Exploring dual solutions and thermal conductivity in hybrid nanofluids: a comparative study of Xue and Hamilton–Crosser models

Hybrid nanofluids show great potential for heat transport applications such as solar thermal systems, car cooling systems, heat sinks, and thermal energy storage. They possess better thermal stability and properties compared to standard nanofluids. In this study, a base fluid, methanol, is injected into an electrically conducting heat-generating/absorbing disk of permeable boundary, and dual solutions are obtained. Two alternative models, Xue and Hamilton–Crosser are considered, and their thermal conductivities are contrasted. Furthermore, thermal radiation and ohmic heating are also considered, and convective boundary conditions are utilized to simulate overall heat gains or losses resulting from conduction, forced or natural convection between nearby objects of nearly constant temperature. Using a similarity transform, the governing equations are obtained and numerically solved via bvp4c, a finite difference method. It is observed that the presence of a magnetic field and the shrinking of the disk elevate the energy transport rate and wall stress. Additionally, the skin friction coefficient and thermal distribution rate increase with wall transmission constraint while fluid flow and energy transport diminish. Furthermore, particle clustering and nano-layer creation suggest that the Hamilton–Crosser model exhibits better thermal conductivity than the Xue model.


Introduction
Studies on nanouid ow frequently prioritize the ow's thermal conductivity and heat transmission characteristics, while its thermophysical properties are oen overlooked.However, including multiple nanoparticles in a base uid can signicantly increase the heat transfer phenomena.Hybrid nanouids, composed of metal, polymeric, or non-metallic composite nanoparticles or a combination of different nanoparticles distributed in a base uid, outperform traditional nanouids with better pressure drop and heat transfer properties.The efficacy of ternary nanouids is strongly inuenced by the types, shapes, sizes, and percentages of the nanoparticles utilized.The revolutionary concept of carbon nanotubes was rst introduced by Iijima 1 through experimentation, resulting in the discovery of microscopic carbon layer straws known as carbon nanotubes.Although the essential components of the liquid were still complex hydrocarbon molecules, Choi and Eastman 2 used the term "nanouid" to describe xed nanoscale particles oating in a uid medium (nano-lubricants).Hybrid nanouids possess exceptional mechanical, electrical, and thermal properties, as well as high electrical and thermal conductivities, making them strong and lightweight.Several studies on the applications of hybrid nanouids are addressed in ref. 3-11.Moldoveanu et al. 12 scrutinized the thermal conductivity of two nanouids and their hybrid, and the experimental outcomes were presented at various temperatures and volume fractions, including room temperature.
Convective heat transfer models can present non-linearity issues and dual solutions, which can result in complex behaviors.Therefore, it is essential to understand both stable and unstable states.To address this, several studies have been conducted on different scenarios.For instance, Zheng et al. 13 explored the radiation effect on velocity and temperature elds in a quiescent micropolar uid with nonlinear power-law surface velocity and temperature distributions.In a similar study, Mahapatra et al. 14 investigated dual solutions in magnetohydrodynamic stagnation-point ow and heat transfer over a shrinking surface with partial slip, while Rostami et al. 15 developed an analytical solution for the steady laminar MHD mixed convection boundary layer ow of a SiO 2 -Al 2 O 3 /water hybrid nanouid near the stagnation point on a vertical at plate.Moreover, Mousavi et al. 16 studied the dual solutions for Casson hybrid nanouid ow due to a stretching/shrinking sheet with suction, radiation, and convective boundary condition effects.Asjad et al. 17 introduced a new fractional operator to model memory effects and solved analytically for temperature and velocity elds using the Laplace transform approach.Studies of heat transport are discussed in further studies, as reported in ref. 18-28.Drawing inspiration from the aforementioned studies, it is formulated that the premise that the base uid, methanol (CH 3 OH), is comprised of a blend of silica (SiO 2 ) and alumina (Al 2 O 3 ), forming a disk with a porous boundary capable of generating and absorbing heat.The problem's coordinates take the form of a cylinder (r,q,z), with the thermal conductivities of the two alternative models: Xue and Hamilton-Crosser, compared.The surface is seen to radially expand and contract with time, with convective boundary conditions provided.A uniform magnetic eld is applied perpendicularly to the z-axis, with minimal interference from the electric eld.Furthermore, both thermal radiation and ohmic heating effects are taken into consideration.The dimensionless equations are obtained through similarity proles and numerically solved using bvp4c.The novelty of this study is in using these nanomaterials and the effects taken.Methanol-based hybrid nanouids incorporating silica and alumina nanoparticles have demonstrated a signicant increase in heat transport efficiency, attributed to their high thermal conductivities and extensive surface area.These hybrid nanomaterials have garnered attention for superior rheological, thermal, and economic performance compared to monotype nanouids.They enhance convective heat transfer, reduce boundary layer thickness, and provide thermal stability, making them indispensable for diverse applications such as solar thermal systems, car cooling, and heat sinks.
The article is structured into four distinct sections.In Section 2, the mathematical problem at hand is comprehensively addressed, providing a detailed insight into the complex computations and analytical techniques employed to obtain the desired results.Section 3 is devoted to graphical illustrations, accompanied by their corresponding physical signicance, which helps readers visualize the intricate details of the problem and facilitates better understanding.Finally, Section 4 presents a concise summary of the ndings, highlighting the key takeaways from the study and providing insights for future research in this area.

