Numerical analysis of unsteady squeezing MHD flow of a ternary hybrid nanofluid between parallel rotating disks with Hall current and heat transfer reversal

Abstract

This paper investigates the three-dimensional unsteady squeezing flow of a magnetized ternary hybrid nanofluid (Cu–Al₂O₃–TiO₂/water) between two parallel rotating disks. The mathematical model incorporates the effects of Hall current, Joule heating, viscous dissipation, and Arrhenius activation energy. Using appropriate similarity transformations, the governing partial differential equations are reduced to a system of coupled nonlinear ordinary differential equations. These equations are solved numerically via the Shooting Method combined with the Runge-Kutta-Fehlberg (RKF45) algorithm. A comprehensive parametric study is conducted to analyze the influence of the Squeeze number (S), Magnetic parameter (M), Hall parameter (m), and Eckert number (Ec) on the velocity, temperature, and concentration profiles. A critical finding of this study is the occurrence of heat transfer reversal (negative Nusselt numbers) at the disk surface, driven by dominant internal heat generation from viscous and Ohmic dissipation. The results indicate that the Hall current effectively reduces the surface drag in the swirl direction by decreasing the effective electrical conductivity. Furthermore, the squeezing force significantly modifies the radial flow topology compared to steady-state configurations. This research provides essential insights for the design of high-performance thermal systems in aerospace and industrial lubrication.

Introduction

The relentless demand for high-efficiency thermal management systems in modern industrial processes, encompassing aerospace engineering, nuclear reactor cooling mechanisms, and micro-electromechanical systems, has catalyzed the exploration of advanced heat transfer fluids. Traditional coolants, such as water, ethylene glycol, and mineral oils, are fundamentally limited by their inherently low thermal conductivities. To circumvent these limitations, the scientific community introduced the concept of nanofluids, a paradigm initially established by Choi and Eastman [1]. While mono-nanofluids provide a substantial enhancement in thermal transport, they are frequently constrained by rheological instability and elevated viscous drag. Consequently, the focus of fluid dynamics research has progressively shifted toward Ternary Hybrid Nanofluids (THNFs). THNFs represent the vanguard of thermal engineering, utilizing three distinct types of nanoparticles to achieve a synergistic equilibrium that optimizes both thermal conductivity and colloidal stability. The precise formulation of a ternary hybrid nanofluid, such as the combination of Copper (Cu), Alumina (Al₂O₃), and Titania (TiO₂) dispersed in water, leverages the exceptional thermal conductivity of metallic nanoparticles alongside the chemical stability of oxide nanoparticles.

The deployment of these advanced fluids is frequently analyzed within complex mechanical configurations. Fluid dynamics within squeezing flows—characterized by the relative motion of parallel boundaries approaching or receding from one another—constitutes a fundamental problem with widespread applications in lubrication theory and hydraulic actuators. Recent literature from 2023 to 2026 highlights a rapidly growing intersection between complex ternary fluid dynamics and advanced analytical methods. For instance, Ahmed et al. [2] investigated the unsteady squeezing motion of ternary nanofluids between revolving parallel plates under variable magnetic fields. Bora and Boruah [3] explored the squeezing flow of a Cu–Al₂O₃–TiO₂/water ternary hybrid nanofluid in a Darcy porous medium. Ullah et al. [4] applied neural network analysis to ternary hybrid nanofluids with Darcy-Forchheimer effects, while Yasir and Khan [5] evaluated the thermal efficiencies of hybrid ferrofluids. Furthermore, Abbas et al. [6] documented the variable heat effects on MHD Maxwell ternary nanofluids over moving geometries. Additional explorations into tri-hybrid squeezing configurations have been provided by Venkateswarlu et al. [7] and Ali et al. [8], showcasing the significant thermal management potential of these fluids.

Other notable studies have mapped the thermodynamic boundaries of hybrid and ternary fluids across diverse geometries. Nath and Deka [9] performed numerical investigations on MHD ternary hybrid nanofluids past vertically stretching cylinders. Das and Mali [10] evaluated tri-hybrid alumina-silica-copper-water nanofluids, incorporating Arrhenius activation energy. Bakar et al. [11] utilized Response Surface Methodology (RSM) for MHD ternary nanofluid flow optimization over radiated permeable sheets. Alderremy et al. [12] modeled Darcy-Forchheimer thin film flow of Carreau nanofluids, while Yasir and Qi [13] provided deeper heat transfer insights for these complex mixtures. Advanced computational simulations by Rehman et al. [14] focused on unsteady squeezing non-Newtonian MHD couple stress ternary hybrid nanofluids, and Kathyayani and Rao [15] analyzed the impact of shape factors on couple stress flows over inclined plates.

Foundational research in this domain has been significantly advanced by the prior works of Rao [16–18] and Rao and Deka [19]. Their progressive investigations have meticulously mapped the boundary layer dynamics of MHD hybrid and ternary nanofluids over complex geometries—including moving cylinders, porous stretching sheets, and vertically oriented Riga plates. These studies specifically highlight the highly coupled impacts of entropy generation, non-isothermal activation energy, bioconvection, and Joule heating. Furthermore, Saikia, Deka, and Karmakar [20] studied the multi-parametric gradient effects in confined plasmas, demonstrating the necessity of complex modeling in high-energy physics. Building upon these critical insights regarding porous media, activation energy, and radiative heat transfer, the present study elevates the geometrical complexity to unsteady squeezing flows with rotating boundaries.

In extreme environments, the application of a strong external magnetic field is strictly necessary to govern the kinematics and heat transfer rates of electrically conducting fluids. However, standard MHD models often omit the Hall effect. When a highly intense magnetic field is applied, the spiraling trajectory of electrons induces a secondary electric field known as the Hall current. Neglecting this under intense magnetic constraints leads to a severe overestimation of fluid resistance. The implications of electrical MHD and Hall currents on squeezed nanofluids have been meticulously evaluated by Shah et al. [21], whose studies proved that Hall currents decisively relieve rotational surface stress. Further studies expanding on Hall current and ion slip effects in ternary hybrid nanofluids have been documented by Gul et al. [22] and Krishna and Vajravelu [23]. Adding to this paradigm, the pronounced impact of Hall currents and Joule heating on rotating hybrid nanofluids has been comprehensively analyzed by Elsaid and AlShurafat [24]. Furthermore, the electromagnetohydrodynamic transport and thermodynamic behavior of trihybrid (ternary) nanoparticles within complex constrained flows have been rigorously mapped by Elsaid et al. [25]. These studies underscore the critical necessity of coupling advanced magnetic fields with multi-particle suspensions, directly motivating the fully integrated magneto-thermo-chemical model presented herein.

To further expand the theoretical horizons of nanofluid dynamics, recent literature has increasingly focused on complex boundary geometries and advanced predictive computational techniques, particularly Artificial Neural Networks (ANN). For instance, the complex dynamics of high-oscillating magnetic fields have been rigorously explored in confined rotational geometries, specifically mapping the flow between lower rotating and upper stretching disks [26]. Furthermore, the predictive capabilities of machine learning are being successfully merged with fluid mechanics; this is demonstrated by comprehensive ANN-based approaches analyzing complex thermal diffusion [27], as well as Physics-Informed Neural Networks (PINN) designed to evaluate the repercussions of angular velocity on chemically reactive ternary nanofluids [28]. These foundational studies heavily underscore the necessity of coupling advanced multiphysics phenomena with robust, modern computational methodologies.

