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

Abstract

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.

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1. Introduction

Studies on nanofluid flow frequently prioritize the flow's thermal conductivity and heat transmission characteristics, while its thermophysical properties are often overlooked. However, including multiple nanoparticles in a base fluid can significantly increase the heat transfer phenomena. Hybrid nanofluids, composed of metal, polymeric, or non-metallic composite nanoparticles or a combination of different nanoparticles distributed in a base fluid, outperform traditional nanofluids with better pressure drop and heat transfer properties. The efficacy of ternary nanofluids is strongly influenced by the types, shapes, sizes, and percentages of the nanoparticles utilized. The revolutionary concept of carbon nanotubes was first introduced by Iijima¹ 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 Eastman² used the term “nanofluid” to describe fixed nanoscale particles floating in a fluid medium (nano-lubricants). Hybrid nanofluids 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 nanofluids are addressed in ref. 3–11. Moldoveanu et al.¹² scrutinized the thermal conductivity of two nanofluids 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 fields in a quiescent micropolar fluid with nonlinear power-law surface velocity and temperature distributions. In a similar study, Mahapatra et al.¹⁴ investigated dual solutions in magnetohydrodynamic stagnation-point flow 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 flow of a SiO₂–Al₂O₃/water hybrid nanofluid near the stagnation point on a vertical flat plate. Moreover, Mousavi et al.¹⁶ studied the dual solutions for Casson hybrid nanofluid flow 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 fields 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 fluid, methanol (CH₃OH), is comprised of a blend of silica (SiO₂) and alumina (Al₂O₃), forming a disk with a porous boundary capable of generating and absorbing heat. The problem's coordinates take the form of a cylinder (r,θ,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 field is applied perpendicularly to the z-axis, with minimal interference from the electric field. Furthermore, both thermal radiation and ohmic heating effects are taken into consideration. The dimensionless equations are obtained through similarity profiles and numerically solved using bvp4c. The novelty of this study is in using these nanomaterials and the effects taken. Methanol-based hybrid nanofluids incorporating silica and alumina nanoparticles have demonstrated a significant 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 nanofluids. 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 significance, which helps readers visualize the intricate details of the problem and facilitates better understanding. Finally, Section 4 presents a concise summary of the findings, highlighting the key takeaways from the study and providing insights for future research in this area.

2. Problem formulation

In this study, hybrid nanofluids 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,θ,z), and the velocity field V is characterized by the components [ũ(r,t), w̃(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 In the far-field region near the stagnation point, the velocity is given by A uniform magnetic field B = [0, 0, B₀] is applied perpendicular to the z-axis, and the effects of the electric field 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 fluid with temperature T̃_w and heat transfer coefficient h_w, while the ambient temperature distribution is represented by T̃_∞.

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.

Fig. 1 Flow mechanism.

Table 1 Numerical values for the thermophysical features

Physical properties	Base fluid	Nanoparticles
	CH3OH	SiO2	Al2O3
ϱ f	792	2650	3970
c p	2545	730	765
k	0.2035	1.5	40
Pr at 25 °C	6.83	—	—
s	3.7 (spherical)

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The governing equations are (see ref. 16)

where is radiative heat flux.

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)

Introducing stream function where and local Reynolds number with the following similarity ansatz

The negative sign in eqn (9) indicates that the fluid at the far-field is being pulled towards the surface, rather than being pushed away from it. Using eqn (11)

where for ternary hybrid nanofluids

The dimensionless parameters are

The quantities of engineering interest, C̃_f and Nu_r are given as

3. 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 C̃_f. 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 finite 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 solver. The estimated error count is constantly monitored during the computation, ensuring the quality of the solution. During the calculations, some parameters are held constant, and the behavior of the first (stable) and second (unstable) solutions is examined. The considered values and ranges of the parameters are Re = 0.6, 1 ≤ Pr ≤ 10, 1 ≤ β_i ≤ 3, 0.01 ≤ δ* ≤ 0.05, 0.5 ≤ f̃_w ≤ 4, 0.01 ≤ Ec ≤ 0.08, 0.01 ≤ M ≤ 2, and 0.1 ≤ λ ≤ 2, and spherical shaped nanoparticles are used. The behavior of a mixture of nanoparticles, namely silica and alumina, submerged into a solution of methanol, i.e., (SiO₂–Al₂O₃/CH₃OH), is examined for two different models, Xue and Hamilton–Crosser models. Solid lines represent the behavior of the first solution, while dot-dashed lines depict the second solution.

