Electroosmotic magnetohydrodynamic flow stability and heat transfer of gravity-driven ternary hybrid nanofluids over a Riga plate

Abstract

The study of ternary composite nanofluids has been given noteworthy attention because of their flow enhancement and thermal properties, making them a useful thermal transport application. The modulation of electroosmotic flow and integration of gravity with a Riga device in fluid transport control and energy dissipation are essential for optimizing the performance of cooling systems, microfluidics, and energy applications. Thus, this research focuses on the parametric sensitivities of gravity and electroosmotic-driven tri-composite nanoparticles of aluminum oxide (Al₂O₃), graphite, and carbon nanotube (CNT) propagated in a water-base fluid flowing past a Riga plate. The combined effects of gravity variation, electroosmotic force, and magnetohydrodynamic (MHD) control via the Riga plate are studied for flow stability optimization and effective heat transfer. A hybrid numerical-analytical technique solves the invariant nonlinear dimensionless equations. Sensitivity analyzes revealed that gravity variation momentously influences thermal boundary layer formation and nanoparticle distribution, while a rising electroosmotic term inspires velocity profiles and discourages viscous drag. The ternary hybrid nanofluid augments thermal conductivity, with graphite and CNT propelling thermal dispersion and Al₂O₃ supporting nanoparticle stability. The findings give an understanding of the optimal electrokinetic tuning and MHD parameters applications in energy harvesting, biomedical microfluidics, and advanced cooling technologies.

Introduction

A specialized plate entrenched with a coupling of permanent magnet for transverse magnetic field generation and electrode for electric field induction to stimulate a Lorentz force of exponential-type, affecting fluid velocity in magnetohydrodynamic flows described as a Riga plate or device. This configuration design makes the Riga plate a unique device from convectional magnetic field-based control devices or heated flat plates, as it gives effective flow management with low mechanical interference, Pang and Choi [1]. In extensive ranges, the Riga plate is utilized in therapy and drug delivery, heat exchangers, renewable energy systems, aerodynamics, and biomedical engineering [2, 3]. As such, Pantokratoras [4] discussed the influence of Lorentz force on the nanofluid flow stability across a Riga plate for a heat transfer system. It was revealed that the thermal productivity and performance of a heat exchanger and cooling system are improved with enhanced electromagnetic force. Zhang et al. [5] examined the nanofluid bioconvective flow of oxytactic microorganisms over a Riga device. As observed, the flow dynamics are precisely controlled resulting in a rising temperature profile owing to an induced viscosity. Salawu et al. [6] presented heat and mass transfer of tangent hyperbolic Arrhenius reacting fluid in porous media with Ohmic heating past a Riga plate. The fluid tiny particle’s reaction and interaction is boosted with rising internal heating, which correspondingly promotes thermal distribution. More investigations on the flow over a Riga device under the influence of electromagnetic force can be obtained from [7–10], as these studies offered substantive contributions to the Riga surface flow with different thermodynamical properties.

In nanotechnology, the flow of nanofluid along a Riga surface has several engineering, industrial, biomedical technology, and biotechnology applications. The study of advanced nanomaterials has been a point of discussion in scientific gathering due to its propelling thermal propagation and fluid flow characteristics as applied in biomedical cooling technologies, energy systems, and microfluidics, Nasir et al. [11]. Composite ternary nanofluids involve the dispersion of three different nanoparticles in a base fluid, which have demonstrated higher thermal properties compared to unitary or hybrid nanofluids as reported and are useful in cancer detection, nuclear reactors, automobile, refrigeration systems, and so on [12–14]. Owing to the usefulness of nanofluids, Suresh et al. [15] worked on synthesizing aluminum oxide and copper nanoparticles by dispersing them in deionized water of 0.1%–2.0% for enhanced thermal conductivity. The study shows that the fluid non-Newtonian characteristics and conductivity are improved by 3.16% compared to a unitary nanoparticle. Zainal et al. [16] analyzed the hybrid nanofluid thermal stability in the presence of an induced magnetic field and viscous dissipation. The flow and heat transfer phenomena were reported to have been precisely influenced for the accurate prediction of thermal science activities. Madhukesh et al. [17] considered ternary hydromagnetic non-Newtonian ternary nanofluid influenced by inclined device and viscosity. It was seen that the mass distribution rate increased with rising thermophoretic numbers but decreased with rising Brownian motion values. Sarada et al. [18] discussed the impact of an internal heating source on a ternary water-based hybridized nanofluid flow in a stretching plate employing a non-Fourier thermal flux. A high improvement of about 4.03% in thermal reaction-diffusion and conduction was observed compared to a single or hybrid nanomaterial. More valuable results on the ternary composite nanoparticles are offered by [19–22]. Meanwhile, in a thermal fluid system, the fluid flow rate and heat distribution can be influenced by various thermodynamic terms under different conditions, among is gravity, porosity, viscous dissipation, electroosmosis, and many more.