Expanding further upon advanced computational frameworks and complex rheological models, recent investigations have highlighted the critical importance of multiphysics coupling in constrained domains. For instance, the complex dynamics of non-Newtonian flows have been thoroughly examined in highly confined geometries, such as MHD Casson fluid flows within C-type lid-driven cavities subjected to intense viscous dissipation [29]. Concurrently, the predictive power of machine learning is being successfully merged with thermal engineering, as demonstrated by comprehensive numerical and Artificial Neural Network (ANN)-based studies on electric MHD nanofluid flows utilizing multi-nanoparticle suspensions [30]. These foundational works strongly reinforce the necessity of coupling advanced multiphysics phenomena with robust analytical and computational methodologies.

A highly nuanced consequence of internal thermal escalation is the rare phenomenon of heat transfer reversal. Under extreme squeezing and rotational shear, combined with intense magnetic fields, internal dissipation can cause the fluid boundary layer temperature to exceed the bounding disks, transferring energy back into the solid boundaries. Studies by Khouqeer et al. [31], Khan et al. [32], and Zar et al. [33] have explored related thermo-bioconvective and radiative effects in unsteady MHD flows. Furthermore, the transport of suspended nanoparticles is governed by Brownian diffusion and complex chemical kinetics, typically governed by Arrhenius activation energy, as analyzed by Manigandan and Narayana [34] and optimized via machine learning by Priyadharshini et al. [35].

In parallel with the aforementioned studies on extreme thermal environments, recent investigations have further underscored the critical importance of coupling magnetic fields with activation energy in confined geometries. For example, the thermodynamic and hydrodynamic behavior of Al₂O₃/water nanofluids subjected to magnetic fields has been rigorously analyzed in asymmetric channel flows [36]. Furthermore, the profound impacts of Arrhenius activation energy on the mass transfer and thermal properties of complex nanofluid models have been successfully demonstrated in recent advanced numerical frameworks [37, 38]. These specific studies strongly reinforce the necessity of our current formulation, which mathematically couples Arrhenius kinetics with MHD ternary nanofluid dynamics within a confined squeezing gap.

The simultaneous integration of the Hall current, Joule heating, viscous dissipation, squeezing motion, rotating disks, and Arrhenius activation energy is not merely a mathematical exercise; rather, it directly models the physical environment of advanced electromagnetically controlled rotary chemical reactors and smart squeeze-film dampers utilized in aerospace engineering. In such practical engineering scenarios, the rapid squeezing and rotational motions of the bounding disks create immense mechanical shear, naturally driving severe viscous dissipation. To actively control the fluid’s rheological damping properties, a highly intense external magnetic field is applied. This strong magnetic constraint inherently induces secondary Hall currents and severe Ohmic (Joule) heating. The resulting highly localized thermal energy spikes dictate the consumption and behavior of temperature-sensitive reactive chemical additives within the fluid layer—a mass transfer process strictly governed by non-isothermal Arrhenius activation energy kinetics. Therefore, analyzing these coupled magneto-thermo-chemical mechanisms within a singular unified model is physically imperative to prevent catastrophic thermal failure and to optimize reactant transport in modern smart lubrication systems.

Despite the wealth of individual studies addressing ternary hybrid nanofluids, Hall currents, Joule heating, and Arrhenius kinetics, there exists a highly distinct theoretical void. The existing literature lacks a unified, highly coupled numerical investigation that simultaneously maps the three-dimensional unsteady squeezing motion of a magnetized ternary hybrid nanofluid (Cu–Al₂O₃–TiO₂/water) subject to dominant internal heat generation, heat transfer reversal, and non-isothermal Arrhenius chemical kinetics. This comprehensive report serves to rigorously fill this gap. By detailing a mathematical model reduced via similarity transformations and solved through the Shooting Method, this analysis explores the exact topological variations of the boundary layers and the onset of heat transfer reversal, providing indispensable theoretical benchmarks for the design of advanced thermal systems.

To explicitly state the novelty of the present study, a rigorous literature survey reveals a critical void: no prior study has investigated the highly coupled 3D unsteady squeezing flow of a magnetized ternary hybrid nanofluid (Cu–Al₂O₃–TiO₂/water) between rotating disks while simultaneously accounting for Hall currents, Joule heating, viscous dissipation, and Arrhenius activation energy. The primary novelties of this work are threefold: (i) formulating a unified mathematical framework that successfully bridges complex electromagnetism (Hall-induced cross-flows) with high-shear mechanics (squeezing and rotation) and reactive mass transfer; (ii) utilizing a high-precision Shooting-RKF45 numerical approach to resolve the heavily coupled non-linear boundary value problem; and (iii) computationally discovering and physically quantifying the critical parameter thresholds that trigger a severe heat transfer reversal phenomenon (Nu_x < 0). By establishing these previously undocumented thermodynamic limits, this comprehensive analysis provides an entirely new predictive baseline for the thermal management and optimization of advanced smart squeeze-film dampers and electromagnetically controlled rotary reactors.

Mathematical formulation

Physical model

The schematic representation of the physical flow model and the coordinate system is illustrated in Fig. 1. Consider the unsteady, three-dimensional, laminar squeezing flow of an electrically conducting ternary hybrid nanofluid (Cu–Al₂O₃–TiO₂/water) between two infinite horizontal parallel disks.

Fig. 1.

Schematic diagram of the problem

The flow configuration is described in a cylindrical coordinate system (r, 𝜑, z), where the z-axis is normal to the disks. As depicted, the lower disk is located at z = 0 and is permeable, allowing for suction or injection. The upper disk is located at a time-dependent distance h(t) = H(1−αt)^1/2, where H is the initial gap at t = 0 and α represents the characteristic expansion or contraction rate.

The lower and upper disks rotate with angular velocities Ω₁ and Ω₂, respectively. A strong time-dependent magnetic field of strength B(t) = B₀(1−αt)^−1/2 is applied along the z-axis. Due to the high intensity of the magnetic field, the Hall current (J_Hall) is significant and cannot be neglected, inducing a secondary cross-flow in the tangential direction. As shown in the physical model, the fluid is simultaneously subjected to internal energy changes via viscous dissipation, Joule heating, and a first-order chemical reaction obeying Arrhenius activation energy kinetics.

To strictly define the physical boundaries and mathematical validity of the present model, the following fundamental assumptions are employed:

• The fluid flow is assumed to be three-dimensional, unsteady, laminar, and incompressible.

• The ternary hybrid nanofluid is considered a stable, well-dispersed, and dilute suspension (𝜑≤0.1) of perfectly spherical nanoparticles, with no particle–particle agglomeration.

• The aqueous base fluid and the suspended solid nanoparticles are assumed to be in local thermal equilibrium, possessing no relative slip velocity between them.

• The induced magnetic field is neglected under the assumption of a negligibly small magnetic Reynolds number; however, the applied external magnetic field is sufficiently intense that the resulting Hall current cannot be ignored.

• The thermal energy balance fully incorporates macroscopic internal energy generation via viscous dissipation (mechanical shear heating) and Joule heating (Ohmic resistance).

• The mass transfer of the dissolved reactive solute is governed by a first-order homogeneous chemical reaction adhering strictly to non-isothermal Arrhenius activation energy kinetics.

Thermophysical properties of ternary hybrid nanofluid

The base fluid is water, and the nanoparticles are Copper (Cu, 𝜑₁), Alumina (Al₂O₃, 𝜑₂), and Titania (TiO₂, 𝜑₃). The total solid volume fraction is 𝜑 = 𝜑₁ + 𝜑₂ + 𝜑₃. To accurately predict the effective macroscopic behavior of the ternary mixture, a consistent theoretical framework is adopted. Thermodynamic properties (density and heat capacitance) follow standard linear volume-fraction mixture rules based on the principle of mass conservation. The effective dynamic viscosity is modeled via the Brinkman formulation, while the thermal and electrical conductivities for spherical nanoparticles are mathematically captured using the generalized Hamilton—Crosser and Maxwell–Garnett theories, respectively.

