Computational investigation of MHD stagnation point flow of Eyring-Powell nanofluid with microorganisms over a cylinder under Cattaneo-Christov heat and mass flux

Abstract

This study investigates magnetohydrodynamic stagnation point flow with non-Fourier heat and mass transfer in an Eyring-Powell nanofluid containing gyrotactic microorganisms over an inclined porous cylindrical surface. The model simultaneously incorporates the Buongiorno nanofluid formulation, nonlinear thermal radiation, Arrhenius activation energy, Brownian motion, thermophoresis, Darcy porous medium resistance, and dual Cattaneo-Christov heat and mass flux model, thereby establishing a unified framework for analysing coupled momentum, thermal, concentration, and bioconvective transport phenomena. The transformed coupled nonlinear governing equations are reduced to a system of ordinary differential equations using suitable similarity transformations and solved numerically using the MATLAB bvp4c solver. The results reveal that the Eyring-Powell parameter suppresses the velocity field by increasing nonlinear rheological resistance, whereas thermal radiation, Brownian motion, thermophoresis, and curvature enhance thermal energy transport. In contrast, thermal relaxation weakens heat transfer by delaying thermal propagation, while Arrhenius activation energy suppresses the chemical reaction rate, thereby preserving nanoparticle concentration. Furthermore, increasing the bioconvection Peclet number reduces the motile microorganism distribution because advective transport dominates microorganism diffusion. Quantitatively, the skin-friction coefficient increases from 0.75 to 0.84, corresponding to an enhancement of about 12% with increasing chemical reaction parameter and decreasing activation energy. Moreover, the local Nusselt number rises from 1.4 to 3.6 (about 157%) with increasing radiation parameter and decreasing heat generation parameter, whereas the local Sherwood number increases from 0.782 to 0.802 (about 2.6%) with increasing chemical reaction parameter and decreasing activation energy. These findings provide new insights into the coupled transport mechanisms governing reactive bioconvective non-Newtonian nanofluids and offer useful guidance for the design and optimization of porous thermal management systems, catalytic reactors, geothermal energy technologies, and microorganism-assisted transport processes.

Introduction

Nanofluids, first introduced by Choi and Eastman [1], are engineered colloidal suspensions of nanoparticles (1-100 nm) dispersed in conventional base fluids to improve thermal conductivity and heat-transfer performance. To provide a more realistic description of nanoparticle transport, Buongiorno [2] proposed a non-homogeneous two-component nanofluid model in which Brownian diffusion and thermophoresis were identified as the dominant slip mechanisms governing nanoparticle transport. Owing to its sound physical basis, the Buongiorno model has become one of the most widely adopted frameworks for analyzing heat and mass transfer in nanofluid systems. Consequently, numerous studies have employed or extended this model to investigate nanofluid transport under diverse physical mechanisms and geometric configurations. Within this framework, Jawad et al. [3] examined the bioconvective Darcy-Forchheimer flow of an electromagnetic Reiner-Philippoff nanofluid over a porous stretching sheet by incorporating nonlinear thermal radiation and Arrhenius activation energy, demonstrating that thermal radiation enhanced heat transfer, whereas porous medium resistance reduced the fluid velocity. Basit et al. [4] investigated thermally radiative bioconvective nanofluid flow over a wavy cylinder and showed that Brownian motion, thermophoresis, and mixed convection strongly affect heat, mass, and microorganism transport. Zahoor Raja et al. [5] employed the Buongiorno model to analyze nanofluid flow over a stretching sheet and reported that Brownian motion, thermophoresis, and nanoparticle volume fraction play important roles in thermal transport. Likewise, Rafique et al. [6] investigated micropolar nanofluid flow over an inclined surface with Soret diffusion and thermal radiation, demonstrating enhanced thermal and concentration transport characteristics. In a related study, Faizan et al. [7] analyzed Darcy-Forchheimer radiative Cross tri-hybrid nanofluid flow over a stretching porous cylinder with entropy generation and showed that the incorporation of tri-hybrid nanoparticles substantially improved heat-transfer performance, highlighting the growing interest in advanced hybrid nanofluid models for thermal management applications.Among the various non-Newtonian fluid models, the Eyring-Powell fluid has attracted considerable attention because it effectively characterizes shear-thinning behavior while being derived from the kinetic theory of liquids rather than empirical constitutive relations. An important advantage of this model is its ability to recover Newtonian fluid behavior under limiting conditions, making it suitable for describing complex transport phenomena encountered in polymer processing, lubrication, biomedical engineering, and other industrial applications. Owing to these favorable rheological characteristics, the Eyring-Powell model has been widely employed to investigate nanofluid transport under diverse thermal and flow conditions. Karthik et al. [8] numerically investigated magnetohydrodynamic Powell-Eyring fluid flow through a porous medium by incorporating the Cattaneo-Christov heat flux model, chemical diffusion, and heat sink effects, demonstrating that the magnetic field suppresses fluid velocity, whereas the heat source enhances the temperature distribution. Patil and Benawadi [9] investigated the bioconvective flow of an Eyring-Powell nanofluid over a vertical slender cylinder using the Buongiorno nanofluid model with entropy generation and demonstrated that Brownian motion, thermophoresis, and bioconvection markedly influence heat, mass, and microorganism transport. Likewise, Narasimha Raju et al. [10] analyzed magnetohydrodynamic Eyring-Powell nanofluid flow over a wedge with convective boundary conditions and thermal radiation based on the Buongiorno model, reporting that Brownian motion, thermophoresis, thermal radiation, and heat generation considerably modify the thermal and concentration fields. More recently, Ahmed et al. [11] examined radiative bioconvective Eyring-Powell nanofluid flow by incorporating Joule heating, viscous dissipation, chemical reaction, Brownian diffusion, and thermophoresis within the Buongiorno framework, highlighting the coupled influence of these mechanisms on fluid flow, heat transfer, and nanoparticle transport.Thermal radiation is one of the principal heat-transfer mechanisms governing energy transport in high-temperature engineering systems and plays a vital role in determining the thermal behavior of non-Newtonian nanofluids. Its incorporation into mathematical models provides a more realistic description of heat transport in applications such as thermal processing, energy conversion, polymer manufacturing, and biomedical engineering. In a recent review, Odebowale et al. [12] highlighted the fundamental principles, recent advances, and emerging applications of radiative heat transfer, emphasizing its increasing importance in advanced thermal management and energy systems. Motivated by these developments, considerable research has focused on the interaction between thermal radiation and Eyring-Powell nanofluid flows under diverse physical conditions. In this regard, Lawal et al. [13] investigated unsteady Eyring-Powell nanofluid flow over a stretching sheet with nonlinear thermal radiation, variable viscosity, and variable thermal conductivity, demonstrating that Brownian motion and thermophoresis modify the thermal and concentration fields. Khan et al. [14] examined three-dimensional magnetohydrodynamic Eyring-Powell nanofluid flow with nonlinear thermal radiation and convective heating, reporting that thermal radiation enhances the temperature distribution and thermal boundary layer thickness. Similarly, Salawu et al. [15] analyzed magnetized Eyring-Powell binary reactive fluid flow with thermal radiation and generalized Arrhenius kinetics, showing that thermal radiation promotes heat transfer and facilitates suspended-particle transport. More recently, Salahuddin and Awais [16] investigated thermo-solutal convection in an Eyring-Powell Buongiorno nanofluid using the Cattaneo-Christov heat and mass flux model, revealing that Brownian motion and thermophoresis enhance the thermal field, whereas thermal and solutal relaxation suppress heat and mass transport.

