1}\right)\times \left[\begin{array}{c}-\left(\frac{m+1}{2}\right)(y_4{y}_8+y_1{y}_{10})+(y_2{y}_9-y_4{y}_8)+\zeta \hfill \\ \multicolumn{1}{c}{\times (y_5{y}_9-y_4{y}_{10})-\left(\frac{A}{E}\right)(m+1)}\\ \multicolumn{1}{c}{\mathrm{Ec}(m{\zeta }^{2m-1}{\left(y_3\right)}^2+(m+1){\zeta }^{2m}{y}_3{y}_6)}\end{array}\right],\hfill \end{array}$$

$$\begin{array}{c}{y}_{11}\to h_c,y_{12}\to {h\prime }_c,y{y}_5\to {h″}_c,\hfill \\ y{y}_5=\mathrm{Sc}\left(\frac{2}{m+1}\right)\times \left[\begin{array}{c}-\left(\frac{m+1}{2}\right)\left(y_1{y}_{12}\right)+\zeta (y_2{y}_{13}-y_4{y}_{12})+k_{\mathrm{sv}}{\zeta }^{1-m}{y}_{11}\hfill \\ \multicolumn{1}{c}{+k_{\mathrm{HO}}{\zeta }^{1-m}{y}_{11}{(1-y_{11})}^2}\end{array}\right],\hfill \end{array}$$

$$\begin{array}{c}{y}_{13}\to R,y_{14}\to R\prime ,y{y}_6\to R″\hfill \\ y{y}_6=\mathrm{Sc}\left(\frac{2}{m+1}\right)\times \left[\begin{array}{c}-\left(\frac{m+1}{2}\right)(y_4{y}_{12}+y_1{y}_{14})+(y_2{y}_{13}-y_4{y}_{12})+\zeta (y_5{y}_{13}-y_4{y}_{14})\hfill \\ \multicolumn{1}{c}{+k_{\mathrm{HO}}(1-y_{11})\left(\begin{array}{c}(1-m){\zeta }^{-m}{y}_{11}(1-y_{11})+\hfill \\ \multicolumn{1}{c}{{\zeta }^{1-m}\left(y_{13}\right)(1-y_{11})-2y_{11}{y}_{13}}\end{array}\right)}\\ \multicolumn{1}{c}{+k_{\mathrm{sv}}({\zeta }^{1-m}{y}_{13}+(1-m){\zeta }^{-m}{y}_{11})}\end{array}\right],\hfill \end{array}$$

(44)

having boundary conditions enumerated below:

$$\begin{array}{c}\eta =0:y_2(\zeta ,0)-\delta ,\left(\frac{m+1}{2}\right)y_1(\zeta ,0)+\zeta y_4(\zeta ,0),y_8(\zeta ,0)+\hfill \\ \sqrt{\frac{2}{m+1}}{\zeta }^{\frac{1-m}{2}}{B}_i(1+y_7(\zeta ,0)),y_{12}(\zeta ,0)-k_{\mathrm{HE}}\sqrt{\frac{2}{m+1}}{\zeta }^{\frac{1-m}{2}}{y}_{11}(\zeta ,0),\hfill \\ y_{14}-k_{\mathrm{HE}}\sqrt{\frac{2}{m+1}}({\zeta }^{\frac{1-m}{2}}{y}_{13}(\zeta ,0)+\left(\frac{1-m}{2}\right){\zeta }^{-\left(\frac{m+1}{2}\right)}{y}_{11}(\zeta ,0)),\hfill \\ \eta \to \infty :y_2(\zeta ,\infty )\to 1,y_7(\zeta ,\infty ),y_5(\zeta ,\infty ),y_9(\zeta ,\infty ),\hfill \\ y_{11}(\zeta ,\infty )\to 1,y_{13}(\zeta ,\infty )\to 0.\hfill \end{array}$$

(45)

To handle the boundary value problem for ODEs, the bvp4c is a MATLAB solver that adopts the collocation technique that uses the finite difference method with adaptive mesh refinement. It implies the fourth-order accurate scheme to ensure reliable convergence. Through a residual control mechanism, stability is guaranteed, ensuring the local error within the specified tolerance. In this study, we have followed the same yardstick. Additionally, consistency between solution accuracy and mesh refinement validates the stability and robustness of the numerical outcomes obtained via the bvp4c solver.