Problem formulation
In this study, hybrid nanouids consisting of silica and alumina nanoparticles in methanol are considered, which have been shown to exhibit excellent cooling properties.The disk's boundary is assumed to be porous, allowing heat generation and absorption.The problem is analyzed using cylindrical coordinates (r,q,z), and the velocity eld V is characterized by the components [ũ(r,t), w(r,t)].The thermal conductivity of two models is considered, namely Xue and Hamilton-Crosser.The surface of the disk is assumed to be radially expanding or contracting, which leads to a velocity component along the boundary of ũw ¼ Cr 1 À at : In the far-eld region near the stagnation point, the velocity is given by ũN ¼ ar 1 À at : A uniform magnetic eld B = [0, 0, B 0 ] is applied perpendicular to the zaxis, and the effects of the electric eld are assumed to be negligible.The study also considers the effects of ohmic heating and thermal radiation.The surface of the disk is heated by convection from a hot uid with temperature Tw and heat transfer coefficient h w , while the ambient temperature distribution is represented by TN .
To assist in understanding, Fig. 1 displays the geometry of the problem, while Table 1 presents the thermophysical characteristics of both the methanol and alumina-silica nanoparticles.
The governing equations are (see ref. 16) (1) Nanoscale Advances Paper where

The thermophysical properties are given as (see ref. 7)
For Hamilton-Crosser model, the thermal conductivity is given as The value of s vary for different shapes, such as for spherical s = 3.0.For Xue model, the thermal conductivity is given as The boundary conditions (BCs) are (see ref. 16) The negative sign in eqn (9) indicates that the uid at the far-eld is being pulled towards the surface, rather than being pushed away from it.Using eqn (11) where for ternary hybrid nanouids The dimensionless parameters are

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The quantities of engineering interest, Cf and Nu r are given as (15) (16)

Interpretation and analysis of results
This section presents an analysis of the numerical results obtained for various relevant parameters, such as velocity and temperature distributions, and the effects of control parameters on Nu r and Cf .This study presents the solution of boundary value problems (BVPs) for ordinary differential equations using MATLAB's bvp4c function.Our approach is based on using a nite difference method to discretize the BVP, which allows us to solve the resulting system of equations using a collocation method.To achieve the desired level of accuracy, the number of mesh points is adjusted, and the initial guess is provided to the   In general, the Nu r increases with increasing Biot number.This is because as the b i increases, the thermal resistance of the solid becomes smaller compared to the thermal resistance of the uid, and the temperature gradient in the uid becomes more uniform.This leads to increased heat transfer from the solid to the uid, and therefore a higher convective heat transfer coefficient, resulting in a higher Nu r .The impression of f w is noted in Fig. 5(a) and (b) on Cf and Nu r , respectively.The mass transmission constraint shows the effect of suction ( f w > 0) and injection ( f w < 0).When f w > 0, uid suction at the surface lowers the viscous effects closer to the wall.Suction has the potential to lower uid velocity f ′ (z), whereas blowing or injection has the opposite effects.As a result, greater suction velocities yield higher entertainment velocities.Thus, suction causes the uid's velocity to decrease, while blowing causes it to increase, as shown in Fig. 6(a).Therefore, Cf improves for f w > 0, as shown in Fig. 5(a).On the other hand, energy of the system is declined as given in The behavior of shrinking/stretching parameter c is noted in Fig. 7(a) and (b) on ow and energy distribution.The parameter shows the effect of stretching (c > 0) and shrinking (c < 0).It is illustrated that shrinking causes the ow eld to decline for both solutions, as shown in Fig. 7(a).However, the temperature prole is boosted signicantly for both models, as given in Fig. 7(b).Whereas it is seen that due to particle clustering and nano-layer creation, the thermal conductivity of Hamilton-Crosser is better than Xue model.Fig. 8(a) and (b) provide an insight into the impact of Prandtl number Pr and heat generation/absorption parameter d* on the system's behavior.The Prandtl number is a ratio of momentum to thermal diffusivity, indicating the dominance of thermal diffusion mechanism for a given uid.The value of Pr for methanol at 25 °C is 6.83, much higher than that of air (Pr = 0.71).A lower value of Pr signies the prevalence of heat Fig. 6 (a and b) Influence of fw on f′ (z) and q(z).