Activation energy plays a crucial role in chemically reactive nanofluid flows by regulating chemical reaction rates and species transport, thereby influencing concentration distributions and mass transfer characteristics. Its incorporation through the Arrhenius formulation provides a more realistic representation of reactive transport phenomena and has received considerable attention because of its broad applications in chemical engineering, geothermal systems, food processing, and other thermochemical processes. Consequently, the coupled effects of activation energy and chemical reaction have been extensively explored in non-Newtonian nanofluid flows. In this context, Reddy et al. [17] investigated chemically reacting Eyring-Powell nanofluid flow over an inclined stretching cylinder by incorporating Arrhenius activation energy, nonlinear thermal radiation, the Cattaneo-Christov heat flux model, binary chemical reaction, and the Buongiorno nanofluid model, demonstrating that activation energy strongly influences reactive mass transport and concentration characteristics. Abbas et al. [18] examined Powell-Eyring nanomaterial flow through a ciliated curved channel with mixed convection, nonlinear thermal radiation, viscous dissipation, chemical reaction, activation energy, and thermophoresis, showing that activation energy and reactive transport substantially affect heat and mass transfer. Similarly, Azam [19] analyzed magnetohydrodynamic Maxwell nanofluid flow by incorporating the Cattaneo-Christov heat flux model, nonlinear thermal radiation, and Arrhenius activation energy, highlighting the important role of activation energy in governing nanoparticle concentration and reactive mass transport. More recently, Anjum et al. [20] investigated modified Eyring-Powell nanofluid flow with thermal radiation, activation energy, Brownian motion, thermophoresis, and bioconvection, demonstrating that activation energy plays a key role in controlling concentration transport and chemically reactive nanofluid behavior.Bioconvection induced by motile gyrotactic microorganisms has emerged as an effective mechanism for enhancing the stability of nanoparticle suspensions and improving heat and mass transport in nanofluid systems. The collective swimming behavior of microorganisms strongly influences the thermal, concentration, and microorganism distributions, making bioconvection an important consideration in advanced non-Newtonian nanofluid models. In this context, Farooq and Imran [21] investigated bioconvective Casson hybrid nanofluid flow over a cylinder by incorporating Arrhenius activation energy, thermal radiation, and the Cattaneo-Christov heat and mass flux model, demonstrating the important role of motile microorganisms in governing heat, mass, and microorganism transport. Aboel-Magd et al. [22] examined bioconvective Casson nanofluid flow over a rotating disk with activation energy, nonlinear thermal radiation, and gyrotactic microorganisms, reporting that the bioconvection Peclet and bioconvective Lewis numbers suppress the microorganism distribution. Similarly, Usman Ali and Irfan [23] analyzed Eyring-Powell nanofluid flow with gyrotactic microorganisms under Arrhenius activation energy, Joule heating, and thermal radiation, showing that activation energy enhances nanoparticle concentration, whereas higher bioconvection Peclet and bioconvective Lewis numbers reduce microorganism density. Kiran Batool et al. [24] investigated bioconvective Eyring-Powell nanofluid flow over a porous cylinder by incorporating entropy generation, activation energy, nonlinear thermal radiation, and chemical reaction, highlighting the influence of microorganism transport on heat, mass, and entropy characteristics. Similar findings were reported by Shah et al. [25], who studied magnetohydrodynamic Prandtl nanofluid flow over a stretching cylinder with Brownian motion, thermophoresis, thermal radiation, activation energy, and bioconvection, demonstrating enhanced thermal transport together with reduced microorganism density at higher bioconvective Lewis numbers. Bilal et al. [26] further examined bioconvective Williamson nanofluid flow over a cylindrical surface by incorporating activation energy, magnetic field, and convective boundary conditions, highlighting the coupled influence of these mechanisms on reactive transport behavior. More recently, Nisar et al. [27] investigated bioconvective transport in a fourth-grade nanofluid under nonlinear thermal radiation using the Buongiorno framework, emphasizing the combined roles of Brownian motion, thermophoresis, and motile gyrotactic microorganisms in transport processes. Likewise, Farhan Ali et al. [28] examined entropy generation in mixed convective Casson nanofluid flow containing gyrotactic microorganisms under the Cattaneo-Christov dual diffusion model, demonstrating the coupled effects of thermal radiation, chemical reaction, Brownian motion, and thermophoresis on transport characteristics. Haq et al. [29] further investigated mixed convective magnetohydrodynamic Eyring-Powell nanofluid flow over a stretching cylinder with gyrotactic microorganisms, highlighting the important contributions of Brownian motion and thermophoresis to heat, mass, and microorganism transport.

Despite the considerable progress achieved in modeling non-Newtonian nanofluid transport under the influences of thermal radiation, Arrhenius activation energy, and bioconvection, several important research challenges remain. Previous investigations have predominantly examined these physical mechanisms either individually or in limited combinations, with most studies focusing on flat plates, stretching sheets, rotating disks, wedges, and simplified cylindrical configurations. Although recent studies have incorporated Eyring-Powell rheology, the Buongiorno nanofluid model, Cattaneo-Christov heat and mass flux, Arrhenius activation energy, thermal radiation, and gyrotactic microorganisms, their combined interaction within a unified Eyring-Powell nanofluid framework over an inclined cylindrical stagnation point surface embedded in a Darcy porous medium has remained insufficiently explored. Moreover, the coupled effects of thermal and concentration relaxation, Brownian motion, thermophoresis, Arrhenius activation energy, and motile gyrotactic microorganisms on the momentum, thermal, concentration, and microorganism fields have received comparatively limited attention. Addressing these aspects is essential for developing a more comprehensive understanding of reactive bioconvective transport in realistic non-Newtonian nanofluid systems.Motivated by the research gap identified above, the present study develops a mathematical model for the steady stagnation point flow of an Eyring-Powell nanofluid over an inclined cylindrical surface embedded in a Darcy porous medium by simultaneously incorporating the Buongiorno nanofluid model, nonlinear thermal radiation, Arrhenius activation energy, gyrotactic microorganisms, and the dual Cattaneo-Christov heat and mass flux model. The novelty of the present work lies in integrating these interacting physical mechanisms within a unified Eyring-Powell nanofluid framework for an inclined cylindrical stagnation point configuration, thereby enabling a systematic evaluation of their coupled effects on momentum, thermal, concentration, and microorganism transport characteristics. The incorporation of Arrhenius activation energy provides a more realistic description of chemically reactive mass transport by accounting for the influence of reaction kinetics on species diffusion and concentration fields. Likewise, the inclusion of gyrotactic microorganisms captures bioconvective transport arising from the collective swimming behavior of motile microorganisms, thereby improving the representation of suspension stability and its influence on heat, mass, and microorganism transport. Furthermore, the Cattaneo-Christov heat and mass flux model accounts for finite thermal and concentration relaxation times, overcoming the limitations of the classical Fourier and Fick diffusion laws in describing non-Fourier transport phenomena. The resulting coupled nonlinear governing equations are transformed into a system of coupled ordinary differential equations using suitable similarity transformations and solved numerically using the MATLAB bvp4c solver to investigate the coupled transport characteristics of reactive bioconvective Eyring-Powell nanofluids over inclined cylindrical geometries.