The nanofluid flow characteristics for an enhanced industrial material and nanotechnology advancement can be improved when subjected to fluid dynamical terms influence. The integration of electroosmotic and hydromagnetic flow mechanisms strengthens thermal performance and fluid distribution, particularly in microchannel or confined environments, Tripathi et al. [23]. When the Riga plate is coupled with other factors and electroosmotic force, it provides the system with a novel mechanism for managing thermal boundary and velocity layers, Sharma et al [24]. Mishra et al. [25] considered ternary hydromagnetic Jeffery (Au–TiO₂–Ag/PVA) nanofluid influenced by gyrotactic microorganisms and electroosmotic force. It was obtained from the analysis that the ternary nanoparticles collision and propagation is encouraged as the electroosmotic force stimulated Ohmic and internal heating to raise the temperature field. Oni and Jha [26] explored electroosmotic and Joule heating impacts on the gravity-driven mixed convective fluid with viscous dissipation in a microchannel and heat flux. It was found that the flow velocity is opposed to the rising electrokinetic term due to the induced electric and magnetic fields. Alsharif et al. [27] examined the efficiency of the Cattaneo heat flux model of hybrid nanofluid driven by electroosmosis force along a DC-operated micropump. The thermal performance of the DC-operated micropump is found to be enhanced as the fluid viscosity is prompted to inspire the current-carrying hybrid nanofluid. Scholars [28–30] give more insightful reports on the combined effect of electroosmotic and gravity on composite nanofluids with industrial and technology applications. Mahabaleshwar et al. [31] investigated the influence of viscosity ratio and time-dependent stretching velocity on ternary liquid film flow over an unsteady permeable stretching surface using a mathematical modeling approach. Mahabaleshwar et al. [31] presented exact multiple solutions for the Darcy-Brinkman flow of Bingham sodium alginate suspensions driven by a two-dimensional bidirectional moving plate, incorporating Navier slip and Biot boundary conditions. Sachhin et al. [32] examined the heat and mass transfer characteristics of a Bingham ternary nanofluid over stretching/shrinking sheets, considering the effects of thermal radiation and Stefan blowing. Sachhin et al. [33] analyzed the entropy generation and thermodynamic behavior of a chemically reactive Maxwell hybrid nanofluid, accounting for thermophoresis, Brownian motion, and irreversibility effects, with applications in nanotechnology. Sachhin et al. [34] investigated the irreversible mechanism and thermodynamic behavior of chemically reactive Maxwell hybrid nanofluid flow by incorporating thermophoresis and Brownian motion effects. Sachhin and Mahabaleshwar [35] examined the influence of bioconvection on chemically reactive Prandtl-Eyring nanofluid flow through a Darcy-Forchheimer porous medium using Buongiorno’s nanofluid model. Devi and Kumar [36] investigated the thermohydrodynamic behavior of tetrahybrid nanofluids flowing through a porous stretching cylinder using an artificial neural network (ANN)assisted approach. Singla, Sharma, and Kumar [37] conducted a comparative investigation of magnetohydrodynamic convective flow characteristics of mono, binary hybrid, and ternary hybrid nanofluids considering thermal radiation and multiple velocity slip effects. Taneja, Sharma, and Kumar [38] investigated the irreversibility characteristics and thermal efficiency of ternary nanofluids flowing through converging and diverging channels. Taneja, Sharma, and Kumar [39] analyzed the thermal efficiency and entropy generation of ternary nanofluids under different flow conditions. Kaswan et al. [40] investigated the bioconvection flow of magnetocross nanofluids containing gyrotactic microorganisms under activation energy and employed an artificial neural network (ANN) to predict the flow characteristics. Kumar et al. [41] examined the Cattaneo-Christov double diffusion effects on third-grade nanofluid flow over a stretching Riga plate and analyzed the associated entropy generation. Kumar et al. [42] analyzed entropy generation in microrotating Casson nanofluid flow through a Darcy-Forchheimer porous medium using the Levenberg-Marquardt neural computing algorithm. Kumar et al. [43] numerically investigated entropy generation in MHD hybrid nanofluid flow considering nonlinear thermal radiation and melting heat transfer. Kaswan et al. [44] studied the stability and multiple solutions of ternary hybrid nanofluid flow through a Darcy-Forchheimer porous medium over a stretching/shrinking surface. Kaswan et al. [45] investigated the thermal characteristics of hybridized AA7072-AA7075 alloy nanomaterials in MHD Darcy-Forchheimer flow over a moving thin needle. Ajithkumar and Makinde [46] investigated the thermofluidic behavior of magneto-radiative viscoelastic nanofluid flow in a chemically reactive porous duct under double-diffusive convection, focusing on heat and mass transfer characteristics. Maddina and Shah [47] examined the thermally dissipative peristaltic pumping of hydromagnetic nanofluids in an oblique micro-wavy conduit and employed an artificial neural network (ANN) for flow optimization and thermal performance prediction. Ajithkumar [48] studied the combined thermal and chemical interaction dynamics of electroactive nanofluid flow in a slanted porous microduct for bioinspired pumping systems, analyzing the effects on fluid flow and heat transfer. Imran et al. [49] employed a machine learning-based Bayesian regularization algorithm to predict the thermal behavior of tri-hybrid nanofluid flow over a stretching sheet and evaluated its heat transfer performance. Hussain et al. [50] investigated magnetized ternary nanofluid flow by integrating a graph neural network with the Keller-Box numerical method to analyze thermal transport characteristics. Hussain et al. [51] developed an artificial intelligence-driven framework to investigate blood-based ternary nanofluid flow for enhanced hemorheological and biomedical applications. Hussain et al. [52] analyzed radiative heat transfer in ternary nanofluid flow using a hybrid graph neural network and finite difference framework based on the Cattaneo-Christov heat flux model. Hussain et al. [53] optimized the thermal efficiency and safety of solar collectors by comparing the performance of non-toxic hybrid nanofluid mixtures using machine learning techniques. Akbar et al. [54] performed a thermodynamic analysis of nanofluids containing aggregated and non-aggregated nanoparticles to evaluate their effects on heat transfer and entropy generation. Akbar et al. [55] developed intelligent models based on response surface methodology and artificial neural networks to predict thermal transport in peristaltic nanofluid flow through porous media. Akbar et al. [56] investigated blood-based nanofluid flow induced by the metachronal beating of artificial cilia to examine its flow and thermal characteristics for biomedical applications. Iqbal et al. [57] analyzed the effects of Hall current and entropy generation on the peristaltic transport of hybrid nanofluids, emphasizing the associated heat transfer behavior. Iqbal et al. [58] examined the thermal enhancement and entropy minimization characteristics of dihydrogen oxide-based nanofluids to improve energy efficiency and heat transfer performance.

The study of hybridized nanofluids has attracted scientist’s attention owing to their thermal performance and flow characteristics, making them significant in advanced thermal science applications. Hence, this study aims to examine the parametric sensitivities of electroosmotic and gravity-driven hybrid ternary nanoparticles flowing along a Riga device with convective cooling and heat loss. Despite the usefulness, the combined influences of electroosmotic force, gravity, and MHD actuation on hybridized ternary nanofluids with heat loss and convective cooling along a Riga surface, remain under-examined. Meanwhile, understanding the parametric sensitivities of such systems is essential for improving their efficiency in biomedical engineering, cooling systems, microfluidics, and energy applications. This analysis is motivated by the fact the integration of gravity-driven and electroosmotic flow in a hydromagnetic Riga plate provides unique control of energy dissipation and fluid transport. Such studies will give ways for the development of higher effective and controllable solutions to thermal management in advanced technological applications. As such, the research outcomes are expected to help in designing efficient system thermal management in engineering.

Safe disposal of nanoparticles at the end of life

The end-of-life management of systems that use nanofluids requires proper disposal of nanoparticles to avoid environmental damage and human health hazards. The following approaches serve to manage nanofluids responsibly:

1. Containment and Recycling: The reuse of nanoparticles through recovery and recycling methods should be implemented to decrease manufacturing waste and minimize dependence on raw materials:

2. Physical Filtration: Nanoparticles can be separated safely through the use of filtration systems including centrifuges and membrane filters

3. Safe Landfilling: Nanoparticles which cannot be reused should be carefully placed in specialized landfill facilities to stop environmental contamination.

4. Neutralization or Deactivation: Environmental safety and human health protection can be achieved by treating specific nanoparticles through chemical processes that render them harmless.

5. Monitoring and Regulation: Systems need continuous environmental and health regulation monitoring to guarantee proper disposal techniques of nanoparticles.

Formulation theoretical equation

Physical assumptions

Consider a laminar, steady, and incompressible flow of a ternary composite nanofluid uniformly dispersed in a water-based solvent under the influence of gravity and electroosmotic forces. The Riga plate surface is assumed to be semi-infinite in extent (x≥0, y ∈ [0, ∞)). This semi-infinite plate assumption simplifies the mathematical formulation by neglecting edge and end effects at the physical boundaries, thereby allowing the boundary layer approximations to remain valid along the length of the plate. The nanofluid flows along a Riga device in the x-direction containing alternating electrodes and permanent magnets, which induces an exponential Lorentz force, as portrayed in Fig. 1. Helmholtz-Smoluchowski velocity is adopted to model the electroosmotic factor, and the electric field is assumed uniform and applied parallel to the Riga plate surface with a suction velocity of φ₀. The ternary hybrid nanofluid is made up of Al₂O₃, Graphite, and CTNs of diverse shapes suspended in water, which is presumed to be very small to prevent particle agglomeration but big enough to display non-Newtonian characteristics. The vertical wall is exposed to convective cooling and surface temperature ϑ_w is considered higher than the ambient temperature ϑ_∞. The energy equation is modeled to capture viscous dissipation, radiation, Joule heating, and heat loss, and to allow linearization within a small range, the nanofluid thermal properties are taken to be temperature-dependent. Following [59–61], the theoretical model is developed in alignment with the mentioned assumptions to promote heat-effective management in thermal energy applications such as solar collectors and cooling devices, this is necessary for augmenting energy efficiency and preserving system failures due to overheating.

