Investigation of Thermal Performance of Ternary Hybrid Nanofluid Flow in a Permeable Inclined Cylinder/Plate

Abstract

This article comprehensively investigates the thermal performance of a ternary hybrid nanofluid flowing in a permeable inclined cylinder/plate system. The study focuses on the effects of key constraints such as the inclined geometry, permeable medium, and heat source/sink on the thermal distribution features of the ternary nanofluid. The present work is motivated by the growing demand for energy-efficient cooling systems in various industrial and energy-related applications. A mathematical model is developed to describe the system’s fluid flow and heat-transfer processes. The PDEs (partial differential equations) are transformed into ODEs (ordinary differential equations) with the aid of suitable similarity constraints and solved numerically using a combination of the RKF45 method and shooting technique. The study’s findings give useful insights into the behavior of ternary nanofluids in permeable inclined cylinder/plate systems. Further, important engineering coefficients such as skin friction and Nusselt numbers are discussed. The results show that porous constraint will improve thermal distribution but declines velocity. The heat-source sink will improve the temperature profile. Plate geometry shows a dominant performance over cylinder geometry in the presence of solid volume fraction. The rate of heat distribution in the cylinder will increase from 2.08% to 2.32%, whereas in the plate it is about 5.19% to 10.83% as the porous medium rises from 0.1 to 0.5.

1. Introduction

Nanofluids are a form of a fluid composed of a base fluid (such as water, oil or ethylene glycol) with nanoscale particles (usually with sizes less than 100 nm) scattered within it. The incorporation of nanoparticles into the base liquid can increase its thermal characteristics, such as thermal conductivity and heat capacity, resulting in enhanced thermal efficiency. Nanofluids have been proven to have better thermal characteristics than conventional coolants, making them appealing for application in cooling systems for electronic devices, power plants, and cars. Nanofluids can be utilized as heat-transfer fluids in solar thermal systems, which utilize solar energy to generate heat. The increased thermal characteristics of nanofluids can aid in the efficiency of these systems, resulting in more energy production and reduced costs. Nanofluids can be utilized as drug-delivery systems, allowing medications to be administered directly to specific tissues or cells. Moreover, nanofluids have several potential uses in a variety of sectors, including lubrication systems and heat exchangers. Their distinct features, including increased thermal conductivity, physical qualities, and targeting capabilities, make them appealing for various industrial and biological applications. Experiments have demonstrated that nanofluids can have much greater thermal conductivities than the base fluid alone, with up to several-fold improvements recorded in some circumstances. The precise improvement in thermal performance will be determined by a number of parameters, including the kind and concentration of nanoparticles, the characteristics of the base fluid, and the operating circumstances. Many reports are made in this view, some of them listed as [1,2,3,4,5]. These works conclude that nanofluids exhibit better thermal performance than base fluids.

A hybrid nanofluid is a type of fluid that mixes nanoparticles with additional additives, such as agents or polymers, to improve its thermal efficiency beyond that of a normal nanofluid. Babu et al. [6] reported that hybrid nanofluid shows better thermal conductivity than nanofluid and viscous fluids. Yang et al. [7] and Esfahani et al. [8] also reviewed hybrid nanofluid and agree with the results of [6]. Atashafrooz et al. [9] showed an improvement in the thermal performance of hybrid nanofluid over viscous fluid. Recently, Adun et al. [10] reviewed the synthesis, stability, thermophysical properties, and thermal distribution of ternary nanofluids. The study revealed that the overall performance of ternary nanofluids is better than hybrid and nanofluids.

Ternary nanofluids (TNFs) are fluids composed of a base liquid and three types of nanoparticles. These fluids have distinct characteristics that can be used for a wide range of possible applications. Ternary nanofluids can have higher thermal conductivity than their base fluid, making them helpful for cooling electronic devices and converting solar thermal energy. Ternary nanofluids, for example, can be utilized in electronic cooling to move heat away from sensitive components, reducing overheating and damage. Ternary nanofluids may be utilized to capture and transport heat from the sun in solar thermal energy conversion, enhancing system efficiency. Ternary nanofluids have several potential uses, including heat transmission, power storage, increased oil extraction, medicinal applications, and environmental services. Adun et al. [10] in 2021 conducted a review of the preparation, thermophysical properties, and stability of TNFs. The study provided a thorough examination of the latest advancements in the field and offered insights into potential future research directions. Alharbi et al. [11] investigated the flow of a TNF over an expanding cylinder while accounting for induction effects using computational approaches. Using a non-Fourier heat flux concept, Sarada et al. [12] studied the consequence of exponential internal heat production on the movement of a TNF. Sharma et al. [13] inspected the movement of TNFs through parallel plates with Nield boundary conditions using numerical and LMBNN. Yogeesha et al. [14] explored the TNF circulation around an unsteady permeable stretched sheet with Dufour and Soret impacts.

