Investigating thermal conduction dynamics in ternary Carreau–Yasuda nanofluids under convective boundary conditions and exponential heat source/sink effects

Abstract

The analysis of the Carreau–Yasuda nanofluid (CYNF) across a stretching surface has practical applications in several fields, such as heat exchange, engineering, and material science. Scholars may discover novel perspectives for increasing heat transfer performance, establishing advanced materials, and enhancing the efficiency of multiple engineering systems by studying the behavior of CYNF in this particular instance. The energy transfer through trihybrid CYNF flow with the effect of magnetic dipole across a stretching sheet is examined in the present study. The ternary hybrid nanofluid (THNF) has been prepared by the addition of ternary nanoparticles (NPs) in the water (50%) and ethylene glycol (50%). Titanium dioxide (TiO₂), silicon dioxide (SiO₂), and aluminum oxide (Al₂O₃) are used in base fluid. The fluid velocity and heat transfer are examined under the impact of Darcy–Forchheimer, chemical reaction, convective condition, activation energy, and exponential heat source. The fluid flow has been stated in form of velocity, energy, and concentration equations. The set of modeled equations is simplified to non-dimensional form of ordinary differential equations (ODEs) by using similarity substitutions. Numerically, the system of lowest order ODEs is calculated through the parametric continuation method. It has been noticed that the temperature field augments against the variation of viscous dissipation and heat source term. Furthermore, the magnetic dipole has significant impact to enhance the thermal curve of THNF whereas falloffs the velocity profile. It can be noticed that the energy propagation rate enhances with the rising numbers of ternary NPs, from 1.22% to 5.98% (nanofluid), 2.30% to 9.61% (hybrid nanofluid), and 3.23% to 11.12% (trihybrid nanofluid).

Introduction

The analysis of fluid motion over an enlarging surface has great importance in the field of fluid dynamics, given its diverse applications across various sectors (Daniel et al., 2017b; Mahmood et al., 2024a). Daniel et al. (2020) examined slip conditions in 2D unsteady mixed convection flow of electrical magnetohydrodynamic (MHD) nanofluid over a stretched sheet. Kandasamy et al. (2005) investigated the homotopy analysis technique to explore the phenomena of free convective 2D hydromagnetic fluid pattern over an extended vertical surface within an absorbent medium. This investigation included the consideration of irregular magnetization and thermal radiation. The unsteady mixed convection flow of nanofluids across permeable stretching sheets was quantitatively investigated in a work by Daniel et al. (2019, 2020). A 2D axisymmetric boundary layer flow over a moving sheet is the primary focus of Rafique et al.’s (2024b) research, which aims to determine the heat transfer mechanism and the conditions necessary to achieve high cooling speeds. The Boussinesq approximation and MHD effects were used by Islam et al. (2024) to explore incompressible triple diffusive fluid flow across a linearly stretched surface. Nanofluids over deformable surfaces subjected to nonlinear thermal radiation were the subject of the research of Ullah et al. (2024). Prasad et al. (2018) investigated the flow and heat transmission properties of a Casson liquid on a perforated vertical stretched sheet while systematically varying liquid parameters. Wahid and Said (2018) studied the resolution of the problem involving consequence of thermal emission and magnetization on temperature exchange in a microcosmic liquid along a vertical stretched surface.

The hybrid nanofluid involves the combination of two distinct types of nanomaterials distributed in a base fluid (Rafique et al., 2024c). Consequently, a well-chosen combination of these nanoparticles (NPs) can enhance each other’s positive attributes while modifying the drawbacks associated with a singular material. For instance, introducing distinct NPs has the potential to elevate the overall mixture’s thermal conductivity. Numerous scholars (Mahmood et al., 2024b; Rafique et al., 2024d) have empirically explored hybrid nanofluids and demonstrated an augmented heat transfer rate in these combinations. Khan et al. (2024) studied the flow of a 2D MHD boundary layer over a stretched sheet in a porous medium. Their work focused on hybrid nanofluids’ innovative usage. Mishra (2024) studied the electromagnetic and hydrothermal characteristics of a hybrid nanofluid $({\mathrm{Fe}}_3{\mathrm{O}}_4-{\mathrm{CoFe}}_2{\mathrm{O}}_4/\mathrm{EG}-\mathrm{water})$ moving over a porous plate, taking into consideration suction, injection, Troian, and Thompson slips. The second-grade hybrid nanofluid with MHDs was used by Kezzar et al. (2024) to model Jeffery Hamel’s channel flow.

A perfect ternary hybrid nanofluid (THNF) is created by three nanostructures interacting. The THNF conducts heat better than nanofluid and hybrid nanofluid (Rafique et al. 2024a, 2024e). Sahoo and Kumar (2020) calculated the effect of heat source on the dynamic viscosity of THNF. Boroomandpour et al. (2020) accompanied a study on the heat capacity of THNF. Ahmed et al. (2021) directed a study on heat transference in turbulent flow using nanofluids with varying concentrations in a quadrilateral thermal exchanger subject to consistent thermal flux. Sepehrnia et al. (2022) experimentally studied the rheological characteristics and kinetic viscosity of 10W40 motor oil utilizing ternary hybrid nanocomponents. Algehyne et al. (2022) reported the trihybrid nanoliquid over an enlarging surface. Jakeer et al. (2023) assessed the upshot of Darcy–Forchheimer effects on the hydromagnetic flow of THNF with radiation effect over a vertical stretchable surface. The study emphasized the increasing thermal competence, offering valuable insights for applications in vehicle cooling, especially with the utilization of hybrid nanofluids. Ramzan et al. (2023) studied the hydromagnetic flow of THNF over cone and wedge geometries, prevalent in polymer data processing applications. The inclusion of MHD in the ternary hybrid nanoliquid had exhibited improved heat transport when compared to conventional nanoliquid. Animasaun et al. (2022) reviewed the THNF, aiming to offer insights into convective linked with thermal boundary layer. Khan et al. (2023) observed the flow characteristics of nanoliquid inclosing ternary hybrid NPs in a Sutterby fluid.

