Unsteady three-dimensional nodal stagnation point flow of polymer-based ternary-hybrid nanofluid past a stretching surface with suction and heat source

Abstract

As a result of the many real-world applications that may be derived from understanding stagnation point flow in designing, such as the coolant of nuclear reactors, there has been a great deal of interest in the topic. Consequently, the purpose of this research was to offer a numerical analysis of an unstable three-dimensional (3D) nodal stagnation point flow of polymer-based $A{l}_2{O}_3-CuO-Ti{O}_2/polymer$ ternary nanofluid past a stretching surface with mass suction and heat source effects. In order to simplify the underlying partial differential equations, an appropriate similarity transformation is applied to them. This simplifies the ordinary differential equations. The shooting with the Runge-Kutta approach is used by the MATHEMATICA software to do the numerical calculation. Suction, stretching, unsteadiness, heat source, and nanoparticle volume fractions are other elements that play a role in regulating the flow and heat transfer as well as drag force profiles and how they affect the problem. The amount of heat transferred, and the friction coefficient increased in both directions when the suction parameter values were raised. In a ternary-hybrid nanofluid, the overall heat transfer rate decreases as the value of the heat source increases. Variations in the nanoparticles’ volume fraction parameter cause an intensification in skin friction in both directions. Expanding the unstable and nanoparticles volume fraction parameters also reduces the Nusselt number. Furthermore, the heat transfer presentation of ternary-hybrid nanofluid has superior to the hybrid nanofluid and the normal nanofluid for the suction parameter. When the results of the current research were compared to those of a study that had already been done and published, they were found to be in good agreement.

Introduction

Mechanical systems rely heavily on heat transfer, making it a crucial technological and industrial phenomenon. There are several technical and industrial procedures that express the function of heat transfer. These include lamination operations and refrigeration as well as the refinement sector, mini heat pumps, and superior thermal oil.^1,2 Many heavy industries rely on base fluids as their principal source of energy. In contrast, many base liquids have limited heat conductivity and so cannot be employed in industrial processes. The study of nanofluids as a potential source of alternative energy has gained a lot of attention in a fleeting period of time. In terms of potential applications, the thermal dynamics of nanoparticles are very intriguing and groundbreaking. For example, nanoparticles are being employed in exothermic reactions, resource efficiency, motorsport, ejection process, polymer coating, slender solar energy, and oil refining. They are also being used in automobile emission controls and air fresheners as well as biochemical mechanisms.

Over the course of the past several decades, polymer nanoparticles have evolved a wide variety of uses because of their low toxicity and high biocompatibility, both of which make them friendly to organisms. As a result, polymer nanoparticles have become more popular. ³ The control of contamination and the cure of medical ailments are two examples of these applications. Additionally, the surface of polymer nanoparticles is amenable to easy modification, especially when a carboxyl group is present. ⁴ Because of this, it is now possible to change the surface to increase the particle's resistance to the effects of temperature and salt. As a result of this, the polymer nanoparticles were fantastic options for the part of carriers. ⁴

The stagnation point flow has been extensively explored by researchers, with a primary emphasis on its applications in technological fields and industries, including hot forming, processing and production, investment casting, and a variety of others. Using the stagnation point flow field and heat exchange, it is possible to determine whether or not certain commodities are consistent. For the first time, Hiemenz ⁵ proposed the concept of stagnation point flow in 1911. The similarity variables were employed in order to cope with two-dimensional (2D) stagnation point flow, and this ultimately led to the discovery of the correct solution. Following that, Libby ⁶ carried out an investigation into the boundary layer to determine how heat and mass were transferred in a 3D stagnation point flow. Chiam ⁷ disclosed the stagnation point dynamic analysis against a stretched sheet and expanded his work on heat transfer modeling utilizing the regular perturbation approach in stagnation point flow. Both of these contributions were made in the context of the stagnation point flow.

It is essential to include consequences in the flow in order to determine whether or not the effect makes a positive contribution to the flow, thus providing good relevance. One phenomenon that has piqued the curiosity of academics is suction, and they want to learn more about it. Suction may be used to manage the boundary layers and bring about a reduction in potential flow or power dissipation. ⁸ Through the use of this approach, boundary layer separation might be avoided or deferred. To deliver the suction impact, the surface has to be flawed in some way, including fissures, holes, permeability zones, or perforations. Make use of these funnels to extract air from the boundary layer, which travels at a more leisurely pace near the wall. The velocity distribution of the boundary layer is now full and reliable from the perspective of separation as a direct result of this, which occurred as a direct outcome of this. Suction or injection was highlighted as necessary by Hartnett ⁹ as a tool for boundary layer control. In most cases, lowering the coefficient of drag and increasing the elevation values may be accomplished by rapidly inhibiting boundary layer separation. This is the case even though the suction or injection of fluid through a surface may significantly change the flow field, even in the case of mass transfer cooling. Because of the incredible potential of this idea, several researchers in the academic community have taken an active part in research on the benefits of suction.^10,11

When it comes to the management of heat transmission, one of the most essential elements is the heat source/sink impact and it exerts a significant amount of its impact on heat transmission in industrial processes. Because of this, several studies have been conducted to explore the impacts of heat sources and sinks on the flow of nanofluid boundary layers and the transmission of heat from a variety of perspectives.^12,13 Sharma and Gupta's research ¹⁴ shows that the boundary layer flow and heat transfer of a Jeffrey nanofluid can be changed by a number of different things. These factors include heat sources and sinks, magnetic fields, and viscous dissipation. Jamaluddin et al. ¹⁵ conducted research to determine the influence that heat sources and sinks had on the flow of mixed convection nanofluids over a moving surface while thermal radiation was serving as the driving force.

In recent years, hybrid nanofluid computational research has been performed to get a deeper understanding of the characteristics of heat transfer and the degree to which this innovative nanofluid category achieves its thermal efficiency. A base fluid plus two or more solid nanoparticles are the components that go into the creation of hybrid nanofluids in the field of colloids, which are combined to form a colloid. A hybrid nanofluid has proven effective in a wide range of heat transfer procedures to date. Turbine conditioning, structural ventilation and air conditioning, inverter refrigeration, cooling systems, the solvent in soldering, coolant in milling, automotive condensation, and uranium reactor evaporation are just a few of the notable uses. These fluids are more effective than nanofluids in these areas.

Because of the importance of these applications, a significant number of academics have been driven to make use of hybrid nanofluids to deliver problems with heat transmission in the real-world. It was previously shown that a single nanofluid may significantly improve the thermal conductivity of a liquid by Choi. ¹⁶ The nanofluid boundary layer flow pattern has since been the subject of several experimental and basic research. It was discovered by Usman et al. ¹⁷ that growth in the permeability velocity constraint enhanced the Nusselt number while lowering the drag force. Nusselt number and drag force for spinning fluids containing $A{l}_2{O}_3$ nanoparticles were found to be higher than those for other nanoparticles ( $CuO,Cu,Ag,Ti{O}_2$ ). ¹⁸ For technical applications like heat exchangers and heating, ventilation, and air conditioning systems, as well as building insulation, fluid dynamics concerns the flow of a hybrid nanofluid through a cavity. Hybrid nanofluids containing silver and magnesium oxide nanoparticles were studied numerically by Ghalambaz et al. and Mehryan et al.^19,20 Considering stretching/shrinking surfaces to evaluate the hybrid nanofluid's thermal conductivity and dynamic viscosity with different conditions were also investigated by the researchers.^21–23

Researchers developed nanofluids in response to the increased need for cooling agents that could transfer heat more effectively in the industry. As a direct result of this, the hybrid nanofluid was developed, and an improvement in heat transmission was discovered. Experiments on ternary nanofluids are being carried out in the hopes of increasing heat transfer rates. The ternary-hybrid nanofluid suspends nanoparticles of three distinct sorts, each with a unique set of physical and chemical connections. As an example, when sulfuric acid is used to treat $Ti{O}_2$ , it generates a covalently bound surface sulfatase, such as $TiOS{O}_4$ . The reaction between aluminum oxide $A{l}_2{O}_3$ and sulfuric acid results in the formation of surface sulfatase, which is analogous to aluminum ionic ions $A{l}_2(S{O}_4{)}_3$ . Because of the interaction between alumina and sulfuric acid, weaker acid sites are formed. ²⁴ The nanoparticle composition will be stable and chemically inert thanks to the acid sites generated. Copper oxide $CuO$ , another essential nanomaterial, is used as a reagent and combined with a range of resources to enhance mechanical properties. $CuO$ finds widespread use in the fields of transistors, metal catalysts, and thermal performance liquids in manufacturing equipment. ²⁵ The goal of this research is to use $Ti{O}_2$ as a coolant. Consequently, to construct a coolant based on the use of $Ti{O}_2$ , $A{l}_2{O}_3$ , and $CuO$ are used as supporting agents.