Geometries embedded in a porous material, such as in this topic, are of interest for scientific and topographical purposes such as geothermal reservoirs, thermal insulation, nuclear reactor cooling, treatment of water, carbon capture, sensors, actuators, and increased oil recovery. Ullah et al. [15] utilized the Keller box tactic to investigate the impact of MHD and temperature slip on viscous liquid movement across a symmetrically vertically hot plate in a permeable material. Rekha et al. [16] investigated the effect of TPD on temperature transmission and nanofluid circulation in the context of permeable media. Using local thermal non-equilibrium conditions for non-Newtonian liquid circulation integrating hybrid nanoparticles, Alsulami et al. [17] examined heat transfer in permeable media. Yu et al. [18] explored the movement of a nanofluid through a saturated porous surface positioned on a horizontal plane while accounting for the effects of generalized slip in a three-dimensional stagnation-point stream. Rawat et al. [19] investigated HNF movement in a Darcy–Forchheimer porous medium among two parallel spinning discs using a non-uniform HS–S and the Cattaneo–Christov model.

Because of their unique thermal characteristics, nanofluids have been investigated as a possible coolant for heat sources and sinks (HS–S). The nanoparticles in a nanofluid can improve the fluid’s thermal conductivity and heat convection, making it more efficient in removing heat. Interior heat generation or absorption is difficult to accurately calculate; nonetheless, certain simple numerical models may represent its regular behavior for utmost physical situations. As a result, the external heat generation or absorption factor (or heat source/sink) must be considered. This heat source/sink is widely used in cooling electronic equipment, in many industrial processes, in the conversion of solar thermal energy, and in automotive engines. Ahmad et al. [20] explored the motion of mixed convection in a radiative Oldroyd B nano liquid in the attendance of HS–S. Khan et al. [21] examined the stability of the buoyancy magneto circulation of a hybrid nanofluid over a shrinkable/stretchable perpendicular surface generated by a micropolar liquid and subject to a nonlinear HS–S. Kumar et al. [22] looked into the movement of carbon nanotubes floating in dusty nanofluid through a stretched permeable rotating disc with a non-uniform HS–S. Ramesh et al. [23] inspected the impacts of ternary nano liquid with HS–S and permeable media in a stretched divergent/convergent path. Waqas et al. [24] conducted the quantitative examination for 3-dimensional bioconvection circulation of Carreau nanofluid with HS–S and motile microorganisms.

The cylinder/plate geometry provides a large surface area for heat transfer to occur, which can lead to an improvement in the overall thermal transfer performance. The addition of nanofluid particles to the fluid further enhances heat transfer by providing additional thermal energy pathways and by growing the effective thermal conductivity of the fluid. This combination provides a promising approach for improving the temperature-transfer performance of a wide range of systems, including those used in energy generation and cooling processes. There are several real-world applications in various sectors. Refrigeration and air conditioning, electricity production, solar energy systems, thermal energy storage, and industrial operations all rely on these systems. In a Darcy–Forchheimer movement of NF containing gyrotactic organisms under the influence of Wu’s slip over a stretched cylinder/plate, Waqas et al. [25] investigated bio-convection heat radiation. Ali et al. [26] made a study to discover the effects of the C-C double diffusions theory on the bioconvective slip movement of a magneto-cross-nanomaterial on a stretching cylinder/plate. Waqas et al. [27] achieved a numerical model of the bio-convection circulation of a non-Newtonian NF related to a stretched cylinder/plate containing floating motile microorganisms. Selimefendigil and Öztop [28] examined the mixed convection nanoliquid circulation inside a cubic container that was separated by an inner revolving cylinder and a plate. Waqas et al. [29] explored the behavior of NF of type magneto-Burgers flowing along with swimming motile microorganisms, which was organized by a stretching cylinder/plate and characterized by dual variables’ conductivity.

The present study provides a novel contribution to the field of thermal management by investigating the thermal performance of a TNF flowing in a permeable inclined cylinder/plate system. The study focuses on the effects of the inclined geometry, porous medium, and heat source/sink on the thermal distribution characteristics of the ternary nanofluid, which has not been well-studied in the literature. This study represents a significant advancement in the field of thermal management and contributes to a better understanding of the thermal performance of ternary nanofluids in porous media over an inclined cylinder/plate with a heat source/sink.