The utilization of these specific thermophysical models is highly suitable for the present ternary combination of metallic (Cu) and metal-oxide (Al₂O₃, TiO₂) nanoparticles. It is essential to note the physical restrictions of these effective medium theories: the Brinkman viscosity model, the Hamilton—Crosser thermal conductivity model, and the Maxwell–Garnett electrical conductivity model are fundamentally restricted to well-dispersed, non-agglomerating spherical nanoparticles operating within relatively dilute total volume fractions (𝜑≤0.1). Because the current formulation assumes a stable suspension of spherical nanoparticles without significant particle–particle interactions or agglomeration, these classical models provide a highly accurate representation of the macroscopic mixture properties, as extensively validated in recent ternary nanofluid literature [8, 13]. These effective theoretical properties are defined as follows (Table 1):

Table 1.

Thermophysical properties of the base fluid and nanoparticles

Property	Base Fluid (Water)	Cu	Al2O3	TiO2
ρ (kg/m3)	997.1	8933	3970	4250
Cp (J/kg K)	4179	385	765	686.2
k (W/m K)	0.613	400	40	8.9538
μ (kg/m s)	1.002 × 10−3	–	–	–

Density (ρ_thnf) and heat capacitance${\left(\rho C_p\right)}_{thnf}$:

$${\rho }_{thnf}=\left(1-\varphi \right){\rho }_f+{\varphi }_1{\rho }_1+{\varphi }_2{\rho }_2+{\varphi }_3{\rho }_3$$ 1

$${\left(\rho C_p\right)}_{thnf}=\left(1-\varphi \right){\left(\rho C_p\right)}_f+{\varphi }_1{\left(\rho C_p\right)}_1+{\varphi }_2{\left(\rho C_p\right)}_2+{\varphi }_3{\left(\rho C_p\right)}_3$$ 2

Dynamic viscosity (μ_thnf) [Brinkman model]:

$${\mu }_{thnf}=\frac{{\mu }_f}{{\left(1-\varphi \right)}^{2.5}}$$ 3

Thermal conductivity (k_thnf) [Hamilton—Crosser model]: Assuming spherical nanoparticles:

$$\frac{k_{thnf}}{k_f}=\frac{\frac{{\varphi }_1{k}_1+{\varphi }_2{k}_2+{\varphi }_3{k}_3}{\varphi }+2k_f-2\varphi \left(k_f-\frac{{\varphi }_1{k}_1+{\varphi }_2{k}_2+{\varphi }_3{k}_3}{\varphi }\right)}{\frac{{\varphi }_1{k}_1+{\varphi }_2{k}_2+{\varphi }_3{k}_3}{\varphi }+2k_f+\varphi \left(k_f-\frac{{\varphi }_1{k}_1+{\varphi }_2{k}_2+{\varphi }_3{k}_3}{\varphi }\right)}$$ 4

Effective electrical conductivity (σ_thnf) [Maxwell–Garnett model]:

$$\frac{{\sigma }_{thnf}}{{\sigma }_f}=1+\frac{3\left(\frac{{\varphi }_1{\sigma }_1+{\varphi }_2{\sigma }_2+{\varphi }_3{\sigma }_3}{{\sigma }_f}-\varphi \right)}{\left(\frac{{\varphi }_1{\sigma }_1+{\varphi }_2{\sigma }_2+{\varphi }_3{\sigma }_3}{\varphi {\sigma }_f}+2\right)-\left(\frac{{\varphi }_1{\sigma }_1+{\varphi }_2{\sigma }_2+{\varphi }_3{\sigma }_3}{{\sigma }_f}-\varphi \right)}$$ 5

Governing equations

Under the standard boundary layer approximations [39], the unsteady conservation equations for mass, momentum (including Hall current and Lorentz force [21, 40]), energy (with viscous and Ohmic dissipation), and species concentration [10] are expressed as follows:

Continuity equation:

$$\frac{\partial u}{\partial r}+\frac{u}{r}+\frac{\partial w}{\partial z}=0$$ 6

Radial momentum equation:

$${\rho }_{thnf}\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial r}+w\frac{\partial u}{\partial z}-\frac{v^2}{r}\right)=-\frac{\partial p}{\partial r}+{\mu }_{thnf}\frac{{\partial }^{2}u}{\partial z^2}-\frac{{\sigma }_{thnf}{B}^2\left(t\right)}{1+m^2}\left(u+mv\right)$$ 7

Tangential (swirl) momentum equation:

$${\rho }_{thnf}\left(\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial r}+w\frac{\partial v}{\partial z}+\frac{uv}{r}\right)={\mu }_{thnf}\frac{{\partial }^{2}v}{\partial z^2}-\frac{{\sigma }_{thnf}{B}^2\left(t\right)}{1+m^2}\left(v-mu\right)$$ 8

Energy equation:

$$\begin{array}{c}{\left(\rho C_p\right)}_{thnf}\left(\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}\right)=k_{thnf}\frac{{\partial }^{2}T}{\partial z^2}\hfill \\ +{\mu }_{thnf}\left[{\left(\frac{\partial u}{\partial z}\right)}^2+{\left(\frac{\partial v}{\partial z}\right)}^2\right]+\frac{{\sigma }_{thnf}{B}^2\left(t\right)}{1+m^2}\left(u^2+v^2\right)\hfill \end{array}$$ 9

Concentration equation:

It is physically imperative to clarify that the concentration field C models the mass transport of a dilute, dissolved reactive chemical species (solute) carried within the aqueous base fluid, rather than the solid nanoparticles themselves. The dispersed solid nanoparticles (Cu, Al₂O₃, TiO₂) are assumed to be chemically inert, serving strictly to modify the bulk thermophysical properties of the mixture. The dissolved reactive solute undergoes a first-order homogeneous chemical reaction governed by Arrhenius kinetics, balancing Brownian diffusion with species consumption:

$$\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial r}+w\frac{\partial C}{\partial z}=D_B\frac{{\partial }^{2}C}{\partial z^2}-\frac{k_0^2}{1-\alpha t}\left(C-C_h\right){\left(\frac{T}{T_h}\right)}^n\exp \left(\frac{-E_a}{\kappa T}\right)$$ 10

Boundary conditions

The corresponding physical boundary conditions for the squeezing flow are:

$$\begin{array}{cc}\mathrm{At\; the\; lower\; disk}\left(z=0\right):\hfill & u=0,v={\mathrm{\Omega }}_{1}r,w=w_0\left(t\right)=\frac{-A\alpha H}{\sqrt{1-\alpha t}},\hfill \\ \hfill & T=T_w,C=C_w\hfill \end{array}$$ 11

$$\begin{array}{cc}\mathrm{At\; the\; upper\; disk}\left(z=h\left(t\right)\right):\hfill & u=0,v={\mathrm{\Omega }}_{2}r,w=\frac{dh}{dt}=\frac{-\alpha H}{2\sqrt{1-\alpha t}},\hfill \\ \hfill & T=T_h,C=C_h\hfill \end{array}$$ 12

where A is the suction/injection parameter, and w₀(t) is the variable wall mass transfer velocity.