Mathematical formulation

The present study investigates the steady, axisymmetric, incompressible, laminar boundary layer flow of an Eyring-Powell nanofluid over an inclined cylindrical surface of radius $R$, embedded in a Darcy porous medium and subjected to a gravitational field inclined at an angle $\alpha$ to the vertical. The mathematical model incorporates magnetohydrodynamic (MHD) effects, nonlinear thermal radiation, Arrhenius activation energy, Brownian motion, thermophoresis, chemical reaction, dual Cattaneo-Christov heat and mass fluxes, and bioconvective transport due to gyrotactic microorganisms. The inclination angle $\alpha$ represents the component of gravity acting along the cylinder and influences the momentum and thermal transport characteristics of the flow. A cylindrical coordinate system $\left(r,x\right)$ is employed, where $x$ denotes the axial direction along the cylinder and $r$ is the radial direction normal to its surface. The corresponding velocity components in the axial and radial directions are represented by $u_1$ and $v_1$, respectively. The physical configuration and coordinate system considered in the present study are illustrated in Fig. 1.

Fig. 1.

Physical geometry of the MHD stagnation point flow of an yring-Powell nanofluid over an inclined cylindrical surface embedded in a Darcy porous medium

To simplify the mathematical formulation while retaining the essential transport physics, the following assumptions are adopted:

• The flow is considered steady, laminar, incompressible, and axisymmetric.

• The nanofluid obeys the Eyring-Powell non-Newtonian constitutive model.

• The flow develops over an inclined cylindrical surface embedded in a Darcy porous medium.

• The induced magnetic field is neglected due to the assumption of a small magnetic Reynolds number.

• The boundary layer approximations are assumed to be valid throughout the flow domain.

• The thermophysical properties of the base fluid and nanoparticles are assumed constant, except for density variations appearing in the buoyancy terms.

• Heat and mass transport are governed by the dual Cattaneo-Christov non-Fourier heat and mass flux models.

• Nonlinear thermal radiation effects are incorporated through the Rosseland diffusion approximation.

• Nanoparticle transport is described by Brownian motion and thermophoretic diffusion according to the Buongiorno nanofluid model.

• The chemical reaction follows Arrhenius activation energy kinetics.

• The nanofluid contains a homogeneous suspension of gyrotactic motile microorganisms, and the associated bioconvective transport is incorporated into the model.

• The effects of viscous dissipation and Joule heating are neglected.

• The cylindrical surface is maintained at prescribed wall velocity, temperature, nanoparticle concentration, and microorganism density.

Based on the foregoing assumptions and the boundary layer approximations, the governing equations for the conservation of mass, momentum, thermal energy, nanoparticle concentration, and motile microorganism density are formulated in cylindrical coordinates $\left(r,x\right)$. The Eyring-Powell constitutive model is employed to describe the non-Newtonian rheology of the nanofluid, while nonlinear thermal radiation, dual Cattaneo-Christov heat and mass fluxes, Brownian motion, thermophoresis, Arrhenius reaction kinetics, chemical reaction, and gyrotactic microorganisms are incorporated into the transport model.

Under the foregoing assumptions, the governing conservation equations are formulated using the standard boundary layer approximations applicable to non-Newtonian nanofluid transport over cylindrical geometries [26, 30].

$$\frac{\partial \left(r{u}_1\right)}{\partial x}+\frac{\partial \left(r{v}_1\right)}{\partial r}=0$$ 1

$$\begin{array}{l}{u}_1\frac{\partial u_1}{\partial x}+v_1\frac{\partial u_1}{\partial r}=u_e\frac{d{u}_e}{dx}+\frac{\cos \alpha }{{\rho }_f}\left\{\begin{array}{l}\left(1-C_{\infty }\right){\rho }_{f}g{\beta }_f\left(T-T_{\infty }\right)-g\left({\rho }_p-{\rho }_f\right)\\ \left(C-C_{\infty }\right)-{\gamma }_{1}g\left({\rho }_m-{\rho }_p\right)\left(N-N_{\infty }\right)\end{array}\right\}\\ +{\upsilon }_f\left(\frac{{\partial }^2{u}_1}{\partial r^2}+\frac{1}{r}\frac{\partial u_1}{\partial r}+\frac{1}{{\mu }_f\beta c}\left(\frac{{\partial }^2{u}_1}{\partial r^2}+\frac{1}{r}\frac{\partial u_1}{\partial r}\right)-\frac{1}{6{\mu }_f\beta c^3}\left(3\frac{{\partial }^2{u}_1}{\partial r^2}{\left(\frac{\partial u_1}{\partial r}\right)}^2+\frac{1}{r}{\left(\frac{\partial u_1}{\partial r}\right)}^3\right)\right)\\ -\frac{{\sigma }_f{B}_0^2}{{\rho }_f}\left(u_1-u_e\right)-\frac{{\mu }_f}{{\rho }_f{K}_0}{u}_1\end{array}$$ 2

Equation (2) represents the momentum transport equation governing the MHD stagnation point flow of an Eyring-Powell nanofluid over an inclined cylindrical surface embedded in a Darcy porous medium. The term $u_e\frac{d{u}_e}{dx}$ denotes the inviscid acceleration associated with the external stagnation point flow, while the remaining terms represent the combined effects of buoyancy, non-Newtonian viscous stresses, the applied magnetic field, and Darcy porous medium resistance. The buoyancy terms account for thermal, solutal, and microorganism density differences induced by temperature, nanoparticle concentration, and gyrotactic microorganisms. The term $-\frac{{\sigma }_f{B}_0^2}{{\rho }_f}\left(u_1-u_e\right)$ represents the Lorentz force generated by the interaction between the electrically conducting nanofluid and the externally applied uniform magnetic field $B_0$. This resistive force opposes the fluid motion and suppresses the velocity profile within the boundary layer. The Darcy resistance term $-\frac{{\mu }_f}{{\rho }_f{K}_0}{u}_1$ represents the resistance offered by the porous medium due to its finite permeability, thereby reducing the fluid velocity. Furthermore, the nonlinear higher-order derivative terms arise from the Eyring-Powell constitutive model and characterize the non-Newtonian shear-dependent rheological behaviour of the fluid. The buoyancy force acting along the inclined cylindrical surface is proportional to $g\cos \alpha$, where $\alpha$ denotes the inclination angle of the cylinder and $\cos \alpha$ represents the component of gravity acting along the flow direction.