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conduction over convection, where heat diffuses faster than uid velocity.In general, Pr for gases ranges around 0.7, while it varies between 1 and 10 for uids.Fig. 8(a) indicates that the thermal transport decreases with increasing Pr, implying that a higher Pr value leads to a signicant reduction in temperature/energy transport.The presence of a heat source or sink has a signicant effect on the temperature distribution in a system.A heat source increases the temperature in its vicinity, while a heat sink reduces it.This effect propagates through the system via heat transfer mechanisms such as conduction, convection, and radiation.The magnitude of the effect depends on the strength and location of the heat source or sink, as well as the thermal properties and boundary conditions of the system.The effect of a heat source or sink can be used to control or optimize temperature distribution in a system q(z), as shown in Fig. 8(b).By contrasting the existing numerical values Cf with the prior results, the current model in Table 2 is validated.The variation of c < 0 is presented in the absence of nanoparticles and magnetic eld.It is seen that the coefficient of skin friction enhanced signicantly due from c = −0.25 to −0.95.

Stability analysis
The current research used a stability analysis to validate the obtained solution.This is of utmost signicance, mainly when the governing system permits multiple-branch solutions.Identifying all probable solutions emerging from the governing boundary layer problems is essential for determining the solution.Based on previous literature studies, [31][32][33][34][35] the smallest eigenvalues a were plotted against c, and the resulting Fig. 9 was obtained.The physical interpretation of this gure is that a positive value of a indicates an initial deterioration of disturbance, implying that the ow is in a stable mode.Conversely, a negative value of a, as s / N, implies that the ow is in an unstable state due to the early increase of disturbance.It should be noted that as c approaches the critical value, c c = −1.2431,both the stable and unstable branches converge to a = 0.This behavior indicates that the solutions bifurcate at the critical values, and it has critical implications for the study of boundary layer problems.

