Computational analysis of shear dependent heat and mass transfer in MHD hybrid nanofluid within a porous medium

Abstract

This work numerically investigates enhanced thermal and mass diffusion in an ionized hybrid nanofluid flowing through a permeable medium. The model incorporates modified formulations of thermal and molecular diffusion to address the limitations of their classical counterparts, accounting for non-local and relaxation effects in heat and mass transfer. The hybrid nanofluid consists of copper (Cu) and alumina (Al₂O₃) nanoparticles dispersed in a propylene glycol base fluid. The formulation uniquely combines several advanced mechanisms: shear-dependent diffusivities, ion-slip current and Hall effect, Joule heating, and modified convective boundary conditions. A key innovation is the implementation of velocity gradient-dependent thermal and mass diffusivities, making both heat and solute transport explicitly dependent on local flow conditions. The validity of the generalized diffusion models is confirmed through their reduction to classical laws in limiting cases. A numerical solution to the similarity-transformed equations is obtained via the bvp4c solver in MATLAB. Higher nanoparticle volume fraction and shear-dependent diffusivities enhance heat and mass transfer, whereas magnetic and porous-medium effects suppress them. The Nusselt and Sherwood numbers rise with nonlinear diffusion, emphasizing shear-dominated transport. Results for the local skin friction coefficient, heat transfer rate, and mass transfer rate are presented in tabular form. These results demonstrate how transport processes are governed by the interplay of nonlinear diffusion, magnetic effects, and porous-medium resistance. The application of this work is particularly relevant to systems where magnetic fields interact with fluid flow, such as in cooling systems or filtration devices, where the Hall effect plays a significant role in modifying transport behavior.

Introduction

Classical Ohm’s law inadequately models electrically conducting fluids under strong magnetic fields. In such regimes, the generalized Ohm’s law must incorporate additional effects, notably the ion-slip and Hall terms [1–3]. Two critical phenomena in magnetized flows are the Hall and ion-slip impacts. The former describes how a Lorentz force deflects electrons, creating a perpendicular current and a three-dimensional flow field. The latter occurs due to collisions between ions and neutral particles, causing the ions to “slip" relative to the electron motion. This effect introduces a resistive correction to the electromagnetic force, altering the current density and reducing an acting force on the fluid viz. Lorentz force. These non-ideal effects are particularly important in plasma physics, astrophysical environments, and high-temperature nanofluid applications, where precise modeling of MHD behavior is critical. When electric and magnetic fields intersect orthogonally, a transverse current called the Hall is generated. This phenomenon underpins the operation of the whole transport (i.e., MHD, therm-solutal) systems including cooling coils, Hall sensors, MHD generators, electric converters, in which the magnetic field strength is determined through the generated current called Hall [4–7]. The combined effects of ion-slip and Hall currents on the flow characteristics of a hybrid nanofluid in a permeable medium were examined by Ahmed and Nadeem [8], while Farooq et al. [9] investigated that the Hall entity plays a decisive role in governing flow irreversibility. Building upon these contributions, Bhatti et al. [10, 11] also advanced the conceptual framework of MHD flows. Enhancements in magnetic field intensity, along with the incorporation of magnetic nanofluids, have been shown to significantly improve heat transfer performance in both refrigeration and heating systems. Consequently, MHD-based guided techniques have found extensive applications across various engineering and industrial domains. Recent investigations, Ishaq et al. [12], Anwar et al. [13], and Mary et al. [14] report new insights into MHD flow behavior of power-law fluids.

The concept of incorporating nanometer-sized particles into conventional base fluids was first introduced by Choi and Eastman [15]. The primary purpose of this process is to enhance the thermal transport capabilities of fluids, marking a pivotal advancement in heat transfer research. Since then, extensive research has been undertaken to elucidate the behavior of nanofluids for use in various fields, including industrial processing, nuclear power systems, transportation technologies, electronic cooling, biomedical engineering, and food preservation [16, 17]. Buongiorno et al. [18] addressed fundamental challenges and research gaps concerning the performance advantages of nanofluids, particularly emphasizing their potential to improve nuclear reactors’ operational efficiency and safety. Furthermore, the comprehensive properties and wide-ranging industrial applications of nanofluids is prsented by Smaisim et al. [19]. Hybrid nanofluids offer superior thermal conductivity and rheological characteristics over mono-nanofluids [12, 13, 20]. In this study, a hybrid nanofluid is formed by dispersing aluminum oxide (Al₂O₃) and copper (Cu) nanoparticles in propylene glycol (C₃H₈O₂). Aluminum oxide offers high thermal stability and chemical resistance, while copper provides superior thermal conductivity. Propylene glycol is selected as the base fluid due to its non-toxic nature, excellent thermal stability, low volatility, antifreeze characteristics, compatibility with various additives, and suitability for industrial cooling and filtration systems. The dispersion of Al₂O₃ and Cu nanoparticles in the base liquid produces a notable synergistic effect, enhancing heat transport efficiency, rheological response, and thermal diffusion characteristics. These hybrid nanofluids are synthesized by blending two dissimilar nanoparticle species in a single carrier fluid, leading to improved thermophysical behavior compared to ordinary nanofluids with a single particulate phase. Consequently, they have gained growing prominence in cutting-edge thermal control technologies, micro-scale fluidic devices, and energy systems requiring precise temperature regulation. Experimental observations consistently confirm that interactive effects between mixed nanoparticles intensify thermal diffusivity and elevate heat transfer performance. As a result, hybrid nanofluids are now widely utilized in diverse thermal management contexts, even in the absence of magnetic fields.

Hybrid nanofluids are used in applications ranging from electronic cooling and heat exchangers to photovoltaic thermal management and nuclear reactor systems. Substantial theoretical and empirical progress in this domain has been reported by Qayyum et al. [21], who examined the unsteady motion of hybrid nanofluids across the rotating disks along with chemical kinetics and activation energy. Dinarvand et al. [22], Kumar [23], and Sarkar et al. [24] further expanded the understanding of hybrid nanofluid transport phenomena and their practical implementations.

Power-law fluids exhibit shear-dependent viscosity, enabling the description of shear-thinning (e.g., blood, polymers) and shear-thickening (e.g., suspensions) behaviors common in industrial and biological flows. While Newtonian fluids like air and water are characterized by a linear increase of shear stress rate, power-law fluids are defined by a nonlinear relationship between these quantities. In shear-thinning (pseudoplastic) fluids—including blood, polymeric solutions, paints, and shampoos—viscosity decreases as the shear rate rises, facilitating easier flow under stress. Conversely, shear-thickening (dilatant) fluids, for example, concentrated colloidal dispersions and cornstarch suspensions, display an increase in viscosity with higher shear rates. Owing to these unique features, power-law fluids are extensively utilized in sectors such as manufacturing, defense, and biotechnology. The foundational power-law formulation was introduced by Bird et al. [25], while subsequent analytical and numerical explorations of their flow characteristics around circular cylinders were presented by Bharti et al. [26] and Patnana et al. [27].

To date, few studies have simultaneously examined generalized Fourier heat conduction, a generalized nonlinear Fick’s law, ion-slip effects, and Joule heating within a porous framework for power-law hybrid nanofluids. This work addresses this gap by modeling viscous flow along a continuously stretched surface immersed in a propylene glycol-based hybrid nanofluid containing copper (Cu) and aluminum oxide (Al₂O₃) nanoparticles, subject to convective thermal and solutal boundary conditions. Such a coupled framework is relevant to magnetically influenced transport in complex fluids, including porous cooling systems and non-Newtonian nanofluid applications.

In plasma-assisted cooling and electromagnetic flow control, Hall and ion-slip effects play a crucial role in modifying flow structure, momentum transport, and heat transfer characteristics. These mechanisms are particularly important in high-temperature environments, such as aerospace thermal protection systems and energy conversion devices. The inclusion of porous media resistance further extends the model’s applicability to compact heat exchangers and filtration-based cooling systems.

The simultaneous inclusion of multiple physical mechanisms is motivated by the need to capture interacting effects in realistic multiphysics environments. Transport properties in hybrid nanofluids and non-Newtonian systems depend on local shear conditions. To account for this, shear-dependent thermal and mass diffusivities are introduced using a phenomenological formulation that provides a tractable representation of flow-dependent diffusion. The normalization of weighting coefficients ensures bounded and physically consistent behavior, and the selected parameter ranges lie within realistic limits reported in the literature.

The primary contributions of this study are:

• The incorporation of shear-dependent thermal and mass diffusivities via generalized Fourier and Fick formulations to investigate hybrid nanofluid transport within a power-law framework.

• A detailed analysis of the combined influence of Hall and ion-slip currents, magnetic field strength, and Joule heating on flow and transport characteristics.