The study of MHD in convection processes is essential in thermal systems, particularly when using ternary nanofluids (Ullah et al., 2024). Heat transfer and unsteady mixed convection electrical MHD flow induced by nanofluids over a porous stretched sheet were studied using the Buongiorno model by Daniel et al. (2017a). Daniel et al. (2018b) numerically investigate the effects of thermal stratification, applied electric and magnetic fields on a boundary layer flow of an electrically conducting nanofluid over a nonlinearly stretching sheet of variable thickness. A 3D magnetized hybrid nanofluid was studied by Mohyud-Din et al. (2020). The researchers employed thermal radiations and the novel Cattaneo–Christov model. Fatunmbi et al. (2024) studied magneto hybrid nanofluid flow dynamics, melting heat transfer, and entropy production in porous media.

In the field of fluid dynamics, an exponential heat source or sink is conventionally denoted as a component within the governing equations that summarizes a heat generation or absorption phenomenon exhibiting an exponential functional form. This term elucidates the modulation of the rate of heat generation/absorption concerning spatial coordinates or temporal evolution. The utilization of an exponential function in this context serves to articulate the dynamic variation in the intensity of the thermal influence relative to spatial dimensions or temporal progression, thereby affording a sophisticated representation of the thermal dynamics within the fluidic system under consideration (Ali et al., 2023). Ali et al. (2024b) numerically assessed the 3D Darcy hybrid nanoliquid flow with partial slip and heat source/sink effects over a turning disk. Farooq et al. (2021) examine the occurrence of bioconvection in a Carreau nanofluid flow, considering different temperature influences. This flow was produced using an elongated cylinder. Chu et al. (2022) numerically analyzed hybrid nanofluids with quartet of nanoparticulate varieties, considering a tilted magnetization effect. The study examined a 2D unsteady flow over an absorbent elongating surface. Ali and Jubair (2023) reported the rheological features of Darcy nanoliquid flow with heat source and thermal emission and across a curved surface. Alharbi et al. (2023) studied the Maxwell–Sutterby nanofluid flow over a plan sheet. The investigation took into account various factors such as bioconvection, swimming microbes, Brownian motion, exponential heat sources/sinks, and the influence of thermal conductivity, all under convective boundary conditions (BCs). Rawat et al. (2023) reported the nanofluid flow over an extended surface with the influence of exponential heat source/sink.

Hamid and Khan (2018) investigated the interaction between changeable magnetization, thermal production, and assimilation in the transient motion of a Williamson fluid. Zeeshan et al. (2018) conducted a study on the MHD radiative Couette–Poiseuille flow of nanoliquid over a horizontal plane. Dhlamini et al. (2019) investigated the special effects of activation energy and binary chemical processes in a time-dependent mixed convective flow across an indefinitely elongated boundary. Ali et al. (2024) and Raza et al. (2022) reported the hybrid nanoliquid flow over a permeable surface. Li et al. (2023) scrutinized the impacts of activation energy on the MHD flow of a Casson fluid thru a permeable medium within a horizontal conduit.

Fluid movement through porous medium is essential to many industries’ product creation. Examples include porous bearings, biosensors, kerosene, thermal insulation, drainage, and irrigation (Gorla et al., 2010; Gorla and Chamkha, 2011). Many efforts have been made to improve fluid flow into permeable zones using Darcy’s law. This guideline does not apply to faster and more permeable circumstances. Most physical difficulties include higher flow rates and imbalanced porosity conditions (Chamkha, 1997; Chamkha and Khaled, 2000). To get around this limitation, Forchheimer (1901) modified the momentum equation by using a quadratic velocity element. Theoretically, buoyancy and heat radiation affect MHD flow over a stretched porous sheet, which Daniel and Daniel (2015) examined. Theoretically studying the continuous 2D electrical MHD nanofluid flow over nonlinear expanding or contracting sheet in porous media is the goal of Daniel et al. (2018a). Daniel (2015) used homotopy analysis (HAM) to explore a steady MHD laminar flow of an incompressible viscous and electrically conducting fluid close to a porous stretched sheet. Recently, Mahmood et al. (2024c) want to study how thermal conductivity models affect important parameters in Darcy–Forchheimer flow and heat transmission of a hybrid nanofluid of ${\mathrm{Al}}_2{\mathrm{O}}_3-\mathrm{Cu}$ and water over a permeable surface.

In this study, the heat transfer through trihybrid CYNF flow with the effect magnetic dipole across a stretching sheet is observed. The THNF was made by dispersing ternary NPs in a mixture of 50% water and 50% ethylene glycol. TiO₂, Al₂O₃, and SiO₂ are used as NPs. The fluid velocity and heat transfer are examined under the influence of Darcy–Forchheimer, chemical reaction, convective condition, activation energy, and exponential heat source. The fluid flow has been designed in form of velocity, energy, and concentration equations. The set of modeled equations is simplified into dimensionless form of ordinary differential equations (ODEs) by using similarity substitutions. Numerically, the system of lowest order ODEs is calculated through the PCM. Furthermore, some core novelties of the current analysis are:

To study the CYNF over a stretching sheet. The unique property of Carreau–Yasuda is its constant viscosity, which makes it prominent from other types of non-Newtonian fluid and also applicable to several industrial applications.

To elucidate the effects on energy transmission of an exponential heat source or sink.

To scrutinize the impacts of Arrhenius activation energy on mass propagation.

To determine how a magnetic dipole influences the flow of fluid through a porous medium.

To find the numerical solution of system of nonlinear partial differential equations.

Applications

This breakthrough improves cooling systems in high-performance electronics, which need perfect temperature management to prevent overheating and ensure operational stability. Carreau–Yasuda nanofluids’ unique rheology may also increase heat transmission in chemical reactors, energy storage systems, and microfluidic devices. This is especially useful for controlling fluid behavior at varied shear rates.