Verification of the assumed model

A comparison of the assumed system with previously published work verifies the high accuracy of the mathematical outcomes achieved, hence validating the numerical technique and conclusions. Table 3 portrays a comparison of the skin friction coefficient $C_f\sqrt{\mathrm{Re}}$ with Hussain et al. ⁴⁹ by letting $\delta =\phi =r_1=G_r=$ $G_R=\mathrm{Sc}=B_i=k_{\mathrm{HE}}=0$ for numerous estimates of $m.$

Table 3.

Comparison of $C_f\sqrt{\mathrm{Re}}$ for different estimates of $\left(m\right)$ in a limiting case.

$m$	Hussain et al. 49	Present
0	0.490140	0.490203
1/11	0.662268	0.662149
1/5	0.805224	0.805205
1/3	0.929150	0.929101
1/2	1.039654	1.039600
1	1.232815	1.232790

Analysis of results

In the current section, the graphical and numerical results are discussed. Each graph illustrates the comparison between Ethylene-glycol-based ternary and hybrid nanoliquids. The default ranges are assessed in such a way that they match the convergence requirements $0.1\le \delta \le 0.5,$ $\frac{1}{3}\le m\le \frac{1}{2},\pm 0.5\le G_r\le \pm 1.5,1.5\le r_1\le 5.5,0.1\le \mathrm{Ec}\le 6.1,$ $0.1\le k_{\mathrm{HO}}\le 6.1,0.1\le k_{\mathrm{sv}}\le 0.5,4.0\le B_i\le 4.4,$ $0.030\le {\phi }_{\mathrm{THNF}}\le 0.040$ , and $0.5\le G_R\le 0.7$ . Figure 2 is delineated to present the trend of the velocity profile $f\prime \left(\eta \right)$ against the Falkner-Skan power law parameter $\left(m\right)$ . It is observed that the velocity profile $f\prime \left(\eta \right)$ enhances for the escalating values of $\left(m\right)$ . The Falkner-Skan power law parameter is associated with the Hartree pressure gradient $\left(\epsilon \right)$ . An increment in $\left(m\right)$ signifies a positive pressure gradient that aids the flow, causing fluid velocity to rise. Figure 3 demonstrates the consequences of the velocity ratio parameter $\left(\delta \right)$ over the velocity distribution. It is inferred that mounting values of $\left(\delta \right)$ promote the velocity profile $f\prime \left(\eta \right)$ more significantly for hybrid nanofluid than for ternary nanofluid. It is due to the definition $\delta =\frac{U_w}{U_{\infty }},$ that is, the ratio of stretching velocity to the ambient. When $\left(\delta \right)$ increases, the stretching velocity surges faster than the ambient, which consequently upsurges the velocity profile. Figure 4 reports that for rising positive values of the thermal convection parameter $(G_r>0),$ the velocity profile diminishes. Moreover, Figure 5 is displayed to analyze the consequence of negative estimates of the thermal convection parameter $(G_r<0)$ against the velocity distribution. It is seen that velocity distribution is enhanced for the numerous estimates of $(G_r<0)$ . It is also demonstrated that the velocity distribution of ternary nanofluid is more dominant than that of hybrid nanoliquid. Figure 6 portrays the consequences of the porosity parameter $\left(r_1\right)$ against the velocity profile $f\prime \left(\eta \right)$ . It is revealed from the graph that $f\prime \left(\eta \right)$ increases against higher estimates of $\left(r_1\right)$ . This is due to the medium’s significant porosity that is, for mounting $\left(r_1\right)$ , the fluid experiences more space to flow and, consequently, its velocity is enhanced. Further, Figure 7 is sketched to illustrate the trend of the temperature profile $\theta \left(\eta \right)$ against the Falkner-Skan power law $\left(m\right)$ . It is delineated that the $\theta \left(\eta \right)$ enhances for the escalating values of $\left(m\right)$ . The prevailing enhancement in $\theta \left(\eta \right)$ owing to varied estimates of $\left(m\right)$ is because of convective condition at the surface of the wedge. Figure 8 reveals the impact of the Eckert number $\left(\mathrm{Ec}\right)$ on the temperature distribution $\theta \left(\eta \right).$ It is recognized that for the escalating estimations of $\left(\mathrm{Ec}\right),$ the temperature distribution upsurges for mounting Eckert number. The reason is that viscous dissipation increments because of frictional heating, which causes the fluid to consume more heat, and thus the temperature of the hybrid nanofluid rises. Figure 9 is sketched to analyze the trend of temperature distribution $\theta \left(\eta \right)$ against the Newtonian heating parameter $B_i$ . It is noted that for higher counts of $B_i,$ the temperature distribution enhanced significantly. The reason for this is that greater Newtonian heating parameter values lead to larger heat transfer coefficients, which raise the fluid temperature. In addition, Figure 10 is displayed to see the effects of the homogeneous parameter $k_{\mathrm{HO}}$ against the concentration profile $h_c\left(\eta \right).