The properties of nanostructures and their ability to affect the pace at which heat is transmitted through a host fluid have recently been revealed by scientific innovations about the nanofluids and their possible uses and implications. Thermal stability and permittivity of the $4:4:2$ mass ratio, which creates ternary carbonate nanofluid and carbon nanotubes, are important factors to consider when designing supercritical solar power plants and have been shown to be helpful via thermal study. ²⁶ Copper oxide, magnesium oxide, and titanium oxide were all examined by Mousavi et al. ²⁷ Despite decreasing with temperature, the viscosity of the five different ternary-hybrid nanofluids studied in this work increased with concentration (the amount of nanoparticle volume). As the temperature rises, the ternary-hybrid nanofluids’ density drops linearly. The inclusion of various kinds of nanoparticles may improve the base fluid's specific heat capacity. The studies by Abbasi et al., Animasaun et al., Cao et al., Elnaqeeb et al., Saleem et al., and Xiu et al.^28–33 are related publications on ternary-hybrid nanofluid. The thermal and mechanical properties of a ternary-hybrid nanofluid may be enhanced.^3,34,35

The performance of ternary-hybrid nanofluids has mostly been studied using inorganic and oxide nanoparticles. The authors have not yet investigated the nanofluids that include aluminum oxide $(A{l}_2{O}_3)$ , copper oxide $(CuO)$ , and titanium dioxide $(Ti{O}_2)$ nanoparticles for unsteady stagnation point flow. Research on unsteady 3D nodal stagnation point flow of polymer-based ternary-hybrid nanofluids across a stretching surface with a heat source and mass suction was the inspiration for this study. The focus of this study is purely theoretical. To find out how the ternary-hybrid nanofluid influences the performance of the polymer base fluid, we’ll utilize aluminum $(A{l}_2{O}_3)$ , copper oxide nanoparticles $(CuO)$ , and titanium dioxide $(Ti{O}_2)$ . Suction with a heat source is one of the model's many creative and customizable features. After the models and simulations have been finished, the numerical computations are done using the shooting technique. The flow, thermal heat transfer, and skin frictional sketches of a fluid are strongly influenced by its flow properties. This is being investigated with the help of several different schematics. Because of this, the purpose of this investigation is to provide plausible explanations for the following research questions (Figure 1):

How does friction in both directions and heat transmission throughout the dynamics of polymer-based ternary-hybrid nanofluid of stagnation point flow change as time goes on when the volume of nanoparticles with fixed blade shapes varies?

How does the distribution of velocity and temperature alter when the unsteady stagnation points flow of polymer-based ternary-hybrid nanofluid changes depending on the volume of the blade-shaped particles?

What role does heat transfer throughout the dynamics of polymer-based ternary-hybrid nanofluids play in the increase in instability, suction, and heat source parameters, and is this effect more pronounced for a smaller or larger number of blade-like microscopic particles in a stagnation point flow?

In what way does the rising velocity that relates to the suction influence the local skin friction in both directions below the dynamics of the unsteady nodal stagnation point flow of polymer base ternary-hybrid nanofluid?

Figure 1.

Methodology for producing ternary-hybrid nanoparticle mixes has been devised.

Mathematical modeling

The present study takes into account unstable, incompressible flow close to stagnation points in the 3D flow of ( $A{l}_2{O}_3,CuO,Ti{O}_2$ ) ternary-hybrid nanoparticles with polymer as the base fluid, which is dependent on mass suction and a heat source toward the stretching surface. As shown in Figure 2, the nodal stagnation point N is the origin of the coordinate system, and $u,v$ , and w are the velocity components along the x, y and z axes, respectively. The ambient fluid temperature is denoted by $T_{\mathrm{\infty }}$ . Specifically, $T_w$ is thought to be the constant surface temperature of the sheet.

Figure 2.

Physical flow coordination model and mechanism.

The outer flow is assumed to be ${\tilde{u}}_e=\frac{ax}{1-\alpha t}$ and ${\tilde{v}}_e=\frac{by}{1-\alpha t}$ in x and $y$ axes, respectively. There is an unsteadiness of the issue toward t in this parameter, which is shown by the symbol $\alpha$ . When the flow is constant and inviscid, $\alpha =0$ is used to be it, while $\alpha >0$ is used to increase the potential flow and $\alpha <0$ corresponds to the reverse flow.

It is important to note that $c=\frac{b}{a}$ , where c represents the 3D stagnation points parameter or the ratio of velocity gradients at the edge of the boundary layer, a and b are the major curvatures characteristic at N or the velocity gradients at the boundary layer's edge along the x and y axes.^36,37 Both a and b are positive values for the constant.

In addition, the axisymmetric situation is represented by $b=a$ , whereas the planar stagnation flow issue is represented by $b=0$ . If both a and b are positive, then the solution to the relevant equations will provide nodal points of attachment, which will be represented by the notation $0\le c\le 1$ . On the other hand, the saddle points of adhesion are announced if both a and b are negative, or more specifically, if $-1.0\le c\le 0.0$ . In addition, it is important to notice that the issue will change into a 2D instance when $c=0,$ but that it will return to its axisymmetric form when $c=1$ . ${\tilde{u}}_w$ and ${\tilde{v}}_w$ stands for the velocity of the stretching/shrinking surface. Because the surface is permeable, there is a mass flow velocity, represented by $w_{{}^{\circ }}>0$ , and the plane of the body remains stationary. If $w_{{}^{\circ }}>0$ is more than zero, the situation is one of the injection, and if $w_{{}^{\circ }}>0$ is less than zero, the condition is one of suction. Several assumptions are also investigated for the physical model, including the following:

Flow is laminar.

It was made certain that the temperature of the conventional fluids and the nanomaterials was always maintained in a state of balance.

Since it is thought that the nanofluid is stable, it is not possible for aggregation and sedimentation of nanoparticles to influence the nanofluid.

The nanoparticles have a consistent shape that resembles a blade.