In the impending “Mathematical analysis” section, the flow problem is modeled. In the “Numerical solution” section, the flow equations are numerically solved. In the “Results and discussion” section, the graphical results are explained in detail.

Mathematical analysis

Consider the mass and heat conduction through 2D CYNF comprises of ternary hybrid NPs over an elongating sheet. Three different sorts of NPs (TiO₂, Al₂O₃, and SiO₂) are dispersed in the water (50%) and ethylene glycol (50%). Furthermore, the main assumptions are:

The 2D flow is supposed to be an incompressible.

The fluid flow is due to uniform stretching of sheet along x-axis.

Magnetic dipole is located horizontally and is placed at the center as displayed in Figure 1(a) and (b).

The y-axis is normal to the sheet surface.

The significances of Darcy effect, convective condition, and magnetic dipole are studied on the fluid flow.

Figure 1.

The physical sketch of the flow model. (a) The physical sketch of the flow model. (b) Preparation of ternary hybrid nanofluid.

The basic equations are specified as (Wang et al., 2022):

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

(1)

$$\begin{array}{rl}{\rho }_{Thnf}\left(u{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}}\right)=& -{\displaystyle \frac{\mathrm{\partial }P}{\mathrm{\partial }x}+{\mu }_{0}M{\displaystyle \frac{\mathrm{\partial }H}{\mathrm{\partial }x}+{\nu }_{Thnf}\left({\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+{\mathrm{\Lambda }}^d\left({\displaystyle \frac{m-1}{d}}\right)(d+1){\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}{\left({\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}\right)}^d}}\right)}}\\ & -{\displaystyle \frac{{\mu }_{Thnf}}{{\rho }_{Thnf}K}u-{\displaystyle \frac{C_b}{\sqrt{K}}{u}^2,}}\end{array}$$

(2)

$$\begin{array}{rl}& (\rho C_p{)}_{Thnf}\left(u{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }y}}}\right)+{\mu }_0\left(u{\displaystyle \frac{\mathrm{\partial }H}{\mathrm{\partial }x}+v{\displaystyle \frac{\mathrm{\partial }H}{\mathrm{\partial }y}}}\right)T{\displaystyle \frac{\mathrm{\partial }M}{\mathrm{\partial }T}=K_{Thnf}{\displaystyle \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}+{\displaystyle \frac{Q_e*}{{(\rho C_p)}_{hnf}}(T-T_{\mathrm{\infty }})}}}\\ & \exp \left(-\sqrt{{\displaystyle \frac{a}{{\upsilon }_f}ny}}\right),\end{array}$$

(3)

$$u{\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }x}+v{\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }y}=D_B{\displaystyle \frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }{y}^2}+{\displaystyle \frac{D_T}{T_{\mathrm{\infty }}}{\displaystyle \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}-k_r^2(C-C_0){\left({\displaystyle \frac{T}{T_{\mathrm{\infty }}}}\right)}^n\exp \left(-{\displaystyle \frac{E_a}{{\kappa }_T}}\right).}}}}}$$

(4)

Here, H, P, and $\mathrm{\Lambda }$ are the magnetic field, pressure, and Carreau–Yasuda fluid term; $Q_0$ , ${\mu }_0$ , $E_a$ , and K are the exponential heat source/sink, magnetic permeability, activation energy, and porosity factor.

The BCs are (Wang et al., 2022):

$$\begin{array}{rl}& u=sx,-k{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }y}=k_1(T_f-T)=0,v=0,C=C_w\mathrm{as}\eta =0,}\\ & u=0,T=T_{\mathrm{\infty }},C=C_{\mathrm{\infty }}\mathrm{as}\eta \to \mathrm{\infty }.\end{array}$$

(5)

Here, $\beta$ is the magnetic force, $H_x\mathrm{and}{H}_y$ are the magnetic inductions along and x- and y-axes expressed as (Wang et al., 2022):

$$\beta ={\displaystyle \frac{\delta }{2\pi }\left({\displaystyle \frac{x}{x^2+{(y+d)}^2}}\right),H_x={\displaystyle \frac{-\mathrm{\partial }\delta }{\mathrm{\partial }x}={\displaystyle \frac{\delta }{2\pi }\left({\displaystyle \frac{x-{(d+y)}^2}{{(x^2+{(d+y)}^2)}^2}}\right),H_y={\displaystyle \frac{-\mathrm{\partial }\delta }{\mathrm{\partial }y}={\displaystyle \frac{\delta }{2\pi }\left({\displaystyle \frac{2x(d+y)}{{(x^2+{(d+y)}^2)}^2}}\right).}}}}}$$

(6)

The magnetic induction magnitude is:

$$H={\left[{\left({\displaystyle \frac{\mathrm{\partial }\delta }{\mathrm{\partial }y}}\right)}^2+{\left({\displaystyle \frac{\mathrm{\partial }\delta }{\mathrm{\partial }x}}\right)}^2\right]}^{12},H_a={\displaystyle \frac{\mathrm{\partial }\delta }{\mathrm{\partial }b}\left({\displaystyle \frac{-2a}{{(d+b)}^4}}\right),H_b={\displaystyle \frac{\mathrm{\partial }\delta }{\mathrm{\partial }b}\left({\displaystyle \frac{-2}{{(d+b)}^3}+{\displaystyle \frac{4a^2}{{(d+b)}^5}}}\right).}}$$

(7)

The similarity variables are defined as (Gul et al., 2020):

$$v(\eta )=-(s{v}_f{)}^{\frac{1}{2}}f,u(\eta )=sx{f}^{\mathrm{\prime }},\theta (\eta )={\displaystyle \frac{T_{\mathrm{\infty }}-T}{T_{\mathrm{\infty }}-T_w},\eta =x\sqrt{{\displaystyle \frac{{\rho }_{f}s}{{\mu }_f}}},\phi (\eta )={\displaystyle \frac{C_{\mathrm{\infty }}-C}{C_{\mathrm{\infty }}-C_w}.}}$$