Fig. 2(a) and (b) elucidate the impact of the magnetic field parameter M on the key parameters, namely C̃_f and Nu_r. Both figures represent the first and second solutions for control parameters χ and M = 0.5, 1.0, 1.5. Fig. 2(a) demonstrates that both solutions' values of C̃_f escalate with an increase in M. The critical values, χ_c = −3.73484, −3.84431, −3.95486, are observed at M = 0.5, 1.0, 1.5, respectively. This enhancement in the behavior of C̃_f can be attributed to the linear increment in C̃_f with the magnetic field's increase. The magnetic field induces Lorentz forces that alter the velocity profile of the fluid, resulting in changes to the skin friction coefficient. Specifically, the skin friction coefficient decreases with an increasing magnetic field due to the development of a boundary layer with lower momentum. The Lorentz force causes the flow field to decline, magnifying the frictional forces, thus augmenting C̃_f. On the 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 fluid molecules' physical energy and their greater intermolecular vibrations caused by increased kinetic energy. The magnetic field enhances fluid 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, χ_c = −3.73484, −3.84431, −3.95486, are observed at M = 0.5, 1.0, 1.5, respectively.

Fig. 2 (a and b) Influence of M on C̃_f and Nu_r.

Fig. 3(a) and (b) demonstrate the influence of unsteadiness parameter λ on C̃_f and Nu_r, respectively. Critical values χ_c = −4.097, −3.925155, −3.73484 at λ = 0.5, 0.71, 0.9, respectively, are observed. As depicted in Fig. 3(a), an increase in λ leads to an enhancement in C̃_f (for both solutions), resulting in a linear increment over time. Physically, unsteadiness can cause a delay in fluid flow, leading to an increase in frictional forces. Conversely, Fig. 3(b) illustrates the impact λ on Nu_r, with critical values of χ_c = −4.097, −3.925155 at λ = 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 first and second solutions, with Hamilton–Crosser models exhibiting better thermal conductivity than the Xue model.

Fig. 3 (a and b) Influence of λ on C̃_f and Nu_r.

Fig. 4(a) and (b) depict the influence 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 fluid due to dissipation effects and temperature gradients. Notably, the critical value remains the same for different values of Ec, i.e., χ_c = −3.73484 for Ec = 0.01, 0.05. On the other hand, the Biot number β_i relates the rate of heat transfer inside a solid to the rate of heat transfer at the solid's surface. It is defined as the ratio of the internal thermal resistance to the external thermal resistance. Fig. 4(b) demonstrates that an increase in β_i leads to an increase in thermal transport rate. The critical value is χ_c = −3.079495 for β_i = 0.01, 0.05. In general, the Nu_r increases with increasing Biot number. This is because as the β_i increases, the thermal resistance of the solid becomes smaller compared to the thermal resistance of the fluid, and the temperature gradient in the fluid becomes more uniform. This leads to increased heat transfer from the solid to the fluid, and therefore a higher convective heat transfer coefficient, resulting in a higher Nu_r.

Fig. 4 (a and b) Influence of Ec and β_i on Nu_r.

The impression of f̃_w is noted in Fig. 5(a) and (b) on C̃_f and Nu_r, respectively. The mass transmission constraint shows the effect of suction (f̃_w > 0) and injection (f̃_w < 0). When f̃_w > 0, fluid suction at the surface lowers the viscous effects closer to the wall. Suction has the potential to lower fluid velocity f̃′(ζ), whereas blowing or injection has the opposite effects. As a result, greater suction velocities yield higher entertainment velocities. Thus, suction causes the fluid's velocity to decrease, while blowing causes it to increase, as shown in Fig. 6(a). Therefore, C̃_f improves for f̃_w > 0, as shown in Fig. 5(a). On the other hand, energy of the system is declined as given in Fig. 6(b). However, Fig. 5(b) depicts that Nu_r is an increasing function of f̃_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 χ_c = −3.73484, −4.35887, −5.05495 at f̃_w = 1, 2, 3, respectively.

Fig. 5 (a and b) Influence of f̃_w on C̃_f and Nu_r.

Fig. 6 (a and b) Influence of f̃_w on f̃′(ζ) and [small theta, Greek, tilde](ζ).