The aim of this investigation is to answer the following research insight questions regarding a fluid system:

• How does the solid volume fraction affect the velocity and temperature profile of the system?
• How does the fluid profile change as the porous constraint value is increased?
• What is the influence of the heat source/sink on the thermal profile of the system?

To answer these questions, the investigation will analyze the behavior of the fluid system under varying conditions and parameters. The findings of this investigation will provide valuable insights into the impact of different factors on the behavior of the system, which can be useful in optimizing the performance and efficiency of the system in various practical applications. By understanding the complex dynamics of the fluid system, we can develop more effective solutions for a wide range of industrial, scientific, and engineering applications.

2. Mathematical Formulation

The current study examines a steady, laminar flow of a ternary nanofluid in two dimensions over an inclined cylinder/plate system that includes a porous medium and a heat source/sink. The coordinates of the physical model are presented in Figure 1. $V{z}_{1w}$ denotes the reference velocity; $r_1$ and $z_1$ represent the radial and axial coordinates of the cylinder. The temperature of the system is denoted by $T_{w1}$ and far-field temperature is represented by $T_{\infty }\left(T_{w1}>T_{\infty }\right)$. Further, external forces and pressure gradients are assumed to have no influence on the system. Under these assumptions, the mathematical expressions for continuity, momentum, and thermal transfer for the ternary nanofluid flow in the presence of a porous medium and a heat source/sink are stated as follows (see [26,27,29,30,31,32]).

$$\frac{\partial \left(r_{1}V{z}_1\right)}{\partial z_1}+\frac{\partial \left(r_{1}V{r}_1\right)}{\partial r_1}=0$$

(1)

$$\begin{array}{l}V{z}_1\frac{\partial \left(V{z}_1\right)}{\partial z_1}+V{r}_1\frac{\partial \left(V{z}_1\right)}{\partial r_1}={\nu }_{mnf}\left(\frac{{\partial }^{2}V{z}_1}{\partial r_1^2}+\frac{1}{r_1}\frac{\partial V{z}_1}{\partial r_1}\right)+\frac{{\left(\rho \beta \right)}_{mnf}g\left(T_1-T_{\infty }\right)\cos \varsigma }{{\rho }_{mnf}}\\ -\frac{{\nu }_{mnf}}{K_1^{*}}V{z}_1\end{array}$$

(2)

$$V{z}_1\frac{\partial \left(T_1\right)}{\partial z_1}+V{r}_1\frac{\partial \left(T_1\right)}{\partial r_1}={\alpha }_{mnf}\left(\frac{{\partial }^2{T}_1}{\partial r_1^2}+\frac{1}{r_1}\frac{\partial T_1}{\partial r_1}\right)+\frac{Q_1}{{\left(\rho Cp\right)}_{mnf}}\left(T_1-T_{\infty }\right)$$

(3)

With boundary conditions (see [25]).

$$\left.\begin{array}{l}V{z}_1=V{z}_1{}_w=\frac{U_0^{*}{z}_1}{l}\\ V{r}_1=0\\ T_1=T_{w1}\end{array}\right\} \mathrm{at} r_1=R_1$$

(4)

$$V{z}_1\to 0,T_1\to T_{\infty }{r}_1\to \infty$$

(5)

In the above expressions, $V{z}_1,V{r}_1$ are the velocity component along $z_1$ and $r_1$ direction; $\nu$ is the kinematic viscosity defined by $\nu =\frac{\mu }{\rho }$($\mu$- Dynamic viscosity and $\rho$ is density); $K_1^{*}$ denotes porous medium permeability; $\beta$ is the thermal expansion factor; $g$ acceleration due to gravity; $\varsigma$ is the inclination angle; $\alpha =\frac{k}{\rho Cp}$ is the thermal diffusivity of the fluid; ($k$ is thermal conductivity and $Cp$ is the specific heat); $Q_1$ is the heat generation/absorption coefficient.

For similarity variables (see [33]).

$$\left.\begin{array}{l}\psi =\sqrt{V{z}_{1w}{\nu }_f{z}_1}{R}_{1}f\left(\eta \right),V{z}_1=\frac{1}{r_1}\frac{\partial \psi }{\partial r_1},V{r}_1=-\frac{1}{r_1}\frac{\partial \psi }{\partial z_1}\\ \eta =\sqrt{\frac{V{z}_{1w}}{{\nu }_f{z}_1}}\left(\frac{r_1^2-R_1^2}{2R_1}\right),\theta =\frac{T_1-T_{\infty }}{T_{w1}-T_{\infty }}\end{array}\right\}$$

(6)