(8)

We obtain, from equation (8):

$$f^{\mathrm{″}\mathrm{\prime }}+{\displaystyle \frac{(d+1)(m-1)}{d}(We{)}^d(f^{\mathrm{″}}{)}^d{f}^{\mathrm{″}\mathrm{\prime }}-\left({\displaystyle \frac{2\theta {\rho }_f\beta }{{(\eta +\gamma )}^4}+{f^{\mathrm{\prime }}}^2-f{f}^{\mathrm{″}}}\right){\displaystyle \frac{v_{Thnf}}{v_f}-kr{f}^{\mathrm{\prime }}-Fr(f^{\mathrm{\prime }}{)}^2,}}$$

(9)

$$\begin{array}{rl}{\theta }^{\mathrm{″}}& +{\displaystyle \frac{{(\rho C_p)}_{Thnf}{k}_f}{{(\rho C_p)}_f{k}_{Thnf}}Pr(f{\theta }^{\mathrm{\prime }}-2f^{\mathrm{\prime }}\theta )+{\displaystyle \frac{{(\rho C_p)}_{Thnf}{k}_f}{{(\rho C_p)}_f{k}_{Thnf}}{\displaystyle \frac{2\beta \lambda f(\theta -\epsilon )}{{(\eta +\gamma )}^3}+{\displaystyle \frac{k_f(Pr\theta )Q_e\exp (-n\eta )}{k_{Thnf}}}}}}\\ & -{\displaystyle \frac{k_f}{k_{Thnf}}{\displaystyle \frac{4\lambda {(1-{\varphi }_2)}^{-2.5}}{{(1-{\varphi }_1)}^{2.5}{(1-{\varphi }_3)}^{2.5}}(f^{\mathrm{″}}{)}^2=0,}}\end{array}$$

(10)

$${\phi }^{\mathrm{″}}+{\displaystyle \frac{Nt}{Nb}{\theta }^{\mathrm{″}}-ScKr(1+\epsilon \delta {)}^n\phi \exp \left(-{\displaystyle \frac{E}{1+\epsilon \delta }}\right)=0}$$

(11)

The simplified BCs are:

$$\begin{array}{c}{f}^{\mathrm{\prime }}(0)=1,f(0)=0,{\theta }^{\mathrm{\prime }}(\eta )=-B_i(1-\theta (\eta )),\phi (0)=1,\\ f^{\mathrm{\prime }}(\mathrm{\infty })=0,\theta (\mathrm{\infty })=0,\phi (\mathrm{\infty })=0.\end{array}\}$$

(12)

The dimensionless parameters that appeared in equations (9) to (12) are expressed as (Gul et al., 2020):

$$\begin{array}{c}Pr={\displaystyle \frac{\upsilon }{\alpha },\beta ={\displaystyle \frac{\gamma }{2\pi }{\displaystyle \frac{{\mu }_0(T_{\mathrm{\infty }}-T_w)\rho }{{\mu }^2},\epsilon ={\displaystyle \frac{T_{\mathrm{\infty }}}{T_{\mathrm{\infty }}-T_w},\gamma =\sqrt{{\displaystyle \frac{s\rho c^2}{\mu }}},}}}}\\ \lambda ={\displaystyle \frac{su}{\rho K(T_{\mathrm{\infty }}-T_w)},Q_e={\displaystyle \frac{Q_e*l}{a{(\rho C_p)}_f},\sigma ={\displaystyle \frac{k_T^2}{c},E={\displaystyle \frac{E_a}{\kappa T_{\mathrm{\infty }}}.}}}}\end{array}\}$$

(13)

Equations (9) to (11) perform as a non-Newtonian fluid model, which can be reduced to the Newtonian fluid model by placing $\beta =0$ and We = 0.

The thermophysical properties of trihybrid nanofluid are expressed as (Bilal et al., 2022; Rafique et al., 2024e):