Because of these premises, the boundary layer equations that regulate it may be stated as follows: ^{23, 37,38}

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

(1)

$${\displaystyle \frac{\mathrm{\partial }\tilde{u}}{\mathrm{\partial }\mathrm{t}}+\tilde{u}{\displaystyle \frac{\mathrm{\partial }\tilde{u}}{\mathrm{\partial }\mathrm{x}}+\tilde{v}{\displaystyle \frac{\mathrm{\partial }\tilde{u}}{\mathrm{\partial }\mathrm{y}}+\tilde{w}{\displaystyle \frac{\mathrm{\partial }\tilde{u}}{\mathrm{\partial }\mathrm{z}}={\displaystyle \frac{\mathrm{\partial }{\tilde{u}}_e}{\mathrm{\partial }\mathrm{t}}+{\tilde{u}}_e{\displaystyle \frac{\mathrm{\partial }{\tilde{u}}_e}{\mathrm{\partial }\mathrm{x}}+{\displaystyle \frac{{\mu }_{mnf}}{{\rho }_{mnf}}{\displaystyle \frac{{\mathrm{\partial }}^2\tilde{u}}{\mathrm{\partial }{z}^2},}}}}}}}}$$

(2)

$${\displaystyle \frac{\mathrm{\partial }\tilde{v}}{\mathrm{\partial }\mathrm{t}}+\tilde{u}{\displaystyle \frac{\mathrm{\partial }\tilde{v}}{\mathrm{\partial }\mathrm{x}}+\tilde{v}{\displaystyle \frac{\mathrm{\partial }\tilde{v}}{\mathrm{\partial }\mathrm{y}}+\tilde{w}{\displaystyle \frac{\mathrm{\partial }\tilde{v}}{\mathrm{\partial }\mathrm{z}}={\displaystyle \frac{\mathrm{\partial }{\tilde{v}}_e}{\mathrm{\partial }\mathrm{t}}+{\tilde{v}}_e{\displaystyle \frac{\mathrm{\partial }{\tilde{v}}_e}{\mathrm{\partial }\mathrm{y}}+{\displaystyle \frac{{\mu }_{mnf}}{{\rho }_{mnf}}{\displaystyle \frac{{\mathrm{\partial }}^2\tilde{v}}{\mathrm{\partial }{z}^2},}}}}}}}}$$

(3)

$${\displaystyle \frac{\mathrm{\partial }\mathrm{T}}{\mathrm{\partial }\mathrm{t}}+\tilde{u}{\displaystyle \frac{\mathrm{\partial }\mathrm{T}}{\mathrm{\partial }\mathrm{x}}+\tilde{v}{\displaystyle \frac{\mathrm{\partial }\mathrm{T}}{\mathrm{\partial }\mathrm{y}}+\tilde{w}{\displaystyle \frac{\mathrm{\partial }\mathrm{T}}{\mathrm{\partial }\mathrm{z}}={\displaystyle \frac{k_{mnf}}{{(\rho C_p)}_{mnf}}{\displaystyle \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{z}^2}+{\displaystyle \frac{Q}{{(\rho C_p)}_{mnf}}(T-T_{\mathrm{\infty }}).}}}}}}}$$

(4)

$\tilde{u},\tilde{v}$ and $\tilde{w}$ symbolizes the velocity section along the x, y and z axes separately. Next, the boundary conditions are ^{23, 37,38}:

$$t<0:\tilde{u}=\tilde{v}=\tilde{w}=0,T=T_{\mathrm{\infty }}\mathrm{for}\mathrm{any}x,y,z=0,$$

(5)

$$t\ge 0:\tilde{u}={\tilde{u}}_w,\tilde{v}={\tilde{v}}_w,\tilde{w}=w_{{}^{\circ }},T=T_w\mathrm{at}z=0,$$

(6)

$$\tilde{u}\to {\tilde{u}}_e(x),\tilde{v}\to {\tilde{v}}_e(x),T\to T_{\mathrm{\infty }}\mathrm{as}z\to \mathrm{\infty }.$$

(7)

The dynamic viscosity of the ternary-hybrid nanofluid is symbolized by the notation ${\mu }_{mnf}$ , the density of the ternary-nanofluid is symbolized by the symbol ${\rho }_{mnf}$ , the thermal conductivity of the ternary-nanofluid is designated by the symbol $k_{mnf}$ , and the heat capacity of the nanofluid is signified by the symbol $(\rho C_p{)}_{mnf}$ . The temperature of the ternary-nanofluid is denoted by T. The symbol Q denotes the heat source/sink. In order to get the simplified version of the controlling boundary layer system (1)–(7), the terms ${\tilde{u}}_w$ and ${\tilde{v}}_w$ are interpreted as follows ³⁹ : ${\tilde{u}}_w=\frac{ax}{1-\alpha t}$ , ${\tilde{v}}_w=\delta \frac{by}{1-\alpha t}$ .

The thermophysical characteristics of aluminum oxide, copper oxide, titanium dioxide, and polymer are listed in Table 1. The thermal properties of nanofluid, hybrid nanofluid, and ternary-hybrid nanofluid are all represented mathematically in Table 2. In addition, the data points for the various form factors depending on the shape parameters m are shown in Table 3. In Table 1, the value referred to as ${\varphi }_1$ represents the volume fraction of $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$ , ${\varphi }_2$ ‘2' represents the volume fraction of $\mathrm{CuO}$ , and ${\varphi }_3^{\mathrm{\prime }}$ represents the volume percentage of $\mathrm{Ti}{\mathrm{O}}_2$ nanoparticles. Subscripts $s1,s2,$ and $s3$ denotes the properties of $A{l}_2{O}_3$ , $CuO$ , and $Ti{O}_2$ nanoparticles.

Table 1.

Numerical values of base polymer and ternary-hybrid nanoparticles.^3,35

Properties	$A{l}_2{O}_3({\varphi }_1)$	$CuO({\varphi }_2)$	$Ti{O}_2({\varphi }_3)$	Base polymer
$\rho (kg/m^3)$	$3970$	$6500$	$4250$	$1060$
$Cp(J/kgK)$	$765$	$533$	$686.2$	$3770$
$k(W/mK)$	$0.40$	$17.65$	$8.538$	$0.429$
$Pr$				$6.2$

Table 2.

Thermal property of ternary-hybrid nanofluid. ³⁴

Properties	Ternary-hybrid nanofluid
Density	${\rho }_{mnf}=(1-{\varphi }_3)\{(1-{\varphi }_2)[{(1-{\varphi }_1)}_{{\rho }_f}+{\varphi }_1{\rho }_{s1}]+{\varphi }_2{\rho }_{s2}\}+{\varphi }_3{\rho }_{s3}$
Dynamic viscosity	${\mu }_{mnf}=\frac{{\mu }_f}{{(1-{\varphi }_3)}^{2.5}{(1-{\varphi }_2)}^{2.5}{(1-{\varphi }_1)}^{2.5}}$
Thermal conductivity	${\displaystyle \frac{k_{mnf}}{k_{hnf}}=\frac{k_{s3}+(m-1)k_{hnf}-(m-1){\varphi }_3(k_{hnf}-k_{s3})}{k_{s3}+(m-1)k_{hnf}+{\varphi }_3(k_{hnf}-k_{s3})},}$ where ${\displaystyle \frac{k_{hnf}}{k_{nf}}=\frac{k_{s2}+(m-1)k_{nf}-(m-1){\varphi }_2(k_{nf}-k_{s2})}{k_{s2}+(m-1)k_{nf}+{\varphi }_2(k_{nf}-k_{s2})},}$
	and ${\displaystyle \frac{k_{nf}}{k_f}=\frac{k_{s1}+(m-1)k_f-(m-1){\varphi }_1(k_f-k_{s1})}{k_{s1}+(m-1)k_f+{\varphi }_1(k_f-k_{s1})},}$
Heat capacity	$\begin{array}{rl}& (\rho C_p{)}_{mnf}=(1-{\varphi }_3)\{(1-{\varphi }_2)[(1-{\varphi }_1){(\rho C_p)}_f+{\varphi }_1{(\rho C_p)}_{s1}]+{\varphi }_2{(\rho C_p)}_{s2}\}\\ & +{\varphi }_3(\rho C_p{)}_{s3}.\end{array}$

Table 3.