The behavior of shrinking/stretching parameter χ is noted in Fig. 7(a) and (b) on flow and energy distribution. The parameter shows the effect of stretching (χ > 0) and shrinking (χ < 0). It is illustrated that shrinking causes the flow field to decline for both solutions, as shown in Fig. 7(a). However, the temperature profile is boosted significantly 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. 7 (a and b) Influence of χ on f̃′(ζ) and [small theta, Greek, tilde](ζ).

Fig. 8(a) and (b) provide an insight into the impact of Prandtl number Pr and heat generation/absorption parameter δ* 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 fluid. 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 signifies the prevalence of heat conduction over convection, where heat diffuses faster than fluid velocity. In general, Pr for gases ranges around 0.7, while it varies between 1 and 10 for fluids. Fig. 8(a) indicates that the thermal transport decreases with increasing Pr, implying that a higher Pr value leads to a significant reduction in temperature/energy transport. The presence of a heat source or sink has a significant 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 [small theta, Greek, tilde](ζ), as shown in Fig. 8(b).

Fig. 8 (a and b) Influence of Pr and δ* on [small theta, Greek, tilde](ζ).

By contrasting the existing numerical values C̃_f with the prior results, the current model in Table 2 is validated. The variation of χ < 0 is presented in the absence of nanoparticles and magnetic field. It is seen that the coefficient of skin friction enhanced significantly due from χ = −0.25 to −0.95.

Table 2 Comparison of C̃_f for various values of χ in the absence of nanoparticles and magnetic field

χ	Khan et al.^30	Wang^29	Present study
			First solution
−0.25	1.4566365	1.456640	1.456640
−0.5	1.4900104	1.490010	1.490041
−0.75	1.3528399	1.352840	1.352838
−0.95	0.9469034	0.946900	0.946904

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3.1 Stability analysis

The current research used a stability analysis to validate the obtained solution. This is of utmost significance, 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,^31–35 the smallest eigenvalues α were plotted against χ, and the resulting Fig. 9 was obtained. The physical interpretation of this figure is that a positive value of α indicates an initial deterioration of disturbance, implying that the flow is in a stable mode. Conversely, a negative value of α, as τ → ∞, implies that the flow is in an unstable state due to the early increase of disturbance. It should be noted that as χ approaches the critical value, χ_c = −1.2431, both the stable and unstable branches converge to α = 0. This behavior indicates that the solutions bifurcate at the critical values, and it has critical implications for the study of boundary layer problems.

Fig. 9 Smallest eigenvalues α for χ.

4. Conclusion

An electrically conducting heat-generating/absorbing disk was assumed with a porous boundary immersed in a mixture of silica (SiO₂) and alumina (Al₂O₃) into methanol (CH₃OH) which acts as a base fluid. 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 field.

❖ Unsteadiness declined the fluid 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., χ_c = −3.73484 and −3.079495, respectively.

❖ Coefficient of skin friction and thermal distribution rate was an increasing function of wall transmission constraint, whereas the fluid flow and energy transport diminished.

❖ Incrementing Prandtl number reduced the temperature and energy transport significantly.

❖ The thermal transport was dominant due to the heat generation/absorption parameter.

Nomenclature

(ũ,ṽ,w̃)	Velocity components (m s^−1)
k	Thermal conductivity (W m^−1 K^−1)
ν	Kinematic viscosity (m^2 s^−1)
ϱ	Density (kg m^−3)
p̃	Pressure (N m^−2)
α	Unsteadiness rate (s^−1)
μ	Absolute viscosity (kg m^−1 s^−1)
σ	Electric conductivity (Ω m)
B 0	Magnetic field strength (N s C^−1 m^−1)
Rer	Reynolds number
σ*	Stefan–Boltzmann coefficient
h w	Heat transfer coefficient
τ w	Wall-shear stress
T̃	Temperature (K)
f̃′,[small theta, Greek, tilde]	Dimensionless velocity and temperature
f̃ w	Wall mass transfer coefficient
λ	Unsteadiness parameter
p̃	Dimensionless pressure
M	Magnetic field parameter
γ 1	Biot number
R d	Radiation parameter
ϕ	Nanoparticle volume fraction
ζ	Dimensionless variable
k*	Mean absorption coefficient
Pr	Prandtl number
Nur	Nusselt number

Subscripts

r,z	Derivative w.r.t r and z
∞	Far-field condition
w	Wall boundary condition
f	Base fluid
n	Nanofluid
hn	Hybrid nanofluid

Superscripts

′	Derivative w.r.t ζ

Conflicts of interest

There are no conflicts to declare.

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