$$\begin{array}{c}{\displaystyle \frac{{\mu }_{Thnf}}{{\mu }_f}={\displaystyle \frac{1}{{(1-{\varphi }_{Si{O}_2})}^{2.5}{(1-{\varphi }_{A{l}_2{O}_3})}^{2.5}{(1-{\varphi }_{Ti{O}_2})}^{2.5}},{\displaystyle \frac{{\rho }_{Thnf}}{{\rho }_f}=(1-{\varphi }_{A{l}_2{O}_3})}}}\\ \left[(1-{\varphi }_{A{l}_2{O}_3})\left\{(1-{\varphi }_{Ti{O}_2})+{\varphi }_{Ti{O}_2}{\displaystyle \frac{{\rho }_{Ti{O}_2}}{{\rho }_f}}\right\}+{\varphi }_{A{l}_2{O}_3}{\displaystyle \frac{{\rho }_{A{l}_2{O}_3}}{{\rho }_f}}\right]+{\varphi }_{Si{O}_2}{\displaystyle \frac{{\rho }_{Si{O}_2}}{{\rho }_f},}\\ {\displaystyle \frac{{(\rho cp)}_{Thnf}}{{(\rho cp)}_f}={\varphi }_{Si{O}_2}{\displaystyle \frac{{(\rho cp)}_{Si{O}_2}}{{(\rho cp)}_f}+(1-{\varphi }_{Si{O}_2})\left[(1-{\varphi }_{Ti{O}_2})\left\{(1-{\varphi }_{Ti{O}_2})+{\varphi }_{Ti{O}_2}{\displaystyle \frac{{(\rho cp)}_{Ti{O}_2}}{{(\rho cp)}_f}}\right\}+{\varphi }_{Ti{O}_2}{\displaystyle \frac{{(\rho cp)}_{Ti{O}_2}}{{(\rho cp)}_f}}\right]}}\},\\ \begin{array}{c}{\displaystyle \frac{k_{Thnf}}{k_{hnf}}=\left({\displaystyle \frac{k_{Ti{O}_2}+2k_{hnf}-2{\varphi }_{Ti{O}_2}(k_{hnf}-k_{Ti{O}_2})}{k_{Ti{O}_2}+2k_{hnf}+{\varphi }_{Ti{O}_2}(k_{hnf}-k_{Ti{O}_2})}}\right),{\displaystyle \frac{k_{hnf}}{k_{nf}}=\left({\displaystyle \frac{k_{A{l}_2{O}_3}+2k_{nf}-2{\varphi }_{A{l}_2{O}_3}(k_{nf}-k_{Zn})}{k_{A{l}_2{O}_3}+2k_{nf}+{\varphi }_{A{l}_2{O}_3}(k_{nf}-k_{Zn})}}\right)}}\\ \end{array}\\ {\displaystyle \frac{{\sigma }_{Thnf}}{{\sigma }_{hnf}}=\left(1+{\displaystyle \frac{3\left({\displaystyle \frac{{\sigma }_{Ti{O}_2}}{{\sigma }_{hnf}}-1}\right){\varphi }_{Ti{O}_2}}{\left({\displaystyle \frac{{\sigma }_{Ti{O}_2}}{{\sigma }_{hnf}}+2}\right)-\left({\displaystyle \frac{{\sigma }_{Ti{O}_2}}{{\sigma }_{hnf}}-1}\right){\varphi }_{Ti{O}_2}}}\right),{\displaystyle \frac{{\sigma }_{hnf}}{{\sigma }_{nf}}=\left(1+{\displaystyle \frac{3\left({\displaystyle \frac{{\sigma }_{A{l}_2{O}_3}}{{\sigma }_{nf}}-1}\right){\varphi }_{A{l}_2{O}_3}}{\left({\displaystyle \frac{{\sigma }_{A{l}_2{O}_3}}{{\sigma }_{nf}}+2}\right)-\left({\displaystyle \frac{{\sigma }_{A{l}_2{O}_3}}{{\sigma }_{nf}}-1}\right){\varphi }_{A{l}_2{O}_3}}}\right).}}\end{array}\}$$

(14)

In the above equation, $\mu$ is the viscosity, $\rho$ is the density, k is the thermal conduction, and $\sigma$ is the electrical conductivity.

The engineering interest physical quantities are expressed as:

$$(Re{)}^{\frac{1}{2}}{C}_f={\displaystyle \frac{-\left(1+{\displaystyle \frac{m-1}{d}{({f^{\mathrm{\prime }}}^{\mathrm{\prime }}(0))}^{d}W{e}^2}\right){f^{\mathrm{\prime }}}^{\mathrm{\prime }}(0)}{{(1-{\varphi }_2)}^{2.5}{(1-{\varphi }_1)}^{2.5}{(1-{\varphi }_3)}^{2.5}},(Re{)}^{\frac{-1}{2}}Nu={\displaystyle \frac{-K_f}{K_{Thnf}}{\theta }^{\mathrm{\prime }}(0).}}$$

(15)

Numerical solution

The numerical solution of the non-dimensional set of ODEs is obtained through PCM. The detailed methodology is followed (Wang et al., 2022):

Step 1: introducing variables for redoing system of ODEs to lowest order

$${\mathrm{\Re }}_1=f(\eta ),{\mathrm{\Re }}_2=f^{\mathrm{\prime }}(\eta ),{\mathrm{\Re }}_3=f^{\mathrm{″}}(\eta ),{\mathrm{\Re }}_4=\theta (\eta ),{\mathrm{\Re }}_5={\theta }^{\mathrm{\prime }}(\eta ),{\mathrm{\Re }}_6=\phi (\eta ),{\mathrm{\Re }}_7={\phi }^{\mathrm{\prime }}(\eta ).$$

(16)

By placing equation (15) in equations (9) to (11) and equation (12), we get the first-order ODEs as:

$${\mathrm{\Re }}_3^{\mathrm{\prime }}+{\displaystyle \frac{(d+1)(m-1)}{d}(We{)}^d{\mathrm{\Re }}_3^{\mathrm{\prime }}({\mathrm{\Re }}_3{)}^d-\left({\mathrm{\Re }}_2^2-{\mathrm{\Re }}_1{\mathrm{\Re }}_3+{\displaystyle \frac{2\beta {\rho }_f{\mathrm{\Re }}_4}{{(\eta +\gamma )}^4}}\right){\displaystyle \frac{v_{Thnf}}{v_f}-kr{\mathrm{\Re }}_2-Fr({\mathrm{\Re }}_2{)}^2.}}$$

(17)

$$\begin{array}{rl}{{\mathrm{\Re }}^{\mathrm{\prime }}}_5& +{\displaystyle \frac{{(\rho C_p)}_{Thnf}{k}_f}{{(\rho C_p)}_f{k}_{Thnf}}Pr({\mathrm{\Re }}_1{\mathrm{\Re }}_5-2{\mathrm{\Re }}_2{\mathrm{\Re }}_4)+{\displaystyle \frac{{(\rho C_p)}_{Thnf}{k}_f}{{(\rho C_p)}_f{k}_{Thnf}}{\displaystyle \frac{2\beta \lambda {\mathrm{\Re }}_1({ƛ}_4-\epsilon )}{{(\eta +\gamma )}^3}+{\displaystyle \frac{k_f(Pr{\mathrm{\Re }}_4)Q_e\exp (-n\eta )}{k_{Thnf}}}}}}\\ & -{\displaystyle \frac{k_f}{k_{Thnf}}{\displaystyle \frac{4\mathrm{\Re }{(1-{\varphi }_2)}^{-2.5}}{{(1-{\varphi }_1)}^{2.5}{(1-{\varphi }_3)}^{2.5}}({\mathrm{\Re }}_3{)}^2=0,}}\end{array}$$