$$\frac{\partial \varphi }{\partial x}+\frac{\partial \phi }{\partial y}=0,$$ 1

$$\begin{array}{ccc}\hfill \varphi \frac{\partial \varphi }{\partial x}+\phi \frac{\partial \varphi }{\partial y}& \hfill =\hfill & \frac{{\mu }_{th}}{{\rho }_{th}}\frac{{\partial }^2\varphi }{\partial y^2}+\frac{J_0{M}_0\pi }{8{\rho }_{th}}\exp \left(\frac{-y\pi }{c}\right)+\frac{1}{{\rho }_{th}}{\rho }_e{E}_x-\frac{{\mu }_{th}}{K{\rho }_{th}}\varphi \hfill \\ \hfill & \hfill \hfill & +\frac{g{\left(\beta \rho \right)}_{th}}{{\rho }_{th}}\left(\vartheta -{\vartheta }_{\infty }\right),\hfill \end{array}$$ 2

$$\begin{array}{ccc}\hfill \varphi \frac{\partial \vartheta }{\partial x}+\phi \frac{\partial \vartheta }{\partial y}& \hfill =\hfill & \frac{k_{th}}{{\left(\rho Cp\right)}_{th}}\frac{{\partial }^2\vartheta }{\partial y^2}+\frac{{\mu }_{th}}{{\left(\rho Cp\right)}_{th}}{\left(\frac{\partial \varphi }{\partial y}\right)}^2+\frac{H}{{\left(\rho Cp\right)}_{th}}\left(\vartheta -{\vartheta }_{\infty }\right)\hfill \\ \hfill & \hfill \hfill & +\frac{16{\sigma }_0{\vartheta }_{\infty }^3}{3{\left(\rho Cp\right)}_{th}{k}_0}\frac{{\partial }^2\vartheta }{\partial y^2}+\frac{{\mu }_{th}}{K{\left(\rho Cp\right)}_{th}}{\varphi }^2,\hfill \end{array}$$ 3

the applied boundary constraints are taken as [61, 62]

$$\begin{array}{c}\varphi ={\varphi }_w+\tau \frac{\partial \varphi }{\partial y},\phi ={\phi }_0,-k_{th}\frac{\partial \vartheta }{\partial y}=h_f\left({\vartheta }_w-\vartheta \right)\mathrm{at}y=0,\hfill \\ \varphi \to 0,\vartheta \to {\vartheta }_{\infty }\mathrm{as}y\to \infty .\hfill \end{array}$$ 4

In the energy equation, the term  + δ(ϑ − ϑ_∞) represents internal volumetric heat generation (heat source) when δ > 0. To ensure consistency between the mathematical formulation and physical interpretation, δ is strictly designated as the heat generation/source parameter. (Alternatively, if heat absorption/sink behavior is considered, the term enters with a negative sign as  − δ(ϑ − ϑ_∞)). The entrenched terms ϕ and φ are the velocity modules, x and y are the dimensional coordinates, J₀ electrodes current density, ϑ presents fluid temperature, H dimensional heat sink, k₀ is the coefficient mean absorption, q_r is the radiation, M₀ defines magnetization of magnet, ρ_e describes ionic energy density, c is electrodes and magnets width, ϑ_w and ϑ_∞ are the wall and ambient temperature, ρ_th, μ_th, Cp, and k_th are the ternary nanofluid density, viscosity, heat capacity, thermal conductivity, and σ₀ denotes Stefan-Boltzmann constant. Note that the streamwise pressure gradient term ($-\frac{\partial p}{\partial x}$) is absent from the momentum Eq. (2) because the ambient fluid outside the boundary layer is quiescent (u_∞ → 0), which yields a constant free-stream pressure and renders $\frac{\partial p}{\partial x}=0$ throughout the boundary layer region. The nanofluid medium is assumed to be optically thick, justifying the application of the Rosseland approximation for thermal radiation. This assumption is physically grounded because the presence of suspended solid nanoparticles (Al₂O₃, Graphite, and CNTs) substantially increases the mean extinction coefficient (k₀) of the fluid, causing radiative heat transfer to behave locally in a diffusion-like manner. It is worth emphasizing the distinct physical roles of the electroosmotic parameters λ and M_e to avoid double counting of the electrical forcing. The electroosmotic velocity parameter (λ) scales the magnitude of the Helmholtz–Smoluchowski electroosmotic drift velocity, representing the strength of the external axial electric field. Conversely, the electroosmotic parameter (M_e = κL) denotes the non-dimensional inverse Debye length, which dictates the dimensionless thickness of the Electric Double Layer (EDL) and governs the spatial distribution of net electric charge density away from the plate surface.

Fig. 1.

Variation of Conventional, nanofluid, hybrid and ternary-hybrid nanofluids

For a thermodynamically applicable model, an invariant dimensionless of the fluid flow governing equations are carried out using the following quantities as employed by Afridi et al. [60]:

$$\begin{array}{c}\eta =\frac{x}{l},\xi =\frac{y}{l},u=\frac{\varphi }{{\varphi }_w},v=\frac{\phi }{{\phi }_0},L=\frac{c}{\pi },l=\frac{L^2{\varphi }_w}{{\nu }_f},{\phi }_0=\frac{\pi {\nu }_f}{c},T=\frac{\vartheta -{\vartheta }_{\infty }}{{\vartheta }_w-{\vartheta }_{\infty }},\hfill \\ {\rho }_e=\frac{ϵ}{{\chi }_e^2}{\gamma }_1,{\gamma }_1=\zeta \gamma ,{\chi }_e^2=\frac{{\vartheta }_{av}{K}_bϵ}{2{\zeta }_0{\left(z,e\right)}^2}.\hfill \end{array}$$ 5

The transformation of the two-dimensional governing partial differential equations into a system of ordinary differential equations depending solely on the dimensionless similarity variable ξ implies that the normal velocity component reduces to a constant value (v = φ₀) across the boundary layer. This self-similar mathematical reduction remains physically valid under the assumptions of a flat semi-infinite plate geometry, a quiescent free-stream (u_∞ → 0), and uniform wall boundary conditions, including constant wall suction/injection (φ₀ = const) and uniform parallel electric field application along the streamwise direction.