$$\begin{array}{l}{u}_1\frac{\partial T}{\partial x}+v_1\frac{\partial T}{\partial r}=\frac{\alpha }{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)+\tau D_B\left(\frac{\partial C}{\partial r}\frac{\partial T}{\partial r}\right)+\frac{\tau D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial r}\right)}^2\\ -{\lambda }_1\left[{u_1}^2\frac{{\partial }^{2}T}{\partial x^2}+{v_1}^2\frac{{\partial }^{2}T}{\partial r^2}+v_1\frac{\partial v_1}{\partial r}\frac{\partial T}{\partial r}+v_1\frac{\partial u_1}{\partial r}\frac{\partial T}{\partial x}+u_1\frac{\partial v_1}{\partial x}\frac{\partial T}{\partial r}+u_1\frac{\partial u_1}{\partial x}\frac{\partial T}{\partial x}+2u_1{v}_1\frac{{\partial }^{2}T}{\partial r\partial x}\right]\\ -\frac{1}{{\left(\rho C_p\right)}_f}\frac{\partial }{\partial r}\left(r{q}_r\right)+\frac{Q_0}{\rho C_{\rho }}\left(T-T_{\infty }\right)\end{array}$$ 3

Equation (3) governs the thermal energy transport within the MHD Eyring-Powell nanofluid system. The Brownian diffusion term $\tau D_B\left(\frac{\partial C}{\partial r}\frac{\partial T}{\partial r}\right)$ and the thermophoresis term $\frac{\tau D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial r}\right)}^2$ originate from the Buongiorno nanofluid model and represent nanoparticle diffusion caused by random molecular motion and temperature-gradient-induced particle migration, respectively. The terms associated with the thermal relaxation parameter ${\lambda }_1$ represent the Cattaneo-Christov heat flux effect, which accounts for finite thermal propagation speed and addresses the unrealistic assumption of instantaneous heat propagation in Fourier’s law. The radiative heat flux term $-\frac{1}{{\left(\rho C_p\right)}_f}\frac{\partial }{\partial r}\left(r{q}_r\right)$ accounts for thermal radiation effects, while the volumetric heat source term$\frac{Q_0}{\rho C_{\rho }}\left(T-T_{\infty }\right)$ represents internal heat generation for $Q_0>0$ and heat absorption for $Q_{0}0$ within the fluid domain.

$$\begin{array}{l}{u}_1\frac{\partial C}{\partial x}+v_1\frac{\partial C}{\partial r}=\frac{D_B}{r}\left(r\frac{\partial C}{\partial r}\right)+\frac{D_T}{r{T}_{\infty }}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)-K_r^2\left(C-C_{\infty }\right){\left(\frac{T}{T_{\infty }}\right)}^{n_1}\exp \left(-\frac{E_a}{k^{⁎}T}\right)\\ -{\lambda }_2\left[{u_1}^2\frac{{\partial }^{2}C}{\partial x^2}+{v_1}^2\frac{{\partial }^{2}C}{\partial r^2}+v_1\frac{\partial v_1}{\partial r}\frac{\partial C}{\partial r}+v_1\frac{\partial u_1}{\partial r}\frac{\partial C}{\partial x}+u_1\frac{\partial v_1}{\partial x}\frac{\partial C}{\partial r}+u_1\frac{\partial u_1}{\partial x}\frac{\partial C}{\partial x}+2u_1{v}_1\frac{{\partial }^{2}C}{\partial r\partial x}\right]\end{array}$$ 4

Equation (4) governs nanoparticle concentration transport within the MHD Eyring-Powell nanofluid system. The Brownian diffusion term $\frac{D_B}{r}\left(r\frac{\partial C}{\partial r}\right)$ represents nanoparticle diffusion caused by random molecular motion, whereas the thermophoretic diffusion term $\frac{D_T}{r{T}_{\infty }}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)$ models nanoparticle migration induced by temperature gradients. The nonlinear reaction term incorporates Arrhenius activation energy kinetics, whereby increasing the activation energy suppresses reactive species consumption by increasing the energy barrier required for chemical reactions. The terms associated with the concentration relaxation parameter ${\lambda }_2$ incorporate the Cattaneo-Christov mass flux model, which accounts for finite-speed mass diffusion and overcomes the limitations of Fick’s classical diffusion law.

$$u_1\frac{\partial N}{\partial x}+v_1\frac{\partial N}{\partial r}+\frac{b{W}_c}{\left(C_w-C_{\infty }\right)}\frac{\partial }{\partial r}\left(N\frac{\partial C}{\partial r}\right)=\frac{D_m}{r}\frac{\partial }{\partial r}\left(r\frac{\partial N}{\partial r}\right)$$ 5

Equation (5) governs the transport of gyrotactic motile microorganisms within the MHD Eyring-Powell nanofluid system. The microorganism diffusion term $\frac{D_m}{r}\frac{\partial }{\partial r}\left(r\frac{\partial N}{\partial r}\right)$ represents the diffusion of motile microorganisms within the fluid, while the bioconvective transport term $\frac{b{W}_c}{\left(C_w-C_{\infty }\right)}\frac{\partial }{\partial r}\left(N\frac{\partial C}{\partial r}\right)$ accounts for microorganism migration induced by nanoparticle concentration gradients. The inclusion of gyrotactic microorganisms enhances suspension stability and significantly influences the bioconvective transport characteristics of the Eyring-Powell nanofluid system. Such mechanisms are relevant to biomedical transport, microbial reactors, and microscale thermal systems involving microorganism-assisted nanofluids.

The corresponding boundary conditions for the present problem are prescribed as follows [25, 30]:

$$\begin{array}{l}{u}_1=U_w,v_1=0,T=T_w,C=C_w,N=N_{W}atr=R,\\ u_1\to u_e=\frac{U_{\infty }x}{l},T\to T_{\infty },C\to C_{\infty },N\to N_{\infty },atr\to \infty \end{array}$$ 6

To satisfy the continuity Eq. (1) identically, the stream function $\psi$ is introduced such that the axial and radial velocity components are expressed as [17, 31]:

$$u_1=\frac{1}{r}\frac{\partial \psi }{\partial r},v_1=-\frac{1}{r}\frac{\partial \psi }{\partial x}.$$ 7

These relations satisfy the continuity equation identically and facilitate the similarity reduction of the governing partial differential equations.