$ It is noted that for escalating $k_{\mathrm{HO}},$ the concentration profile is dwindled. This is because increasing the homogeneous parameter elevates the rate of homogeneous reaction in the working fluid, leading to a decline in the concentration of species. Figure 11 is drawn to describe the impact of the surface-catalyzed parameter $k_{\mathrm{sv}}$ against the concentration profile $h_c\left(\eta \right).$ It is shown that the concentration distribution dwindled, logically. The surface-catalyzed reaction rate is enhanced with an advancing reaction interface on porous surfaces, while the concentration of species progressively declines to the lowest at the fixed point. This indicates that a porous medium made of an identical catalyst reduces the reaction time. Figure 12 is displayed to witness the effects of the heterogeneous parameter $k_{\mathrm{HE}}$ against the concentration profile $h_c\left(\eta \right)$ . It is noted that for escalating $k_{\mathrm{HE}},$ the concentration profile dwindled due to the consumed reactant in the reactions. Figure 13 is sketched to elucidate the impact of the Falkner-Skan power law parameter $m$ against the concentration profile $h_c\left(\eta \right)$ . It is noted that the concentration profile diminishes against the increasing values of $m$ . Figure 14 is plotted to compute the role of Schmidt number $\mathrm{Sc}$ over the concentration distribution $h_c\left(\eta \right)$ . It is seen that concentration distribution is diminishes for greater counts of $\mathrm{Sc}$ . The reason for this is that Sc is defined as the kinematic viscosity divided by the diffusion coefficient. When Sc improves species velocity diffusion predominates, causing the reactant particles to accelerate, which tends to reduce the concentration distribution. Figure 15 is sketched to illustrate the role of the Falkner-Skan power law parameter $\left(m\right)$ , the Quadratic convection parameter $\left(G_R\right)$ , over the skin friction coefficient $C_f\sqrt{\mathrm{Re}}$ . It is seen that the skin friction coefficient increases against both parameters. Moreover, Figure 16 is drawn to determine the consequences of the Falkner-Skan power law parameter $\left(m\right)$ , the Quadratic convection parameter $\left(G_R\right)$ , over the Nusselt number $\mathrm{Nu}{\left(\mathrm{Re}\right)}^{-1/2}$ . It is seen that the heat transmission rate increases significantly for ternary nanofluid against both parameters. The numerical values of the surface drag coefficient and heat transfer rate are computed and tabulated in Tables 4 and 5. Table 4 illustrates the consequences of particle volume fraction $\left(\phi \right)$ , Falkner-Skan power law parameter $\left(m\right)$ , Quadratic convection parameter $\left(G_R\right)$ , and Newtonian heating parameter $\left(B_i\right)$ over the surface drag coefficient $C_f\sqrt{\mathrm{Re}}$ . It is revealed from the numerical data values that the surface drag coefficient is more convincing for Ternary nanofluid flow than hybrid nanofluid against higher estimations of $\phi ,m,G_R$ , and $B_i$ . This is a result of the greater nanoparticle counts in ternary nanofluid, which have three distinct nanoparticles, which tend to enhance the surface resistance. Moreover, Table 5 demonstrates the impacts of particle volume fraction $\left(\phi \right)$ , Falkner-Skan power law parameter $\left(m\right)$ , Quadratic convection parameter $\left(G_R\right)$ , and Newtonian heat parameter $\left(B_i\right)$ on the surface heat transfer rate $N_u{\left(\mathrm{Re}\right)}^{\frac{-1}{2}}.$ It is revealed from the results that a higher heat transfer rate is prominent for Ternary nanofluid flow than for hybrid nanofluid against greater values of $\phi ,m,G_R$ and $B_i.$ This is due to the fact that ternary nanofluids, which contain three distinct types of nanoparticles, have complex heat exchange networks that may improve the thermal conductivity.