By introducing Equation (6) into the Equations (1)–(3) and boundary conditions (4)–(5). It is reduced into the following form:

$$\frac{\left[\left(1+\left(2{\delta }_1\right)\eta \right)f^{″\prime }+\left(2{\delta }_1\right)f^{″}\right]}{B_1{B}_2}-{\left(f^{\prime }\right)}^2+f{f}^{″}-\frac{P_m}{B_1{B}_2}{f}^{\prime }+\frac{B_3}{B_2}{\gamma }_1\theta \cos \zeta =0$$

(7)

$$\frac{k_{mnf}\left[\left(1+2{\delta }_1\eta \right){\theta }^{″}+2{\delta }_1{\theta }^{\prime }\right]}{k_f\mathrm{Pr}{B}_4}+f{\theta }^{\prime }+\frac{H_{SS}}{B_4}\theta =0$$

(8)

And reduced boundary conditions:

$$f\left(\eta \right),f^{\prime }\left(\eta \right),\theta \left(\eta \right)=0,1,1\mathrm{at} \eta =0$$

(9)

$$f^{\prime }\left(\infty \right)=0,\theta \left(\infty \right)=0\mathrm{as} \eta \to \infty$$

(10)

From the Equations (7)–(10), the controlling parameter ${\delta }_1=\sqrt{\frac{{\nu }_{f}l}{U_0^{*}{R}_1^2}}$ is the Curvature parameter (${\delta }_1=0$ denotes Plate geometry and ${\delta }_1>0$ denotes Cylinder geometry); $P_m=\frac{{\nu }_{f}l}{U_0^{*}{K}_1^{*}}$ is the porosity constraint; ${\gamma }_1=\frac{Gr}{{\mathrm{Re}}^2}=\frac{g\beta \left(T_{w1}-T_{\infty }\right)l}{V{z}_{1w}{U}_0^{*}}$ is the Buoyancy parameter/Mixed convection parameter; $\zeta$ is the inclined angle; $\mathrm{Pr}=\frac{{\nu }_f}{{\alpha }_f}$ is the Prandtl number; $H_{SS}=\frac{Q_{1}l}{U_0^{*}{\left(\rho Cp\right)}_f}$ is the heat source/sink parameter; $Gr=\frac{g\beta \left(T_{w1}-T_{\infty }\right)z_1{}^3}{{\nu }_f^2}$ is the local Grashof number; and $\mathrm{Re}=\frac{V{z}_{1w}{z}_1}{{\nu }_f}$ is the local Reynolds number. Further, $B_1={\left(1-{\varphi }_1\right)}^{2.5}{\left(1-{\varphi }_2\right)}^{2.5}{\left(1-{\varphi }_3\right)}^{2.5},$$B_2=\left(1-{\varphi }_3\right)\left(\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right)+\frac{{\rho }_{S1}{\varphi }_1}{{\rho }_f}\right]+\frac{{\rho }_{S2}{\varphi }_2}{{\rho }_f}\right)+\frac{{\rho }_{S3}{\varphi }_3}{{\rho }_f},$$B_3=\left(1-{\varphi }_3\right)\left(\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right)+\frac{{\rho }_{S1}{\varphi }_1{\beta }_{{}_{S1}}}{{\beta }_f{\rho }_f}\right]+\frac{{\rho }_{S2}{\varphi }_2{\beta }_{{}_{S2}}}{{\beta }_f{\rho }_f}\right)+\frac{{\rho }_{S3}{\varphi }_3{\beta }_{{}_{S3}}}{{\beta }_f{\rho }_f},$ and $B_4=\left(1-{\varphi }_3\right)\left(\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right)+\frac{{\rho }_{S1}C{p}_{{}_{S1}}{\varphi }_1}{{\rho }_{f}C{p}_f}\right]+\frac{{\rho }_{S2}C{p}_{{}_{S2}}{\varphi }_2}{{\rho }_{f}C{p}_f}\right)+\frac{{\rho }_{S3}C{p}_{{}_{S3}}{\varphi }_3}{{\rho }_{f}C{p}_f}.$ For engineering coefficients and their reduced form:

$$Cf={\left.\frac{\partial V{z}_1}{\partial r_1}\right|}_{r_1=R_1}\frac{{\mu }_{mnf}}{{\rho }_{f}V{z}_{1w}{}^2}\mathrm{and}Nu={\left.\frac{\partial T_1}{\partial r_1}\right|}_{r_1=R_1}\frac{-z_1{k}_{mnf}}{k_f\left(T_{w1}-T_{\infty }\right)}$$

(11)

$$Cf=\frac{f^{″}\left(0\right)}{B_1{\left(\mathrm{Re}\right)}^{0.5}} \mathrm{and} Nu=\frac{-k_{mnf}{\theta }^{\prime }\left(0\right){\left(\mathrm{Re}\right)}^{0.5}}{k_f}$$