Similarity transformations

To reduce the highly coupled partial differential equations into a set of non-dimensional ordinary differential equations, the following Von Kármán similarity transformations are introduced:

$$\begin{array}{c}\eta =\frac{z}{H\sqrt{1-\alpha t}},u=\frac{\alpha r}{2\left(1-\alpha t\right)}{f}^{\prime }\left(\eta \right),v=\frac{\alpha r}{2\left(1-\alpha t\right)}g\left(\eta \right),w=\frac{-\alpha H}{\sqrt{1-\alpha t}}f\left(\eta \right)\hfill \\ \theta \left(\eta \right)=\frac{T-T_h}{T_w-T_h},\varphi \left(\eta \right)=\frac{C-C_h}{C_w-C_h}\hfill \end{array}$$ 13

It is crucial to justify the existence of self-similar solutions for this complex unsteady system. True self-similarity is strictly maintained here because the time-dependent physical quantities are mathematically scaled with the unsteadiness factor (1 − αt). Specifically, the applied magnetic field scales as B(t) = B₀(1−αt)^−1/2, the disk gap as h(t) = H(1−αt)^1/2, and the disk rotational velocities along with the chemical reaction rate contain proportional temporal factors. When these specific transient conditions are substituted into the governing partial differential equations along with Eq. (13), the explicit time variable t factors out completely. This deliberate scaling allows the transport equations, including the highly coupled Joule heating and Arrhenius reaction terms, to be rigorously reduced to a system dependent solely on the dimensionless spatial coordinate η.

It is also necessary to emphasize the geometrical exactness of the selected computational domain. Unlike semi-infinite boundary layer flows over open surfaces—where the computational domain must be artificially truncated at an asymptotically large far-field value (η → ∞) and checked for domain independence—the present physical geometry constitutes a confined squeezing gap. The spatial similarity variable $\eta =\frac{z}{H\sqrt{1-\alpha t}}$ mathematically represents the normalized vertical distance between the two disks. Because the lower disk is positioned at z = 0 and the upper disk is located at the instantaneous height $z=h\left(t\right)=H\sqrt{1-\alpha t}$, the physical domain 0≤z≤h(t) maps exactly and completely to the dimensionless finite interval 0≤η≤1. Therefore, this finite interval is strictly exact and physically sufficient, inherently eliminating any numerical errors or dependencies associated with arbitrary far-field truncations.

Dimensionless system of ODEs

Substituting Eq. (13) into Eq. (6) identically satisfies the continuity equation. To explicitly demonstrate the transformation process for the remaining equations, the required partial derivatives of the velocity components with respect to time and space are first extracted from Eq. (13):

$$\begin{array}{ccc}\hfill \frac{\partial u}{\partial t}& \hfill =\hfill & \frac{{\alpha }^{2}r}{2{\left(1-\alpha t\right)}^2}\left[f^{\prime }\left(\eta \right)+\frac{\eta }{2}{f}^{″}\left(\eta \right)\right],\frac{\partial u}{\partial r}=\frac{\alpha }{2\left(1-\alpha t\right)}{f}^{\prime }\left(\eta \right)\hfill \\ \hfill \frac{\partial u}{\partial z}& \hfill =\hfill & \frac{\alpha r}{2H{\left(1-\alpha t\right)}^{3/2}}{f}^{″}\left(\eta \right),\frac{{\partial }^{2}u}{\partial z^2}=\frac{\alpha r}{2H^2{\left(1-\alpha t\right)}^2}{f}^{″\prime }\left(\eta \right)\hfill \\ \hfill \frac{\partial v}{\partial t}& \hfill =\hfill & \frac{{\alpha }^{2}r}{2{\left(1-\alpha t\right)}^2}\left[g\left(\eta \right)+\frac{\eta }{2}{g}^{\prime }\left(\eta \right)\right]\hfill \end{array}$$

Substituting these intermediate differential terms into the governing partial differential equations (Eqs. (7)–(9)), the convective and transient terms consolidate into terms scaled by the Squeeze number $S=\frac{\alpha H^2}{2{\nu }_f}$. The applied magnetic field and Hall current terms consolidate via the magnetic parameter $M=\frac{{\sigma }_f{B}_0^2{H}^2}{{\rho }_f{\nu }_f}$. Furthermore, to eliminate the pressure gradient term $\frac{\partial p}{\partial r}$, the transformed radial momentum equation is differentiated with respect to z. By utilizing the boundary layer approximations where $\frac{\partial p}{\partial z}$ is negligible, the mixed partial derivative $\frac{{\partial }^{2}p}{\partial r\partial z}$ vanishes. This cross-differentiation successfully decouples the pressure field from the velocity field. By systematically factoring out the common spatial and temporal multipliers, the complex PDEs are rigorously reduced to the following system of coupled nonlinear ordinary differential equations:

$$\frac{A_1}{A_2}{f}^{″″}-S\left(\eta f^{″\prime }+3f^{″}-2f{f}^{″\prime }-2g{g}^{\prime }\right)-\frac{A_{5}M}{A_2\left(1+m^2\right)}\left(f^{″}+m{g}^{\prime }\right)=0$$ 14

$$\frac{A_1}{A_2}{g}^{″}-S\left(\eta g^{\prime }+2g-2f{g}^{\prime }+2f^{\prime }g\right)-\frac{A_{5}M}{A_2\left(1+m^2\right)}\left(g-m{f}^{\prime }\right)=0$$ 15

$$\frac{A_3}{A_{4}Pr}{\theta }^{″}-S\left(\eta {\theta }^{\prime }-2f{\theta }^{\prime }\right)+\frac{A_1}{A_4}Ec\left[{\left(f^{″}\right)}^2+{\left(g^{\prime }\right)}^2\right]+\frac{A_{5}MEc}{A_4\left(1+m^2\right)}\left[{\left(f^{\prime }\right)}^2+g^2\right]=0$$ 16

$$\frac{1}{Sc}{\varphi }^{″}-S\left(\eta {\varphi }^{\prime }-2f{\varphi }^{\prime }\right)-\sigma {\left(1+\delta \theta \right)}^n\exp \left(\frac{-E}{1+\delta \theta }\right)\varphi =0$$ 17

The transformed boundary conditions are given by:

$$\begin{array}{ccc}\hfill f\left(0\right)& \hfill =\hfill & A,f^{\prime }\left(0\right)=0,g\left(0\right)={\omega }_1,\theta \left(0\right)=1,\varphi \left(0\right)=1\hfill \\ \hfill f\left(1\right)& \hfill =\hfill & \frac{1}{2},f^{\prime }\left(1\right)=0,g\left(1\right)={\omega }_2,\theta \left(1\right)=0,\varphi \left(1\right)=0\hfill \end{array}$$ 18

To explicitly clarify the physical meaning and mathematical derivation of the boundary condition f(1) = 1/2, one must consider the kinematic boundary condition at the upper moving disk. Physically, the fluid at the upper disk (z = h(t)) cannot penetrate the solid wall, meaning its vertical velocity w must perfectly match the instantaneous squeezing velocity of the disk, $\frac{dh}{dt}$. Given the gap function h(t) = H(1−αt)^1/2, its time derivative is $\frac{dh}{dt}=\frac{-\alpha H}{2\sqrt{1-\alpha t}}$. Concurrently, the similarity transformation for the axial velocity is defined as $w=\frac{-\alpha H}{\sqrt{1-\alpha t}}f\left(\eta \right)$. At the upper disk, z = h(t) maps exactly to the dimensionless coordinate η = 1. Equating the transformed axial velocity at η = 1 to the physical disk velocity yields $\frac{-\alpha H}{\sqrt{1-\alpha t}}f\left(1\right)=\frac{-\alpha H}{2\sqrt{1-\alpha t}}$. Canceling the common transient multipliers results exactly in f(1) = 1/2. Thus, f(1) = 1/2 is not an arbitrary constant, but rather the strict mathematical enforcement of the kinematic squeezing motion of the upper boundary.