Figure 2.

Upshot of Falkner-Skan power law parameter $m$ versus $f\prime \left(\eta \right)$ with $\delta =0.1,r_1=1.5,\mathrm{Ec}=0.1,G_r=0.1,k_{\mathrm{HO}}=0.1,k_{\mathrm{sv}}=0.1,B_i=4.0,\phi =0.03,G_R=0.5.$

Figure 3.

Upshot of velocity ratio parameter $\delta$ versus $f\prime \left(\eta \right)$ with $m=\frac{1}{3},r_1=1.5,\mathrm{Ec}=0.1,G_r=0.1,k_{\mathrm{HO}}=0.1,k_{\mathrm{sv}}=0.1,B_i=4.0,\phi =0.03,G_R=0.5.$

Figure 4.

Upshot of thermal convection parameter $(G_r>0)$ versus $f\prime \left(\eta \right)$ with $\delta =0.1,m=\frac{1}{3},r_1=1.5,\mathrm{Ec}=0.1,k_{\mathrm{HO}}=0.1,k_{\mathrm{sv}}=0.1,B_i=4.0,\phi =0.03,G_R=0.5.$

Figure 5.

Upshot of thermal convection parameter $(G_r<0)$ versus $f\prime \left(\eta \right)$ with $\delta =0.1,m=\frac{1}{3},r_1=1.5,\mathrm{Ec}=0.1,k_{\mathrm{HO}}=0.1,k_{\mathrm{sv}}=0.1,B_i=4.0,\phi =0.03,G_R=0.5$ .

Figure 6.

Upshot of porosity parameter $r_1$ versus $f\prime \left(\eta \right)$ with $\delta =0.1,m=\frac{1}{3},\mathrm{Ec}=0.1,G_r=0.1,k_{\mathrm{HO}}=0.1,k_{\mathrm{sv}}=0.1,B_i=4.0,\phi =0.03,G_R=0.5$ .

Figure 7.

Upshot of Falkner–Skan parameter $m$ versus $\theta \left(\eta \right)$ with $\delta =0.1,r_1=1.5,\mathrm{Ec}=0.1,G_r=0.1,k_{\mathrm{HO}}=0.1,k_{\mathrm{sv}}=0.1,B_i=4.0,\phi =0.03,G_R=0.5$ .

Figure 8.

Upshot of Eckert parameter $\mathrm{Ec}$ versus $\theta \left(\eta \right)$ with $\delta =0.1,m=\frac{1}{3},r_1=1.5,G_r=0.1,k_{\mathrm{HO}}=0.1,k_{\mathrm{sv}}=0.1,B_i=4.0,\phi =0.03,G_R=0.5$ .