(18)

$${\mathrm{\Re }}_7^{\mathrm{\prime }}+{\displaystyle \frac{Nt}{Nb}{\mathrm{\Re }}_5^{\mathrm{\prime }}-ScKr(1+\epsilon \delta {)}^n{\mathrm{\Re }}_6\exp \left(-{\displaystyle \frac{E}{1+\epsilon \delta }}\right)=0,}$$

(19)

with the corresponding BCs:

$$\begin{array}{rl}& {\mathrm{\Re }}_1(0)=0,{\mathrm{\Re }}_2(0)=1,{\mathrm{\Re }}_5=-B_i(1-{\mathrm{\Re }}_4),{\mathrm{\Re }}_6(0)=1,\\ & {\mathrm{\Re }}_2(\mathrm{\infty })=0,{\mathrm{\Re }}_4(\mathrm{\infty })=0,{\mathrm{\Re }}_6(\mathrm{\infty })=0.\end{array}$$

(20)

Step 2: Presenting parameter p:

$$\begin{array}{rl}{{\mathrm{\Re }}^{\mathrm{\prime }}}_3+& {\displaystyle \frac{(d+1)(m-1)}{d}(We{)}^d{{\mathrm{\Re }}^{\mathrm{\prime }}}_3(({\mathrm{\Re }}_3-1)p{)}^d}\\ & -\left({\mathrm{\Re }}_2^2-{\mathrm{\Re }}_1{\mathrm{\Re }}_3+{\displaystyle \frac{2\beta {\rho }_f{\mathrm{\Re }}_4}{{(\eta +\gamma )}^4}}\right){\displaystyle \frac{v_{Thnf}}{v_f}-kr{\mathrm{\Re }}_2-Fr({\mathrm{\Re }}_2{)}^2.}\end{array}$$

(21)

$$\begin{array}{rl}{{\mathrm{\Re }}^{\mathrm{\prime }}}_5& +{\displaystyle \frac{{(\rho C_p)}_{Thnf}{k}_f}{{(\rho C_p)}_f{k}_{Thnf}}Pr({\mathrm{\Re }}_1({\mathrm{\Re }}_5-1)p-2{\mathrm{\Re }}_2{\mathrm{\Re }}_4)+{\displaystyle \frac{{(\rho C_p)}_{Thnf}{k}_f}{{(\rho C_p)}_f{k}_{Thnf}}{\displaystyle \frac{2\beta \lambda {\mathrm{\Re }}_1({ƛ}_4-\epsilon )}{{(\eta +\gamma )}^3}}}}\\ & +{\displaystyle \frac{k_f(Pr{\mathrm{\Re }}_4)Q_e\exp (-n\eta )}{k_{Thnf}}-{\displaystyle \frac{k_f}{k_{Thnf}}{\displaystyle \frac{4\mathrm{\Re }{(1-{\varphi }_2)}^{-2.5}}{{(1-{\varphi }_1)}^{2.5}{(1-{\varphi }_3)}^{2.5}}({\mathrm{\Re }}_3{)}^2=0,}}}\end{array}$$

(22)

$${\mathrm{\Re }}_7^{\mathrm{\prime }}+{\displaystyle \frac{Nt}{Nb}{\mathrm{\Re }}_5^{\mathrm{\prime }}+{\mathrm{\Re }}_7({\mathrm{\Re }}_6-1)p-ScKr(1+\epsilon \delta {)}^n{\mathrm{\Re }}_6\exp \left(-{\displaystyle \frac{E}{1+\epsilon \delta }}\right)=0.}$$

(23)

Step 3: Solving the Cauchy problems

By using numerical implicit system as:

$${\displaystyle \frac{U^{i+1}-U^i}{\mathrm{\Delta }\eta }=A{U}^{i+1}\mathrm{and}{\displaystyle \frac{W^{i+1}-W^i}{\mathrm{\Delta }\eta }=A{W}^{i+1}.}}$$

(24)

The final iterative form is obtained as:

$$U^{i+1}={\displaystyle \frac{U^i}{(I-\mathrm{\Delta }\eta A)}\mathrm{and}{W}^{i+1}={\displaystyle \frac{(W^i+\mathrm{\Delta }\eta R)}{(I-\mathrm{\Delta }\eta A)}.}}$$

(25)

Results and discussion

In this section, the graphical results for $f^{\mathrm{\prime }}(\eta )$ , energy $\theta (\eta )$ , and mass $\phi (\eta )$ profiles are plotted and discussed. The values of trihybrid NPs are taken among 0.01 to 0.04 and represented through $\varphi ,$ which means ${\varphi }_1={\varphi }_2={\varphi }_3.$

Velocity interpretation

Figures 2 to 7 display the demonstration of velocity $f^{\mathrm{\prime }}(\eta )$ field against the variation of $\beta ,$ porous surface factor kr, Weissenberg numbers We, Darcy–Forchheimer factor Fr, trihybrid NP factor $\phi ,$ and power law factor m correspondingly. Figures 2 and 3 display that the fluid velocity develops with the impact of $\beta$ whereas falloffs with the upshot of porosity parameter. Figure 2 illustrates that the velocity profile rises with the ferro-hydrodynamic interaction parameter $(\beta )$ . A magnetic field influences the magnetic NPs in the fluid. $\beta$ measures the fluid’s magnetic field’s interaction with NPs’ magnetic properties. As $\beta$ increases, the fluid’s magnetic forces increase, aligning the magnetic NPs in the magnetic field. Aligning particles reduces fluid flow barriers and enhances momentum transfer, increasing the velocity profile. Magnetic NPs and magnetic fields enhance the effect in nanofluids, hybrid nanofluids, and THNFs. As ferro-hydrodynamic interactions increase, fluid velocity increases, making flow more efficient due to the magnetic field’s lower drag and increased propulsion effect. This effect’s intensity depends on the fluid’s NP type and concentration.