• An investigation of effective convective thermal and solutal diffusivities in the presence of solid boundaries using generalized forms of Fourier’s and Fick’s laws.

A central innovation is the use of a shear-dependent generalized nonlinear Fick’s law, in which the effective mass diffusivity depends explicitly on local shear gradients, capturing the shear-sensitive nature of diffusion in power-law hybrid nanofluids. An analogous formulation is adopted for heat transfer through a generalized Fourier law with gradient-dependent thermal diffusivity. This dual approach yields a coupled nonlinear framework wherein transport processes are directly influenced by the underlying shear field.

By constructing vector-based expressions for these diffusivities, the model accounts for both primary and cross-flow shear effects. Applying similarity transformations, the governing partial differential equations are reduced to a dimensionless system of coupled nonlinear ordinary differential equations, solved numerically using MATLAB’s bvp4c solver. Results are presented in terms of velocity, temperature, and concentration profiles, along with skin frictions, Nusselt number, and Sherwood number, to quantify the influence of key physical parameters on flow and transport characteristics.

Model formulation

This paper analyzes the steady, incompressible boundary layer flow induced by a linearly stretching surface in a permeable medium. The study focuses on the forced convection of a power-law hybrid nanofluid, accounting for the influence of Hall and ion-slip effects. Copper (Cu) and aluminum oxide (Al₂O₃) nanoparticles are dispersed within a propylene glycol (C₃H₈O₂) base fluid. The surface velocity is given by u_w = λx, with λ specifying the stretching rate (s⁻¹). The flow is described using a Cartesian coordinate system where the xy-plane contains the stretching sheet. The primary velocity u is directed along the x-axis, while v acts along the y-axis. A magnetohydrodynamically driven secondary flow w is defined along the z-axis, normal to the sheet. A uniform magnetic field B = (0, B₀, 0) is applied in the y-direction. The flow is three-dimensional, yet the variables remain independent of the z-coordinate (∂/∂z = 0), characterizing a quasi-three-dimensional boundary layer flow. This configuration, along with the associated velocity components, magnetic field, and growing boundary layers, is depicted schematically in Fig. 1. The thermal and molar diffusions depend on the local shear rates, ${∣\frac{\partial u}{\partial y}∣}^{m-1}$ and ${∣\frac{\partial w}{\partial y}∣}^{m-1}$, which introduce nonlinear coupling of momentum, energy, and species transport equations. The size and shape of nanoparticles are assumed to be uniform, and thermal equilibrium is maintained between the base fluid and particles. Additionally, the effective thermal diffusivity (α_eff) and mass diffusivity (D_eff) vary with local velocity, temperature, and concentration gradients, further enhancing the nonlinearity of the transport processes. This model couples shear-dependent thermal and mass diffusivities with Hall and ion-slip effects and porous-medium resistance. Although this increases model complexity, the configuration reflects practical scenarios such as microfluidic devices, electronic cooling, and porous media heat exchangers, and the chosen parameter ranges are physically realistic and experimentally relevant.

Fig. 1.

Schematic of the physical model and coordinate system

Following Bird et al. [25], Gope et al. [28], and Zhang et al. [29], the rheological behavior of the power-law hybrid nanofluid is described by the constitutive relation

$$\mathit{\tau }=-K_c\mathbf{D}{∣\sqrt{\frac{\mathbf{D}:\mathbf{D}}{2}}∣}^{m-1},$$ 1

where $\mathbf{D}=\frac{1}{2}\left(\nabla \mathbf{u}+{\left(\nabla \mathbf{u}\right)}^T\right)$ is the rate-of-strain tensor, K_c is the consistency index, and m is the power-law index. The cases m = 1, m < 1, and m > 1 correspond to Newtonian, shear-thinning, and shear-thickening behaviors, respectively.

Under boundary layer assumptions, the dominant shear components are ∂u/∂y and ∂w/∂y, yielding the approximate shear stresses

$${\tau }_{yx}\approx K_c{∣\frac{\partial u}{\partial y}∣}^{m-1}\frac{\partial u}{\partial y},{\tau }_{yz}\approx K_c{∣\frac{\partial w}{\partial y}∣}^{m-1}\frac{\partial w}{\partial y}.$$

The steady incompressible Cauchy momentum equation is written as

$${\rho }_{\mathrm{hnf}}\left(\mathbf{u}\cdot \nabla \mathbf{u}\right)=\nabla \cdot \mathbf{T}+\mathbf{f},$$ 2a

where T = − pI + τ is the total stress tensor and f represents body forces arising from magnetic and porous medium effects.

The Lorentz force components incorporating Hall and ion-slip currents are

$$f_x=-{\sigma }_{\mathrm{hnf}}{B}_0^2\left({\alpha }_{1}u-{\alpha }_{2}w\right),f_z=-{\sigma }_{\mathrm{hnf}}{B}_0^2\left({\alpha }_{1}w+{\alpha }_{2}u\right),$$

where

$${\alpha }_3={\beta }_e^2+{\left(1+{\beta }_i{\beta }_e\right)}^2,{\alpha }_1=\frac{1+{\beta }_i{\beta }_e}{{\alpha }_3},{\alpha }_2=\frac{{\beta }_e}{{\alpha }_3},$$

and β_e and β_i denote the Hall and ion-slip parameters, respectively. The porous medium contributes the resistance terms  − μ_hnfu/K and  − μ_hnfw/K.

Applying the boundary layer approximations (∂p/∂x = ∂p/∂z = 0), retaining dominant shear stresses, and introducing the kinematic viscosity ν_hnf = K_c/ρ_hnf, the governing equations reduce to

$$u_x+v_y=0,$$ 2

$$\begin{array}{ccc}\hfill u{u}_x+v{u}_y& \hfill =\hfill & {\nu }_{\mathrm{hnf}}\frac{\partial }{\partial y}\left({∣u_y∣}^{m-1}{u}_y\right)-\frac{{\sigma }_{\mathrm{hnf}}{B}_0^2\left[{\alpha }_{1}u-{\alpha }_{2}w\right]}{{\rho }_{\mathrm{hnf}}}\hfill \\ \hfill & \hfill \hfill & -\frac{{\nu }_{\mathrm{hnf}}}{K}u,\hfill \end{array}$$ 3

$$\begin{array}{ccc}\hfill u{w}_x+v{w}_y& \hfill =\hfill & {\nu }_{\mathrm{hnf}}\frac{\partial }{\partial y}\left({∣w_y∣}^{m-1}{w}_y\right)-\frac{{\sigma }_{\mathrm{hnf}}{B}_0^2\left[{\alpha }_{1}w+{\alpha }_{2}u\right]}{{\rho }_{\mathrm{hnf}}}\hfill \\ \hfill & \hfill \hfill & -\frac{{\nu }_{\mathrm{hnf}}}{K}w,\hfill \end{array}$$ 4

$$u{T}_x+v{T}_y=\frac{\partial }{\partial y}\left({\alpha }_{\mathrm{eff}}\left(y\right)T_y\right)+\frac{{\sigma }_{\mathrm{hnf}}{B}_0^2\left(u^2+w^2\right)}{{\left(\rho c_p\right)}_{\mathrm{hnf}}{\alpha }_3}+\frac{{\mu }_{\mathrm{hnf}}\left(u^2+w^2\right)}{{\left(\rho c_p\right)}_{\mathrm{hnf}}K},$$ 5

$$u{C}_x+v{C}_y=\frac{\partial }{\partial y}\left(D_{\mathrm{eff}}\left(y\right)C_y\right)-k_r\left(C-C_{\infty }\right),$$ 6

Transport properties in hybrid nanofluids are strongly influenced by local shear due to microstructural rearrangements and shear-induced particle migration. Experimental and theoretical studies in suspensions [28, 30] show that shear-induced diffusion enhances effective particle transport, even at low Reynolds numbers, by promoting particle interactions and microscale mixing. In nanofluids, this implies that local shear can significantly increase effective thermal and mass transport by redistributing nanoparticles and enhancing microscale convection.

Motivated by these observations, the present model treats the effective thermal and mass diffusivities, α_eff and D_eff, as shear-dependent functionals. The power-law form (∣∂ψ_i/∂y∣^m−1) ensures dimensional consistency with the rheology, reduces to constant base diffusivities (α_hnf, D_hnf) for Newtonian flows (m = 1), and allows weighting coefficients ξ_i and ζ_i to reflect the relative influence of velocity and scalar gradients.