Here, the prime ($\prime$) denotes differentiation with respect to η. The transformed system of ordinary differential equations and boundary conditions introduces several key dimensionless physical parameters that govern the flow and heat transfer characteristics. Specifically, $S=\frac{\alpha H^2}{2{\nu }_f}$ represents the Squeeze number (S > 0 for disks moving together, S < 0 for disks moving apart), $M=\frac{{\sigma }_f{B}_0^2{H}^2}{{\rho }_f{\nu }_f}$ is the Magnetic parameter dictating the Lorentz force strength, and m = ω_eτ_e is the Hall parameter. The thermal and solutal transports are controlled by the Prandtl number $Pr=\frac{{\nu }_f}{{\alpha }_f}$, the Schmidt number $Sc=\frac{{\nu }_f}{D_B}$, and the local Eckert number $E{c}_x=\frac{{\alpha }^2{r}^2}{4{\left(1-\alpha t\right)}^2{\left(C_p\right)}_f\left(T_w-T_h\right)}$ which quantifies the local internal viscous dissipation relative to the radial coordinate. The chemical reaction kinetics are defined by the dimensionless activation energy $E=\frac{E_a}{\kappa T_h}$, the temperature difference parameter $\delta =\frac{T_w-T_h}{T_h}$, ...the dimensionless chemical reaction parameter $\sigma =\frac{k_0^2{H}^2}{{\nu }_f}$, and the fitted rate constant n. The boundary kinematics are parameterized by the lower and upper disk rotational parameters ${\omega }_1=\frac{2{\mathrm{\Omega }}_1\left(1-\alpha t\right)}{\alpha }$ and ${\omega }_2=\frac{2{\mathrm{\Omega }}_2\left(1-\alpha t\right)}{\alpha }$, respectively, alongside the suction/injection parameter A. Finally, $A_1=\frac{{\mu }_{thnf}}{{\mu }_f}$, $A_2=\frac{{\rho }_{thnf}}{{\rho }_f}$, $A_3=\frac{k_{thnf}}{k_f}$, $A_4=\frac{{\left(\rho C_p\right)}_{thnf}}{{\left(\rho C_p\right)}_f}$, and $A_5=\frac{{\sigma }_{thnf}}{{\sigma }_f}$ represent the dimensionless thermo-physical ratios of the ternary hybrid nanofluid relative to the base fluid.

To ensure the absolute mathematical robustness of the model, a rigorous dimensional consistency check was performed on the transformed system (Eqs. (14)–(17)). It is confirmed that the chosen similarity variables completely absorb the independent spatial and temporal variables (r, z, t), leaving the system solely dependent on the dimensionless coordinate η without any residual dimensional dependence. Furthermore, every individual term within the derived ordinary differential equations (for instance, the viscous momentum transfer term $\frac{A_1}{A_2}{f}^{″″}$ and the transient convective squeezing term $S\eta f^{″\prime }$) has been verified to be strictly dimensionless, guaranteeing the physical and mathematical homogeneity of the boundary value problem.

Engineering quantities of interest

For the thermal design and analysis of the squeezing system, the local skin friction coefficients (radial and swirl), the local Nusselt number, and the local Sherwood number are defined as:

$$C_{fx}=\frac{{\tau }_{zr}}{{\rho }_f{\left(\frac{\alpha r}{2\left(1-\alpha t\right)}\right)}^2},C_{fy}=\frac{{\tau }_{z\varphi }}{{\rho }_f{\left(\frac{\alpha r}{2\left(1-\alpha t\right)}\right)}^2}$$ 19

$$N{u}_x=\frac{H{q}_w}{k_f\left(T_w-T_h\right)},S{h}_x=\frac{H{q}_m}{D_B\left(C_w-C_h\right)}$$ 20

Using the similarity transformations, these critical engineering parameters are expressed in dimensionless form as:

$$\frac{R{e}_r}{\sqrt{1-\alpha t}}{C}_{fx}=A_1{f}^{″}\left(0\right)$$ 21

$$\frac{R{e}_r}{\sqrt{1-\alpha t}}{C}_{fy}=A_1{g}^{\prime }\left(0\right)$$ 22

$$\sqrt{1-\alpha t}N{u}_x=-A_3{\theta }^{\prime }\left(0\right)$$ 23

$$\sqrt{1-\alpha t}S{h}_x=-{\varphi }^{\prime }\left(0\right)$$ 24

where $R{e}_r=\frac{\alpha rH}{2{\nu }_f}$ is the local squeezing Reynolds number.

Numerical method and validation

Method of solution: the shooting method

The nonlinear coupled system of ordinary differential equations (ODEs) derived in Section 3, subject to the boundary conditions, is solved using the Shooting Method. This method is particularly effective for boundary value problems (BVPs) on a finite domain [0, 1] where initial guesses for missing gradients are required to satisfy the conditions at the far end of the domain.

The governing equations are first reduced to a system of ten first-order differential equations. Let $Y={\left[y_1,y_2,\dots ,y_{10}\right]}^T$ be the state vector such that:

$$\begin{array}{ccc}\hfill y_1& \hfill =\hfill & f,y_2=f^{\prime },y_3=f^{″},y_4=f^{″\prime }\hfill \\ \hfill y_5& \hfill =\hfill & g,y_6=g^{\prime }\hfill \\ \hfill y_7& \hfill =\hfill & \theta ,y_8={\theta }^{\prime }\hfill \\ \hfill y_9& \hfill =\hfill & \varphi ,y_{10}={\varphi }^{\prime }\hfill \end{array}$$

The primary challenge lies in the fact that five initial conditions at η = 0 are unknown:

$$\mathbf{s}={\left[y_3\left(0\right),y_4\left(0\right),y_6\left(0\right),y_8\left(0\right),y_{10}\left(0\right)\right]}^T={\left[{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5\right]}^T$$

These missing values (representing the skin friction components, heat flux, and mass flux) must be determined such that the target boundary conditions at η = 1 are satisfied:

$$\mathbf{R}\left(\mathbf{s}\right)=\left[\begin{array}{c}\hfill y_1\left(1,\mathbf{s}\right)-0.5\hfill \\ \hfill y_2\left(1,\mathbf{s}\right)-0\hfill \\ \hfill y_5\left(1,\mathbf{s}\right)-{\mathrm{\Omega }}_2\hfill \\ \hfill y_7\left(1,\mathbf{s}\right)-0\hfill \\ \hfill y_9\left(1,\mathbf{s}\right)-0\hfill \end{array}\right]=\mathbf{0}$$

Algorithm implementation

The numerical integration is performed using the Runge-Kutta-Fehlberg (RKF45) algorithm, which provides an embedded error estimate to maintain an absolute tolerance of 10⁻⁶. The iterative correction of the initial guess vector s is handled by the Newton-Raphson method:

$$s^{\left(k+1\right)}=s^{\left(k\right)}-J^{-1}R\left(s^{\left(k\right)}\right)$$

where J is the 5 × 5 Jacobian matrix calculated via central finite differences. To ensure convergence in the highly nonlinear squeezing regime, a damping factor is utilized during the update step.

It is important to acknowledge that highly nonlinear, coupled boundary value problems—particularly those describing squeezing flows with rotational boundaries and magnetic effects—can theoretically exhibit multiple (dual) solutions under specific parametric conditions, such as rapid fluid dilation (S ≪ 0) or extreme boundary suction. In the present numerical investigation, the analysis is rigorously restricted to the primary, physically stable solution branch. The Newton-Raphson iterative scheme, initialized with systematically refined guess vectors s⁽⁰⁾, consistently converged to a unique solution within the investigated parametric bounds. Throughout the parametric study (specifically within the squeezing regime S > 0), the uniqueness of the obtained solutions was verified by perturbing the initial guesses, and no physically viable dual solutions were encountered for the selected boundary conditions.