Analysis of electro-osmotic fluid flow

Following the Poisson distribution, the electric potential γ₁ along the double electrical layer across the electrolytic solution is given as Vijayaragavan et al. [62]

$${\mathrm{\Delta }}^2{\gamma }_1=-\frac{{\rho }_e}{ϵ},$$ 6

where ϵ present the dielectric fluid ionic permittivity. Applying the estimated boundary layer, the Eq. (6) gives

$$\frac{d^2{\gamma }_1}{d{y}^2}=-\frac{{\rho }_e}{ϵ}.$$ 7

The net charge ionic species density ρ_e is taken from Boltzmann distribution as employed by Vijayaragavan et al. [62]

$${\rho }_e=\left(n^{-}-n^{+}\right)e{z}_v,$$ 8

such that z_v indicates the ionic valence charges, n⁻ os the negative ions, and n⁺ denotes the positive ions. Therefore,

$$n^{+}=n_0{e}^{+\frac{z_{v}e}{{\vartheta }_{av}{K}_b}{\gamma }_1},n^{-}=n_0{e}^{-\frac{z_{v}e}{{\vartheta }_{av}{K}_b}{\gamma }_1},$$ 9

where the Boltzmann constant is K_b, average temperature is denoted by ϑ_av, and bulk density is n₀ respectively. The wall zeta potential (≤25 mV) is assumed insignificant. Thus, by the Debye-Huckel linear technique, the Boltzmann-Poisson equation becomes

$$\frac{{\partial }^2{\gamma }_1}{\partial y^2}=\frac{1}{{\chi }_e^2}{\gamma }_1.$$ 10

Introducing the variables $\gamma =\frac{{\gamma }_1}{\zeta }$ and $\xi =\frac{y}{L}$ to obtain

$$\frac{{\partial }^2\gamma }{\partial {\xi }^2}=M_e^2\gamma ,$$ 11

here, $M_e=z_{v}Le{\left(\frac{2n_0}{K_bϵ{\vartheta }_{av}}\right)}^{1/2}$ represents the electroosmotic term. By applying the boundary constraints γ = 1 at ξ = 0, and γ = 0 as ξ → ∞ leads to:

$$\gamma =\exp \left(-M_e\xi \right).$$ 12

Hence, to have an invariant dimensionless model, the quantities defined in Eq. (5) and the fluid electroosmotic analysis are utilized on the flow governing equations, these give:

$$\frac{\partial u}{\partial \eta }+\frac{\partial v}{\partial \xi }=0$$ 13

$$h_{1}u\frac{\partial u}{\partial \eta }+h_{1}v\frac{\partial u}{\partial \xi }=h_0\frac{{\partial }^{2}u}{\partial {\xi }^2}+Q\exp \left(-\xi \right)+\lambda M_e^2\exp \left(-M_e\xi \right)+h_2\alpha T-h_0\mathrm{\Lambda }u,$$ 14

$$Pr{h}_3\left(u\frac{\partial T}{\partial \eta }+v\frac{\partial T}{\partial \xi }\right)=\left(h_4+\frac{4}{3}R\right)\frac{{\partial }^{2}T}{\partial {\xi }^2}+h_{0}EcPr{\left(\frac{\partial u}{\partial \xi }\right)}^2+Pr\delta T+h_{0}PrEc\mathrm{\Lambda }{u}^2,$$ 15

the corresponding invariant boundary constraints give

$$\begin{array}{c}u=1+\omega \frac{\partial u}{\partial \xi },v=v_w,h_4\frac{\partial T}{\partial \xi }=-Bi\left(1-T\left(\xi \right)\right),\mathrm{at}\xi =0,\hfill \\ u\to 0,T\to 0,\mathrm{as}\xi \to \infty .\hfill \end{array}$$ 16

To maintain explicit notation separation within the solution scheme, the dimensionless electroosmotic velocity ratio is denoted as λ_eo, whereas λ_i represents the exponential decay parameters associated with the spatial basis expansion. The dynamically entrenched parameters in the dimensionless equations are the velocity slip $\omega =\frac{\tau }{l}$, Biot number $Bi=\frac{l{h}_f}{k_f}$, Eckert number $Ec=\frac{{\varphi }_w^2}{{\left(Cp\right)}_f\left({\vartheta }_w-{\vartheta }_{\infty }\right)}$, Helmholtz-Smoluchowski velocity $\lambda =-\frac{ϵ{\vartheta }_{av}{K}_b{E}_x}{{\mu }_{f}c{z}_v{x}_0}$, medium porosity $\mathrm{\Lambda }=\frac{l^2}{K}$, thermal convection $\alpha =\frac{{\beta }_f{l}^{3}g\left({\vartheta }_w-{\vartheta }_{\infty }\right)}{{\nu }_f^2}$, heat loss $\delta =\frac{H{l}^2}{{\mu }_f{\left(Cp\right)}_f}$, Prandtl number $Pr=\frac{{\mu }_f{\left(Cp\right)}_f}{k_f}$, radiative heat $R=\frac{4{\sigma }_0{\vartheta }_{\infty }^3}{k_0{k}_f}$, Hartmann modified number $Q=\frac{J_0{M}_0{L}^2\pi }{8{\nu }_f{\rho }_f{\varphi }_w}$, and wall porosity $v_w=\frac{V_w}{V_0}$, respectively. Here, x₀ denotes a constant characteristic streamwise reference length, ensuring that λ and all associated parameters remain strictly constant and independent of the spatial coordinate x. The ternary nanofluid ratios are defined as $h_0=\frac{{\mu }_{th}}{{\mu }_f}$, $h_1=\frac{{\rho }_{th}}{{\rho }_f}$, $h_2=\frac{{\left(\beta \rho \right)}_{th}}{{\left(\beta \rho \right)}_f}$, $h_3=\frac{{\left(\rho Cp\right)}_{th}}{{\left(\rho Cp\right)}_f}$, and $h_4=\frac{k_{th}}{k_f}$. Following [61, 63, 64] the ternary nanofluid viscosity with CNTs and graphite nanoparticles, and platelet shape aluminum oxide in H₂O base solvent is defined The thermophysical density of the ternary hybrid nanofluid is derived strictly from the law of mixture volume conservation:

$${\rho }_{thnf}=\left(1-{\psi }_a-{\psi }_b-{\psi }_c\right){\rho }_f+{\psi }_a{\rho }_a+{\psi }_b{\rho }_b+{\psi }_c{\rho }_c$$ 17

where the volume fraction of the base fluid is given by (1 − ϕ_t) with ϕ_t = ψ_a + ψ_b + ψ_c ≤ t which results in velocity re 0.04, ensuring that total mixture volume is identically preserved.

Furthermore, the effective dynamic viscosity is evaluated using a linear cross-weighting superposition model:

$${\mu }_{thnf}={\mu }_f\left(\frac{{\psi }_a}{{\left(1-{\psi }_a\right)}^{2.5}}+\frac{{\psi }_b}{{\left(1-{\psi }_b\right)}^{2.5}}+\frac{{\psi }_c}{{\left(1-{\psi }_c\right)}^{2.5}}\right)$$ 18