To formalize this, we define the vectors:

$$\mathit{\psi }=\left[u,w,T\right],\mathit{\varphi }=\left[u,w,C\right],$$

$$\mathit{\xi }=\left[{\xi }_1,{\xi }_2,{\xi }_3\right],\mathit{\zeta }=\left[{\zeta }_1,{\zeta }_2,{\zeta }_3\right],$$

where ψ and ϕ contain the primary flow variables influencing thermal and mass transport, and ξ, ζ are positive weighting constants satisfying ∑_iξ_i = ∑_iζ_i = 1. The effective diffusivities are then:

$${\alpha }_{\mathrm{eff}}\left(y\right)={\alpha }_{\mathrm{hnf}}{\sum }_{i=1}^3{\xi }_i{∣\frac{\partial {\psi }_i}{\partial y}∣}^{m-1},$$ 7

$$D_{\mathrm{eff}}\left(y\right)=D_{\mathrm{hnf}}{\sum }_{i=1}^3{\zeta }_i{∣\frac{\partial {\varphi }_i}{\partial y}∣}^{m-1}.$$ 8

This shear-dependent mechanism governs the effective transport at the macro-scale. For the constituent material properties (e.g., the density or specific heat of the base fluid and nanoparticles), constant values are assumed. This standard simplification allows the present study to focus on the primary, coupled effects of shear-enhanced transport, Hall and ion-slip currents, and porous-medium resistance, providing a tractable and insightful first-order model of the hybrid nanofluid system.

The boundary conditions for the governing model are (see [28, 29]):

$$\begin{array}{c}u=\lambda x,v=0,w=0,\hfill \\ {\sum }_{i=1}^3{\xi }_i{∣\frac{\partial {\psi }_i}{\partial y}∣}^{m-1}{T}_y=\frac{h_{f}T-h_f{T}_w}{k_{\mathrm{hnf}}},\mathrm{at}y=0,\hfill \\ {\sum }_{i=1}^3{\zeta }_i{∣\frac{\partial {\varphi }_i}{\partial y}∣}^{m-1}{C}_y=\frac{k_{f}C-k_f{C}_w}{D_{\mathrm{hnf}}},\mathrm{at}y=0,\hfill \\ w\to 0,u\to 0,C\to C_{\infty },T\to T_{\infty }\mathrm{as}y\to \infty .\hfill \end{array}$$ 9

From the above equations, the quantities ${\left(\rho c_p\right)}_{\mathrm{hnf}},{\mu }_{\mathrm{hnf}},k_{\mathrm{hnf}},{\rho }_{\mathrm{hnf}}$ and σ_hnf correspond to the heat capacity, viscosity, thermal conductivity, density and electrical conductivity of hybrid nanofluid (HNF), accordingly and λ defines the stretching rate of the surface. Finally, ξ_i and ζ_i are normalized weights, and K defines the permeability medium.

The governing equations are reduced to a self-similar form by introducing the following similarity transformations and dimensionless variables(see[3, 28]):

$$\begin{array}{ccc}\hfill \psi \left(x,y\right)& \hfill =\hfill & {\lambda }^{\frac{2m-1}{m+1}}{\nu }_f^{\frac{1}{m+1}}{x}^{\frac{2m}{m+1}}f\left(\eta \right),u=\frac{\partial \psi }{\partial y}=\lambda xf\prime \left(\eta \right),w=\lambda xg\left(\eta \right),\hfill \\ \hfill v& \hfill =\hfill & -\frac{\partial \psi }{\partial x}=-{\lambda }^{\frac{2m-1}{m+1}}{\nu }_f^{\frac{1}{1+m}}{x}^{\frac{m-1}{1+m}}\left[\frac{2m}{1+m}f\left(\eta \right)+\eta \left(\frac{1-m}{1+m}\right)f\prime \left(\eta \right)\right],\hfill \\ \hfill \theta \left(\eta \right)& \hfill =\hfill & \frac{T-T_{\infty }}{T_w-T_{\infty }},\eta ={\lambda }^{\frac{2-m}{m+1}}{\nu }_f^{-\frac{1}{m+1}}{x}^{\frac{1-m}{m+1}}y,\varphi \left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\hfill \end{array}$$ 10

The dimensionless variables used in the analysis are η (similarity variable), f (stream function), $f^{\prime }$ (primary velocity), g (secondary velocity), θ (temperature), and ϕ (concentration).

To model the hybrid nanofluid, the following thermophysical properties are employed [28, 30] (Table 1):

$$\begin{array}{ccc}\hfill \frac{{\rho }_{hnf}}{{\rho }_f}& \hfill =\hfill & \left(1-{\varphi }_1-{\varphi }_2+{\varphi }_1{\varphi }_2\right)+{\varphi }_1\left(1-{\varphi }_2\right)\frac{{\rho }_{s1}}{{\rho }_f}+{\varphi }_2\frac{{\rho }_{s2}}{{\rho }_f},\hfill \\ \hfill D_{\mathrm{hnf}}& \hfill =\hfill & D_{\mathrm{f}}\left(1-\left({\varphi }_1+{\varphi }_2\right)\right),{\mu }_{hnf}={\mu }_f\cdot {\left[\left(1-{\varphi }_1\right)\left(1-{\varphi }_2\right)\right]}^{-5/2},\hfill \\ \hfill \frac{{\left(\rho C_p\right)}_{hnf}}{{\left(\rho C_p\right)}_f}& \hfill =\hfill & \left(1-{\varphi }_1-{\varphi }_2+{\varphi }_1{\varphi }_2\right)+{\varphi }_1\left(1-{\varphi }_2\right)\frac{{\left(\rho C_p\right)}_{s1}}{{\left(\rho C_p\right)}_f}+{\varphi }_2\frac{{\left(\rho C_p\right)}_{s2}}{{\left(\rho C_p\right)}_f},\hfill \\ \hfill \frac{{\sigma }_{hnf}}{{\sigma }_f}& \hfill =\hfill & \left[\frac{-2{\varphi }_2\left({\sigma }_{bf}-{\sigma }_{s2}\right)+{\sigma }_{s2}+2{\sigma }_{bf}}{{\varphi }_2\left({\sigma }_{bf}-{\sigma }_{s2}\right)+{\sigma }_{s2}+2{\sigma }_{bf}}\right]\left[\frac{2{\sigma }_f+{\sigma }_{s1}+{\varphi }_1\left({\sigma }_f-{\sigma }_{s1}\right)}{2{\sigma }_f+{\sigma }_{s1}-2{\varphi }_1\left({\sigma }_f-{\sigma }_{s1}\right)}\right],\hfill \\ \hfill \frac{k_{hnf}}{k_f}& \hfill =\hfill & \left(\frac{k_{s1}+\left(j-1\right)k_f+{\varphi }_1\mathrm{\Delta }{k}_1}{k_{s1}+\left(j-1\right)k_f-\left(j-1\right){\varphi }_1\mathrm{\Delta }{k}_1}\right)\left(\frac{k_{s2}+\left(j-1\right)k_{bf}-\left(j-1\right){\varphi }_2\mathrm{\Delta }{k}_2}{k_{s2}+\left(j-1\right)k_{bf}+{\varphi }_2\mathrm{\Delta }{k}_2}\right)\hfill \end{array}$$ 11

where Δk₁ = k_f − k_s1 and Δk₂ = k_bf − k_s2. The volume fraction of copper (Cu) is specified by ϕ₁ and the volume fraction of aluminum oxide (Al₂O₃) is denoted by ϕ₂. The dynamic viscosity of the base fluid (C₃H₈O₂) and the hybrid nanofluid are represented by μ_f and μ_hnf. σ_s1, σ_s2, and σ_f are symbols of electrical conductivities for Cu, Al₂O₃ and (C₃H₈O₂), respectively. The densities of Cu, and Al₂O₃ are ρ_f, ρ_s1, and ρ_s2, while their respective heat capacities are ${\left(\rho C_p\right)}_{s1}$, ${\left(\rho C_p\right)}_{s2}$, and ${\left(\rho C_p\right)}_f$. The thermal conductivities of the base fluid,Cu, and Al₂O₃ are denoted by k_f, k_s1, and k_s2. Finally, j represents an empirical shape factor for the nanoparticles.

Table 1.