Conclusion
An electrically conducting heat-generating/absorbing disk was assumed with a porous boundary immersed in a mixture of silica (SiO 2 ) and alumina (Al 2 O 3 ) into methanol (CH 3 OH) which acts as a base uid.The thermal conductivity of two models, Xue and Hamilton-Crosser were considered and compared.The effects of MHD, heat source/sink, suction/injection, thermal radiation, and convective boundary conditions were scrutinized over a radially shrinking/stretching surface.The primary outcomes are given below.❖ Due to particle clustering and nano-layer creation, the thermal conductivity of the Hamilton-Crosser was better than the Xue model.
❖ Energy transport rate and wall stress were elevated due to shrinking and the presence of the magnetic eld.
❖ Unsteadiness declined the uid motion, which caused the frictional forces to incline.
❖ Due to the increment in Biot number, the surface heat resistance declined, which dominated the convection mechanism resulting in a higher thermal transport rate.
❖ The critical value was the same for variation of Eckert and Biot numbers, i.e., c c = −3.73484and −3.079495, respectively.
❖ Coefficient of skin friction and thermal distribution rate was an increasing function of wall transmission constraint, whereas the uid ow and energy transport diminished.
❖ Incrementing Prandtl number reduced the temperature and energy transport signicantly.
Fig. 2 (a and b) Influence of M on Cf and Nu r .
other hand, Fig.2(b) demonstrates the impact of M on Nu r , which indicates that heat energy gains result from an increase in magnetic force, leading to chaotic particle motion.The system temperature increases due to the uid molecules' physical energy and their greater intermolecular vibrations caused by increased kinetic energy.The magnetic eld enhances uid mixing and promote heat transfer, leading to an increase in the Nusselt number.It is noteworthy that the behavior of Xue and Hamilton-Crosser models is depicted here.The results indicate that the magnetic parameter has improved the outcomes of Nu r .Moreover, the values for the Hamilton-Crosser models are more pronounced than the Xue model, indicating that they have better thermal conductivity than the Xue model.The critical values, c c = −3.73484,−3.84431, −3.95486, are observed at M = 0.5, 1.0, 1.5, respectively.Fig. 3(a) and (b) demonstrate the inuence of unsteadiness parameter l on Cf and Nu r , respectively.Critical values c c = −4.097,−3.925155, −3.73484 at l = 0.5, 0.71, 0.9, respectively, are observed.As depicted in Fig. 3(a), an increase in l leads to an enhancement in Cf (for both solutions), resulting in a linear increment over time.Physically, unsteadiness can cause a delay in uid ow, leading to an increase in frictional forces.Conversely, Fig. 3(b) illustrates the impact l on Nu r , with critical values of c c = −4.097,−3.925155 at l = 0.5, 0.71, respectively.Augmentation in thermal transport rate occurs as a result of an increase in unsteadiness, which leads to chaotic particle motion, greater intermolecular vibrations, and increased kinetic energy, resulting in elevated system temperatures.The behavior of Xue and Hamilton-Crosser models is depicted for rst and second solutions, with Hamilton-Crosser models exhibiting better thermal conductivity than the Xue model.Fig. 4(a) and (b) depict the inuence of Eckert number and Biot number on Nu r .It can be observed that Nu r increases with an increase in Ec, which represents the self-heating of the uid due to dissipation effects and temperature gradients.Notably, the critical value remains the same for different values of Ec, i.e., c c = −3.73484for Ec = 0.01, 0.05.On the other hand, the Biot number b i relates the rate of heat transfer inside a solid to the rate of heat transfer at the solid's surface.It is dened as the ratio of the internal thermal resistance to the external thermal

Fig. 3 (
Fig. 3 (a and b) Influence of l on Cf and Nu r .

Fig. 6 (
Fig. 6(b).However, Fig. 5(b) depicts that Nu r is an increasing function of f w as well.It is seen that the thermal conductivity of Hamilton-Crosser's model is more than as compared to Xue's model.The critical values for Fig. 5(a) and (b) are c c = −3.73484,−4.35887, −5.05495 at f w = 1, 2, 3, respectively.The behavior of shrinking/stretching parameter c is noted in Fig.7(a) and (b) on ow and energy distribution.The parameter shows the effect of stretching (c > 0) and shrinking (c < 0).It is illustrated that shrinking causes the ow eld to decline for both solutions, as shown in Fig.7(a).However, the temperature prole is boosted signicantly for both models, as given in Fig.7(b).Whereas it is seen that due to particle clustering and nano-layer creation, the thermal conductivity of Hamilton-Crosser is better than Xue model.Fig.8(a) and (b) provide an insight into the impact of Prandtl number Pr and heat generation/absorption parameter d* on the system's behavior.The Prandtl number is a ratio of momentum to thermal diffusivity, indicating the dominance of thermal diffusion mechanism for a given uid.The value of Pr for methanol at 25 °C is 6.83, much higher than that of air (Pr = 0.71).A lower value of Pr signies the prevalence of heat

Fig. 5 (
Fig. 5 (a and b) Influence of fw on Cf and Nu r .

Fig. 7 (
Fig. 7 (a and b) Influence of c on f′ (z) and q(z).

Fig. 8 (
Fig. 8 (a and b) Influence of Pr and d* on q(z).

Fig. 9
Fig. 9 Smallest eigenvalues a for c.

Table 1
Numerical values for the thermophysical features

Table 2
Comparison of Cf for various values of c in the absence of nanoparticles and magnetic field © 2023 The Author(s).Published by the Royal Society of Chemistry Nanoscale Adv., 2023, 5, 6695-6704 | 6701