(12)

The thermophysical properties of ternary nanofluid are provided below (see [34]).

$${\mu }_{mnf}={\mu }_f/{\left(1-{\varphi }_1\right)}^{2.5}{\left(1-{\varphi }_2\right)}^{2.5}{\left(1-{\varphi }_3\right)}^{2.5}$$

(13)

$${\rho }_{mnf}=\left(\left(1-{\varphi }_3\right)\left(\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right)+\frac{{\rho }_{S1}{\varphi }_1}{{\rho }_f}\right]+\frac{{\rho }_{S2}{\varphi }_2}{{\rho }_f}\right)+\frac{{\rho }_{S3}{\varphi }_3}{{\rho }_f}\right){\rho }_f$$

(14)

$${\left(\rho \beta \right)}_{mnf}=\left(\left(1-{\varphi }_3\right)\left(\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right)+\frac{{\rho }_{S1}{\beta }_{{}_{S1}}{\varphi }_1}{{\rho }_f{\beta }_f}\right]+\frac{{\rho }_{S2}{\beta }_{{}_{S2}}{\varphi }_2}{{\beta }_f{\rho }_f}\right)+\frac{{\rho }_{S3}{\beta }_{{}_{S3}}{\varphi }_3}{{\beta }_f{\rho }_f}\right){\beta }_f{\rho }_f$$

(15)

$${\left(\rho Cp\right)}_{mnf}=\left(\left(1-{\varphi }_3\right)\left(\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right)+\frac{{\rho }_{S1}C{p}_{{}_{S1}}{\varphi }_1}{{\rho }_{f}C{p}_f}\right]+\frac{{\rho }_{S2}C{p}_{{}_{S2}}{\varphi }_2}{{\rho }_{f}C{p}_f}\right)+\frac{{\rho }_{S3}C{p}_{{}_{S3}}{\varphi }_3}{{\rho }_{f}C{p}_f}\right){\rho }_{f}C{p}_f$$

(16)

$$\begin{gathered} k_{nf}=\frac{1}{k_f{}^{-1}}\left(\frac{k_{s1}+2k_f-{\varphi }_{1}2\left[k_f-k_{s1}\right]}{k_{s1}+2k_f+{\varphi }_1\left[k_f-k_{s1}\right]}\right) \\ {k}_{hnf}=\frac{1}{k_{nf}{}^{-1}}\left(\frac{k_{s2}+2k_{nf}-{\varphi }_{2}2\left[k_{nf}-k_{s2}\right]}{k_{s2}+2k_{nf}+{\varphi }_2\left[k_{nf}-k_{s2}\right]}\right) \\ {k}_{mnf}=k_{hnf}\left(\frac{k_{s3}+2k_{hnf}-2{\varphi }_3\left[k_{hnf}-k_{s3}\right]}{k_{s3}+2k_{hnf}+{\varphi }_3\left[k_{nf}-k_{s3}\right]}\right) \end{gathered}$$

(17)

3. Numerical Procedure and Validation

The governing Equations (1)–(3) along with the boundary conditions (4) and (5) can be very challenging to solve. To simplify the problem, similarity variables (6) are often used to reduce the governing equations into a set of ordinary differential equations (ODEs) without compromising the originality of the equations. However, the reduced ODEs (7) and (8) and boundary conditions (9) and (10) remain highly nonlinear and two-point in nature, making them difficult to solve directly. One effective approach to overcome this challenge is to transform the reduced ODEs into a first-order system. This technique simplifies the problem and provides a more straightforward method to solve the equations. This approach has been widely used in various fields of science and engineering and has been proven to be effective in solving complex problems. By converting the reduced ODEs into a first-order system, we can also analyze the stability and behavior of the system more effectively. This is particularly useful in understanding the dynamics of complex systems, where the behavior can be affected by various parameters and boundary conditions. For this consider, $\begin{array}{l}\left[f,f^{\prime },f^{″}\right]=\left[p,q,r\right]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left[\theta ,{\theta }^{\prime }\right]=\left[p_1,q_1\right]\end{array}$.