To transform the governing coupled partial differential Eqs. (1)-(5), together with the boundary conditions (6), into a self-similar system of coupled nonlinear ordinary differential equations, the following similarity transformations are introduced. These transformations are formulated for the cylindrical stagnation point flow configuration and are widely employed in the analysis of non-Newtonian boundary layer transport problems [17, 30].

$$\begin{array}{l}\eta =\frac{r^2-R^2}{2R}{\left(\frac{U_0}{{\upsilon }_{f}l}\right)}^{1/2},\\ \psi ={\left(U_w{\upsilon }_{f}x\right)}^{1/2}Rf\left(\eta \right),\\ \theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\varphi \left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\chi \left(\eta \right)=\frac{N-N_{\infty }}{N_w-N_{\infty }}.\end{array}$$ 8

Reduced similarity equations

$$\begin{array}{l}\left(1+2\gamma \eta \right)\left(1+\epsilon \right)f‴+2\gamma \left(1+\epsilon \right)f″-\frac{4}{3}\epsilon \delta \gamma \left(1+2\gamma \eta \right){\left(f″\right)}^3-\epsilon \delta {\left(1+2\gamma \eta \right)}^2{\left(f″\right)}^2\left(f‴\right)\\ +ff″-{\left(f\prime \right)}^2+A^2+Ri\cos \alpha \left\{\theta -Nr\varphi -Nc\chi \right\}-M\left(f\prime -A\right)-K_{p}f\prime =0\end{array}$$ 9

Equation (9) represents the dimensionless momentum equation governing the reduced boundary layer flow. The dimensionless parameters appearing in this equation characterize the combined effects of non-Newtonian rheology, the applied magnetic field, porous medium resistance, mixed convection, and the external stagnation point flow on momentum transport.

Among the Eyring-Powell fluid parameters, the parameter $\epsilon$ intensifies the nonlinear rheological contribution in the momentum equation, thereby increasing effective viscous resistance and suppressing axial momentum transport within the boundary layer. Physically, larger values of $\epsilon$ strengthen the nonlinear shear-dependent behavior of the Eyring-Powell fluid, which produces greater resistance to fluid motion and weakens axial momentum transport. Consequently, the velocity profile and momentum boundary layer thickness decrease with increasing $\epsilon$. In addition, the mixed convection contribution appears in the reduced momentum equation in the form $Ri\cos \alpha$, where $\cos \alpha$ represents the component of gravity acting along the inclined cylindrical surface.

$$\begin{array}{l}\frac{1}{\mathrm{Pr}}\left(1+\frac{4}{3}Rd\right)\left(\left(1+2\gamma \eta \right)\theta ″+2\gamma \theta \prime \right)+f\theta \prime +\left(1+2\gamma \eta \right)\left(Nb\theta \prime \varphi \prime +Nt{\left(\theta \prime \right)}^2\right)\\ -{\Lambda }_1\left(ff\prime \theta \prime +\theta ″f^2\right)+Q\theta =0\end{array}$$ 10

$$\begin{array}{l}\frac{1}{Sc}\left(\left(1+2\gamma \eta \right)\varphi ″+2\gamma \varphi \prime \right)+f\varphi \prime +\frac{1}{Sc}\frac{Nt}{Nb}\left(\left(1+2\gamma \eta \right)\theta ″+2\gamma \theta \prime \right)\\ -\Gamma \varphi {\left[1+\theta \left({\theta }_w-1\right)\right]}^{n_1}{e}^{\frac{-E}{1+\theta \left({\theta }_w-1\right)}}-{\Lambda }_2\left(ff\prime \varphi \prime +f^2\varphi ″\right)=0\end{array}$$ 11

$$\frac{1}{Lb}\left(\left(1+2\gamma \eta \right)\chi ″+2\gamma \chi \prime \right)+f\chi \prime -\frac{Pe}{Lb}\left(\left(1+2\gamma \eta \right)\varphi \prime \chi \prime +\left[\chi +\Omega \right]\left[\left(1+2\gamma \eta \right)\varphi ″+\gamma \varphi \prime \right]\right)=0.$$ 12

Substituting the similarity transformations given in Eq. (8) into the governing Eqs. (1)-(5) yields the coupled system of nonlinear ordinary differential Eqs. (9)-(12). The transformation converts the spatially dependent transport equations into self-similar equations governing momentum, thermal energy, nanoparticle concentration, and motile microorganism transport.

The corresponding transformed boundary conditions are given by:

$$\begin{array}{l}f\prime =1,f=0,\theta =1,\varphi =1,\chi =1,\mathrm{at}\eta =0,\\ f\prime \to A,\theta \to 0,\varphi \to 0,\chi \to 0,\mathrm{as}\eta \to \infty .\end{array}$$ 13

The dimensionless parameters appearing in Eqs. (9)-(13) represent the relative influences of the governing physical mechanisms, including the applied magnetic field, porous medium resistance, mixed convection, nonlinear thermal radiation, Brownian diffusion, thermophoresis, activation energy, bioconvection, and non-Newtonian rheology, and are defined as follows:

$$\begin{array}{l}A=\frac{U_{\infty }}{U_0},E=\frac{E_a}{k^{⁎}{T}_{\infty }},E=\frac{E_a}{k^{⁎}{T}_{\infty }},K_p=\frac{l{\mu }_f}{{\rho }_f{U}_0{K}_0},Lb=\frac{{\upsilon }_f}{D_m},M=\frac{l{\sigma }_f{B}_0^2}{U_0{\rho }_f},Nb=\frac{\tau D_B\left(C_w-C_{\infty }\right)}{{\upsilon }_f}\\ Nc=\frac{{\gamma }_1\left({\rho }_m-{\rho }_f\right)g{\beta }_f\left(N_w-N_{\infty }\right)}{\left(1-C_{\infty }\right){\rho }_f{\beta }_f\left(T_w-T_{\infty }\right)},Nr=\frac{\left({\rho }_p-{\rho }_f\right)g{\beta }_f\left(C_w-C_{\infty }\right)}{\left(1-C_{\infty }\right){\rho }_f{\beta }_f\left(T_w-T_{\infty }\right)},Nt=\frac{\tau D_T\left(T_w-T_{\infty }\right)}{{\upsilon }_f{T}_{\infty }},Pe=\frac{b{W}_c}{D_m}\\ \mathrm{Pr}=\frac{{\left(\mu c_p\right)}_f}{k_f},Q=\frac{l{Q}_0}{U_0{\rho }_f{C}_{{\rho }_f}},Rd=\frac{4{\sigma }^{⁎}{T}_{\infty }^3}{k^{⁎}{k}_f},Ri=\frac{\left(1-C_{\infty }\right)g{\beta }_f\left(T_w-T_{\infty }\right)}{U_w^2},{\Lambda }_1=\frac{U_0{\lambda }_1}{l},{\Lambda }_2=\frac{U_0{\lambda }_2}{l}\\ Sc=\frac{{\upsilon }_f}{D_B},\Gamma =\frac{l{K}_r^2}{U_0},{\theta }_w=\frac{T_w}{T_{\infty }},\gamma =\frac{1}{R}\sqrt{\frac{{\upsilon }_{f}l}{U_0}},\Omega =\frac{N_{\infty }}{N_w-N_{\infty }},\delta =\frac{U_0{U}_w^2}{2{\upsilon }_{f}l{c}^2}and\epsilon =\frac{1}{{\mu }_f\beta c},\end{array}$$