Figure 2.

Ferro-hydrodynamic interaction parameter $\beta$ versus velocity $f^{\mathrm{\prime }}(\eta )$ curve.

Figure 3.

Effect of porous surface parameter on the velocity $f^{\mathrm{\prime }}(\eta )$ curve.

Figure 4.

Weissenberg number We versus velocity $f^{\mathrm{\prime }}(\eta )$ curve.

Figure 5.

Darcy–Forchheimer's term Fr versus velocity curve $f^{\mathrm{\prime }}(\eta )$ .

Figure 6.

Trihybrid nanoparticle $\phi$ versus velocity $f^{\mathrm{\prime }}(\eta )$ curve.

Figure 7.

Power law term m versus velocity $f^{\mathrm{\prime }}(\eta )$ curve.

Figure 3 shows a decrease in velocity as the porous surface parameter $(kr)$ rises due to the porous material’s fluid flow resistance. The parameter $(kr)$ commonly expresses the permeability or resistance of porous media. Higher $kr$ values indicate less permeable or more resistant material. In porous mediums, solid structures drag fluid flow. As the permeability coefficient $(kr)$ increases, the fluid flow resistance through the porous medium also increases, leading to a decrease in fluid velocity. More obstacles lead to energy loss and a decrease in flow velocity.

Figure 4 illustrates a downward trend in the velocity profile as the Weissenberg number $(We)$ increases. Viscoelastic fluid mechanics may explain this. The Weissenberg number is calculated by multiplying the fluid’s relaxation time $(\lambda )$ by its characteristic shear rate. It measures elastic effects in moving fluids. As (We) increases, the fluid resists deformation better and “relax” faster. High-viscoelasticity fluids have a low velocity profile because they are less readily deformed.

Figure 5 illustrates that increasing the Darcy–Forchheimer term (Fr) lowers the velocity profile. Because of the porous media and inertial effects, flow resistance increases. A fluid’s flow resistance through a porous medium may be characterized using the Darcy–Forchheimer model, which accounts for both drag forces. As (Fr) increases, the Forchheimer term becomes more influential, implying that inertial effects and viscous drag in the porous medium become more significant. Drag forces cumulatively impede fluid motion, increasing flow resistance.

The velocity profile in Figure 6 shows an increasing trend as the NP volume fraction $(\phi )$ rises, indicating enhanced thermal and mechanical properties of the fluid. Increasing $\phi$ increases NP concentration in the base fluid, improving thermal conductivity and viscosity. The thermal barrier layer is thinner because increased thermal conductivity improves heat transmission. A more constant temperature distribution may reduce the fluid’s resistance to flow and improve the velocity profile. A higher NP volume fraction increases viscosity, which has an impact on flow dynamics. NPs may reduce momentum diffusion at the boundary, resulting in a smaller momentum border layer and higher velocity in the flow’s center portion.

Figure 7 shows that the velocity profile falls as the power law term (m) grows due to the fluid's non-Newtonian properties. In non-Newtonian fluids, the power law model describes the link between shear stress and shear rate. When $m<1$ , the fluid becomes easily flowable, as its apparent viscosity decreases with increasing shear rate. As the shear rate increases, the fluid thickens and thins less, resulting in an increase in apparent viscosity. Fluid flow resistance rises as (m) goes from 0.0 (showing shear thinning) to higher values like 0.3 and 0.7. Greater shear rates increase fluid viscosity due to the shear-thickening behavior of certain fluids. This additional resistance slows the fluid, reducing its velocity profile.

Temperature interpretation

Figure 8 illustrates the influence of Biot number Bi on the energy field. It can be perceived that the energy field enriches with the impact of Bi. Figures 9 and 10 elucidate that the fluid temperature enhances with the impact of $\beta$ and Q_e. Physically, the magnetic dipole fascinates the particles of fluid near the surface of sheet and this fascination creates friction amid the fluid layers and sheet surface, which consequences in the upgrading of energy curve $\theta (\eta )$ as stated in Figure 9. The escalating impact of heat source parameter Q_e offers further energy to the fluid, which speed up the thermal field $\theta (\eta )$ as presented in Figure 10. Figures 11 and 12 described that energy sketch of THNF upraises with the enhancement of Eckert number and accumulation of trihybrid NPs (TiO₂, Al₂O_3, and SiO₂). As we have deliberated before in Figure 6, the mounting number of NPs in EG/water declines its heat absorbing proficiency, which reasons in the enrichment of fluid energy $\theta (\eta )$ curve as revealed in Figure 11.

Figure 8.

Biot number $Bi$ .versus the energy curve $\theta (\eta )$ .

Figure 9.

Heat source term $Q_e$ versus the energy curve $\theta (\eta )$ .

Figure 10.

Ferro-hydrodynamic interaction term versus the energy curve $\theta (\eta )$ .

Figure 11.

Eckert number Ec versus the energy curve $\theta (\eta )$ .

Figure 12.

Ternary hybrid nanoparticle $\phi$ versus the energy curve $\theta (\eta )$ .

Concentration interpretation

Figures 13 to 15 indicate the exposition of concentration outline $\phi (\eta )$ against the impact of E, Sc, and Kr. Figure 13 defines that the consequence of E increases the mass conduction rate. The Arrhenius equation may describe reactions in fluid flows as a function of temperature, explaining the increase in concentration with higher activation energy parameter $(E)$ . The parameter $(E)$ represents an energy barrier that molecules must overcome. Increasing E increases the energy required for diffusion–reaction processes, reducing the number of particles with sufficient energy to react or diffuse. As reaction or diffusion rates drop further from the surface, species concentrations rise. NPs in nanofluids, hybrid nanofluids, and THNFs alter energy distribution and interactions. Higher activation energy reduces thermal energy for mass diffusion, producing a larger concentration boundary layer. Due to complex NP–base fluid interactions, hybrid and THNFs are more affected by this phenomenon. These interactions may slow diffusion, raising the concentration profile.