Shape factor of distinct nanoparticles. ³⁴

Types of nanoparticles	Shape factor
Blades	$8.6$
Platelets	$5.7$
Bricks	$3.7$
Cylinder	$4.9$

Similarity transformations have been proposed in response to the findings of solution ³⁸ :

$$\begin{array}{rl}\tilde{u}=& {\displaystyle \frac{ax{f}^{\mathrm{\prime }}(\eta )}{1-\alpha t},\tilde{v}={\displaystyle \frac{by{h}^{\mathrm{\prime }}(\eta )}{1-\alpha t},\tilde{w}=-\sqrt{{\displaystyle \frac{\upsilon a}{1-\alpha t}}}(f(\eta )+ch(\eta )),\theta (\eta )={\displaystyle \frac{T-T_{\mathrm{\infty }}}{T_w-T_{\mathrm{\infty }}},}}}\\ \eta & =\sqrt{{\displaystyle \frac{a}{\upsilon (1-\alpha t)}}}z.\end{array}$$

(8)

Here prime denotes differentiation in reference to $\eta$ . Equations (1)–(4) and (8) are included into the unsteady-state equations to produce the following set of ordinary differential equations.

$$f^{\mathrm{″}\mathrm{\prime }}+{\displaystyle \frac{{\rho }_{mnf}/{\rho }_f}{{\mu }_{mnf}/{\mu }_f}\left[\left({\displaystyle \frac{1}{2}\epsilon \eta +f+ch}\right)f^{\mathrm{″}}-f^{{}^{\mathrm{\prime }}2}+\epsilon +1\right]=0,}$$

(9)

$$h^{\mathrm{″}\mathrm{\prime }}+{\displaystyle \frac{{\rho }_{mnf}/{\rho }_f}{{\mu }_{mnf}/{\mu }_f}\left[\left({\displaystyle \frac{1}{2}\epsilon \eta +f+ch}\right)h^{\mathrm{″}}-c{h}^{{}^{\mathrm{\prime }}2}+\epsilon +c\right]=0,}$$

(10)

$${\displaystyle \frac{1}{Pr}{\displaystyle \frac{k_{mnf}/k_f}{{(\rho C_p)}_{mnf}/{(\rho C_p)}_f}{\theta }^{\mathrm{″}}+\left({\displaystyle \frac{1}{2}\epsilon \eta +f+ch}\right){\theta }^{\mathrm{\prime }}+\lambda \theta =0.}}$$

(11)

Alteration of boundary conditions (5)–(7) to become

$$f(0)=S,h(0)=0,f^{\mathrm{\prime }}(0)=h^{\mathrm{\prime }}(0)=\delta ,\theta (0)=1,$$

(12)

$$f^{\mathrm{\prime }}(\eta )\to 1,h^{\mathrm{\prime }}(\eta )\to 1,\theta (\eta )\to 0,as\eta \to \mathrm{\infty }.$$

(13)

In the equations that came before, the parameter that signifies instability is written as $\epsilon =\alpha /a$ shows steady-state flow when $\epsilon =0$ , a decelerating flow is indicated when $\epsilon <0$ , and an accelerating flow is indicated when $\epsilon >0$ and $Pr={\nu }_f/{\mu }_f$ symbolizes the Prandtl number. $S=-\frac{w_{{}^{\circ }}}{\sqrt{a{\nu }_f}}$ characterizes the mass flux parameter $S>0$ for suction. In this context, $\delta$ refers to the stretch/shrink parameter, in which a sheet is stretched if $\delta >0$ while $\delta <0$ indicates that the sheet is shrinking. $\lambda =\frac{Q}{{(\rho C_p)}_f}$ represents heat source/sink. To this inquiry, the friction coefficients along the x and y axes, denoted by $C{f}_x$ and $C{f}_y$ , respectively, and the local Nusselt number, $N{u}_x$ are of utmost significance.

$$C_{fx}={\displaystyle \frac{{\tau }_{\tilde{w}x}}{{\rho }_f{u}_e^2},C_{fy}={\displaystyle \frac{{\tau }_{\tilde{w}y}}{{\rho }_f{v}_e^2},N{\tilde{u}}_x={\displaystyle \frac{x{q}_{\tilde{w}}}{k_f(T_w-T_{\mathrm{\infty }})}.}}}$$

(14)

Here, The shear stresses along the x and y axes are denoted by the symbols ${\tau }_{\tilde{w}x}$ and ${\tau }_{\tilde{w}y}$ , respectively, and the heat flow is denoted by $q_{\tilde{w}}$ . One possible explanation for each of these phrases is as follows:

$${\tau }_{\tilde{w}x}={\mu }_{mnf}{\left({\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }z}}\right)}_{z=0},{\tau }_{\tilde{w}y}={\mu }_{mnf}{\left({\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }z}}\right)}_{z=0},q_{\tilde{w}}=-k_{mnf}\left[{\left({\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }z}}\right)}_{z=0}\right].$$

(15)

By provoking the equations (8), (14), and (15) we get the following:

$$R{e}_x^{1/2}{C}_{fx}={\displaystyle \frac{{\mu }_{mnf}}{{\mu }_f}{f}^{\mathrm{″}}(0),cR{e}_y^{1/2}{C}_{fy}={\displaystyle \frac{{\mu }_{mnf}}{{\mu }_f}{h}^{\mathrm{″}}(0),R{e}_x^{-1/2}N{u}_x=-{\displaystyle \frac{K_{mnf}}{k_f}{\theta }^{\mathrm{″}}(0),}}}$$

(16)

where $R{e}_x=\frac{(a)x^2}{(1-\alpha t){\upsilon }_f}$ and $R{e}_y=\frac{(b)y^2}{(1-\alpha t){\upsilon }_f}$ signify the local Reynolds numbers along the x and y axes, respectively.

Numerical procedure

Equations of ordinary differentials (9–11) that are nonlinear and connected with related boundary conditions (12, 13) have a character that is both nonlinear and exceedingly difficult. It is strongly recommended that a numerical method be used to address these challenges. The use of Runge-Kutta IV (RK-IV) in conjunction with various shooting procedures makes it possible to record ternary-hybrid nanofluid flow paths for a wide range of distinct physical parameters. To get the numerical solution with a maximum accuracy of ${\eta }_{max}=10$ and precision to the fifth decimal place as the criterion of convergence, the step size $\mathrm{\Delta }\eta =0.001$ is used. It is necessary to perform a few transformations, the specifics of which depend on the sequence in which the self-similar momentum and energy equations (9)–(11) are used, as well as the boundary domains contained in the equations (12) and (13). Figure 3 depicts the process that is followed for various shooting approaches.

Figure 3.

Schematic diagram for shooting method.

Because of these changes, the model was able to be simplified so that it could solve a first-order initial value problem. This problem can then be solved as follows:

$$\begin{array}{rl}{\mathrm{\Xi }}_1=& f,{\mathrm{\Xi }}_2=f^{\mathrm{\prime }},{\mathrm{\Xi }}_3=f^{\mathrm{″}},{\mathrm{\Xi }}_4=f^{\mathrm{″}\mathrm{\prime }},{\mathrm{{\rm Y}}}_1=h,{\mathrm{{\rm Y}}}_2=h^{\mathrm{\prime }},{\mathrm{{\rm Y}}}_3=h^{\mathrm{″}},{\mathrm{{\rm Y}}}_4=h^{\mathrm{″}\mathrm{\prime }},\\ {\mathrm{\Lambda }}_1& =\theta ,{\mathrm{\Lambda }}_2={\theta }^{\mathrm{\prime }},{\mathrm{\Lambda }}_3={\theta }^{\mathrm{″}}\end{array}$$

(17)

using the equation (17),

$${{\mathrm{\Xi }}^{\mathrm{\prime }}}_1=f^{\mathrm{\prime }},{{\mathrm{\Xi }}^{\mathrm{\prime }}}_2=f^{\mathrm{″}},{{\mathrm{\Xi }}^{\mathrm{\prime }}}_3=f^{\mathrm{″}\mathrm{\prime }}={\mathrm{\Xi }}_4,{{\mathrm{{\rm Y}}}^{\mathrm{\prime }}}_1=h^{\mathrm{\prime }},{{\mathrm{{\rm Y}}}^{\mathrm{\prime }}}_2=h^{\mathrm{″}},{{\mathrm{{\rm Y}}}^{\mathrm{\prime }}}_3=h^{\mathrm{″}\mathrm{\prime }}={\mathrm{{\rm Y}}}_4,{{\mathrm{\Lambda }}^{\mathrm{\prime }}}_1={\theta }^{\mathrm{\prime }},{{\mathrm{\Lambda }}^{\mathrm{\prime }}}_2={\mathrm{\Lambda }}_3={\theta }^{\mathrm{″}}.$$