Engineering quantities of interest

The dimensionless engineering quantities of interest, namely the skin-friction coefficient, local Nusselt number, local Sherwood number, and local motile microorganism density number, are defined as follows:

$$\begin{array}{l}\frac{1}{2}{C}_f{\mathrm{Re}}_x^{1/2}X=\left(1+\epsilon \right)f\prime \prime \left(0\right)-\frac{\epsilon \delta }{3}{\left(f\prime \prime \left(0\right)\right)}^3,\\ Nu{\mathrm{Re}}_x^{-1/2}{X}^{-1}=-\left(1+\frac{4}{3}Rd\right)\theta \prime \left(0\right),\\ S{h}_x{\mathrm{Re}}_x^{-1/2}{X}^{-1}=-\varphi \prime \left(0\right),\\ N{n}_x{\mathrm{Re}}_x^{-1/2}{X}^{-1}=-\chi \prime \left(0\right).\end{array}$$ 14

The corresponding dimensional engineering quantities are defined, respectively, as:$C_f=\frac{2{\tau }_w}{\rho U_w^2},$$N{u}_x=\frac{x{q}_w}{k\left(T_w-T_{\infty }\right)}$ and $S{h}_x=\frac{x{J}_w}{D\left(C_w-C_{\infty }\right)}.$

where the wall shear stress, wall heat flux, and wall mass flux are given by:${\tau }_w=\mu {\left(\frac{\partial u}{\partial r}+\frac{1}{\beta c}\left(\frac{\partial u}{\partial r}\right)-\frac{1}{6\beta c^3}{\left(\frac{\partial u}{\partial r}\right)}^3\right)}_{r=R}$, $q_w=-k\left(1+\frac{16{\sigma }^{⁎}{T}_{\infty }^3}{3k^{⁎}k}\right){\left(\frac{\partial T}{\partial r}\right)}_{r=R}$ and$J_w=-D{\left(\frac{\partial C}{\partial r}\right)}_{r=R}$

The local motile microorganism density is given by:

$N{n}_x=\frac{x{i}_w}{D_n\left(N_w-N_{\infty }\right)},where.i_w=-D_n\left(\frac{\partial N}{\partial r}\right)$  

The similarity variable and the local Reynolds number are defined as:

$$X=\sqrt{\frac{x}{l}},and.{\mathrm{Re}}_x=\frac{U_{w}l}{{\upsilon }_f}$$

Numerical methodology

The transformed coupled nonlinear ordinary differential Eqs. (9)-(12), together with the associated boundary conditions (13), are solved numerically using the MATLAB built-in boundary value solver bvp4c, which is specifically designed for two-point boundary value problems. Suitable initial estimates for $f\left(3\right),f\left(5\right),f\left(7\right)$ are supplied, since these quantities are not specified by the boundary conditions. To examine the possibility of multiple solution branches, several initial guess profiles and computational domain sizes were tested. In every instance, the numerical procedure converged to the same solution branch, indicating the existence of a unique physically realistic solution within the considered parameter ranges. The higher-order ordinary differential equations are first transformed into an equivalent system of coupled first-order ordinary differential equations by introducing suitable auxiliary variables. The resulting first-order system is then solved iteratively using the collocation-based MATLAB bvp4c solver, which employs adaptive mesh refinement to improve numerical accuracy and ensure convergence. During the iterative solution process, the approximate solution is progressively refined until the governing equations and associated boundary conditions are satisfied within the prescribed numerical tolerance. The bvp4c solver is based on the three-stage Lobatto IIIA collocation formula and provides fourth-order accurate solutions with reliable convergence for nonlinear boundary value problems [32].

An appropriate, smooth initial guess that satisfies the prescribed boundary conditions is supplied over the computational domain to initiate the numerical solution process. The semi-infinite physical domain is truncated to a finite computational domain, where the far-field boundary conditions are satisfied asymptotically within the selected domain range.

The accuracy and reliability of the present numerical procedure are validated through comparison with previously published results, as presented in Table 1. An appropriate finite value of the truncated boundary, ${\eta }_{\infty }$, is chosen to ensure that the asymptotic far-field boundary conditions are accurately satisfied [33]. The initial mesh step size and the error tolerance adopted in the present computations are h = 0.001 and 10^-8, respectively.

Table 1.

Comparison of the present skin-friction coefficient with previously published numerical results for different values of the magnetic parameter $M$.

Magnetic parameter $M$	Jakeer et al. [34]	Vajravelu et al. [35]	Present study
0	1.000000	1.000001	0.999998
0.5	1.224745	1.224745	1.224742
1	1.414214	1.414214	1.414216
1.5	1.581139	1.581139	1.581136
2	1.732051	1.732051	1.732050

The following auxiliary variables are introduced to transform the higher-order ordinary differential equations into an equivalent system of coupled first-order ordinary differential equations:

$$\begin{array}{l}f(\eta )=f\left(1\right),f\prime (\eta )=f\left(2\right),f″\left(\eta \right)=f\left(3\right),f‴\left(\eta \right)=f\prime \left(3\right)\\ \theta =f\left(4\right),\theta \prime =f\left(5\right),\theta ″=f\prime \left(5\right)\\ \varphi =f\left(6\right),\varphi \prime =f\left(7\right),\varphi ″=f\prime \left(7\right)\\ \chi =f\left(8\right),\chi \prime =f\left(9\right),\chi ″=f\prime \left(9\right)\end{array}f\prime \left(3\right)=\frac{1}{\left[\begin{array}{l}\left(1+2⁎\gamma ⁎\eta \right)⁎\left(1+\epsilon \right)\\ -\epsilon ⁎\delta ⁎{\left(1+2⁎\gamma ⁎\eta \right)}^2⁎f{\left(3\right)}^2\end{array}\right]}\left[\begin{array}{l}-2⁎\gamma ⁎\left(1+\epsilon \right)⁎f\left(3\right)+\frac{4}{3}⁎\epsilon ⁎\delta ⁎\gamma ⁎\left(1+2⁎\gamma ⁎\eta \right)⁎{\left(f\left(3\right)\right)}^3\\ -f\left(1\right)⁎f\left(3\right)+f{\left(2\right)}^2-A^2\\ -Ri⁎\cos \alpha ⁎\left(f\left(4\right)-Nr⁎f\left(6\right)-Nc⁎f\left(8\right)\right)\\ +M⁎\left(f\left(2\right)-A\right)+K_p⁎f\left(2\right)\end{array}\right]$$ 15

$$f\prime \left(5\right)=\frac{1}{\left[\begin{array}{l}\frac{1}{\mathrm{Pr}}⁎\left(1+\frac{4}{3}⁎Rd\right)⁎\left(1+2⁎\gamma ⁎\eta \right)\\ +\left(1+2⁎\gamma ⁎\eta \right)⁎Nb⁎f\prime \left(7\right)-{\lambda }_1⁎{\left(f\left(1\right)\right)}^2\end{array}\right]}\left[\begin{array}{l}-\frac{1}{\mathrm{Pr}}⁎\left(1+\frac{4}{3}⁎Rd\right)⁎\left(2⁎\gamma ⁎f\left(5\right)\right)-f\left(1\right)⁎f\left(5\right)\\ +{\lambda }_1⁎f\left(1\right)⁎f\left(2\right)⁎f\left(5\right)-Q⁎f\left(4\right)\\ -\left(1+2⁎\gamma ⁎\eta \right)⁎Nt⁎{\left(f\prime \left(5\right)\right)}^2\end{array}\right]$$ 16