Thermo-physical properties at T = 300 K [31, 32]

Properties	Propylene glycol (C3H8O2)	Copper (Cu)	Aluminium oxide(Al2O3)
ρ(Kg m−3)	1036	8933	3970
κ(W/mK)	0.201	400	40
Cp(J/KgK)	2510	385	765
σ(Ωm)−1	1 × 10−5	5.96 × 107	3.69 × 107

After applying the similarity transformations, the governing equations (2)–(9) reduce to the following system of nonlinear equations:

$$\begin{array}{c}{A}_1\frac{d}{d\eta }\left(\mid f^{″}{\mid }^{m-1}{f}^{″}\right)+\left(\frac{2m}{m+1}\right)f{f}^{″}-A_{2}M\left({\alpha }_1{f}^{\prime }-{\alpha }_{2}g\right)\hfill \\ -{\left(f^{\prime }\right)}^2-A_1{K}_p{f}^{\prime }=0,\hfill \end{array}$$ 12

$$\begin{array}{c}{A}_1\frac{d}{d\eta }\left(\mid g^{\prime }{\mid }^{m-1}{g}^{\prime }\right)+\left(\frac{2m}{m+1}\right)f{g}^{\prime }-M{A}_2\left({\alpha }_{1}g+{\alpha }_2{f}^{\prime }\right)\hfill \\ -A_1{K}_{p}g-f^{\prime }g=0,\hfill \end{array}$$ 13

$$\begin{array}{c}{A}_3{A}_4\frac{d}{d\eta }\left({\sum }_{j=1}^3\frac{\mid {\mathit{\psi }}_j^{\prime }{\mid }^{m-1}}{P{r}_j}\theta \prime \right)+\left(\frac{2m}{m+1}\right)f{\theta }^{\prime }\hfill \\ -Ec\left(\frac{M{A}_5}{{\alpha }_3}-A_6{K}_p\right)\left({\left(f^{\prime }\right)}^2+g^2\right)=0,\hfill \end{array}$$ 14

$${\beta }_3\frac{d}{d\eta }\left({\sum }_{k=1}^3\frac{\mid \mathit{\chi }{\prime }_k{\mid }^{m-1}}{S{c}_k}\varphi \prime \right)-{\kappa }_o\varphi =0.$$ 15

The corresponding boundary conditions are given by:

$$\begin{array}{c}{f}^{\prime }\left(0\right)=1,g\left(0\right)=0,f\left(0\right)=0,\hfill \\ \theta \left(0\right)=1+A_3\left({\sum }_{p=1}^3\frac{\mid {\mathit{\psi }}_p^{\prime }\left(0\right){\mid }^{m-1}}{{\mathrm{Bi}}_{hp}}\right){\theta }^{\prime }\left(0\right),\hfill \\ \varphi \left(0\right)=1+{\beta }_3\left({\sum }_{q=1}^3\frac{\mid {\mathit{\chi }}_q^{\prime }\left(0\right){\mid }^{m-1}}{{\mathrm{Bi}}_{mq}}\right){\varphi }^{\prime }\left(0\right),\hfill \\ f^{\prime }\left(\infty \right),g\left(\infty \right),\theta \left(\infty \right),\varphi \left(\infty \right)\to 0.\hfill \end{array}$$ 16

where

$$\begin{array}{c}{A}_1=\frac{{\mu }_{\mathrm{hnf}}/{\mu }_f}{{\rho }_{\mathrm{hnf}}/{\rho }_f},A_2=\frac{{\sigma }_{\mathrm{hnf}}/{\sigma }_f}{{\rho }_{\mathrm{hnf}}/{\rho }_f},A_3=\frac{k_{\mathrm{hnf}}}{k_f},A_4=\frac{{\sigma }_{\mathrm{hnf}}/{\sigma }_f}{{\left(\rho C_p\right)}_{\mathrm{hnf}}/{\left(\rho C_p\right)}_f},\hfill \\ {\beta }_3=\frac{D_{\mathrm{hnf}}}{D_f},A_5=\frac{{\mu }_{\mathrm{hnf}}/{\mu }_f}{{\left(\rho C_p\right)}_{\mathrm{hnf}}/{\left(\rho C_p\right)}_f},A_6=\frac{{\rho }_{\mathrm{hnf}}/{\rho }_f}{{\left(\rho C_p\right)}_{\mathrm{hnf}}/{\left(\rho C_p\right)}_f},\hfill \\ {\mathit{\psi }}^{\prime }=\left[\begin{array}{c}\hfill f^{″}\hfill \\ \hfill g^{\prime }\hfill \\ \hfill {\theta }^{\prime }\hfill \end{array}\right],{\mathit{\chi }}^{\prime }=\left[\begin{array}{c}\hfill f^{″}\hfill \\ \hfill g^{\prime }\hfill \\ \hfill {\varphi }^{\prime }\hfill \end{array}\right],\hfill \end{array}$$ 17

$$\begin{array}{c}{K}_p=\frac{{\nu }_f}{K\lambda },S{c}_1=\frac{{\nu }_f}{{\zeta }_1{D}_f},S{c}_3=\frac{{\nu }_f{\left(C-C_{\infty }\right)}^{1-m}{\left(\lambda x\right)}^{m-1}}{{\zeta }_3{D}_f},\hfill \\ Ec=\frac{{\lambda }^2{x}^2}{\left(T_w-T_{\infty }\right){\left(C_p\right)}_{\mathrm{f}}},{\kappa }_o=\frac{k_r}{\lambda },P{r}_1=\frac{{\nu }_f}{{\xi }_1{\alpha }_f},P{r}_2=\frac{{\nu }_f}{{\xi }_2{\alpha }_f},\hfill \\ {\mathrm{Bi}}_{m3}=\frac{{\mathrm{k}}_f{\left({\lambda }^{m-2}{\nu }_f{x}^{m-1}\right)}^{\frac{m}{m+1}}{\left(C_w-C_{\infty }\right)}^{1-m}}{D_f{\zeta }_3},S{c}_2=\frac{{\nu }_f}{{\zeta }_2{D}_f},\hfill \\ P{r}_3=\frac{{\nu }_f{\left(T-T_{\infty }\right)}^{1-m}{\left(\lambda x\right)}^{m-1}}{{\xi }_3{\alpha }_f},{\mathrm{Bi}}_{h1}=\frac{{\mathrm{h}}_f{\left({\lambda }^{1-2m}{\nu }_f^m{x}^{1-m}\right)}^{\frac{1}{m+1}}}{k_f{\xi }_1},\hfill \\ {\mathrm{Bi}}_{h2}=\frac{{\mathrm{h}}_f{\left({\lambda }^{1-2m}{\nu }_f^m{x}^{1-m}\right)}^{\frac{1}{m+1}}}{k_f{\xi }_2},{\mathrm{Bi}}_{h3}=\frac{{\mathrm{h}}_f{\left({\lambda }^{m-2}{\nu }_f{x}^{m-1}\right)}^{\frac{m}{m+1}}{\left(T_w-T_{\infty }\right)}^{1-m}}{k_f{\xi }_3},\hfill \\ {\mathrm{Bi}}_{m1}=\frac{{\mathrm{k}}_f{\left({\lambda }^{1-2m}{\nu }_f^m{x}^{1-m}\right)}^{\frac{1}{m+1}}}{D_f{\zeta }_1},M=\frac{{\sigma }_f{B}_0^2}{\lambda {\rho }_f},\hfill \\ {\mathrm{Bi}}_{m2}=\frac{{\mathrm{k}}_f{\left({\lambda }^{1-2m}{\nu }_f^m{x}^{1-m}\right)}^{\frac{1}{m+1}}}{D_f{\zeta }_2}..\hfill \end{array}$$ 18

The relevant engineering outputs for this flow configuration are the skin friction coefficients (C_fx, C_fz) and the gradient-based numbers for heat and mass transfer Nu_x and Sh_x, respectively. Applying the defined similarity transformations yields their dimensionless expressions:

• The local skin friction coefficients C_fx and C_fz, representing surface shear stress in the x- and z-directions [33, 34], are defined as:

  $${\tau }_{w,x}=-{\mu }_{\mathrm{hnf}}{\left[{∣u_y∣}^{m-1}{u}_y\right]}_{y=0},{\tau }_{w,z}=-{\mu }_{\mathrm{hnf}}{\left[{∣w_y∣}^{m-1}{w}_y\right]}_{y=0},$$ 19

  where μ_hnf stands for the dynamic viscosity of the hybrid nanofluid, and m is the power-law index. Using the similarity transformations:

  $$u_y={\lambda }^{\frac{3}{1+m}}{\nu }_f^{-\frac{1}{1+m}}{x}^{\frac{2}{1+m}}{f}^{″}\left(\eta \right),w_y={\lambda }^{\frac{3}{1+m}}{\nu }_f^{-\frac{1}{1+m}}{x}^{\frac{2}{1+m}}{g}^{\prime }\left(\eta \right),$$

  and evaluating at η = 0:

  $${\tau }_{w,x}=-{\mu }_{hnf}{\left({\lambda }^3{\nu }_f^{-1}{x}^2\right)}^{\frac{m}{1+m}}{∣f^{″}\left(0\right)∣}^{m-1}{f}^{″}\left(0\right).$$

  This simplifies to:

  $${\tau }_{w,x}=-{\mu }_{hnf}{\lambda }^{\frac{3m}{m+1}}{\nu }_f^{-\frac{m}{m+1}}{x}^{\frac{2m}{m+1}}{f}^{″}\left(0\right){∣f^{″}\left(0\right)∣}^{m-1}.$$