By substituting these terms in the resultant equations, the equation becomes

$$f^{″\prime }=\frac{-B_1{B}_2}{\left(1+2{\delta }_1\eta \right)}\left(\frac{2{\delta }_{1}r}{B_1{B}_2}+rp-{\left(q\right)}^2-\frac{P_m}{B_1{B}_2}q+\frac{B_3}{B_2}{\gamma }_1{p}_1\cos \zeta \right)$$

(18)

$${\theta }^{″}=\frac{-k_f\mathrm{Pr}{B}_4}{k_{mnf}\left(1+2{\delta }_1\eta \right)}\left(\frac{k_{mnf}2{\delta }_1{q}_1}{k_f\mathrm{Pr}{B}_4}+p{q}_1+\frac{H_{SS}}{B_4}{p}_1\right)$$

(19)

and the boundary conditions become

$$\left.\begin{array}{l}p\left(0\right)=0\\ q\left(0\right)=1\\ r\left(0\right)={\vartheta }_1\\ p_1\left(0\right)=1\\ q_1\left(0\right)={\vartheta }_2\end{array}\right\}$$

(20)

The terms $p,q,r,p_1 \& q_1$ represent $f,f^{\prime },f^{″},\theta \& {\theta }^{\prime }$ respectively; ${\delta }_1,{\gamma }_1,\zeta \hspace{0.17em}$ and $H_{SS}$ denote parameters.

The Runge–Kutta–Fehlberg (RKF-45) technique is implemented to crack the system of equations represented by (18) and (19), as well as the boundary conditions stated in Equation (20). A shooting strategy is used to identify the unknown variables in Equation (20): the thermophysical properties of the nanofluid as indicated in Equations (13) to (17), thermophysical characteristics stated in

Table 1. Thermophysical characteristics of nanoparticles and base liquid are provided by (see [35]).

Properties	${\mathit{H}}_2\mathit{O}$	$\mathit{A}{\mathit{l}}_2{\mathit{O}}_3$	$\mathit{T}\mathit{i}{\mathit{O}}_2$	$\mathit{C}\mathit{u}\mathit{O}$
$\rho \left({\mathrm{kg}/\mathrm{m}}^3\right)\hspace{0.17em}$	$997.1$	$3970$	$4250$	$6320$
$Cp\left(\mathrm{J}/\mathrm{kgK}\right)\hspace{0.17em}$	$4179$	$765$	$686.2$	$531.8$
$k\left(\mathrm{W}/\mathrm{mK}\right)$	$0.613$	$40$	$8.9538$	$76.5$
$\hspace{0.17em}\beta \times {10}^{-5}\left(1/\mathrm{K}\right)$	$21$	$0.85$	$0.9$	$1.8$

, along with the values of the thermophysical properties of the nanoparticles mentioned in Table 1 and setting the initial values of the parameters to $P_m={\gamma }_1=0.1,\zeta ={30}^0,{\varphi }_1={\varphi }_2={\varphi }_3=0.01 \& H_{SS}=0.5$. The step size is set to 0.01, and the error tolerance is set to 10⁻⁶, ensuring that the solution is reliable and well-resolved.

The algorithm of the RKF-45 strategy is given below.

The 6 step sizes of the RKF-45 scheme:

$$h_1{f}_1\left(x_i,y_i\right)=k_1$$

(21)

$$h_1{f}_1\left(x_i+\frac{1}{4}{h}_1,y_i+\frac{1}{4}{k}_1\right)=k_2$$

(22)

$$h_1{f}_1\left(x_i+\frac{3}{8}{h}_1,y_i+\frac{3}{32}{k}_1+\frac{9}{32}{k}_2\right)=k_3$$

(23)

$$h_1{f}_1\left(x_i+\frac{12}{13}{h}_1,y_i+\frac{1932}{2147}{k}_1-\frac{7200}{2147}{k}_2+\frac{7296}{2147}{k}_3\right)=k_4$$

(24)

$$h_1{f}_1\left(x_i+h_1,y_i+\frac{439}{216}{k}_1-8k_2+\frac{3680}{513}{k}_3-\frac{845}{4104}{k}_4\right)=k_5$$

(25)

$$h_1{f}_1\left(x_i+\frac{1}{2}{h}_1,y_i+2k_2-\frac{8}{27}{k}_1-\frac{11}{40}{k}_5-\frac{3544}{2565}{k}_3-\frac{1859}{4104}{k}_4\right)=k_6$$

(26)

$$y_{i+1}=-\frac{1}{5}{k}_5+\frac{25}{216}{k}_1+\frac{1408}{2565}{k}_3+\frac{2197}{4104}{k}_4+y_i$$

(27)

$$z_{i+1}=\frac{2}{55}{k}_6+\frac{16}{135}{k}_1+\frac{6656}{12825}{k}_3+\frac{28561}{56430}{k}_4-\frac{9}{50}{k}_5+y_i$$

(28)

The present numerical scheme is validated with the existing available literature and found to be the best match for each other (see

Table 2. Comparison of present numerical scheme with the work of [36] in the absence of ${\delta }_1$, ${\gamma }_1$, $B_1$, and $B_2$.