$$f\prime \left(7\right)=\frac{1}{\left[\frac{1}{Sc}⁎\left(1+2⁎\gamma ⁎\eta \right)-{\Lambda }_2⁎{\left(f\left(1\right)\right)}^2\right]}\left[\begin{array}{l}\frac{-2⁎\gamma }{Sc}⁎f\left(7\right)-f\left(1⁎f\left(7\right)\right)-\frac{1}{Sc}⁎\frac{Nt}{Nb}\left(\left(1+2⁎\gamma ⁎\eta \right)\right)⁎f\prime \left(5\right)\\ +2⁎\gamma ⁎f\left(5\right)+\Gamma ⁎f\left(6\right){\left[1+f\left(4\right)⁎\left({\theta }_w-1\right)\right]}^{n_1}⁎e^{\frac{E}{1+f\left(4\right)⁎\left({\theta }_w-1\right)}}\\ +{\Lambda }_2⁎f\left(1\right)⁎f\left(2\right)⁎f\left(7\right)\end{array}\right]$$ 17

$$f\prime \left(9\right)=\frac{1}{\left[1+2⁎\gamma ⁎\eta \right]}\left[\begin{array}{l}-2⁎\gamma ⁎f\left(9\right)-Lb⁎f\left(1\right)⁎f\left(9\right)\\ +Pe⁎\left(1+2⁎\gamma ⁎\eta \right)⁎f\left(7\right)⁎f\left(9\right)\left[f\left(8\right)+\Omega \right]⁎\left[\left(1+2⁎\gamma ⁎\eta \right)⁎f^{\left(7\right)}\right]\end{array}\right]$$ 18

The corresponding boundary conditions for the first-order system are given by:

$$\begin{array}{l}{f}_a\left(2\right)=1,f_a\left(1\right)=0,f_a\left(4\right)=1,f_a\left(6\right)=1,f_a\left(8\right)=1\\ f_b\left(2\right)\to A,f_b\left(4\right)\to 0,f_b\left(6\right)\to 0,f_b\left(8\right)\to 0\end{array}$$ 19

These relations complete the first-order system for numerical implementation.

Where $a=0$ and $b=\infty$.

Physical significance and order-of-magnitude analysis

The mathematical model developed in this study is applicable to a range of engineering and biomedical transport processes involving Eyring-Powell nanofluid flow over cylindrical geometries. Representative applications include polymer extrusion, microbial fuel cells, thermal processing systems, geothermal transport devices, and bio-reactive cooling technologies. The physical relevance of the selected dimensionless parameters is demonstrated through order-of-magnitude estimates based on representative thermophysical scales reported in the literature [2, 36].

For representative cylindrical transport systems, characteristic scales such as$R\sim {10}^{-3}-{10}^{-2}m$, stretching and free-stream velocities of $U_w,U_e\sim {10}^{-1}-1m/s$, kinematic viscosity ${\nu }_f\sim {10}^{-6}{m}^2/s$, thermal diffusivity ${\alpha }_f\sim {10}^{-7}{m}^2/s$, nanoparticle diffusivity $D_B\sim {10}^{-11}{m}^2/s$, and microorganism diffusivity $D_n\sim {10}^{-10}{m}^2/s$, the resulting Reynolds number, Peclet number, Brownian motion, and thermophoresis parameters are consistent with the computational ranges used in this study. Furthermore, magnetic field strengths of $B_0\sim 0.1-1T$ and temperature variations of $\Delta T\sim 10-100K$. These characteristic scales yield realistic values of the magnetic and radiation parameters for electrically conducting nanofluid systems.

The selected activation energy parameter is also consistent with Arrhenius-type reactive transport in nanofluid systems. Consequently, the chosen parameter ranges remain physically realistic and suitable for modelling reactive bioconvective Eyring-Powell nanofluid flow over inclined cylindrical surfaces embedded in a Darcy porous medium.

To validate the accuracy of the present numerical methodology, the governing equations are reduced to the limiting case reported by Jakeer et al. [34] and Vajravelu et al. [35] by neglecting the effects of thermal radiation, activation energy, porous medium resistance, Brownian motion, thermophoresis, and bioconvection. The resulting skin-friction coefficient is then compared with the published numerical results, as presented in Table 1. The excellent agreement confirms the accuracy and reliability of the present numerical implementation based on the MATLAB bvp4c solver.

Results and discussion

This section presents and discusses the numerical results for the velocity, thermal, nanoparticle concentration, and bioconvective transport characteristics of Eyring-Powell nanofluid flow over an inclined cylindrical surface embedded in a Darcy porous medium. The influences of the governing dimensionless parameters on the velocity, temperature, nanoparticle concentration, and motile microorganism profiles are analysed through graphical illustrations. Unless otherwise stated, all graphical results are presented for two inclination angles, $\alpha =\frac{\pi }{6}$ and $\alpha =\frac{\pi }{3}$, to examine the influence of cylinder inclination on the transport characteristics. Since the buoyancy force acting along the inclined cylindrical surface is proportional to $\cos \alpha$, increasing the inclination angle reduces the effective buoyancy contribution to the momentum transport. Consequently, noticeable variations in the momentum, thermal, concentration, and microorganism boundary layer characteristics are observed with increasing cylinder inclination.

For clarity, the effects of the governing parameters on the velocity, temperature, nanoparticle concentration, and motile microorganism profiles are discussed sequentially in the following figures.

Figure 2(a-e) illustrates the effects of the velocity ratio parameter $A$, Eyring-Powell parameter $\epsilon$, mixed convection parameter $Ri$, buoyancy ratio parameter $Nr$, and bioconvection Rayleigh number $Nc$ on the velocity profile $f\prime \left(\eta \right)$ for inclination angles $\alpha =\frac{\pi }{6}$ and $\alpha =\frac{\pi }{3}$. As illustrated in Fig. 2(a), increasing the velocity ratio parameter $A$ enhances the fluid velocity and thickens the momentum boundary layer. An increase in the velocity ratio parameter $A$ corresponds to a stronger stretching velocity relative to the external stagnation flow, which accelerates the fluid along the cylinder surface and enhances axial momentum transport. For identical values of $A$, the velocity profiles corresponding to $\alpha =\frac{\pi }{3}$ are slightly higher than those for $\alpha =\frac{\pi }{6}$, indicating that the larger inclination modifies the buoyancy component acting along the cylinder surface and alters the momentum transport characteristics. The velocity profile is suppressed with the enhancement of the Eyring-Powell parameter $\epsilon$, as shown within the boundary layer in Fig. 2(b). The parameter $\epsilon$ enhances the nonlinear rheological effects in the momentum equation, thereby strengthening the nonlinear shear-dependent resistance to fluid motion. Consequently, the fluid experiences greater resistance to deformation, thereby decreasing axial momentum transport and producing a thinner momentum boundary layer. The influence of the Eyring-Powell parameter becomes more pronounced for larger inclination angles because the effective gravitational buoyancy component acting along the inclined cylinder varies with $\cos \alpha$.