(18)

After rearranging the equations and making use of equations (17) and (18) in (9)–(13), we arrive at the following result:

$${\mathrm{\Xi }}_4=-\left[{\displaystyle \frac{{\rho }_{mnf}/{\rho }_f}{{\mu }_{mnf}/{\mu }_f}}\right]\left(\left({\displaystyle \frac{1}{2}\epsilon \eta +{\mathrm{{\rm Y}}}_1+{\mathrm{\Xi }}_1}\right){\mathrm{\Xi }}_3-{\mathrm{\Xi }}_2^2+\epsilon +1\right),$$

(19)

$${\mathrm{{\rm Y}}}_4=-\left[{\displaystyle \frac{{\rho }_{mnf}/{\rho }_f}{{\mu }_{mnf}/{\mu }_f}}\right]\left(\left({\displaystyle \frac{1}{2}\epsilon \eta +\mathrm{c}{\mathrm{{\rm Y}}}_1+{\mathrm{\Xi }}_1}\right){\mathrm{{\rm Y}}}_3-c{\mathrm{{\rm Y}}}_2^2+\epsilon +\mathrm{c}\right),$$

(20)

$${\mathrm{\Lambda }}_3=-\left[Pr{\displaystyle \frac{{(\rho C_p)}_{mnf}/{(\rho C_p)}_f}{k_{mnf}/k_f}\left({\displaystyle \frac{1}{2}\epsilon \eta +{\mathrm{\Xi }}_1+\mathrm{c}{\mathrm{{\rm Y}}}_1}\right){\mathrm{\Lambda }}_2+\lambda {\mathrm{\Lambda }}_1}\right],$$

(21)

$${\mathrm{\Xi }}_1(0)=S,{\mathrm{\Xi }}_2(0)={\mathrm{{\rm Y}}}_2(0)=\delta ,{\mathrm{\Lambda }}_1(0)=1,{\mathrm{\Xi }}_3\to 1,{\mathrm{{\rm Y}}}_3\to 1,{\mathrm{\Lambda }}_2\to 0$$

(22)

To address the subsequent issue with the initial value, we need to include the solutions to equations (17) and (18) into equations (9) through (11), inserting them in the right places each time. To offer analytical results for the equations presented equations (9)–(11), an extremely powerful piece of computational software known as Mathematica 10 is used. It first determines the governing system and then employs the numerical method that is best suited for the system to get proper responses for the system. This is a distinctive feature.

Results and discussion

We utilize the RK-IV shooting method to solve equations (9)–(11) and present the numerical results graphically and tabulated in the computing program Mathematica. Except for the assessment, which can be seen in the figures and tables, the boundary layer thickness ${\eta }_{\mathrm{\infty }}=10$ and $Pr=6.2$ (base polymer) are both fixed values. The control characteristics, on the other hand, fluctuate. The fulfillment of the conditions for the far-field boundary (12, 13) is what establishes these values. Considering that $A{l}_2{O}_3-CuO-Ti{O}_2/polymer$ ternary nanofluid is maintained in the study that is being presented. The study of Turkyilmazoglu, ⁴⁰ which showed that a non-Newtonian fluid may be created if the concentration of nanoparticles is more than 5% to 6%, corresponds to a different set of volume fraction ${\varphi }_3$ values that is confined in between $0.005$ and $0.02$ . Other parameters, such as $-0.0\le \epsilon \le -0.4$ (unsteadiness), $0.0\le \delta \le 2.0$ (stretching), $2.0\le S\le 2.4$ (suction), $0.0\le \lambda \le 1.0$ (source), and $c=0.5$ are fixed for nodal point are selected to ensure the accuracy of the solution.

Physically modeling the flow of a $(A{l}_3{O}_3-CuO-Ti{O}_2/\mathrm{polymer})$ ternary-hybrid nanofluid is demonstrated in Figure 2. In the next subsections, the variation in hydrothermal profiles that is induced by associated flow factors will be discussed in more detail. Several diagrams are produced to illustrate the significant effect that flow parameters have on temperature, velocity, heat transfer, and skin frictional profiles.

Impact of pertinent parameters on velocity ${\mathit{f}}^{\mathbf{\prime }}(\mathit{\eta })$ , ${\mathit{h}}^{\mathbf{\prime }}(\mathit{\eta })$ and temperature $\mathit{\theta }(\mathit{\eta })$

In this subsection, the profiles for $(A{l}_3{O}_3-CuO-Ti{O}_2/\mathrm{polymer})$ ternary-hybrid nanofluid velocities in both direction and temperature $\theta (\eta )$ for numerous estimates of ${\varphi }_3,\epsilon ,S$ , and $\lambda$ are expounded when ${\varphi }_1={\varphi }_2=0.01$ in Figures 4–10. The accuracy of the current solutions is supported by the fulfillment of the far-field boundary requirements. It is essential to emphasize the fact that the Prandtl number of the polymer molecule is $Pr=6.2$ now.

Figure 4.

Variation in $f^{\mathrm{\prime }}(\eta )$ and $h^{\mathrm{\prime }}(\eta )$ with ${\varphi }_3$ .

Figure 5.

Change in $\theta (\eta )$ in contrast to ${\varphi }_3$ .

Figure 6.

Variation in $f^{\mathrm{\prime }}(\eta )$ and $h^{\mathrm{\prime }}(\eta )$ with $\epsilon$ .

Figure 7.

Change in $\theta (\eta )$ in contract to $\epsilon$ .

Figure 8.

Variation in $f^{\mathrm{\prime }}(\eta )$ and $h^{\mathrm{\prime }}(\eta )$ with $S$ .

Figure 9.

Change in $\theta (\eta )$ in contract to S.

Figure 10.

Variation in $\theta (\eta )$ in contract to $\lambda$ .

Figures 4 and 5 demonstrate the upshot of volume fraction ${\varphi }_3=0.005,0.01,0.015,\mathrm{0.02.}$ When $S=2.0,\epsilon =-0.2,c=0.5,\delta =0.7,\lambda =0.2,m=8.6,{\varphi }_1={\varphi }_2=0.005$ for the dimensionless velocities $f^{\mathrm{\prime }}(\eta )$ and $h^{\mathrm{\prime }}(\eta )$ and temperature profiles of ternary-hybrid nanofluid flow for stretching case. These descriptions also demonstrate that when volume friction rises, the velocity profiles become more noticeable both inside the nanofluid and on the boundary layer thickness. This is true regardless of the location of the boundary layer. The relationship between volume fraction and velocity profiles is made abundantly evident by these graphs. Increases in ${\varphi }_3$ resulted in an upsurge in the nanofluid density, which in turn resulted in a slowing of the flow in both directions due to the impact that their respective flow velocity profiles had. With increased ${\varphi }_3$ , the width of the momentum barrier layer diminishes, increasing the velocity of the fluid flow. When there is a greater amount of fluid flow velocity, there is a corresponding rise in the surface shear stress. It was also possible to detect, by looking at Figure 5, that the temperature of the ternary $A{l}_3{O}_3-CuO-Ti{O}_2/\mathrm{polymer}$ hybrid nanofluid rose alongside a rise in the values of ${\varphi }_1,{\varphi }_2$ , and ${\varphi }_3$ . The increases in ${\varphi }_1,{\varphi }_2$ , and ${\varphi }_3$ are shown to have contributed to a rise in the thickness of the thermal boundary layer in addition to an improvement in the heat conductivity of ternary-hybrid nanofluids. From a purely physical point of view, the nanoparticles release heat energy into the surrounding space. As a result, an increase in the number of the nanoparticles might produce more heat, which ultimately causes a rise in the temperature. This suggests that an increase or decrease in ${\varphi }_1,{\varphi }_2$ , and ${\varphi }_3$ might be used to modulate the temperature of the ternary-hybrid nanofluids.