  In the x-direction it can be defined as:

  $$C_{f_x}=\frac{{\tau }_{w,x}}{\frac{1}{2}{\rho }_{hnf}{u}_{1w}^2}=\frac{2{\tau }_{w,x}}{{\rho }_{hnf}{\lambda }^2{x}^2}.$$

  Substituting τ_w,x:

  $$C_{f_x}=-\frac{2{\mu }_{hnf}}{{\rho }_{hnf}}{\lambda }^{\frac{3m}{m+1}-2}{\nu }_f^{-\frac{m}{m+1}}{x}^{\frac{2m}{m+1}-2}{∣f^{″}\left(0\right)∣}^{m-1}{f}^{″}\left(0\right).$$ 20

  Similarly, for the z-direction:

  $$C_{f_z}=-\frac{2{\mu }_{hnf}}{{\rho }_{hnf}}{\lambda }^{\frac{3m}{m+1}-2}{\nu }_f^{-\frac{m}{m+1}}{x}^{\frac{2m}{m+1}-2}{∣g^{\prime }\left(0\right)∣}^{m-1}{g}^{\prime }\left(0\right).$$ 21
• The enhancement of gradient-based surface heat transfer is characterized by local Nusselt number, defined as: $N{u}_x=\frac{q_{w}x}{k_{hnf}\left[{\xi }_1{∣\frac{\partial u}{\partial y}∣}^{m-1}+{\xi }_2{∣\frac{\partial w}{\partial y}∣}^{m-1}+{\xi }_3{∣\frac{\partial T}{\partial y}∣}^{m-1}\right]\left(T_w-T_{\infty }\right)},$ where $q_w=-k_{hnf}\left[{\xi }_1{∣\frac{\partial u}{\partial y}∣}^{m-1}+{\xi }_2{∣\frac{\partial w}{\partial y}∣}^{m-1}+{\xi }_3{∣\frac{\partial T}{\partial y}∣}^{m-1}\right]T_y$ is the surface heat flux. The dimensionless expression of Nu_x is:

  $$N{u}_x=-A_3{\left({\lambda }^{2-m}{\nu }_f^{-1}{x}^2\right)}^{\frac{1}{m+1}}{\theta }^{\prime }\left(0\right).$$ 22
• Mass transfer efficiency at the boundary is characterized by the local Sherwood number, is given by: $S{h}_x=\frac{j_{w}x}{D_{hnf}\left[{\zeta }_1{∣\frac{\partial u}{\partial y}∣}^{m-1}+{\zeta }_2{∣\frac{\partial w}{\partial y}∣}^{m-1}+{\zeta }_3{∣\frac{\partial C}{\partial y}∣}^{m-1}\right]\left(C_w-C_{\infty }\right)},$ where $j_w=-D_{hnf}\left[{\zeta }_1{∣\frac{\partial u}{\partial y}∣}^{m-1}+{\zeta }_2{∣\frac{\partial w}{\partial y}∣}^{m-1}+{\zeta }_3{∣\frac{\partial C}{\partial y}∣}^{m-1}\right]C_y$ is the surface mass flux. The dimensionless expression of Sh_x is:

  $$S{h}_x=-{\beta }_3{\left({\lambda }^{2-m}{\nu }_f^{-1}{x}^2\right)}^{\frac{1}{m+1}}{\varphi }^{\prime }\left(0\right).$$ 23

Concentration model validation

To validate the generalized concentration model, we demonstrate that it reduces to the well-known constant-diffusivity convection-diffusion-reaction equation in the Newtonian pure-fluid limit. Specifically, we start from the generalized concentration Eq. (6), with the effective mass diffusivity and the boundary conditions in (9). For a Newtonian, single-component fluid, the assumptions m = 1 and β₃ = 1 ⇒ D_hnf = D_f apply. Under these conditions, the model collapses to the classical constant-diffusivity form:

$$u{C}_x+v{C}_y=D_f{C}_{yy}-k_{r}C-k_r{C}_{\infty },$$ 24

subject to

$$D_f{C}_y=k_{f}C-k_f{C}_w,y=0,C\to C_{\infty },y\to \infty .$$ 25

Equation (24) together with (25) represents the standard convection-diffusion-reaction model with a homogeneous first-order chemical reaction and a convective (Robin-type) boundary condition, as commonly employed in classical diffusion theory and in modern boundary-layer analyses with surface mass transfer [35–37].

Applying the standard similarity transformation defined in (10) with m = 1, Eq. (24) and its associated boundary conditions (25), reduces to the ordinary differential equation

$${\varphi }^{″}+Scf\left(\eta \right){\varphi }^{\prime }-{\kappa }_{0}Sc\varphi =0,\varphi \left(0\right)=\frac{1}{{\mathrm{Bi}}_m}\varphi \prime \left(0\right)+1,\varphi \left(\infty \right)\to 0,$$ 26

where

$$Sc=\frac{\nu }{D_f},{\kappa }_0=\frac{k_r}{\lambda },{\mathrm{Bi}}_m=\frac{k_f}{D_f}\sqrt{\frac{\nu }{\lambda }}.$$

This reduced ODE is identical in structure to those reported in boundary-layer studies of reactive mass transfer with convective surface exchange [35, 37]. Consequently, the generalized model is validated by its reduction to established constant-diffusivity theory and its consistency with previously published similarity formulations.

Solution method

The governing equations are solved numerically using MATLAB’s bvp4c solver. This algorithm is designed for two-point boundary value problems and utilizes a high-order collocation scheme suitable for stiff systems. For numerical implementation, the system is recast into first-order form through the following variable substitutions:

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

These substitutions allow the original system to be rewritten in a first-order form suitable for use with the bvp4c solver. Appropriate boundary conditions are also reformulated in terms of the new variables.

To enhance methodological clarity, the numerical procedure follows clearly justified steps: first, the physical problem is nondimensionalized and the governing equations are derived; second, the system is transformed into a first-order form for solver compatibility; third, realistic parameter ranges for Prandtl (Pr_j), Schmidt (Sc_k), Biot (Bi_hp, Bi_mq), Hartmann (M), Hall (β_e), ion-slip (β_i), and porous-medium (K_p) numbers are adopted based on literature to ensure engineering relevance; fourth, convergence criteria and limiting cases are tested to validate the numerical method; and finally, the solver is applied to compute velocity, temperature, and concentration profiles for various parameter combinations. This formulation explicitly accounts for shear-dependent thermal and mass diffusivities, Hall and ion-slip effects, and porous-medium resistance, enabling physically meaningful and practically realizable simulations. The method ensures reproducibility and allows direct comparison with previous studies on hybrid nanofluid flows under similar conditions [4, 28, 38].

The first-order system of ODEs is given by:

$$\left\{\begin{array}{c}{y}_1^{\prime }=y_2,\hfill \\ y_2^{\prime }=y_3,\hfill \\ y_3^{\prime }=\frac{1}{m{A}_1\mid y_3{\mid }^{m-1}}\left[-\frac{2m}{m+1}{y}_1{y}_3+A_{2}M\left({\alpha }_1{y}_2-{\alpha }_2{y}_4\right)+y_2^2+A_1{K}_p{y}_2\right],\hfill \\ y_4^{\prime }=y_5,\hfill \\ y_5^{\prime }=\frac{1}{m{A}_1\mid y_5{\mid }^{m-1}}\left[-\frac{2m}{m+1}{y}_1{y}_5+M{A}_2\left({\alpha }_1{y}_4+{\alpha }_2{y}_2\right)+A_1{K}_p{y}_4+y_2{y}_4\right],\hfill \\ y_6^{\prime }=y_7,\hfill \\ y_7^{\prime }=\frac{-A_3{A}_4{y}_7\left(m-1\right)\left(\frac{1}{P{r}_1}\mid y_3{\mid }^{m-2}{y}_3^{\prime }+\frac{1}{P{r}_2}\mid y_5{\mid }^{m-2}{y}_5^{\prime }\right)-\left(\frac{2m}{m+1}{y}_1{y}_7+J_o\right)}{A_3{A}_4\left(S_{\theta }+y_7\frac{m-1}{P{r}_3}\mid y_7{\mid }^{m-2}\right)},\hfill \\ y_8^{\prime }=y_9,\hfill \\ y_9^{\prime }=\frac{{\kappa }_o{y}_8-{\beta }_3{y}_9\left(m-1\right)\left(\frac{1}{S{c}_1}\mid y_3{\mid }^{m-2}{y}_3^{\prime }+\frac{1}{S{c}_2}\mid y_5{\mid }^{m-2}{y}_5^{\prime }\right)}{{\beta }_3\left(S_{\varphi }+y_9\frac{m-1}{S{c}_3}\mid y_9{\mid }^{m-2}\right)},\hfill \end{array}\right)$$ 27

where

$$\begin{array}{ccc}\hfill J_o& \hfill =\hfill & Ec\left(\frac{M{A}_5}{{\alpha }_3}-A_6{K}_p\right)\left(y_2^2+y_4^2\right),\hfill \\ \hfill S_{\theta }& \hfill =\hfill & \frac{\mid y_3{\mid }^{m-1}}{P{r}_1}+\frac{\mid y_5{\mid }^{m-1}}{P{r}_2}+\frac{\mid y_7{\mid }^{m-1}}{P{r}_3},\hfill \\ \hfill S_{\varphi }& \hfill =\hfill & \frac{\mid y_3{\mid }^{m-1}}{S{c}_1}+\frac{\mid y_5{\mid }^{m-1}}{S{c}_2}+\frac{\mid y_9{\mid }^{m-1}}{S{c}_3}.\hfill \end{array}$$