This formulation is physically grounded in dilute multi-phase suspension theory, where individual solid species contribute independently to the effective momentum dissipation rate without inter-particle chemical interference at low total concentrations (ϕ_t≤4%). The ternary nanofluid density is defined as:

$${\rho }_{th}=\left(1-{\psi }_a+{\psi }_b+{\psi }_c\right){\rho }_{sf}+{\rho }_{spa}{\psi }_a+{\rho }_{spb}{\psi }_b+{\rho }_{spc}{\psi }_c,$$ 19

where ρ_spa, ρ_spb, and ρ_spc denotes the densities aluminum oxide, graphite, and CNTs respectively. The ternary composite nanofluid heat capacity is described as:

$${\left(\rho Cp\right)}_{th}={\left(\rho Cp\right)}_{sf}\left(1-{\psi }_a-{\psi }_b-{\psi }_c\right)+{\psi }_a{\left(\rho Cp\right)}_{spa}+{\psi }_b{\left(\rho Cp\right)}_{spb}+{\psi }_c{\left(\rho Cp\right)}_{spc},$$ 20

where (ρCp)_sf denotes the water base solvent heat capacity. The hybrid ternary thermal conductivity of the nanofluid is given as:

$$k_{th}=\left(k_{nfa}{\psi }_c+k_{nfb}{\psi }_a+k_{nfc}{\psi }_b\right){\psi }^{-1},$$ 21

where k_sf is the water fluid solvent thermal conductivity. Thus, the aluminum oxide, graphite and CNTs nanoparticles thermal conductivity are individual defined as:

$$\begin{array}{ccc}\hfill k_{nfa}& \hfill =\hfill & \left(\frac{-2\psi \left(k_{sf}-k_{spa}\right)+2k_{sf}+k_{spa}}{\psi \left(k_{sf}-k_{spa}\right)+2k_{sf}+k_{spa}}\right)k_{sf},\hfill \\ \hfill k_{nfb}& \hfill =\hfill & \left(\frac{-3.9\psi \left(k_{sf}-k_{spb}\right)+3.9k_{sf}+k_{spb}}{\psi \left(k_{sf}-k_{spb}\right)+2k_{sf}+k_{spb}}\right)k_{sf},\hfill \\ \hfill k_{nfc}& \hfill =\hfill & \left(\frac{-4.7\psi \left(k_{sf}-k_{spc}\right)+4.7k_{sf}+k_{spc}}{\psi \left(k_{sf}-k_{spc}\right)+2k_{sf}+k_{spc}}\right)k_{sf}.\hfill \end{array}$$ 22

Likewise, the therm expansion of the composite ternary nanoparticles is described as:

$${\left(\rho \beta \right)}_{th}=\left(1-{\psi }_c\right)\left(1-{\psi }_b\right)\left(\left(1-{\psi }_a\right){\left(\rho \beta \right)}_{sf}+{\psi }_a{\left(\rho \beta \right)}_{spa}\right)+{\psi }_b{\left(\rho \beta \right)}_{spb}+{\psi }_c{\left(\rho \beta \right)}_{spc}.$$ 23

The different shape nanoparticle physical properties of the employed models and the associated dynamical characteristics are presented in Table 1.

Table 1.

Nanoparticles and base solvent dynamical properties and values

Quantities	Cp (J/kgK)	k (W/mK)	ρ (kg/m3)	β (K−1)
H2O	4179	0.613	997.1	21 × 10−5
Al2O3	765	40	3970	0.85
Graphite	410	5000	2200	2.5 × 10−6
CNTs	410	3007.4	2100	44

Based on the suction assumption, Eq. (13) is automatically satisfied while the dimensionless Eqs. (14) and (15) is streamlined and transformed to:

$$h_1{v}_w\frac{\partial u}{\partial \xi }=h_0\frac{{\partial }^{2}u}{\partial {\xi }^2}+Q\exp \left(-\xi \right)+\lambda M_e^2\exp \left(-M_e\xi \right)+h_2\alpha T-h_0\mathrm{\Lambda }u,$$ 24

$$Pr{h}_3\left(v_w\frac{\partial T}{\partial \xi }\right)=\left(h_4+\frac{4}{3}R\right)\frac{{\partial }^{2}T}{\partial {\xi }^2}+h_{0}EcPr{\left(\frac{\partial u}{\partial \xi }\right)}^2+Pr\delta T+h_{0}PrEc\mathrm{\Lambda }{u}^2,$$ 25

the corresponding invariant boundary constraints give

$$\begin{array}{c}u=1+\omega \frac{\partial u}{\partial \xi },v=v_w,h_4\frac{\partial T}{\partial \xi }=-Bi\left(1-T\left(\xi \right)\right),\mathrm{at}\xi =0,\hfill \\ u\to 0,T\to 0,\mathrm{as}\xi \to \infty .\hfill \end{array}$$ 26

The essential engineering dynamical quantities, which is crucial in precise prediction of their activities and applicable in the study of the performance of industrial machines and equipments are the wall friction and wall heat transfer rate. These wall effects are respectively defined mathematically in dimensionless form for the considered flow model as:

$$Wf\sqrt{Re}=-h_0\frac{du}{d\xi },\frac{Wh}{\sqrt{Re}}=-\left(h_4+\frac{4}{3}R\right)\frac{dT}{d\xi },\mathrm{at}\xi =0.$$ 27

Solution methodology

The governing equations is solve using Galerkin Weighted Residual Method (GWRM), the following steps are:

Step 1: Given Differential Equations

$$h_1{v}_w\frac{\partial u}{\partial \xi }-h_0\frac{{\partial }^{2}u}{\partial {\xi }^2}-Q\exp \left(-\xi \right)-\lambda M_e^2\exp \left(-M_e\xi \right)-h_2\alpha T+h_0\mathrm{\Lambda }u=0$$ 28

$$Pr{h}_3\left(v_w\frac{\partial T}{\partial \xi }\right)-\left(h_4+\frac{4}{3}R\right)\frac{{\partial }^{2}T}{\partial {\xi }^2}-h_{0}EcPr{\left(\frac{\partial u}{\partial \xi }\right)}^2-Pr\delta T-h_{0}PrEc\mathrm{\Lambda }{u}^2=0$$ 29

with the boundary conditions:

1. At ξ = 0:

$$u=1+\omega \frac{\partial u}{\partial \xi },v=v_w,h_4\frac{\partial T}{\partial \xi }=-Bi\left(1-T\left(\xi \right)\right)$$ 30

2. As ξ → ∞:

$$u\to 0,T\to 0$$ 31

Step 2: Approximate Solution Expansion

We assume approximate solutions for velocity u(ξ) and temperature T(ξ) as a series expansion using suitable basis functions. For smooth decay as ξ → ∞, we use polynomial-exponential terms:

$$u_N\left(\xi \right)={\sum }_{i=0}^N{a}_i{\varphi }_i\left(\xi \right),T_N\left(\xi \right)={\sum }_{i=0}^N{b}_i{\psi }_i\left(\xi \right)$$ 32

where:

• ϕ_i(ξ) and ψ_i(ξ) are basis functions.

• a_i and b_i are unknown coefficients to be determined.

A common choice for basis functions that satisfy the decay condition is:

$${\varphi }_i\left(\xi \right)=e^{-{\lambda }_i\xi },{\psi }_i\left(\xi \right)=e^{-{\gamma }_i\xi }$$ 33

For a two-term expansion (N = 1):

$$u\left(\xi \right)\approx a_0{e}^{-{\lambda }_0\xi }+a_1{e}^{-{\lambda }_1\xi },T\left(\xi \right)\approx b_0{e}^{-{\gamma }_0\xi }+b_1{e}^{-{\gamma }_1\xi }$$ 34

where λ_i, γ_i > 0 ensure decay.