Figure 9.

Upshot of Newtonian heating parameter $B_i$ versus $\theta \left(\eta \right)$ with $\delta =0.1,m=\frac{1}{3},r_1=1.5,\mathrm{Ec}=0.1,G_r=0.1,k_{\mathrm{HO}}=0.1,k_{\mathrm{sv}}=0.1,\phi =0.03,G_R=0.5$ .

Figure 10.

Upshot of HOM parameter $k_{\mathrm{HO}}$ versus $h_c\left(\eta \right)$ with $\delta =0.1,m=\frac{1}{3},r_1=1.5,\mathrm{Ec}=0.1,G_r=0.1,k_{\mathrm{sv}}=0.1,B_i=4.0,\phi =0.03,G_R=0.5.$

Figure 11.

Upshot of surface catalyzed parameter $k_{\mathrm{sv}}$ versus $h_c\left(\eta \right)$ with $\delta =0.1,m=\frac{1}{3},r_1=1.5,\mathrm{Ec}=0.1,G_R=0.1,k_{\mathrm{HO}}=0.1,B_i=4.0,\phi =0.03,G_R=0.5$ .

Figure 12.

Upshot of HET parameter $k_{\mathrm{HE}}$ versus $h_c\left(\eta \right)$ with $\delta =0.1,m=\frac{1}{3},r_1=1.5,\mathrm{Ec}=0.1,G_r=0.1,k_{\mathrm{HO}}=0.1,k_{\mathrm{sv}}=0.1,B_i=4.0,\phi =0.03,G_R=0.5.$

Figure 13.

Upshot of Falkner–Skan parameter $m$ versus $h_c\left(\eta \right)$ with $\delta =0.1,r_1=1.5,\mathrm{Ec}=0.1,G_r=0.1,k_{\mathrm{HO}}=0.1,k_{\mathrm{sv}}=0.1,B_i=4.0,\phi =0.03,G_R=0.5$ .

Figure 14.

Upshot of Schmidt number $\mathrm{Sc}$ versus $\mathrm{Sc}$ with $\delta =0.1,m=\frac{1}{3},r_1=1.5,\mathrm{Ec}=0.1,G_r=0.1,k_{\mathrm{HO}}=0.1,k_{\mathrm{sv}}=0.1,B_i=4.0,\phi =0.03,G_R=0.5$ .

Figure 15.

Skin friction coefficient against $\left(m\right)$ and $\left(m\right)$ with $\delta =0.1,r_1=1.5,\mathrm{Ec}=0.1,G_r=0.1,k_{\mathrm{HO}}=0.1,k_{\mathrm{sv}}=0.1,B_i=4.0,\phi =0.03$ .

Figure 16.

Nusselt number against $\left(m\right)$ and $\left(G_R\right)$ with $\delta =0.1,r_1=1.5,\mathrm{Ec}=0.1,G_r=0.1,k_{\mathrm{HO}}=0.1,k_{\mathrm{sv}}=0.1,B_i=4.0,\phi =0.03$ .

Table 4.

Estimations for surface drag coefficient $C_f\sqrt{\mathrm{Re}}$ against distinct parameters.

$\phi$	$m$	$G_R$	$\mathrm{Ec}$	$B_i$	Hybrid nanofluid	Ternary nanofluid
					$C_f\sqrt{\mathrm{Re}}$	$C_f\sqrt{\mathrm{Re}}$
0.030	$\frac{1}{3}$	0.5	0.1	4.0	2.11909	2.16392
0.035	-	-	-	-	2.13817	2.18691
0.040	-	-	-	-	2.15757	2.21031
-	$\frac{1}{2.5}$	-	-	-	2.53772	2.59046
-	$\frac{1}{2}$	-	-	-	2.70276	2.75867
-	-	0.6	-	-	2.12618	2.17147
-	-	0.7	-	-	2.13327	2.17903
-	-	-	0.2	-	2.11945	2.16432
-	-	-	0.3	-	2.11984	2.16476
-	-	-	-	4.2	2.11909	2.16391
-	-	-	-	4.4	2.11910	2.16392

Table 5.