Figures 6 and 7 demonstrates the impact of unsteadiness constraint $\epsilon =-0.05,-0.1,-0.2,-0.3$ , when $\lambda =0.2,S=2.2,c=0.5,\delta =0.7,m=8.6,{\varphi }_1={\varphi }_2={\varphi }_3=0.01$ on velocities $f^{\mathrm{\prime }}(\eta )$ and $h^{\mathrm{\prime }}(\eta )$ and temperature profiles, respectively, of polymer base ternary-hybrid nanofluid for stretching case. Clearly this figure show that decreasing the values of unsteadiness parameter decline in both the velocity profiles has been shown. This may be seen in Figure 6 which shows that unsteadiness alters the velocity profiles. A decrease in shear stress causes a fall in fluid velocity and a corresponding expansion of the momentum boundary layer along the wall, as seen in Figure 6. Figure 7 display the stimulus of unsteadiness constraint $\epsilon$ on temperature profile of $(A{l}_3{O}_3-CuO-Ti{O}_2/\mathrm{polymer})$ ternary-hybrid nanofluid. It makes sense that a decrease in the values of $\epsilon$ would cause a rise in the temperature profile of a $(A{l}_3{O}_3-CuO-Ti{O}_2/\mathrm{polymer})$ ternary nanofluid. This is because the spaces between molecules would spread out more in an unsteady flow if there were fewer quantities of an unstable parameter. This would cause a rise in the temperature profile and a decrease in the cooling rate of the fluid. Because of this, the feature of unsteadiness should be highlighted, and serious thought should be given to the ways in which it might be put to an advantage.

Figures 8 and 9 illustrates the consequence of suction constraint $S=2.1,2.2,2.3,2.4$ , when $\lambda =0.2,\delta =0.7,c=0.5,\epsilon =-0.2,m=8.6,{\varphi }_1={\varphi }_2={\varphi }_3=0.01$ on velocities $f^{\mathrm{\prime }}(\eta )$ and $g^{\mathrm{\prime }}(\eta )$ and temperature profiles of the polymer base ternary-hybrid nanofluid flow for stretching case. In this picture, an increase in velocity can be seen along the profile because of increasing quantities of the suction parameter. This is a result of the increased density caused by the inclusion of additional particles. The thickness of the momentum boundary layer is decreased as a result of suction, which results in an increase in the flow of the stretched sheet. Figure 9 illustrates how $\theta (\eta )$ responds to suction parameter S. By cumulative value of S decreasing the temperature of ternary-hybrid nanofluid. The value of S at the surface of a permeable stretching sheet is directly correlated to the level of suction experienced by a permeable stretching sheet. To improve the sluggish flow of the fluid on the expanding sheet, enhancement of the S consisted of slowing down the molecules while they were in the fluid regime. As a result, it has a dampening effect on the momentum border layer.

Figure 10 show the impact of source $\lambda =0.2,0.4,0.6,0.8$ with $\delta =1.0,c=0.5,\epsilon =-0.2,S=2.2,m=8.6,{\varphi }_1={\varphi }_2={\varphi }_3=0.01$ on the temperature profile of $(A{l}_3{O}_3-CuO-Ti{O}_2/\mathrm{polymer})$ ternary-hybrid nanofluid. Figure 10 presents the heat source parameter $\lambda$ in question for the purpose of analyzing the temperature profile variation over a range of different values.

When the heat source parameter $(\lambda >0)$ is raised, a rising temperature profile is seen in $Ti{O}_2-CuO-A{l}_2{O}_3/\mathrm{polymer}$ nanofluids used for stretching sheets.

The temperature rises as more accurate estimates of the parameter $\lambda$ that describes the heat source are used. The source of heat in the system is reflected by the parameter's positive values, and higher values indicate that a greater quantity of heat is being produced. The inclusion of a heat source in Figure 10 substantiates the assertion that polymer base ternary-hybrid nanofluid exhibit thermal increase. The impacts are most noticeable close to the surface.

Skin friction and heat transfer

Important industrial metrics include the local Nusselt number, often known as the simple local heat transfer rate $\left(\frac{1}{\sqrt{R{e}_x}}N{u}_x\right)$ , as well as the skin friction ( $\sqrt{R{e}_x}C{f}_x$ , $c\sqrt{R{e}_y}C{f}_y$ ) in both directions. Because of their applications in industry, the significance of these data cannot be called into question. The skin friction coefficients ( $\sqrt{R{e}_x}C{f}_x$ , $c\sqrt{R{e}_y}C{f}_y$ ) and the local Nusselt number $\left(\frac{1}{\sqrt{R{e}_x}}N{u}_x\right)$ respond differently to different fluctuations in the parameters.

The volume fraction of the nanoparticles is an essential physical parameter that must be considered when determining the effect that nanoparticles have on the rate of fluid flow and heat transfer. Figures 11 and 12 show the effect that the nanoparticles’ volume fraction ${\varphi }_3$ with $\delta >0$ , when $\lambda =0.2,S=2.2,c=0.5,\epsilon =-0.2,m=8.6,$ , ${\varphi }_1={\varphi }_2=0.01$ , respectively, has on the reduced ( $\sqrt{R{e}_x}C{f}_x$ and $c\sqrt{R{e}_y}C{f}_y$ ) in both directions and the reduced local Nusselt number $\left(\frac{1}{\sqrt{R{e}_x}}N{u}_x\right)$ for $Ti{O}_2-CuO-A{l}_2{O}_3-polymer$ ternary-hybrid nanofluid. Figure 11 demonstrates that the rate of ( $\sqrt{R{e}_x}C{f}_x$ and $c\sqrt{R{e}_y}C{f}_y$ ) for ${\varphi }_1={\varphi }_2={\varphi }_3=0.04$ ( $Ti{O}_2-CuO-A{l}_2{O}_3-polymer$ ) is the maximum, when assessed to ${\varphi }_1={\varphi }_2={\varphi }_3=0.01$ ( $Ti{O}_2-CuO-A{l}_2{O}_3-polymer$ ). This trend unmistakably demonstrates the presence of a colloid that is spread in the conventional fluid and contains three solid nanoparticles. As a outcome, the conventional fluid exhibits greater values of ( $\sqrt{R{e}_x}C{f}_x$ and $c\sqrt{R{e}_y}C{f}_y$ ) in both directions than the colloid does. Physically, the intensification in the volume fraction of nanoparticles causes a growth in the viscosity of the polymer base ternary-hybrid nanofluid, which in turn causes an intensification in the fluid velocity as it moves through a permeable stretching sheet (see Figure 4). A closer look at Figure 4 reveals that an intensification in the ${\varphi }_1={\varphi }_2={\varphi }_3$ brings about a decline in the thickness of the boundary layer, which in turn speeds up the fluid velocity and makes the velocity gradient more pronounced. The rise in wall shear stress is a direct result of the decrease in the thickness of the momentum boundary layer, which in turn leads to an improvement in ( $\sqrt{R{e}_x}C{f}_x$ and $c\sqrt{R{e}_y}C{f}_y$ ). Higher ( $\sqrt{R{e}_x}C{f}_x$ and $c\sqrt{R{e}_y}C{f}_y$ ) values indicate an upsurge in the amount of frictional drag employed on the decreasing surface, it is not appropriate for preserving the flow characteristics of a laminar boundary layer. According to the findings, the polymer base ternary-hybrid nanofluid is more difficult to maintain a laminar flow than the hybrid nanofluid and usual nanofluid. When the sheet is not moving $\delta =0$ , Figure 11 further emphasizes the value of ( $\sqrt{R{e}_x}C{f}_x$ and $c\sqrt{R{e}_y}C{f}_y$ ), this suggests that there is no drag caused by frictional forces acting on the surface of the sheet. When compared with ${\varphi }_1={\varphi }_2={\varphi }_3=0.01$ and ${\varphi }_1={\varphi }_2={\varphi }_3=0.04$ , it is clear from Figure 12 that $\left(\frac{1}{\sqrt{R{e}_x}}N{u}_x\right)$ for ${\varphi }_1={\varphi }_2={\varphi }_3=0.0$ has the greatest value. Based on the findings of this important experiment, it can be deduced that the base fluid had a higher $\left(\frac{1}{\sqrt{R{e}_x}}N{u}_x\right)$ than the ternary-hybrid nanofluid. In addition, this data suggests that the encouragement of raising the nanoparticles’ volume fraction might have the effect of lowering the temperature gradient. As a consequence of this, the efficiency of heat transmission drops, which in turn brings $\left(\frac{1}{\sqrt{R{e}_x}}N{u}_x\right)$ down by bringing the values of ${\varphi }_1,{\varphi }_2,{\varphi }_3$ closer. Figure 12 portrays the diminishing trend of $\left(\frac{1}{\sqrt{R{e}_x}}N{u}_x\right)$ as the values of nanoparticles $\varphi$ with $\delta$ when $\lambda =0.2,S=2.2,c=0.5,\epsilon =-0.2,m=8.6$ . When an ordinary nanofluid and a hybrid nanofluid combine to form a ternary-hybrid nanofluid, the rate of heat transmission slows down because of the increased number of nanoparticles. The values of $\left(\frac{1}{\sqrt{R{e}_x}}N{u}_x\right)$ have significantly decreased, which is a notable development. The temperature profiles displayed in Figure 5 lend credence to the general pattern shown in Figure 12, which provides an explanation of the rise in temperature that occurs when an ordinary nanofluid and a hybrid nanofluid combine to form a ternary-hybrid nanofluid. Since there are fewer regions toward the shrinking sheet for heat energy transmission, the fluid may be heated to a higher temperature than it would be at the extended surface, and the nanoparticle volume fraction may be able to physically spread more energy. An increase in the temperature of the ternary-hybrid nanofluid leads to a rise in the thermal conductivity of the ternary-hybrid nanofluid, which in turn causes an increase in the temperature of the ternary-hybrid nanofluid, which in turn causes a reduction in the rate of heat transfer through convection.