Step 3: Computing Derivatives of Approximate Solutions

We substitute these expansions into the differential equations. First, we compute the derivatives:

$$\frac{du}{d\xi }={\sum }_{i=0}^N\left(-{\lambda }_i{a}_i\right)e^{-{\lambda }_i\xi },\frac{d^{2}u}{d{\xi }^2}={\sum }_{i=0}^N\left({\lambda }_i^2{a}_i\right)e^{-{\lambda }_i\xi }$$ 35

$$\frac{dT}{d\xi }={\sum }_{i=0}^N\left(-{\gamma }_i{b}_i\right)e^{-{\gamma }_i\xi },\frac{d^{2}T}{d{\xi }^2}={\sum }_{i=0}^N\left({\gamma }_i^2{b}_i\right)e^{-{\gamma }_i\xi }$$ 36

Step 4: Substituting into Governing Equations

Substituting these expansions into the given differential equations:

$$h_1{v}_w\frac{du}{d\xi }-h_0\frac{d^{2}u}{d{\xi }^2}-Q{e}^{-\xi }-\lambda M_e^2{e}^{-M_e\xi }-h_2\alpha T+h_0\mathrm{\Lambda }u=0$$ 37

$$Pr{h}_3{v}_w\frac{dT}{d\xi }-\left(h_4+\frac{4}{3}R\right)\frac{d^{2}T}{d{\xi }^2}-h_{0}EcPr{\left(\frac{du}{d\xi }\right)}^2-Pr\delta T-h_{0}PrEc\mathrm{\Lambda }{u}^2=0$$ 38

For the First Equation (Momentum Equation)

$$\begin{array}{c}{h}_1{v}_w{\sum }_{i=0}^N\left(-{\lambda }_i{a}_i\right)e^{-{\lambda }_i\xi }-h_0{\sum }_{i=0}^N\left({\lambda }_i^2{a}_i\right)e^{-{\lambda }_i\xi }-Q{e}^{-\xi }-\lambda M_e^2{e}^{-M_e\xi }-h_2\alpha {\sum }_{i=0}^N{b}_i{e}^{-{\gamma }_i\xi }\hfill \\ +h_0\mathrm{\Lambda }{\sum }_{i=0}^N{a}_i{e}^{-{\lambda }_i\xi }=0\hfill \end{array}$$ 39

Rearranging terms:

$${\sum }_{i=0}^N\left[-h_1{v}_w{\lambda }_i{a}_i-h_0{\lambda }_i^2{a}_i+h_0\mathrm{\Lambda }{a}_i-h_2\alpha b_i\right]e^{-{\lambda }_i\xi }=Q{e}^{-\xi }+\lambda M_e^2{e}^{-M_e\xi }$$ 40

For the Second Equation (Energy Equation)

$$\begin{array}{c}Pr{h}_3{v}_w{\sum }_{i=0}^N\left(-{\gamma }_i{b}_i\right)e^{-{\gamma }_i\xi }-\left(h_4+\frac{4}{3}R\right){\sum }_{i=0}^N\left({\gamma }_i^2{b}_i\right)e^{-{\gamma }_i\xi }-h_{0}EcPr{\sum }_{i=0}^N\left({\lambda }_i^2{a}_i^2{e}^{-2{\lambda }_i\xi }\right)\hfill \\ -Pr\delta {\sum }_{i=0}^N{b}_i{e}^{-{\gamma }_i\xi }-h_{0}PrEc\mathrm{\Lambda }{\sum }_{i=0}^N{a}_i^2{e}^{-2{\lambda }_i\xi }=0\hfill \end{array}$$ 41

Rearranging terms:

$$\begin{array}{c}{\sum }_{i=0}^N\left[-Pr{h}_3{v}_w{\gamma }_i{b}_i-\left(h_4+\frac{4}{3}R\right){\gamma }_i^2{b}_i-Pr\delta b_i\right]e^{-{\gamma }_i\xi }=h_{0}EcPr{\sum }_{i=0}^N\left({\lambda }_i^2{a}_i^2{e}^{-2{\lambda }_i\xi }\right)\hfill \\ +h_{0}PrEc\mathrm{\Lambda }{\sum }_{i=0}^N{a}_i^2{e}^{-2{\lambda }_i\xi }\hfill \end{array}$$ 42

Step 5: Applying the Boundary Conditions

$$h_4{\left(\frac{dT}{d\xi }\right.∣}_{\xi =0}=-Bi\left(1-T\left(0\right)\right)$$ 43

Substituting T(ξ):

$$h_4{\sum }_{i=0}^N\left(-{\gamma }_i{b}_i\right)=-Bi\left(1-{\sum }_{i=0}^N{b}_i\right)$$ 44

$${\sum }_{i=0}^N\left(h_4{\gamma }_i{b}_i\right)+Bi{\sum }_{i=0}^N{b}_i=Bi$$ 45

This provides another constraint on b_i.

As ξ → ∞

Since the selected basis functions decay exponentially, they inherently satisfy:

$$u\left(\xi \right)\to 0,T\left(\xi \right)\to 0\mathrm{as}\xi \to \infty$$ 46

Step 6: Approximate Solutions Using Two-Term Expansion

To illustrate, let’s take N = 1 (two terms):

$$u\left(\xi \right)\approx a_0{e}^{-{\lambda }_0\xi }+a_1{e}^{-{\lambda }_1\xi },T\left(\xi \right)\approx b_0{e}^{-{\gamma }_0\xi }+b_1{e}^{-{\gamma }_1\xi }$$ 47

Applying the Boundary Conditions

1. For velocity:

$$a_0+a_1+\omega \left(-{\lambda }_0{a}_0-{\lambda }_1{a}_1\right)=1$$ 48

2. For temperature:

$$h_4\left({\gamma }_0{b}_0+{\gamma }_1{b}_1\right)+Bi\left(b_0+b_1\right)=Bi$$ 49

These equations can be solved for a₀, a₁, b₀, b₁.

To determine the coefficients a_i and b_i, we enforce Galerkin’s condition, which requires the weighted residuals to vanish:

$${\int }_0^{\infty }{R}_u\left(\xi \right){\varphi }_j\left(\xi \right)d\xi =0,j=0,1,\dots ,N$$ 50

$${\int }_0^{\infty }{R}_T\left(\xi \right){\psi }_j\left(\xi \right)d\xi =0,j=0,1,\dots ,N$$ 51

Expanding these integrals results in a system of algebraic equations for a_i and b_i, which can be solved numerically. It can be observed from Table 2 that the numerical results closely align with those reported in previous studies.

Table 2.

Comparison of the u(ξ) and T(ξ) with past published outcomes

Me	Result of [65]		Present result
	u(ξ)	T(ξ)	u(ξ)	T(ξ)
0.00	1.0873	-0.5607	1.087336	-0.560723
0.01	1.0863	-0.5658	1.086323	-0.565813
0.04	1.0833	-0.5810	1.083329	-0.581056
0.25	1.0630	-0.6830	1.063014	-0.683035
1.00	1.0000	-1.0000	1.000000	-1.000000
4.00	0.8311	-1.8968	0.831129	-1.896819
25.00	0.4702	-4.9155	0.470228	-4.915523

Results and discussion

The default parameter values used in the numerical computations are set as Pr = 6.2, ψ_a = ψ_b = ψ_c = 0.02, Q = 0.5, λ = 0.5, M_e = 1.0, v_w = 0.5, ω = 0.2, Bi = 0.2, R = 0.5, δ = 0.1, Ec = 0.01, Λ = 0.2, and α = 0.5. The Table 3 and Fig. 1 presents heat transfer enhancement outcome for conventional fluid and for different nanofluids including nanofluid (NFs) and hybrid (HNFs) and ternary-hybrid nanofluids (THNFs). The outcomes show a consistent improvement in thermal conductivity as additional nanoparticles (NPs) are introduced to the fluid system. However, the nanofluid (NFs) exhibits the lowest thermal conductivity, followed by the hybrid nanofluids, while the ternary-hybrid nanofluids achieves the highest thermal conductivity due to the synergistic interactions with multiple nanoparticles. The percentage upsurge in thermal conductivity is highest for the (THNFs), reaching up to 10.86 percentage, whereas the (HNFs) displays an increase of around 9.55 percentage. The result shows that heat transfer performance increases through nanofluid systems with better results achieved when using mixed nanoparticle combinations.