Numerical estimates for heat transfer rate $N_u{\left(\mathrm{Re}\right)}^{\frac{-1}{2}}$ against varied parameters.

$\phi$	$m$	$G_R$	$\mathrm{Ec}$	$B_i$	Hybrid nanofluid	Ternary nanofluid
					$N_u{\left(\mathrm{Re}\right)}^{\frac{-1}{2}}$	$N_u{\left(\mathrm{Re}\right)}^{\frac{-1}{2}}$
0.030	$\frac{1}{3}$	0.5	0.1	4.0	0.419089	0.421134
0.035	-	-	-	-	0.419213	0.422094
0.040	-	-	-	-	0.419300	0.423012
-	$\frac{1}{2.5}$	-	-	-	0.424884	0.427597
-	$\frac{1}{2}$	-	-	-	0.425831	0.428752
-	-	0.6	-	-	0.419208	0.421251
-	-	0.7	-	-	0.419328	0.421369
-	-	-	0.2	-	0.443994	0.446502
-	-	-	0.3	-	0.467351	0.470247
-	-	-	-	4.2	0.428714	0.428713
-	-	-	-	4.4	0.434949	0.434950

Conclusions

This study has presented a comparative analysis of hybrid and ternary nanofluid flows over a wedge to enhance thermal transport. It has incorporated viscous dissipation, Newtonian heating, and quadratic thermal convection in a permeable medium. Homogeneous reactions have occurred within the fluid, while heterogeneous reactions have been confined to surface interfaces. Velocity and temperature profiles have been computed using the bvp4c method for various parameters. A non-similar solution to the envisioned model is obtained. The findings reveal that increasing values of $\left(m\right),$ lead to higher velocity and temperature, while the concentration profile decreases. Under a positive pressure gradient, fluid velocity increases, but the boundary layer thickness is reduced. Similarly, the temperature profile increases with the Falkner–Skan parameter, with hybrid nanofluids showing superior performance compared to ternary nanofluids. Additionally, both the velocity ratio and porosity parameters boost the velocity distribution, with this effect being more pronounced in hybrid nanofluids than in ternary nanofluids. Furthermore, higher values of the surface-catalyzed parameter lead to a reduction in the concentration profile. An increase in the Eckert number also raises the temperature profile and enhances the heat transfer rate. In both cases, this upward trend is more significant for the ternary nanofluid. Additionally, the heat transfer rate and wall shear stress increase with rising particle volume fraction $({\phi }_{\mathrm{THNF}}={\phi }_1+{\phi }_2+{\phi }_3)$ , with the effect being more prominent in ternary nanofluids than in hybrid nanofluids. The addition of nanoparticles enhances the effective thermal conductivity of the fluid. As the volume fraction of nanoparticles increases, surface roughness also rises, leading to greater drag. Moreover, the quadratic convection parameter significantly boosts both the wall heat transfer rate and surface drag along the wedge surface. Overall, quadratic convection has a more substantial impact on heat transfer and surface drag compared to linear convection. The quadratic buoyancy force primarily drives vortex formation and turbulence, which enhance heat transfer and mixing, while also contributing to increased surface drag.

Future direction

The anticipated model may be extended as follows:

• The thermal radiation amalgamated with Cattaneo-Christov heat flux effects may be considered in the temperature equation.

• The convective condition may be replaced with some appropriate boundary condition.

• The proposed model may be considered for an applied magnetic field.

Application to the modeled problem

The flow of nanofluids over a wedge closely models high-speed boundary layer behavior over aerodynamic surfaces like nose-cones or wing leading edges. Enhancing heat transfer in such regions is critical for protecting structural components from thermal damage due to high temperatures and shear stresses during atmospheric re-entry or supersonic flight.