Boundary conditions are specified as:

$$\left\{\begin{array}{c}{y}_1\left(0\right)=y_4\left(0\right)=0,y_2\left(0\right)=1,\hfill \\ y_6\left(0\right)=1+y_7\left(0\right)A_3\left({\sum }_{p=1}^3\frac{\mid {\mathit{\psi }}_p^{\prime }\left(0\right){\mid }^{m-1}}{{\mathrm{Bi}}_{hp}}\right),\hfill \\ y_8\left(0\right)=1+y_9\left(0\right){\beta }_3\left({\sum }_{q=1}^3\frac{\mid \mathit{\chi }\prime \left(0\right){\mid }^{m-1}}{{\mathrm{Bi}}_{mq}}\right),\hfill \\ y_2\left(\infty \right),y_4\left(\infty \right),y_6\left(\infty \right),y_8\left(\infty \right)\to 0.\hfill \end{array}\right)$$ 28

Validity of the results

The accuracy of the computed solution, which is crucial for the model’s reliability, is verified by comparison with previously published works. First, the present results are compared with the classical study of Cortell [39]. To establish a basis for comparison, the parameters M, K_p, β_i, and β_e are set to zero. Furthermore, to model a Newtonian fluid, the parameter m is set to unity, and the nanoparticle parameters are set to A₁ = A₂ = 1 to represent a pure fluid. These simplifications reduce the current model’s momentum equation for the primary velocity to the classic case studied by Cortell [39] for n = 1 (a linearly stretching sheet). The computed skin friction value,  − f^″(0) = 1.00, is in exact agreement with the value reported by Cortell [39]. This excellent agreement validates the accuracy of the MATLAB program and the numerical model implemented in this study.

In addition, the present Nusselt number results are validated against the existing values of Gope et al. [28], as shown in Table 2. The comparison demonstrates excellent agreement across a wide range of parameters, with mean absolute errors of 1.53 × 10⁻³ for PLHNF (hybrid nanofluid: ϕ₁ = ϕ₂ = 0.01) and 6.80 × 10⁻³ for PLNF (nanofluid: ϕ₁ = 0.01, ϕ₂ = 0). The maximum errors of 0.00644 (PLHNF) and 0.03228 (PLNF) occur at extreme parameter values, while errors decrease to as low as 8 × 10⁻⁵ for PLHNF at β_e = 0.8. These results confirm the robustness and accuracy of the present model in analyzing hybrid nanofluid transport under Hall and power-law effects.

Table 2.

Validation of the present results for $-{\theta }^{\prime }\left(0\right)$ compared with Gope et al. [28]

Pr1	Pr2	βe	Bih1	Bih2	Present study		Gope et al. [28]		Absolute Error
					PLHNF	PLNF	PLHNF	PLNF	PLHNF	PLNF
1	2	0.2	2	2	0.32596	0.31933	0.32694	0.31353	0.00098	0.00580
2	2	0.2	2	2	0.34295	0.33007	0.34307	0.33699	0.00012	0.00692
3	2	0.2	2	2	0.39967	0.33514	0.40193	0.36742	0.00226	0.03228
1	1	0.2	2	2	0.27552	0.24132	0.27635	0.24662	0.00083	0.00530
1	3	0.2	2	2	0.44643	0.42306	0.44446	0.42791	0.00197	0.00485
1	2	0.5	2	2	0.35234	0.34864	0.35226	0.34668	0.00008	0.00196
1	2	0.8	2	2	0.36936	0.35958	0.36970	0.35964	0.00034	0.00006
1	2	0.2	1	2	0.30249	0.30840	0.30122	0.30034	0.00127	0.00806
1	2	0.2	3	2	0.38927	0.37580	0.38283	0.37285	0.00644	0.00295
1	2	0.2	2	1	0.33645	0.32228	0.33693	0.32660	0.00048	0.00432
1	2	0.2	2	3	0.37231	0.36887	0.37428	0.36116	0.00197	0.00771

PLHNF = Hybrid nanofluid (ϕ₁ = ϕ₂ = 0.01), PLNF = Nanofluid (ϕ₁ = 0.01, ϕ₂ = 0).

Fixed parameters: Pr₃ = 2, β_i = 0.2, Bi_h3 = 2, M = 1, Kp = 0.4, Ec = 0.5, m = 1.1.

Absolute Error = ∣Nu_present − Nu_Gope∣.

Results and discussion

A non-Newtonian rheological model, grounded in fundamental conservation laws and incorporating advanced non-Fourier and non-Fickian formulations, is developed for a hybrid nanofluid flow generated by a linearly stretching permeable surface. The hybrid nanofluid consists Copper (Cu) and Aluminum oxide (Al₂O₃) nanoparticles dispersed in a propylene glycol base fluid. The boundary value problem is solved numerically to analyze the complex transport phenomena. A detailed analysis is presented to scrutinize the effects of several physical entities on momentum, thermal, and solutal distribution phenomena. Parameter ranges are: 1≤Pr_j, Sc_k≤10, 0.8≤m≤1.1, 0.1≤β_e, β_i, K_p≤0.8, 0.5≤κ₀≤5, 0.1≤Ec≤0.6, 1≤M≤5, and 1≤Bihp, Bimq≤5.

Increasing the magnetic parameter M strengthens the Lorentz force, reducing primary velocity $f\prime \left(\eta \right)$ (Fig. 2) and, consequently, secondary velocity g(η) (Fig. 3). The stream function f(η) also decreases (Fig. 4), indicating reduced net mass flow.

Fig. 2.

Primary velocity $f^{\prime }\left(\eta \right)$ against M

Fig. 3.

Secondary velocity g(η) against M

Fig. 4.

Stream function f(η) against M

The power-law index is a critical parameter in analyzing non-Newtonian fluids. In boundary layer flows involving convective heat and mass transfer, this index governs the shear-dependent viscosity, thereby significantly influencing the velocity, temperature, and concentration profiles. For shear-thinning fluids (m < 1), the flow becomes flatter away from the wall, while the secondary velocity field near the wall increases. Thus, the secondary velocity decreases close to the plate, as illustrated in Fig. 5.

Fig. 5.

Secondary velocity g(η) against m

The ion slip parameter characterizes the degree to which ions deviate from perfect coupling with the magnetic field. Physically, an increase in ion slip reduces the ability of the magnetic entity to exert a restraining Lorentz force in the system. When this magnetic resistance weakens, momentum transfer between the charged and neutral particles becomes less effective, thereby diminishing the damping influence of the magnetic factor. Hence, the fluid experiences less opposition in the cross-flow direction, which enhances the development of the secondary velocity as shown in Fig. 6.

Fig. 6.

Secondary velocity g(η) against β_i

The inclusion of the Hall effect significantly alters the momentum distribution in the fluid. Physically, the Hall effect arises due to the deflection of charged particles under the exertion of a strong magnet, which generates an additional current component perpendicular to both the magneto–electrical fields. This modification reduces the effective Lorentz force opposing the direction of the main flow, thereby diminishing the electromagnetic resistance to motion of the fluid and causing an increase in the primary velocity field. However, the redistribution of current density caused by the Hall effect enhances the resistive influence acting on the secondary flow component. Consequently, the secondary velocity is suppressed. Thus, the Hall current effect simultaneously promotes the primary flow while attenuating the secondary flow due to the altered balance of electromagnetic forces (See Figs. 7 and 8).

Fig. 7.

Primary velocity $f^{\prime }\left(\eta \right)$ against β_e

Fig. 8.

Secondary velocity g(η) against β_e

Permeability represents the ability of a porous medium to permit fluid to flow through it. In present context, an increase in permeability implies a reduction in the resistance offered by the medium to fluid motion. Hence, the primary velocity increases because the flow encounters less drag from the porous structure. Similarly, the secondary velocity also increases with permeability, since reduced resistance allows more pronounced cross-flow development, particularly in the presence of Hall currents. Overall, higher permeability enhances both primary and secondary velocity magnitudes due to diminished flow obstruction (see Figs. 9 and 10).