Table 3.

Comparison of different fluid types at varying λ values

λ	Conventional	(NFs)	% Increase	(HNFs)	% Increase	(THNFs)	% Increase
1.0	0.59012	0.62534	5.96	0.65487	10.96	0.67045	13.61
2.0	0.60145	0.63891	6.24	0.66792	11.06	0.68512	13.92
3.0	0.61278	0.64923	5.94	0.67915	10.83	0.69687	13.71
4.0	0.62056	0.65879	6.16	0.68734	10.77	0.70521	13.61
5.0	0.63034	0.66825	6.02	0.69642	10.48	0.71567	13.53

Modified Hartmann number (Q): The effect of the Q on the fluid flow is a essential aspect of MHD velocity flow analysis. The Q measures the combined effect of magnetic field as well as viscous force resistance in the flowing fluid. As the modified Hartmann number rises it causes substantial changes to fluid velocity profiles. As the Q rises it causes substantial changes to u (ξ). However, the enhancement of this modified Hartmann number boosts the electromagnetic damping outcome, because it suppresses fluid movement. A rise in magnetic field intensity causes the Lorentz force to intensify and produces stronger opposing resistance to fluid flow direction. An electrically conducting fluid encounters resistance against motion that reduces velocity because of the Lorentz force. As seen in Fig. 2a and b, the increase in the Q exerts a controlling influence on the fluid flow of magnetohydrodynamic fluids via introducing a resistive force that decreases liquid motion. The combination of inertia forces with electromagnetic forces explains the physical behavior that was observed. The fluid attains fast motion at lower modified Hartmann numbers because it faces minimal resistance to flow. An increase in modified Hartmann number causes electromagnetic forces to control flow movement which results in velocity reduction.

Fig. 2.

a: Effect of Q on u(ξ). b: 3D plot of Q on u(ξ)

Helmholtz-Smoluchowski velocity: The Helmholtz-Smoluchowski velocity serves as a powerful factor which influencing the fluid velocity as showed in Fig. 3a and b. As this fluid flow upsurges, a remarkable influence on the velocity distribution is observed, leading to changes in the fluid dynamics within the system. An electrokinetic effect produces the Helmholtz-Smoluchowski velocity because charged surfaces come into contact with electrolyte solutions which creates electroosmotic flow. The importance of this effect becomes crucial for both microfluidics and nanofluidic systems since they require accurate fluid control. A higher speed of Helmholtz-Smoluchowski causes an increase in motion of fluid throughout the domain. The fluid velocity profile increases due to enhanced electroosmotic force that operates on charged fluid particles. The flow pattern becomes uniform and stable due to which velocity gradients decrease which yields enhanced mixing and transport properties.

Fig. 3.

a: Effect of λ on u(ξ). b: 3D plot of λ on u(ξ)

Slip parameter: The velocity slip parameter serves as the important function which determines how fluid velocity function in flow systems. The interaction between fluid and solid boundary becomes altered when velocity slip parameters upsurge which results in surface velocity reduction. Figure 4a and b demonstrates that elevated velocity slip parameters lead to diminished velocity distribution throughout the flow region. The flow velocity decreases when slip occurs because surface shear stress weakens under the no-slip boundary condition. Physically, this phenomenon occurs because the fluid boundary experiences different momentum exchanges with the surrounding fluid. When no-slip conditions occur normally the fluid moves at the same velocity as the solid surface at their contact point. The introduction of slip between the fluid and surface enables the fluid to shift its position relative to the surface thus reducing its resistance. The decreased friction leads to reduced velocity gradient at the boundary which modifies the way velocity distributes through the fluid domain. The velocity slip parameter shows practical uses within different engineering and industrial operations.

Fig. 4.

a: Variation of slip term vs. u(ξ). b: 3D plot of slip term on u(ξ)

Electro-osmotic parameters: The graphical representation in Fig. 5a and b shows how rising electro-osmotic parameters affect velocity distribution for microfluidic and nanofluidic systems. An applied electric field interacts with the electric double layer near the fluid-solid interface to produce electro-osmotic flow. The fluid transport characteristics change due to modified velocity profiles as the electro-osmotic parameter rises. The flow shows enhanced acceleration because the velocity distribution intensifies near charged surfaces. The physical explanation of this phenomenon occurs because of electrokinetic processes operating in the fluid. Fluid motion undergoes electrokinetic modification according to the electro-osmotic parameter which derives from zeta potential and the electric field strength. When this parameter rises, electrical forces become stronger than viscous forces which leads to velocity improvements. The observed changes in velocity profile arise from the combination of fluid nature along with EDL thickness and surface charge distribution characteristics. The increased electro-osmotic parameter causes fundamental changes to velocity profile which results in accelerated fluid movement through enhanced electrokinetic forces. Furthermore, it was observed that the impact of suction parameter improved the fluid flow. The impact of the electroosmotic parameter (M_e) exhibits a non-monotonic, competing physical behavior governed by the product term $M_e^2{e}^{-M_e\xi }$. In the immediate vicinity of the plate surface (ξ → 0), the quadratic coefficient $M_e^2$ dominates because $e^{-M_e\xi }\approx 1$, producing a stronger local electrostatic force that initially accelerates the fluid within the Electric Double Layer (EDL). However, moving outward into the boundary layer (ξ > 0), the exponential decay term $e^{-M_e\xi }$ rapidly suppresses the force due to a compressed Debye length. Consequently, higher values of M_e confine the electroosmotic driving effect to a very thin layer near the surface, rapidly diminishing its influence across the remaining boundary layer thickness.

Fig. 5.

a: Effect of M_e on u(ξ). b: Effect of M_e on u(ξ)

Biot number: When the Biot number increases, the resistance to heat conduction within the material diminishes, making the surface convective heat transfer the dominant factor limiting thermal transfer. Heat moves rapidly throughout the material because of its capability to rapidly distribute thermal energy which produces uniform temperatures inside. The higher external convective resistance creates a temperature gradient which develops near the material surface. A higher Biot number creates greater surface-core temperature differences which enhance material sensitivity to external temperature changes. Higher Biot number values indicate heat conduction becomes more effective than the rate at which heat exchanges through convection with the surroundings. An increasing Biot number indicates heat conduction becomes faster than surface heat exchange with environmental conditions. As the Biot number drops below 0.1 the material operates as a lumped system without appreciable internal temperature gradients. The material exhibits substantial internal temperature gradients when the Biot criterion is greater than unity (Bi > 1) (see Fig. 6a and b).