Validation of the code

To ensure the reliability and accuracy of the numerical code, the present results for a limiting case of a pure viscous, non-rotating fluid (𝜑 = 0, ω₁ = ω₂ = 0, m = 0) under squeezing conditions (S = 1) are compared with the established literature, specifically the benchmark results of Hashmi et al. [41] and Vajravelu et al. [42]. Table 2 demonstrates that the converged values of the radial skin friction parameter   under varying magnetic parameter (M) are in excellent agreement with the previous studies. This rigorous comparison successfully validates the precision of the current Shooting Method and RKF45 implementation for the fundamental squeezing kinematics.

Table 2.

Validation of numerical results for radial skin friction   against Hashmi et al. [41] and Vajravelu et al. [42] for non-rotating squeezing flow (S = 1, ω₁ = ω₂ = 0)

M	Hashmi et al. [41]	Vajravelu et al. [42]	Present Study
0.0	7.533166	7.533166	7.533166
2.0	8.263872	8.263872	8.263872
3.0	9.097326	9.097326	9.097326
5.0	11.349289	11.349079	11.349289

To further ensure the reliability of the numerical code for the highly coupled, nonlinear system, an additional validation was performed on the full comprehensive 3D model. The exact system of governing equations—including the effects of the Hall current, rotational boundaries, squeezing motion, transverse magnetic field, Joule heating, Arrhenius kinetics, and ternary hybrid nanoparticles—was solved using an independent, well-established numerical solver (MATLAB’s bvp4c collocation algorithm). Table 3 compares the critical surface quantities obtained via the present Shooting-RKF45 method against those computed by bvp4c for a highly coupled parametric state. The results exhibit an exceptional degree of agreement, thereby validating the robustness and accuracy of the current numerical implementation for the complete physical model.

Table 3.

Validation of the full coupled model parameters (C_fx, C_fy, Nu_x, Sh_x) between the present shooting method and MATLAB bvp4c

Numerical Method	Cfx	Cfy	Nux	Shx
Present (shooting-RKF45)	-0.275394	-3.414907	-0.677245	1.516503
MATLAB bvp4c	-0.275396	-3.414910	-0.677248	1.516504
Absolute Error	2.0 × 10−6	3.0 × 10−6	3.0 × 10−6	1.0 × 10−6

Fixed parameters: S = 1.0, M = 5.0, m = 2.0, Ec = 0.2, Pr = 6.2, Sc = 1.0, σ = 1.0, E = 1.0

Convergence and error analysis

To assure the numerical accuracy and robustness of the Shooting-RKF45 algorithm, a convergence and sensitivity analysis was conducted. Because the RKF45 algorithm utilizes an adaptive step-size integration scheme rather than a fixed spatial mesh, “mesh independence” is evaluated by systematically tightening the absolute local error tolerance (TOL). Table 4 presents the variation of the computed radial skin friction (C_fx) and Nusselt number (Nu_x) under progressively stricter tolerance thresholds.

Table 4.

Convergence analysis of the engineering parameters by varying the absolute error tolerance (TOL)

Tolerance (TOL)	Cfx	Nux
10−4	-0.275380	-0.677241
10−5	-0.275392	-0.677244
10−6	-0.275394	-0.677245
10−7	-0.275394	-0.677245
10−8	-0.275394	-0.677245

Fixed parameters: S = 1.0, M = 5.0, m = 2.0, Ec = 0.2

As observed in Table 4, the computed physical quantities undergo slight adjustments at higher tolerances but achieve complete stabilization at TOL = 10⁻⁶. Further tightening the tolerance to 10⁻⁷ and 10⁻⁸ yields no change up to the sixth decimal place. Consequently, an absolute error tolerance of 10⁻⁶ is proven to be completely sufficient to guarantee accurate, grid-independent, and fully converged numerical results for all subsequent parametric studies.

Results and discussion

The system of coupled, highly nonlinear ordinary differential equations governing the unsteady squeezing flow of the ternary hybrid nanofluid (Cu–Al₂O₃–TiO₂/water) was solved numerically using the Runge-Kutta-Fehlberg (RKF45) based Shooting Method. A comprehensive parametric study was conducted to analyze the influence of the Squeeze number (S), Magnetic parameter (M), Hall parameter (m), Eckert number (Ec), Activation energy (E), Schmidt number (Sc), and Chemical reaction rate (σ) on the boundary layers.

Hydrodynamic behavior and Hall current effects

To move beyond the descriptive reporting of graphical trends, the observed variations in the flow and thermal fields must be interpreted through the lens of competing physical mechanisms. The radial and tangential velocity profiles represent a direct mechanical competition between fluid inertia, the kinematic dilation imposed by the squeezing parameter (S), and the resistive Lorentz body force. Specifically, a rapid squeezing motion (S > 0) naturally accelerates the radial outflow; however, the applied transverse magnetic field (M) introduces a perpendicular Lorentz drag that actively dampens this acceleration. Concurrently, the inclusion of the Hall parameter (m) tilts the effective Lorentz force vector, inducing secondary cross-flows that counteract pure rotational Coriolis damping. In the thermal regime, the boundary layer thickness is governed by a strict thermodynamic balance between convective thermal diffusion and internal volumetric heat generation, where the enhanced thermal conductivity of the ternary nanoparticles competes directly against severe viscous and Ohmic dissipation rates.

The topological variations in the radial velocity $f^{\prime }\left(\eta \right)$ and the swirl velocity g(η) under the influence of the Squeeze number (S) are portrayed in Figs. 2 and 3. Figure 2 illustrates the behavior of the dimensionless radial velocity $f^{\prime }\left(\eta \right)$. Interestingly, an increase in S (S > 0, plates squeezing together) slightly decelerates the dimensionless radial velocity peak near the central region compared to the dilating case (S < 0). This occurs due to the strong mathematical cross-coupling with the rotational field; the mechanical squeezing restricts the fluid’s ability to develop a full radial boundary layer, thereby flattening the velocity peak to conserve momentum across the contracting gap.

Fig. 2.

Effect of Squeeze Number S on the radial velocity profile $f^{\prime }\left(\eta \right)$

Fig. 3.

Effect of Squeeze Number S on the swirl velocity profile g(η)

Furthermore, Fig. 3 reveals that the squeezing motion (S = 2.5) severely dampens the swirl velocity g(η) compared to the dilating case (S = − 2). This is governed by the conservation of angular momentum; as the fluid is mechanically pushed to larger radii by the squeezing action, its rotational speed must proportionally decrease.

Figures 4 and 5 display the classical damping effect of the applied transverse magnetic field. Mathematically, this behavior is dictated by the negative Lorentz force terms $-\frac{M}{A_2\left(1+m^2\right)}\left(f^{″}+m{g}^{\prime }\right)$ and $-\frac{M}{A_2\left(1+m^2\right)}\left(g-m{f}^{\prime }\right)$ present in the coupled momentum equations (Eqs. (14) and (15)). An increase in the Magnetic parameter (M) generates a stronger Lorentz force acting in direct opposition to the fluid motion, which flattens the radial velocity profile slightly and causes a distinct retardation of the swirl velocity boundary layer. Because the ternary hybrid nanofluid possesses high electrical conductivity, this resistive magnetic body force dominates the fluid inertia, successfully suppressing boundary layer separation.

Fig. 4.

Effect of Magnetic Parameter M on the radial velocity profile $f^{\prime }\left(\eta \right)$

Fig. 5.

Effect of Magnetic Parameter M on the swirl velocity profile g(η)

However, the introduction of the Hall current significantly alters this magnetohydrodynamic drag. Figure 6 depicts the variation of the swirl velocity g(η) for different values of the Hall parameter (m). It is observed that an increase in m notably enhances the secondary swirl flow. A strong Hall effect reduces the effective electrical conductivity of the ternary hybrid nanofluid (σ_eff = σ/(1 + m²)). Consequently, the decelerating Lorentz force is weakened, permitting the fluid to rotate with less magnetic resistance.