Fig. 9.

Primary velocity $f^{\prime }\left(\eta \right)$ against k_p

Fig. 10.

Secondary velocity g(η) against k_p

It is important to emphasize that, although several observed trends (e.g., the influence of magnetic parameter, generalized Prandtl and Schmidt numbers) are qualitatively consistent with classical boundary layer theory, the present model reveals additional behaviors arising from shear-dependent diffusivity. Unlike classical formulations with constant thermal and mass diffusivities, the effective transport coefficients in this study are explicitly coupled to local velocity gradients. As a result, thermal and concentration boundary layers become strongly dependent on the shear field, leading to spatially varying diffusion rates.

This coupling gives rise to deviations from classical scaling laws, particularly in the interpretation of generalized Prandtl and Schmidt numbers, whose effects are no longer independent of the flow structure. Furthermore, under higher shear and magnetic field conditions, the interaction between nonlinear diffusion and momentum transport may produce non-monotonic variations in temperature and concentration distributions. In certain parameter regimes, an initial enhancement in thermal or solutal transport is followed by a reduction as shear-induced suppression of diffusivity becomes dominant.

These findings highlight that the inclusion of shear-dependent diffusivity introduces a flow-regulated transport mechanism that cannot be captured by conventional constant-property models. Consequently, the present framework provides deeper insight into coupled transport processes in non-Newtonian hybrid nanofluids subjected to electromagnetic effects.

The generalized Prandtl numbers (Pr_j, j = 1, 2, 3) significantly influence the temperature profiles in the boundary layer. As shown in Figs. 11, 12, 13, increasing Pr_j reduces thermal diffusivity relative to momentum diffusivity, resulting in steeper temperature gradients near the surface and a thinner thermal boundary layer. Conversely, lower values of Pr_j promote enhanced heat diffusion, yielding broader temperature distributions.

Fig. 11.

Temperature θ(η) for varying Pr₁

Fig. 12.

Temperature θ(η) for varying Pr₂

Fig. 13.

Temperature θ(η) for varying Pr₃

In the present formulation, however, the effect of Pr_j is further modulated by shear-dependent diffusivity. Since thermal diffusivity varies with the local velocity gradient, the reduction in thermal boundary layer thickness is not uniform across the domain. Instead, regions of high shear exhibit suppressed diffusion, leading to localized thermal gradients and slight deviations from classical monotonic decay profiles, particularly at higher parameter values.

The influence of the nonlinear thermal Biot number is illustrated in Figs. 14, 15, 16. As the Biot number increases, convective heat transfer at the boundary becomes more dominant, leading to stronger thermal gradients near the surface. In contrast to classical interpretations, the present model indicates that this effect is coupled with shear-dependent diffusivity, resulting in non-uniform thickening or thinning of the thermal boundary layer depending on the local flow conditions. This highlights the interplay between surface convection and flow-induced diffusion.

Fig. 14.

Temperature θ(η) for varying Bi_h1

Fig. 15.

Temperature θ(η) for varying Bi_h2

Fig. 16.

Temperature θ(η) for varying Bi_h3

The Eckert number (Ec) represents the ratio of kinetic energy to thermal energy and characterizes viscous dissipation effects. An increase in Ec enhances internal heat generation due to viscous effects, leading to a rise in temperature within the boundary layer, as shown in Fig. 17. In the present model, this effect is further amplified in regions of high shear, where both viscous dissipation and shear-dependent diffusivity contribute to elevated thermal energy levels.

Fig. 17.

Temperature θ(η) for varying Ec

The generalized Schmidt numbers Sc_k (k = 1, 2, 3) govern mass diffusion behavior. As shown in Figs. 18, 19, 20, increasing Sc_k reduces mass diffusivity, resulting in thinner concentration boundary layers and steeper gradients near the surface. Conversely, lower Sc_k values yield broader concentration profiles.

Fig. 18.

Concentration ϕ(η) for varying Sc₁

Fig. 19.

Concentration ϕ(η) for varying Sc₂

Fig. 20.

Concentration ϕ(η) for varying Sc₃

However, due to shear-dependent mass diffusivity, the concentration field exhibits additional complexity. Diffusion becomes locally suppressed in high-shear regions, leading to enhanced gradient localization and slight deviations from classical exponential decay behavior. This demonstrates that mass transfer is not solely governed by Sc_k, but also by the underlying flow structure.

The nonlinear mass Biot numbers Bi_mq (q = 1, 2, 3) enhance convective mass transfer at the boundary, leading to sharper concentration gradients and thinner boundary layers, as shown in Figs. 21, 22, 23. In contrast to classical models, the present formulation indicates that the resulting concentration distribution is strongly influenced by shear-dependent diffusivity, producing localized transport intensification near the surface rather than uniform enhancement across the domain.

Fig. 21.

Concentration ϕ(η) for varying Bi_m1

Fig. 22.

Concentration ϕ(η) for varying Bi_m2

Fig. 23.

Concentration ϕ(η) for varying Bi_m3

Consistent with mass action kinetics, Fig. 24 shows that a faster reaction rate depletes reactants more quickly, resulting in a steeper concentration profile. This is a well-established principle in reaction engineering.

Fig. 24.

Concentration ϕ(η) against κ_o

In this study, physically realistic hybrid nanofluid volume fractions are limited to 0≤ϕ≤0.20. Higher values (ϕ > 0.20) are included only to illustrate the mathematical behavior of the model, since such concentrations may cause particle agglomeration, yield-stress effects, or solid-like transitions that are not captured by the present power-law formulation. Therefore, results for ϕ > 0.20 should be interpreted as mathematical extensions rather than physically realizable states.

The influence of several parameters on local transport coefficients are summarized in Table 3. It is obvious that an increase in magnetic parameter M as well as porosity parameter K_p augments both primary and secondary skin friction coefficients. This augmentation stems from the combined effect of a stronger Lorentz force and increased impedance from the porous structure, leading to heightened shear at the boundary. In contrast, augmented β_e and β_i leads to a reduction in skin frictions, as Hall and ion-slip currents tend to weaken the effective electromagnetic resistance within the flow. The influence of Ec, Pr₁, and Sc₁ on these engineering quantities is negligible, suggesting that viscous dissipation and thermal–solutal diffusivities play a minor role in modifying the wall shear stresses. Moreover, the comparative analysis clearly indicates that PLHNF produces higher skin friction than PLNF, highlighting the stronger momentum transport capability of the hybrid nanofluid.

Table 3.

Variations in engineering parameters of interest for selected dimensionless groups

Ec	M	Pr1	Kp	Sc1	βe	βi	− f″(0)		$-g^{\prime }\left(0\right)$		$-{\theta }^{\prime }\left(0\right)$		$-{\varphi }^{\prime }\left(0\right)$
							PLHNF	PLNF	PLHNF	PLNF	PLHNF	PLNF	PLHNF	PLNF
0.1	1	2	0.1	0.1	0.2	0.2	1.3548	1.3684	0.0974	0.0951	0.3072	0.3534	0.3224	0.3083
0.4	–	–	–	–	–	–	1.3548	1.3684	0.0974	0.0951	0.3316	0.3793	0.3228	0.3085
–	2	–	–	–	–	–	1.5165	1.5257	0.1654	0.1622	0.3147	0.3610	0.3189	0.3042
–	4	–	–	–	–	–	1.8004	1.8032	0.2644	0.2601	0.3284	0.3770	0.3142	0.3016
–		3	–	–	–	–	1.5165	1.5257	0.1654	0.1622	0.2873	0.3303	0.3180	0.3040
–		6	–	–	–	–	1.5165	1.5257	0.1654	0.1622	0.3047	0.3505	0.3180	0.3040
–			0.2	–	–	–	1.5165	1.5257	0.1654	0.1622	0.3147	0.3610	0.3189	0.3042
–			0.4	–	–	–	1.5711	1.5801	0.1596	0.1564	0.3087	0.3543	0.3186	0.3042
–				0.2	–	–	1.5165	1.5257	0.1654	0.1622	0.3150	0.3609	0.2730	0.2618
–				0.6	–	–	1.5165	1.5257	0.1654	0.1622	0.3141	0.3609	0.2973	0.2848
–					0.2	–	1.5928	1.6003	0.0948	0.1776	0.3290	0.3775	0.3226	0.3082
–					0.6	–	1.4932	1.5030	0.0931	0.1740	0.3111	0.3578	0.3177	0.3042
–						0.2	1.5468	1.5553	0.1983	0.1944	0.3187	0.3653	0.3175	0.3026
–						0.6	1.5073	1.5168	0.1563	0.1532	0.3133	0.3598	0.3188	0.3046

PLHNF: Power Law Hybrid Nanofluid; PLNF: Power Law Nanofluid

Bold entries represent the varying parameters for each case

Dashes (–) indicate parameters held constant at baseline values

Analysis reveals a directly proportional relationship between Nusselt number and the parameters Ec, M, and Pr₁ in both fluids. The enhancement is attributed to increased thermal energy from viscous and Joule dissipation (Ec, M) and a sharpened wall temperature gradient due to suppressed thermal diffusion at high Pr₁. In contrast, a reduction in gradient-based heat transfer is revealed with rising K_p, β_e, and β_i, where porous resistance and the dissipation of Hall/ion-slip currents impair thermal convection. The thermal transport remains largely unaffected by the Schmidt number (Sc₁), especially in PLNF, underscoring the decoupling of solutal and thermal phenomena in this context.