Fig. 6.

a: Impact of Biot number on T(ξ). b: 3D plot of Biot number on T(ξ)

Radiation parameter: The Fig. 7a and b shows how raising the radiation parameter affects the temperature profile distribution in Fig. 7a. An increase in the radiation parameter leads to noticeable elevations in the temperature profile. A rise in radiation parameter increases heat transfer by radiation throughout the medium creating additional thermal energy in the system. The heat transfer mechanism of radiation proves essential within high-temperature situations while its impact increases as radiation parameters rise. Thermal radiation intensity increases with temperature according to the Stefan-Boltzmann law which enables more efficient energy distribution in physical systems. Higher values of the radiation parameter lead to enhanced radiative heat transfer which decreases thermal resistance throughout the medium. The higher fluid temperature occurs because of the elevated internal energy resulting from this phenomenon. The enlarged thermal boundary layer while exposing to more radiation results in a better distribution of temperatures throughout the medium. The influence of radiation in energy transfer becomes most apparent when working with optically thick fluids in which radiative heat transport dominates.

Fig. 7.

a: Impact of R vs. T(ξ). b: 3D plot of R vs. T(ξ)

Heat generation parameter: Fig. 8a and b shows how increasing the heat generation parameter affects the temperature profile. When the heat generation parameter rises the temperature profile demonstrates significant increases across all domain regions. A rise in internal heat generation produces extra thermal energy that results in higher temperature levels throughout the system. The heat generation parameter plays a vital role in heat transfer analyses because it controls thermal energy provision within systems which affects both their thermal stability and process efficiency. A rise in this parameter requires increased heat insulation from the medium which results in stronger thermal gradients. This behavior fulfills the criteria of energy conservation principles. The internal heat generation introduces a source term to the energy equation that strengthens the temperature field. The conductive heat transfer process works to balance heat redistribution through the material although it leads to increased total temperatures. Higher parameter values of heat generation lead to increased localized thermal heating that produces material stresses that damage both mechanical properties and product lifetime. The phenomenon holds special significance especially when intense heat production results in material deterioration followed by thermal expansion or phase transformation.

Fig. 8.

a: Variation of δ vs. T(ξ). b: 3D plot of δ vs. T(ξ)

Conclusion

In summary, this study presents a theoretical analysis focusing on the parametric sensitivities associated with gravity and electroosmotic-driven ternary composite nanoparticles flowing over a Riga device, considering both heat dissipation and convective cooling effects. The current investigation establishes advanced nanocomposites’ essential position in creating ternary hybrid nanofluids through the display of their exceptional thermophysical properties suitable for renewable energy thermal process optimization. The key outcomes derived from the present analysis are as follows:

1. A higher modified Hartmann number (Q) enhances the fluid velocity profile by providing an accelerating electro-magnetic momentum source from the Riga plate, thereby boosting flow motion.

2. When electrokinetic influences intensify, they lower the effective flow resistance, allowing the liquid to move at a greater velocity.

3. The velocity profile decreases as suction strength increases because stronger boundary suction extracts fluid momentum and thins the velocity boundary layer.

4. The velocity profile decreases as the slip parameter increases because increased surface slip reduces boundary shear transmission.

5. The velocity profile enhances as electro-osmotic parameters increase because stronger electrical body forces accelerate the flow.

6. Higher Biot numbers elevate the temperature profile due to enhanced convective heat transfer from the heated surface.

7. Higher radiation parameters elevate the thermal boundary layer because increased radiative heat flux supplies additional thermal energy to the fluid.

8. The temperature profile rises with an increase in the heat generation parameter because internal volumetric heat sources supply additional thermal energy to the system.

Limitations and future research directions

Despite the significant findings reported in this study, some limitations should be acknowledged. The present analysis is based on a steady, two-dimensional, laminar flow assumption, which may not fully represent the complex transport behavior encountered in practical engineering systems. Additionally, the model assumes homogeneous nanoparticle dispersion and constant thermophysical properties, while possible effects of nanoparticle aggregation, sedimentation, and temperature-dependent properties have not been considered. The influence of transient effects, three-dimensional flow structures, and turbulence-induced transport mechanisms also remains unexplored. Furthermore, although the mathematical model provides valuable insights into the coupled effects of electroosmotic forcing, gravity modulation, and Riga plate-induced electromagnetic control, experimental validation is required to confirm the predicted trends under practical operating conditions.

Future investigations may extend this work by incorporating non-uniform nanoparticle distributions, variable thermophysical properties, and particle-fluid interaction mechanisms using advanced nanofluid models. Experimental studies using microfluidic channels integrated with Riga plate actuators could be performed to validate the theoretical predictions. Moreover, high-fidelity computational fluid dynamics (CFD) simulations, including transient and three-dimensional formulations, may provide further understanding of the flow characteristics. The integration of optimization algorithms and artificial intelligence-based predictive models could also be explored to determine optimal electroosmotic and electromagnetic parameters for enhanced heat transfer efficiency in advanced cooling systems, biomedical microfluidics, and energy transport applications.

Nomenclature

x, y

Cartesian coordinates (m)

ϕ, φ

Velocity components in the x- and y-directions (m s⁻¹)

u, v

Dimensionless velocity components (–)

ϑ

Fluid temperature (K)

ϑ_w

Wall temperature (K)

ϑ_∞

Ambient temperature (K)

T

Dimensionless temperature (–)

J₀

Electrode current density (A m⁻²)

M₀

Magnetization (A m⁻¹)

E_x

Applied electric field strength (V m⁻¹)

ρ_e

Electric charge density (C m⁻³)

ρ_th

Ternary nanofluid density (kg m⁻³)

μ_th

Dynamic viscosity of ternary nanofluid (Pa s)

k_th

Thermal conductivity of ternary nanofluid (W m⁻¹ K⁻¹)

C_p

Specific heat capacity (J kg⁻¹ K⁻¹)

K

Permeability of porous medium (m²)

g

Gravitational acceleration (m s⁻²)

β

Thermal expansion coefficient (K⁻¹)

H

Heat source/sink coefficient (W m⁻³K⁻¹)

k₀

Mean absorption coefficient (m⁻¹)

σ₀

Stefan–Boltzmann constant (W m⁻²K⁻⁴)

c, L

Characteristic lengths (m)

Pr

Prandtl number (–)

Ec

Eckert number (–)

Bi

Biot number (–)

R

Radiation parameter (–)

Q

Modified Hartmann parameter (–)

M_e

Electro-osmotic parameter (–)

Λ

Porosity parameter (–)

λ

Helmholtz–Smoluchowski velocity parameter (–)

α

Thermal convection parameter (–)

δ

Heat generation/absorption parameter (–)

ω

Velocity slip parameter (–)

v_w

Wall suction/injection parameter (–)

Wf

Skin-friction coefficient (–)

Wh

Wall heat transfer rate (Nusselt number) (–)

Re

Reynolds number (–)

Author contributions

LM, HA: Conceptualization, Methodology, Software, Visualization, Investigation, Writing- Original draft preparation. SC: Data curation, Validation, Supervision, Resources, Writing - Review and Editing. MA, MP: Project administration, Supervision, Resources, Writing - Review and Editing.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Data availability

The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request

Declarations

Ethics

Not applicable.

Consent to publish

Not applicable.

Consent to participate

not applicable.

Clinical trial number

Not applicable.

Competing interests

The authors declare no competing interests.