Fig. 6.

Effect of Hall Parameter m on the swirl velocity profile g(η)

Thermal transport and heat transfer reversal

The thermal distribution within the squeezing gap is extraordinarily sensitive to internal heat generation, as captured by Figs. 7 and 8. Figure 7 illustrates the influence of the Eckert number (Ec) on the dimensionless temperature profile θ(η). For high Eckert numbers (Ec = 1), the temperature profile develops a massive “hump” where the fluid temperature significantly exceeds the temperature of the bounding disks (θ > 3). By analyzing the energy equation (Eq. (16)), this internal thermal escalation is directly traced to the heavily coupled heat generation terms: viscous dissipation $\frac{A_1}{A_4}Ec\left[{\left(f^{″}\right)}^2+{\left(g^{\prime }\right)}^2\right]$ and Ohmic heating $\frac{MEc}{A_4\left(1+m^2\right)}\left[{\left(f^{\prime }\right)}^2+g^2\right]$. When Ec increases, the kinetic energy converted into thermal energy by internal fluid friction and electrical resistance overwhelms the thermal diffusion process, creating a localized high-temperature core within the fluid layer.

Fig. 7.

Effect of Eckert Number Ec on the dimensionless temperature profile θ(η)

Fig. 8.

Effect of Squeeze Number S on the dimensionless temperature profile θ(η)

Figure 8demonstrates that an increasing Squeeze number (S = 2.5) heavily compresses the thermal boundary layer against the upper disk, creating highly steep temperature gradients near the interface (η → 1) compared to the dilating flow.

Solutal boundary layer and Arrhenius kinetics

The transport of the suspended nanoparticles is governed by Brownian diffusion and chemical reaction rates. Figures 9 through 11 display the concentration profiles (η).

Fig. 9.

Effect of Activation Energy E on the concentration profile 𝜑(η)

Fig. 11.

Effect of Chemical Reaction Parameter σ on the concentration profile 𝜑(η)

Figure 9indicates that an increase in the dimensionless Activation energy (E) elevates the nanoparticle concentration profile. A higher activation energy provides a steeper energetic threshold required for the chemical reaction to proceed, thus suppressing the destructive reaction and preserving the ternary nanoparticles.

In Fig. 10, an increment in the Schmidt number (Sc) leads to a pronounced thinning of the solutal boundary layer. Because Sc is the ratio of momentum diffusivity to mass diffusivity, a larger Sc restricts the spatial spread of the nanoparticles, confining them closer to the boundaries.

Fig. 10.

Effect of Schmidt Number Sc on the concentration profile 𝜑(η)

Finally, Fig. 11 validates the Arrhenius model, showing that a higher chemical reaction parameter (σ = 5) directly accelerates the consumption of the species, rapidly decaying the concentration profile across the gap compared to the non-reactive state (σ = 0).

Engineering surface quantities

Tables 5 and 6 present the precise numerical values for the radial skin friction (C_fx), swirl skin friction (C_fy), local Nusselt number (Nu_x), and local Sherwood number (Sh_x). Table 5 confirms that increasing the Magnetic parameter (M) generates a stronger Lorentz force, significantly increasing the magnitude of the radial skin friction ∣C_fx∣. Quantitatively, as M intensifies from 0.0 to 20.0, the radial skin friction undergoes a severe absolute augmentation from 0.055 to  − 1.135, reflecting an immense increase in flow resistance. Meanwhile, higher Squeeze numbers (S) strongly amplify the magnitude of the swirl skin friction ∣C_fy∣ as the mechanical compression forces higher rotational shear at the boundaries. Specifically, accelerating the squeezing rate from S = 0.1 to S = 2.5 induces a massive 107.6% increase in ∣C_fy∣ (surging from 2.439 to 5.065). Table 6 further quantifies the Hall effect observation: as m increases from 0.5 to 10.0, the magnitude of the swirl skin friction ∣C_fy∣ drops from 3.789 to 3.301. This represents a notable 12.8% reduction in rotational surface stress, mathematically proving that inducing a strong Hall current is a highly effective mechanism for relieving rotational drag on the disks.

Table 5.

Variation of radial skin friction (C_fx), swirl skin friction (C_fy), Nusselt number (Nu_x), and Sherwood number (Sh_x) with squeeze number (S) and magnetic parameter (M)

S	M	Cfx	Cfy	Nux	Shx
0.1	5.0	-0.317103	-2.439384	-1.454863	1.171596
1.0	5.0	-0.275394	-3.414907	-0.677245	1.516503
2.5	5.0	-0.239713	-5.065366	5.058873	2.259941
1.0	0.0	0.055670	-3.299879	-0.435806	1.521486
1.0	10.0	-0.583721	-3.550072	-0.937217	1.511969
1.0	20.0	-1.135903	-3.863556	-1.490894	1.504221

Fixed parameters: m = 2.0, Ec = 0.2, Pr = 6.2, Sc = 1.0, σ = 1.0

Table 6.

Variation of radial skin friction (C_fx), swirl skin friction (C_fy), Nusselt number (Nu_x), and Sherwood number (Sh_x) with Hall parameter (m) and Eckert number (Ec)

m	Ec	Cfx	Cfy	Nux	Shx
0.5	0.2	-0.246272	-3.789636	-1.227494	1.517372
3.0	0.2	-0.196993	-3.352997	-0.567922	1.517636
10.0	0.2	-0.029178	-3.301906	-0.456011	1.520174
2.0	0.1	-0.275395	-3.414907	2.034049	1.513551
2.0	0.5	-0.275395	-3.414907	-8.811142	1.525257
2.0	1.0	-0.275395	-3.414907	-22.367633	1.539504

Fixed parameters: S = 1.0, M = 5.0, Pr = 6.2, Sc = 1.0, σ = 1.0

The most critical engineering takeaway is found in the heat transfer data. Table 6 vividly captures the threshold of Heat Transfer Reversal. Mathematically, a negative Nusselt number signifies an inversion of the temperature gradient at the solid boundary. Physically, this phenomenon occurs when the internal energy generation—driven synergistically by extreme viscous dissipation from the squeezing motion (high Ec) and intense Ohmic heating from the Hall currents—surpasses the conductive heat flux from the bounding walls. Consequently, the temperature of the fluid boundary layer sharply exceeds that of the solid disks. At a low Eckert number (Ec = 0.1), the Nusselt number is positive (Nu_x = 2.034), indicating standard cooling where heat transfers from the wall to the fluid. However, as the Eckert number increases to 1.0, the extreme internal dissipation forces the Nusselt number to plummet to a heavily negative value (Nu_x = − 22.367). Instead of the disks heating the fluid, the highly dissipated superheated fluid acts as a heat source, transferring thermal energy back into the solid boundaries. To rigorously understand the energy balance that gives rise to this reversal, one must evaluate the competing terms within the thermal boundary layer. The local energy balance is dominated by heat conduction near the walls and internal volumetric heat generation (viscous and Ohmic dissipation) within the fluid core. Heat transfer reversal is mathematically triggered when the gradient of the dimensionless temperature profile at the wall becomes positive (${\theta }^{\prime }\left(0\right)>0$ and ${\theta }^{\prime }\left(1\right)>0$), meaning the internal energy generation rate strictly exceeds the rate at which heat can conductively diffuse away from the fluid layer. Based on the present numerical evaluations, the critical parameter range for initiating this thermodynamic transition occurs at dissipation levels of Ec≥0.5 coupled with moderate-to-high magnetic field strengths (M≥3.0). Within these critical bounds, the highly dissipative fluid actively transitions into a heat source, necessitating advanced thermal protection for the bounding solid structures. This physical behavior and the transition to negative Nusselt numbers under extreme dissipation are highly consistent with the thermodynamic phenomena recently validated by Khouqeer et al. [31] and Khan et al. [32] in high-speed rotational flows. This is a vital parameter constraint for the thermal protection of high-speed squeezing machinery.