Figure 13.

Activation energy E versus mass curve $\phi (\eta )$ .

Figure 14.

Schmidt number versus mass curve $\phi (\eta )$ .

Figure 15.

Chemical reaction parameter Kr versus mass curve $\phi (\eta )$ .

The concentration profile drops as seen in Figure 14 with higher values of the Schmidt number $(Sc)$ due to its definition as the ratio of momentum diffusivity (kinematic viscosity to mass diffusivity). A high Schmidt number results in a smaller momentum diffusivity compared to mass diffusivity. Physically, this means that the concentration boundary barrier is smaller because solutes and NPs have less motivation to spread. The species loses concentration quicker.

The concentration profile in Figure 15 drops as the reaction parameter (Kr) rises due to the quicker chemical reaction. Chemical reactions in the fluid consume diffusing species, and the chemical reaction parameter (Kr) represents their rate. As the rate constant $(Kr)$ rises, the response rate increases, depleting fluid species faster. The faster reduction of species leads to a more significant decrease in their concentration. The rise in response rate accelerates the consumption of species, thereby reducing the concentration of the boundary layer.

Figures 16 and 17 show the percentage valuation for energy and velocity transfer rate. The values of $\varphi$ are taken among 0.01 to 0.04. It can be noticed that the energy propagation rate enhances with the rising numbers of SiO₂, SiO₂+ TiO₂, and SiO₂+ TiO₂+ Al₂O₃ for three different cases, from 1.22% to 5.98% (nanofluid), 2.30% to 9.61% (hybrid nanofluid), and 3.23% to 11.12% (trihybrid nanofluid), respectively. The similar phenomena have been observed in case of velocity graph.

Figure 16.

Percentage energy transfer rate.

Figure 17.

Percentage velocity transfer rate.

Table 1 reveals the numerical values for base fluid and ternary hybrid NPs. The numerical assessment of $-(Re{)}^{1/2}{C}_f$ and $-(Re{)}^{-1/2}Nu$ against the variation of viscous dissipation, power law parameter, heat source, and Weissenberg number are deliberated in Table 2. The energy source factor causes lessening in the fluid flow rate energy transfer rate.

Table 1.

The numerical values of TiO₂, SiO₂, Al₂O₃, and ethylene glycol/water (Bilal et al., 2022; Rafique et al., 2024e).

	$\rho (kg/m^3)$	$Cp(\mathrm{j}/\mathrm{kg}K)$	$k(W/mK)$	$\sigma (\mathrm{S}/m)$
SiO2	2270	3630	1.4013	$3.5\times {10}^6$
TiO2	4250	686.2	8.953	$2.38\times {10}^6$
Al2O3	6310	773	32.9	$5.96\times {10}^7$
C2H6O2-H2O	1063.8	3630	0.387	0.00509

Table 2.

Numerical comparison of the $(Re{)}^{1/2}{C}_f$ and $-(Re{)}^{-\frac{1}{2}}Nu$ with the published study.

Parameters	Values	Wang et al., (2022), $-(Re{)}^{1/2}{C}_f$	Present study, $-(Re{)}^{1/2}{C}_f$	Wang et al. (2022), $-(Re{)}^{-\frac{1}{2}}Nu$	Present study, $-(Re{)}^{-\frac{1}{2}}Nu$
Qe	0.2	0.287472225	0.287476321	0.158246732	0.158246735
	0.4	0.047425935	0.047425937	0.112661776	0.112661777
	0.6	0.065673302	0.065673304	0.265108748	0.265108756
We	0.1	0.181510502	0.181510504	0.516015477	0.516015487
	0.4	0.199382000	0.199382015	0.504825966	0.504825977
	0.9	0.378384841	0.378384844	0.253441241	0.253441243
$\lambda$	0.0	0.163087093	0.163087094	0.527911357	0.527911358
	0.1	0.162497815	0.162497817	0.532030817	0.532030827
	0.2	0.162250018	0.162250028	0.441622587	0.441622671
M	0.1	0.055701071	0.055701074	0.459527833	0.459527841
	0.2	0.138855310	0.138855312	0.609365052	0.609365063
	0.3	0.162911012	0.162911017	0.642710054	0.642710073

Conclusion

We have estimated the energy transfer through trihybrid CYNF flow with the effect of magnetic dipole across a stretching sheet is examined in the present study. The THNF has been equipped by the dispersal of ternary NPs in the water (50%) and ethylene glycol (50%). The fluid velocity and heat transfer are examined under the effect of Darcy–Forchheimer, chemical reaction, convective condition, and exponential heat source. Numerically, the system of lowest order ODEs are deliberated through the PCM. The key judgments are:

The energy propagation rate enhances with the rising numbers of TiO₂, Al₂O₃ and SiO₂ NPs from 1.22% to 5.98% in case of nanofluid and from 2.30% to 9.61% in case of hybrid nanofluid.

The fluid velocity develops with the impact of magnetic dipole parameter, whereas declines with the influence of porosity factor.

The fluid velocity falls with the effect of We, Darcy factor and Power law term m.

The addition of trihybrid NPs in base fluid (EG/water) amplifies the fluid velocity.

The energy transfer rate augments from 3.23% to 11.12% in case of trihybrid nanofluid.

The stimulus of Biot number Bi enriches the fluid temperature.

The fluid temperature enhances with the impact of $\beta$ (Ferro-hydrodynamic interaction parameter) and Q_e.

The consequence of E increases the mass conduction rate, whereas, the effect of Sc and Kr lessens the mass dissemination rate.

The energy sketch of THNF upraises with the enhancement of Eckert number and accumulation of trihybrid NPs.

Future recommendations

Experimental verification of this study’s theoretical and numerical models would strengthen and apply the conclusions. Practical applications may need precise measurements of viscosity, convective heat transfer, and thermal conductivity.