Parameter	[36]		Present Study
$P_m$	Analytical $\left(-f^{″}\left(0\right)\right)$	SRM $\left(-f^{″}\left(0\right)\right)$	RKF-45 $\left(-f^{″}\left(0\right)\right)$
1	1.41421356	1.41421356	1.4142375
2	1.73205081	1.73205081	1.7320517
5	2.44948974	2.44948974	2.4494897
10	3.31662479	3.31662479	3.3166247

).

4. Results and Discussion

The current inquiry focuses on the numerical solution of reduced Ordinary Differential Equations (ODEs) and Boundary Conditions (BCs), as well as the graphical display of the findings. The effects of numerous dimensionless factors on thermophysical characteristic profiles are carefully investigated and assessed. The findings are provided for two different geometries: cylinder $\left({\delta }_1=0.3\right)$ and plate $\left({\delta }_1=0\right)$. The use of efficient mathematical software allows for accurate and precise answers, allowing for the investigation of the impact of the many dimensionless factors on the system’s thermophysical properties. This gives a thorough knowledge of the nanofluid’s behavior under various situations, offering significant insights into its performance and behavior.

Figure 2 and Figure 3 show the variation of velocity and temperature profiles in the presence of porous constraint $P_m$. As seen in Figure 2, increasing the value of the porosity constraint $P_m$ causes the fluid velocity profile to drop. The porous medium acts as a barrier to fluid flow, reducing fluid velocity and increasing the thickness of the Momentum Boundary Layer (MBL). This decrease in velocity emphasizes the porous constraint’s inhibitory influence on fluid flow. Figure 3 shows the fluctuation of the thermal profile in the presence of a porous constraint $P_m$. The inclusion of a porous medium improves system thermal performance by increasing the thickness of the Thermal Boundary Layer (TBL), which advances thermal dispersion. It is detected from the figure that velocity is more in the cylinder case than the plate in the case of $P_m$, but the reverse trend is observed in the thermal profile.

Figure 4 and Figure 5 are plotted to show the influence of a mixed convection constraint ${\gamma }_1$ on velocity and thermal profiles, respectively. The rise in the values of ${\gamma }_1$ will improve the velocity profile (see Figure 4) and decline the thermal profile (see Figure 5). When the magnitude of the mixed convection coefficient is greater, it implies that the thermal buoyancy force in the fluid system is more dominating. This superiority of the thermal buoyancy force results in an increase in the fluid’s velocity profile. It operates perpendicular to the direction of flow. When the thermal buoyancy force is greater, it pushes the fluid movement, resulting in an increase in fluid velocity that decreases the temperature. From the diagram it is observed that velocity is higher in the cylinder than the plate, and temperature is higher in the plate than the cylinder in the presence of ${\gamma }_1$.

The influence of $\zeta$ on velocity and thermal profiles is displayed in Figure 6 and Figure 7, respectively. At $\zeta =0$, the system experiences maximum buoyancy impact. This is due to the presence of ${\gamma }_1$ with $\zeta$ and $\cos \left(\zeta =0\right)\to 1$. As the value of inclination angle reaches to $\frac{\pi }{2}$, the inclined angle term tends to zero. This leads the minimum buoyancy impact on the flow system. When the inclination angle increases, the influence of the buoyancy force on fluid flow decreases, leading to a decrease in fluid velocity. This decrease in fluid velocity enhances the thermal distribution and leads to improved thermal performance. This effect is particularly relevant in applications where thermal management is critical, such as in heat exchangers, cooling systems, and electronic devices. By controlling the inclination angle of the fluid system, it is possible to optimize the thermal performance and enhance the overall efficiency of the system. From the diagrams it is further observed that velocity is less in the plate than the cylinder, and thermal distribution is greater in the case of the plate than the cylinder in the presence of $\zeta$.

Figure 8 represents the nature of thermal profile in the presence of the HS–S constraint $H_{SS}$. The escalation in $H_{SS}$ will enhance the thermal profile. The heat sink $\left(H_{SS}<0\right)$ will remove the temperature from the fluid and as a result, temperature decreases. Heat source $\left(H_{SS}>0\right)$, which generates the temperature from the surface of the geometry, leads to improvement in the temperature. From the illustration, it is further observed that thermal distribution is more in the case of the plate than the cylinder. Figure 9 and Figure 10 show the alteration of velocity and thermal profiles in the presence of solid volume fraction ${\varphi }_3$. The momentum and thermal boundary layers are gradually improving by escalating the ${\varphi }_3$ value. This leads the fluid flow slowly, which leads velocity decreases. The velocity of the plate is very lower than the velocity of the cylinder (see Figure 9). The enhancement in the TML makes the system deliver more temperature with the improved values of ${\varphi }_3$. Distribution of temperature is more in the case of the plate than the cylinder over ${\varphi }_3$ (see Figure 10).