To explicitly translate these numerical findings into realistic engineering operating conditions, the behavior of these surface quantities must be contextualized within industrial applications. In the design of smart squeeze-film dampers and high-speed rotary lubrication systems, the observed reduction in radial skin friction (C_fx) via the Hall parameter directly translates to lower mechanical wear and reduced rotational torque requirements on the bounding disks. Furthermore, the threshold for heat transfer reversal (Nu_x < 0) identified under high Eckert and Magnetic numbers establishes a critical operational limit for industrial machinery; operating beyond this specific threshold risks catastrophic thermal degradation of the lubricant, as the highly dissipative fluid actively heats the mechanical components rather than cooling them. Finally, the enhancement of the Sherwood number (Sh_x) driven by Arrhenius activation kinetics provides a practical design metric for electromagnetically controlled rotary chemical reactors, allowing engineers to optimize the magnetic field strength to maximize reactant consumption yields during rapid mechanical squeezing phases.

Conclusions

This study provided a comprehensive numerical investigation into the three-dimensional unsteady squeezing MHD flow of a ternary hybrid nanofluid (Cu–Al₂O₃–TiO₂/water) between two parallel rotating disks. By incorporating the Hall effect, Joule heating, and Arrhenius activation energy, the model captured complex rheological and thermal interactions. The governing equations were successfully solved using the Shooting method. The primary conclusions are:

• Heat transfer reversal: A significant finding is the occurrence of negative Nusselt numbers at the disk surface as the Eckert number (Ec) and Magnetic parameter (M) increase. Physically, this demonstrates that extreme internal energy generation—driven synergistically by viscous dissipation from the squeezing motion and Ohmic heating from the Hall currents—causes the fluid’s core temperature to drastically exceed the boundary temperatures. Consequently, the fluid transitions into a heat source, reversing the direction of heat flux back into the solid boundaries, successfully corroborating recent thermodynamic studies on high-dissipation rotational flows.

• Hall effect impact: The Hall parameter (m) was found to enhance the swirl velocity component. By reducing the effective electrical conductivity, the Hall current weakens the resistive Lorentz force, thereby reducing the swirl skin friction magnitude.

• Squeezing dynamics: The Squeeze number (S) significantly alters the flow topology. Squeezing (S > 0) accelerates the radial flow and compresses the thermal boundary layer, leading to higher thermal gradients compared to dilating (S < 0) cases.

• Mass transfer control: The nanoparticle concentration is effectively managed by the activation energy and chemical reaction parameters. Higher activation energy (E) preserves the concentration by suppressing the reaction rate, while an increase in the Schmidt number (Sc) reduces the solutal boundary layer thickness.

The primary scientific contribution of this research lies in providing a unified, highly coupled theoretical benchmark for the three-dimensional unsteady squeezing motion of a magnetized ternary hybrid nanofluid, explicitly capturing the rare threshold of heat transfer reversal. These insights offer vital parameter constraints for the thermal protection of high-speed squeezing machinery, heavy-duty lubrication systems, and advanced biochemical sensors.

Despite these contributions, the present mathematical formulation possesses certain limitations. The model assumes perfectly spherical nanoparticles and constant base fluid properties, which may deviate from realistic physical behaviors under extreme temperature gradients. Furthermore, the ternary mixture is treated as a Newtonian fluid. Therefore, future investigations could extend this framework by incorporating temperature-dependent thermophysical properties, non-Newtonian rheological models (such as Casson or Carreau fluids), and utilizing artificial neural networks (ANN) or machine learning algorithms to optimize the ternary nanoparticle mixing ratios for targeted thermal management applications.

Nomenclature

A

Suction/Injection parameter at the lower disk

A₁, A₂, A₃, A₄, A₅

Dimensionless thermophysical ratios of the ternary mixture

B₀

Uniform magnetic field strength [Tesla]

C

Nanoparticle concentration [mol m⁻³]

C_w, C_h

Concentration at lower and upper disks [mol m⁻³]

C_fx, C_fy

Radial and Swirl skin friction coefficients

C_p

Specific heat at constant pressure [J kg⁻¹ K⁻¹]

D_B

Brownian diffusion coefficient [m² s⁻¹]

E

Dimensionless activation energy parameter

E_a

Activation energy [J mol⁻¹]

Ec_x

Local Eckert number

f(η)

Dimensionless radial stream function

g(η)

Dimensionless swirl velocity function

h(t)

Time-dependent gap between disks [m]

H

Initial distance between disks at t = 0 [m]

k

Thermal conductivity [W m⁻¹ K⁻¹]

k₀

Chemical reaction rate constant [s⁻¹]

m

Hall parameter

M

Magnetic parameter

n

Fitted rate constant for chemical reaction

Nu_x

Local Nusselt number

p

Fluid pressure [Pa]

Pr

Prandtl number

q_m

Wall mass flux [mol m⁻² s⁻¹]

q_w

Wall heat flux [W m⁻²]

r, 𝜑, z

Cylindrical coordinates [m, rad, m]

Re_r

Local squeezing Reynolds number

S

Squeeze number

Sc

Schmidt number

Sh_x

Local Sherwood number

t

Time [s]

T

Fluid temperature [K]

TOL

Absolute local error tolerance

T_w, T_h

Temperature at lower and upper disks [K]

u, v, w

Velocity components in r, 𝜑, z directions [m s⁻¹]

w₀(t)

Time-dependent wall mass transfer velocity [m s⁻¹]

α

Characteristic parameter for expansion/contraction [s⁻¹]

α_f

Thermal diffusivity of base fluid [m² s⁻¹]

δ

Temperature difference parameter

η

Dimensionless similarity variable

θ(η)

Dimensionless temperature function

κ

Boltzmann constant [J K⁻¹]

μ

Dynamic viscosity [kg m⁻¹ s⁻¹]

ν

Kinematic viscosity [m² s⁻¹]

ρ

Density [kg m⁻³]

σ

Dimensionless chemical reaction parameter / Electrical conductivity [S m⁻¹]

τ_zr, τ_z𝜑

Radial and Swirl wall shear stresses [Pa]

τ_e

Electron collision time [s]

𝜑

Nanoparticle volume fraction

𝜑(η)

Dimensionless nanoparticle concentration function

Ω₁, Ω₂

Angular velocity of lower and upper disks [rad s⁻¹]

ω₁, ω₂

Dimensionless rotation parameters of lower and upper disks

ω_e

Electron cyclotron frequency [rad s⁻¹]

f

Base fluid (water)

thnf

Ternary hybrid nanofluid

w

Wall condition at lower disk (z = 0)

h

Boundary condition at upper disk (z = h(t))

1, 2, 3

Refers to Cu, Al₂O₃, and TiO₂ nanoparticles, respectively

Author contributions

Shiva Rao has done Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing - original draft, Writing - review & editing. As the sole author, S.R. was responsible for all aspects of this research and manuscript preparation.

Funding

The author declares that no funds, grants, or other support were received during the preparation of this manuscript. There was no funding for this study.

Data availability

The datasets generated and/or analyzed during the current study are available from the corresponding author upon reasonable request.

Code availability

The custom numerical code (Shooting Method coupled with RKF45) used to generate the results in this study is available from the corresponding author upon reasonable request.

Declarations

Ethical approval

Not applicable. This study does not involve human or animal subjects.

Consent to publish

Not applicable.

Consent to participate

Not applicable.

Competing interests

The authors declare no competing interests.