Figure 11.

Variation in $\sqrt{R{e}_x}C{f}_x$ and $c\sqrt{R{e}_y}C{f}_y$ in contract to $\varphi$ with $\delta$ .

Figure 12.

Variation in $\frac{1}{\sqrt{R{e}_x}}N{u}_x$ in contract to $\varphi$ with $\delta$ .

Figure 13 illustrate the impact that the unsteady parameter $\epsilon$ has when the value changes from $-0.2$ to $-1.0$ against $\delta >0$ , when $\lambda =0.2,S=2.2,\varphi =0.01,c=0.5,m=8.6$ of the skin friction coefficients ( $\sqrt{R{e}_x}C{f}_x$ , $c\sqrt{R{e}_y}C{f}_y$ ) of $Ti{O}_2-CuO-A{l}_2{O}_3/polymer$ . Figure 13 illustrates that when the value of $\epsilon$ dropped, the solution exhibited a lower level of ( $\sqrt{R{e}_x}C{f}_x$ , $c\sqrt{R{e}_y}C{f}_y$ ). The reduction brings about an increase in the thickness of the boundary layer, which, in turn, brings about a reduction in the velocity gradient of the permeable sheet. As a result, the difference between $\sqrt{R{e}_x}C{f}_x$ and $c\sqrt{R{e}_y}C{f}_y$ is reduced. Because of the increase in the viscosity of $Ti{O}_2-CuO-A{l}_2{O}_3/polymer$ the presence of nanoparticle volume fraction may potentially cause a decrease in $\sqrt{R{e}_x}C{f}_x$ and $c\sqrt{R{e}_y}C{f}_y$ . Figure 14 illustrates a situation in which various estimates of $\epsilon$ toward $\delta >0$ , when $\lambda =0.2,S=2.2,\varphi =0.01,c=0.5,m=8.6$ . In addition, the findings that were obtained can be shown in Figure 14 and show that $\left(\frac{1}{\sqrt{R{e}_x}}N{u}_x\right)$ in the solution decreases, which is comparable to the heat transfer rate. This happens when $\delta >0$ occurs in the permeable sheet. The authors are able to draw the conclusion that the $\epsilon$ parameter contributes greatly to the worsening of heat transfer based on the data that is now available and that which has previously been gathered. Despite this, the authors would like to assert that the impacts could be different in different situations if several control factors are considered simultaneously.

Figure 13.

Variation in $\sqrt{R{e}_x}C{f}_x$ and $c\sqrt{R{e}_y}C{f}_y$ in contract to $\epsilon$ with $\delta$ .

Figure 14.

Variation in $\frac{1}{\sqrt{R{e}_x}}N{u}_x$ in contract to $\epsilon$ with $\delta$ .

Figure 15 shows the impact of $S=2.1,2.2,2.3,2.4,2.5\mathrm{with}\delta ,\mathrm{when}\lambda =0.2,\varphi =0.01,c=0.5,\epsilon =-0.2,m=8.6$ against skin friction coefficient ( $\sqrt{R{e}_x}C{f}_x$ , $c\sqrt{R{e}_y}C{f}_y$ ) in both the directions of polymer base ternary-hybrid nanofluid. Figure 15 demonstrates that an increase in S will result in a significant increase ( $\sqrt{R{e}_x}C{f}_x$ , $c\sqrt{R{e}_y}C{f}_y$ ) in the solution of polymer base ternary-hybrid nanofluid for stretching sheet case. The suction event may make it easier to maintain the boundary layer's steady state. Additionally, the suction reduces the friction caused by the external flow acting on the bodies, which in turn leads to a lessening in the thickness of the boundary layer and an expansion in the velocity differential of the permeable sheet. This is accomplished by the removal of fluid across the surface with the lowest momentum. Suction is required to enhance the values of skin friction in both directions.

Figure 15.

Variation in $\sqrt{R{e}_x}C{f}_x$ and $c\sqrt{R{e}_y}C{f}_y$ in contrast to S with $\delta$ .

The relative performance of $\left(\frac{1}{\sqrt{R{e}_x}}N{u}_x\right)$ polymer base $Ti{O}_2$ nanofluid, $Ti{O}_2-CuO-$ hybrid nanofluid, and $Ti{O}_2-CuO-A{l}_2{O}_3$ ternary-hybrid nanofluid is compared in Figure 16. When $S=2.0,2.1,2.2,2.3,2.4\mathrm{and}\lambda =0.2,\varphi =0.01,c=0.5,\epsilon =-0.2,m=8.6$ . It has been revealed that the ternary-hybrid nanofluid exhibits improved capabilities in terms of heat transmission in comparison to those of the other nanofluids. This is because the use of various nanoparticles that are bound together by a variety of chemical bonds contributes to an increase in heat transfer. This is because each nanoparticle, along with the chemical bond that binds it, has its own properties that need to be taken into consideration. For instance, in this mixture, the photocatalytic property of $Ti{O}_2$ combined with its high thermal conductivity causes it is primarily responsible for most of the heat conduction. $CuO$ is used to increase the catalytic nature of $Ti{O}_2$ for it to be able to conduct more heat, while the addition of $A{l}_2{O}_3$ ensures that the fluid is chemically inert and stable. This is accomplished using $CuO$ . When seen in Figure 16, the solution carries with it an increase in $\left(\frac{1}{\sqrt{R{e}_x}}N{u}_x\right)$ as S rises over the permeable sheet. This is straightforward evidence of this phenomenon. Consider the fact that the suction impact makes it possible for the molecules of the ternary-hybrid nanofluid to occupy the surface, which in turn helps to physically enhance the heat transfer rate at the permeable sheet. It is possible to find solutions to the problem of expanding surface flows without the use of suction. Suction is required to enhance the values of the Nusselt number.