The rate of mass transfer, quantified by the Sherwood number, is enhanced by rising Ec and β_i values. This enhancement stems from the coupling between viscous dissipation and mass diffusion, alongside the solutal transport promoted by ion-slip currents. In contrast, an increasing magnetic parameter (M) and Hall parameter (β_e) diminish the Sherwood number by strengthening the Lorentz force and Hall currents, which act to flatten solutal gradients at the boundary. The parameters Pr₁ and K_p exhibit a weak influence, suggesting that thermal and porous effects play a subordinate role to the dominant electromagnetic mechanisms in the mass transfer process.

Conclusion

This study presents a comprehensive model for the flow of a power-law hybrid nanofluid over a continuously stretched surface in a permeable medium. The model integrates several advanced mechanisms, including modified Fourier heat conduction, generalized nonlinear Fick’s law, Hall and ion-slip currents, and Joule dissipation, to capture coupled heat and mass transport under surface convective boundaries. A key innovation is the adoption of shear-dependent thermal and mass diffusivities, explicitly linking transport properties to velocity gradients and enabling the characterization of flow-induced diffusive mechanisms.

The model also incorporates Hall and ion-slip effects and porous-medium resistance, reflecting practical applications such as microfluidic devices, electronic cooling, and porous media heat exchangers. The results emphasize the significant interplay between porous resistance, magnetic field strength, nonlinear diffusion, and ion-slip parameters in regulating temperature, velocity, and concentration fields. This study provides a framework for understanding momentum, heat, and mass transfer in power-law hybrid nanofluids with Hall and ion-slip effects, making it particularly relevant for micro- and nano-scale systems, where surface interactions and boundary effects are significant.

The key findings of this study are summarized as follows:

• Stream function, primary, and secondary velocity profiles decrease with increasing magnetic field strength.

• Higher permeability enhances both primary and secondary velocity magnitudes due to reduced flow resistance.

• Hall current causes a reduction in secondary flow intensity.

• Temperature distribution decreases with increasing generalized Prandtl numbers.

• Concentration distribution decreases with increasing generalized Schmidt numbers.

• Higher generalized Biot numbers enhance the corresponding temperature and concentration distributions.

These findings have practical implications for real-world thermal-fluid systems. They suggest that Hall parameters and shear-dependent viscosity effects can be leveraged to enhance heat transfer and regulate flow behavior in applications such as microchannel heat sinks, compact cooling devices, and plasma-assisted cooling systems. Furthermore, the results are applicable to nanofluid-based cooling technologies, where non-Newtonian characteristics and electromagnetic interactions are essential for optimizing system performance.

Limitations of the work

Several limitations of the present study must be acknowledged:

• Local thermal equilibrium: The model assumes local thermal equilibrium between the hybrid nanofluid and the porous matrix, which may not hold under highly non-equilibrium conditions.

• Nanoparticle effects: Nanoparticle aggregation and Brownian motion are neglected, although they may affect thermal transport at low volume fractions.

• Flow assumptions: The flow is assumed to be steady and laminar, with variables independent of the z-coordinate (quasi-three-dimensional). Unsteady, turbulent, or fully three-dimensional effects are not considered.

• Thermal radiation: Thermal radiation effects are neglected, which may be significant in high-temperature applications or systems with large temperature gradients.

• Soret and Dufour effects: The Soret (thermal-diffusion) and Dufour (diffusion-thermo) effects are neglected, though they may influence coupled heat and mass transfer in systems with steep temperature and concentration gradients.

• Idealized diffusivity model: The phenomenological shear-dependent diffusivity formulation represents an idealized version of complex diffusion processes.

Future work

Future investigations should address each of the above limitations. The local thermal equilibrium assumption should be relaxed using a two-temperature model for non-equilibrium porous media flow. Nanoparticle aggregation and Brownian motion should be incorporated through population balance equations or experimental correlations. The steady, laminar, quasi-three-dimensional flow assumption should be extended to unsteady, turbulent, and fully three-dimensional regimes. Thermal radiation effects should be included, particularly for high-temperature applications. The Soret and Dufour effects should be integrated to account for cross-diffusion phenomena in systems with steep temperature and concentration gradients. The phenomenological shear-dependent diffusivity model should be refined using physically based or experimentally validated formulations.

Beyond these theoretical extensions, experimental validation using stabilized Cu–Al₂O₃ nanocomposites is crucial. Future studies should also assess long-term operational stability, including performance degradation under cyclic thermal and magnetic loading, copper ion release kinetics, and surface fouling or corrosion. Finally, more complex geometries, such as microchannel heat sinks, annular configurations, and non-uniform surfaces, should be explored. Additionally, unsteady flow effects, thermal radiation, and turbulent transport should be considered to further enhance the model’s applicability to high-performance thermal devices.

Nomenclature

Symbol

Description (Units)

u, v, w

Velocity components in x, y, z directions (ms⁻¹)

x, y, z

Cartesian coordinates (m)

T

Fluid temperature (K)

C

Species concentration (kgm⁻³)

T_w, T_∞

Wall and ambient temperatures (K)

C_w,C_∞

Wall and ambient concentrations (kgm⁻³)

c_p

Specific heat at constant pressure (J kg⁻¹ K⁻¹)

B₀

Magnetic field strength (T)

K

Permeability of porous medium (m²)

k_r

Chemical reaction rate (s⁻¹)

h_f

Convective heat transfer coefficient (Wm⁻² K⁻¹)

k_f

Convective mass transfer coefficient (ms⁻¹)

λ

Stretching rate (s⁻¹)

μ

Dynamic viscosity (Pa s)

ρ

Density (kgm⁻³)

k

Thermal conductivity (Wm⁻¹ K⁻¹)

σ

Electrical conductivity (Sm⁻¹)

D

Mass diffusivity (m² s⁻¹)

α

Thermal diffusivity (m² s⁻¹)

η

Similarity variable

f(η)

Dimensionless stream function

$f\prime \left(\eta \right)$

Dimensionless primary velocity

g(η)

Dimensionless secondary velocity

θ(η)

Dimensionless temperature

ϕ(η)

Dimensionless concentration

ϕ1, ϕ2

Volume fractions of Cu and Al2O3 nanoparticles

(·)_hnf, (·)_f

Hybrid nanofluid and base fluid properties

ρs₁, ρs₂

Densities of Cu and Al2O3 nanoparticles (kgm⁻³)

ks₁, ks₂

Thermal conductivities of Cu and Al2O3 nanoparticles (Wm⁻¹ K⁻¹)

m

Power-law index

K_c

Consistency coefficient (Pa s^m)

β_e

Hall parameter

β_i

Ion-slip parameter

M

Magnetic parameter

K_p

Porous medium parameter

Ec

Eckert number

Pr_j

Generalized Prandtl numbers (j = 1, 2, 3)

Sc_k

Generalized Schmidt numbers (k = 1, 2, 3)

Bi_hp

Thermal Biot numbers (p = 1, 2, 3)

Bi_mq

Solutal Biot numbers (q = 1, 2, 3)

κ₀

Chemical reaction parameter

ξ_i, ζ_i

Thermal and solutal weighting coefficients

C_fx,C_fz

Skin friction coefficients

Nu_x

Local Nusselt number

Sh_x

Local Sherwood number

τ_w,x, τ_w,z

Wall shear stresses Pa

q_w

Surface heat flux (Wm⁻²)

j_w

Surface mass flux (kgm⁻² s⁻¹)

Author contributions

MFE contributed to the proposed main idea and initial preparation. Moreover, MFE performed all the steps of the solutions in this research. Prof. XZ edited the draft and all the steps. Both authors read and approved the final manuscript.

Funding

The authors did not receive support from any organization for the submitted work.

Data Availability

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

Declarations

Ethics approval

Not applicable.

Consent to participate

Not applicable.

Consent to publish

Not applicable.

Competing interests

The authors declare no competing interests.