Figure 11 shows the important engineering coefficient $Cf$ on $P_m$ for various values of ${\varphi }_3$. The improved values of ${\varphi }_3$ will enhance the MBL thickness, and the presence of $P_m$ will make the fluid flow slowly. The presence of these two factors will enhance the surface drag force in the system. Further, it is observed that $Cf$ is more in the cylinder than the plate.

Figure 12 displays the variation of $Nu$ on $H_{SS}$ for different values of ${\varphi }_3$. The improvement in the ${\varphi }_3$ will thicken the TBL, and the improvement in $H_{SS}$ enhances the thermal distribution rate. Therefore, in the presence of these two factors, the thermal distribution rate enhances.

Table 3. Computational values of $Nu\hspace{0.17em}\%$ for various dimensionless constraints.

Parameters				$\mathbf{Cylinder} \left({\mathit{\delta }}_1 = 0.3\right)$	Plate $\left({\mathit{\delta }}_1 = 0.0\right)$
$P_m$	${\gamma }_1$	$\zeta$	$H_{SS}$	$\left|\frac{{\left.Nu\right|}_{{\varphi }_1={\varphi }_2={\varphi }_3=0.01}-{\left.Nu\right|}_{{\varphi }_1={\varphi }_2={\varphi }_3=0}}{{\left.Nu\right|}_{{\varphi }_1={\varphi }_2={\varphi }_3=0}}\right|\times 100$	$\left|\frac{{\left.Nu\right|}_{{\varphi }_1={\varphi }_2={\varphi }_3=0.01}-{\left.Nu\right|}_{{\varphi }_1={\varphi }_2={\varphi }_3=0}}{{\left.Nu\right|}_{{\varphi }_1={\varphi }_2={\varphi }_3=0}}\right|\times 100$
0.1	0.1	30°	0.5	2.08%	5.19%
0.3				2.20%	7.54%
0.5				2.32%	10.83%
0.1	0.1	30°	0.5	2.08%	5.19%
	0.5			2.03%	3.84%
	1.0			1.98%	2.74%
0.1	0.1	0°	0.5	2.08%	5.13%
		30°		2.08%	5.19%
		60°		2.09%	5.38%
0.1	0.1	30°	−0.2	0.14%	3.37%
			0	0.27%	2.96%
			0.2	0.81%	2.05%

displays the improvement in the Nusselt number percentage in the presence and absence of a solid volume fraction for various dimensionless constraints. From the table it is observed that, as the porous constraint rises from 0.1 to 0.5, the rate of heat distribution in the cylinder will increase from 2.08% to 2.32%, whereas in the plate it is about 5.19 % to 10.83%. In the presence of a mixed convection constraint (0.5 to 1.0), $Nu$ will decline by 2.08% to 1.98% in the cylinder but in the plate, it will decline by 5.19% to 2.74%. As the inclination angle varies from 0° to 60°, $Nu$ will increase slightly from 2.08% to 2.09% in the cylinder, but in the plate it improves from 5.13% to 5.38%. In the presence of a heat source/sink, the rate of heat distribution augments by 0.14% to 0.81% in the cylinder, and in the plate it decreases from 3.37% to 2.05 %. In all the cases, the addition of nanoparticles shows gradual improvement in the process of the rate of thermal distribution than in the absence of nanoparticles.

5. Final Remarks

By exploring the thermal performance of a ternary hybrid nanofluid flowing in a permeable inclined cylinder/plate system, the current work makes a unique addition to the field of thermal management. The research looks at the impacts of inclined geometry, porous media, and heat source/sink.

The major findings in the study disclose that in the presence of ternary nanofluid (contains $A{l}_2{O}_3,Ti{O}_2,CuO$ and base fluid as $H_{2}O$), the velocity of the fluid decreases with improved values of porous factor while the temperature distribution increases. A change in the angle of inclination and heat source/sink will improve the thermal profile. The addition of nanoparticles in the fluid will decrease the velocity but improve the thermal distribution. The rate of thermal distribution percentage will always be greater in the presence of a ternary nanofluid than a base fluid (water) over all the constraints. Further, it is concluded that the rate of thermal distribution percentage is higher in a ternary nanofluid with water as the base fluid compared to pure water in a plate geometry than cylinder geometry. The present work finds its importance in the field of heat exchangers, cooling systems, renewable energy systems, and heating ventilation and air conditioning.