Figure 16.

Comprasion in heat transfer performance of ternary-hybrid nanofluid, hybrid nanofluid, and nanofluid.

Figure 17 depicts the fluctuation of the decreased local Nusselt number $\frac{1}{\sqrt{R{e}_x}}N{u}_x$ with an assortment of estimates of the heat source parameter $(\lambda >0)$ with $\delta$ , when S $=2.2,\varphi =0.01,c=0.5,\epsilon =-0.2,m=8.6$ of polymer-based ternary-hybrid nanofluid of the shrinking sheet. It was discovered that the ratio of $\frac{1}{\sqrt{R{e}_x}}N{u}_x$ became smaller when the value of the heat source parameter increased $(\lambda >0)$ . Based on this finding, if there is a heat source present in the boundary layer, then we can draw the conclusion that there is a possibility for there to be a rise in the thickness of the thermal boundary layer. Because of this, the heat flow is decreased, and consequently, $\frac{1}{\sqrt{R{e}_x}}N{u}_x$ goes down if there is an expansion in the amount of the heat source parameter $(\lambda >0)$ .

Figure 17.

Variation in $\frac{1}{\sqrt{R{e}_x}}N{u}_x$ compared to $\lambda$ with $\delta$ .

Table discussion

The purpose of the comparisons of findings that are shown in Tables 4 and 5 is to verify the numerical approach that was used in the current research for the stable case ( $\epsilon =0$ ) by comparing it to the numerical results that were obtained by Zainal et al. ³⁸ for a different kind of fluid. While Zainal et al. ³⁸ in their earlier works attempted to solve the issue of hybrid nanofluid fluid, the authors of the current study opted to apply ternary-hybrid nanofluid. Additionally, Zainal et al. ³⁸ utilized the bvp4c program included in MATLAB, notwithstanding this, the most recent research carried out a simulation using the Runge-Kutta procedure in Mathematica. These findings are found to be in agreement with the steady-state fluid case solutions that were previously discovered; as a result, this provides us with the confidence that the computational structure used to investigate the ternary-hybrid nanofluid flow behaviors and heat transfer in this study can be utilized with a significant amount of assurance. The most important component in determining the flow behavior of nanofluids and the amount of heat that may be transferred is the configuration of appropriate ternary nanofluids.

Table 4.

$f^{\mathrm{″}}(0)\mathrm{and}{h}^{\mathrm{″}}(0)\mathrm{by}{\varphi }_1={\varphi }_2={\varphi }_3=\epsilon =S=\delta =m=0$ and $Pr=0.7$ generates outcomes for estimates of $c$ .

$c$	Current findings		Zainal et al. 38
	$f^{\mathrm{″}}(0)$	$h^{\mathrm{″}}(0)$	$f^{\mathrm{″}}(0)$	$h^{\mathrm{″}}(0)$
1.0	1.31194	1.31194	1.311938	1.311938
0.75	1.28863	1.16432	1.288629	1.164316
0.5	1.26687	0.99811	1.266866	0.998111
0.25	1.24761	0.80514	1.247612	0.805137
0.00	1.23259	0.57047	1.232588	0.570465
−0.25	1.22513	0.26795	1.225129	0.267950
−0.5	1.23020	−0.11150	1.230195	−0.111500
−0.75	1.24732	−0.48219	1.247319	−0.482131
−1.0	1.27277	−0.80950	1.271539	−0.794493

Table 5.

$-{\theta }^{\mathrm{\prime }}(0)\mathrm{by}{\varphi }_1={\varphi }_2={\varphi }_3=\epsilon =S=\delta =0$ and $Pr=0.7$ generates outcomes for estimates of $c$ .

	Present results	Zainal et al. 38
$c$	$-{\theta }^{\mathrm{\prime }}(0)$	$-{\theta }^{\mathrm{\prime }}(0)$
1.0	0.66538	0.665378
0.75	0.62308	0.623085
0.5	0.57967	0.579670
0.25	0.53621	0.536212
0.00	0.49587	0.495866
−0.25	0.46778	0.467776
−0.5	0.47059	0.470589
−0.75	0.50755	0.507541
−1.0	0.56595	0.562037

Conclusion

The primary takeaway from this line of research is that it may be possible to speed up the rate of heat transfer. This novel study explicates the unsteady 3D nodal stagnation point flow of polymer-based ternary-hybrid nanofluid contains $(A{l}_2{O}_3,CuO,Ti{O}_2)$ nanoparticles past a stretching sheet with heat source and mass suction effects. The RK-IV was used with shooting techniques for the purpose of establishing the conclusions, and the MATHEMATICA program was applied. The influence that a variety of numerous factors have on velocity, temperature, skin frictions, and Nusselt number profiles may be shown graphically. The research questions that were presented in the introduction section are going to be applied to the findings in the following manner:

Due to the presence of a suction parameter, the ternary-hybrid nanofluid has a higher rate of heat transfer than the hybrid nanofluid and the ordinary nanofluid. This could improve how well heat is transferred, making the ternary-hybrid nanofluid better than both the hybrid nanofluid and the regular nanofluid.

As the volume percentage of the polymer-based ternary-hybrid nanofluid is estimated to increase, the velocity and temperature profiles of the nanofluid will also rise.

The opposite trend is exhibited in the temperature profile of the $(A{l}_2{O}_3-CuO-Ti{O}_2/polymer)$ ternary-hybrid nanofluid when the suction parameter is varied. This causes an increase in the velocity profiles in both directions.

The coefficient of skin frictions ( $\sqrt{R{e}_x}C{f}_x$ , $c\sqrt{R{e}_y}C{f}_y$ ) and the local Nusselt number $\left(\frac{1}{\sqrt{R{e}_x}}N{u}_x\right)$ both rise as a consequence of an increase in the strength of the suction parameter from $2.0$ to $2.5$ . This, in turn, outcomes in an improvement in the heat transfer enhancement and a decline in temperature.

A decreasing significance of $\epsilon$ diminishes the velocity profiles and increases the temperature profile of polymer base ternary-hybrid nanofluid.

Subsequently, a decline in the unsteadiness parameter with volume friction diminishes the $\sqrt{R{e}_x}C{f}_x$ , $c\sqrt{R{e}_y}C{f}_y$ , and $\frac{1}{\sqrt{R{e}_x}}N{u}_x$ over the shrinking surface of polymer base ternary-hybrid nanofluid.

Temperature profile boosts up with boosting estimates of heat source parameter of polymer base ternary-hybrid nanofluid.

For stretching sheet, the rate of heat transfer diminishes with improving values of heat source.

It is essential to keep in mind that the results of another research employing other nanoparticles and base fluids might provide conclusions that are dissimilar to those produced in the present investigation. In addition, the present work could be able to assist other researchers with the following:

• Giving an original perspective on the transformation that is applicable to simplify the equations involved in the boundary layer flow. This is important for achieving the goal.

• Choosing the right values for the relevant parameters to achieve optimal performance in the heat transfer process within the context of the current industry.

• However, it is possible that additional researchers will be able to derive ideas from the existing study for future investigations into the heat transfer of the stretching flow with melting, mixed convection, viscous dissipation, thermal radiation, and Joule heating effects on ternary-hybrid nanofluids.

Data availability

The dataset used during the current study are available from the corresponding author on reasonable request.