Fig. 2.

Effects of a velocity ratio parameter $A$, b eyring-powell parameter $\epsilon$, c mixed convection parameter $Ri$, d buoyancy ratio parameter $Nr$, and, e bioconvection rayleigh number $Nc$ on the velocity profile $f\prime \left(\eta \right)$  

From Fig. 2(c), it is observed that the velocity profile decreases as the mixed convection parameter $Ri$ increases. Increasing the mixed convection parameter $Ri$ intensifies the buoyancy interaction between the thermal, concentration, and microorganism fields. Under the present parameter range, this combined buoyancy effect opposes the primary flow direction, thereby suppressing the velocity profile and producing a thinner momentum boundary layer. Consequently, the momentum boundary layer becomes thinner as the resistance to fluid motion increases. Furthermore, the suppression is more pronounced for $\alpha =\frac{\pi }{3}$, where the effective buoyancy component acting along the inclined cylinder is reduced owing to the $\cos \alpha$ dependence. The velocity profile decreases with increasing buoyancy ratio parameter $Nr$ as demonstrated in Fig. 2(d). Higher levels of $Nr$ strengthen the concentration induced buoyancy force relative to thermal buoyancy effects. As a result, the opposing buoyancy force becomes stronger, offering greater resistance to the primary flow. This concentration induced buoyancy force increases the resistance to the primary flow, thereby reducing axial momentum transport and producing a thinner momentum boundary layer. Figure 2(e) shows that the velocity profile decreases as the bioconvection Rayleigh number $Nc$ increases. Due to the accumulation of gyrotactic microorganisms, the increased microorganism concentration enhances the bioconvective buoyancy force, producing additional resistance to the primary flow. Consequently, the axial momentum transport is weakened, resulting in a thinner momentum boundary layer and a lower velocity profile within the boundary layer. Across all parameter variations, the velocity profiles corresponding to $\alpha =\frac{\pi }{3}$ remain lower than those for $\alpha =\frac{\pi }{6}$, owing to the reduction in the effective buoyancy component acting along the inclined cylinder surface.

The influence of the velocity ratio parameter $A$, curvature parameter $\gamma$, thermal relaxation parameter ${\Lambda }_1$, mixed convection parameter $Ri$, radiation parameter $Rd$, Brownian motion parameter $Nb$ and the thermophoresis parameter $Nt$ on the temperature profile $\theta \left(\eta \right)$ is shown in Fig. 3(a-g). From Fig. 3(a), the temperature profile decreases as the velocity ratio parameter $A$ increases. Physically, the stronger stretching motion augments the fluid flow along the cylindrical surface, thereby improving convective cooling and reducing thermal energy accumulation within the boundary layer. Consequently, a thinner thermal boundary layer is formed due to enhanced heat transfer away from the surface. In Fig. 3(b), the temperature profile increases, and the thermal boundary layer becomes thicker as the curvature parameter $\gamma$ increases. The magnitude of the curvature effects increases with larger values of $\gamma$ and alters radial thermal diffusion, leading to the greater retention of thermal energy in the surface region. This promotes the accumulation of thermal energy within the boundary layer, thereby increasing the temperature profile and producing a thicker thermal boundary layer.

Fig. 3.

Effects of a velocity ratio parameter $A$, b curvature parameter $\gamma$, c thermal relaxation parameter ${\Lambda }_1$, d mixed convection parameter $Ri$, e radiation parameter $Rd$, f Brownian motion parameter $Nb$, and g thermophoresis parameter $Nt$ on the temperature profile $\theta \left(\eta \right)$

The influence of the thermal relaxation parameter ${\Lambda }_1$ on the temperature field is illustrated in Fig. 3(c). It is observed that increasing ${\Lambda }_1$ suppresses the temperature profile within the boundary layer. The Cattaneo-Christov heat flux model accounts for finite-speed thermal propagation through the thermal relaxation mechanism, unlike Fourier’s classical law of instantaneous heat diffusion. Therefore, larger values of ${\Lambda }_1$ retard thermal energy transport and weaken conductive heat diffusion within the fluid, thereby producing a thinner thermal boundary layer. The temperature profile increases, and the thermal boundary layer becomes thicker with increasing mixed convection parameter $Ri$, as shown in Fig. 3(d). Higher values of $Ri$ enhance buoyancy-induced thermal transport, thereby increasing thermal energy diffusion and thickening the thermal boundary layer. Consequently, the temperature profile increases within the boundary layer. As demonstrated in Fig. 3(e), the temperature profile increases, and the thermal boundary layer becomes thicker with increasing radiation parameter $Rd$. Physically, thermal radiation provides extra thermal energy to the fluid layer and enhances the effective thermal conductivity of the nanofluid system. This results in greater absorption and storage of radiative energy within the boundary layer, thereby increasing the temperature profile and thickening the thermal boundary layer.

Figure 3(f) illustrates the effect of the Brownian motion parameter $Nb$ on the temperature profile. As $Nb$ increases, the temperature profile increases, and the thermal boundary layer becomes thicker. On the physical side, Brownian motion enhances the random motion and collisions of the nanoparticles with base-fluid molecules, thereby increasing thermal energy and enhancing microscopic thermal diffusion. Increased nanoparticle activity enhances heat transfer in the fluid, thereby thickening the thermal boundary layer. Also, Fig. 3(g) shows that the temperature profile increases, and the thermal boundary layer becomes thicker with increasing thermophoresis parameter $Nt$. Due to temperature gradients, thermophoretic forces push the nanoparticles away from the heated cylinder surface into the cooler parts of the fluid. The migration of nanoparticles transports thermal energy from the heated cylinder surface into the surrounding fluid, thereby increasing the temperature profile and thickening the thermal boundary layer. A similar inclination-angle behavior is observed in the thermal field, where larger inclination angles yield lower temperature profiles and thinner thermal boundary layers.

Figure 4(a-g) illustrates the effect of the velocity ratio parameter $A$, activation energy parameter $E$, curvature parameter $\gamma$, chemical reaction parameter $\Gamma$, and the thermophoresis parameter $Nt$ on the concentration profile $\varphi \left(\eta \right)$. As shown in Fig. 4(a), the concentration profile decreases as the velocity ratio parameter $A$ increases. The stronger stretching motion accelerates the fluid along the cylindrical surface, thereby enhancing axial momentum transport and convective species transport, which reduces nanoparticle accumulation within the concentration boundary layer. As a result, a thinner concentration boundary layer is formed due to intensified species transport away from the surface. Figure 4(b) shows that increasing the activation energy parameter $E$ enhances the concentration profile and thickens the concentration boundary layer. Activation energy is the minimal energy necessary to initiate a chemical reaction. In the Arrhenius reaction mechanism, increasing the activation energy parameter $E$ raises the energy barrier, thereby reducing the chemical reaction rate and decreasing nanoparticle consumption within the fluid layer. Consequently, higher nanoparticle concentrations are retained inside the concentration boundary layer. The finding highlights the importance of the activation energy in controlling the reactive mass transfer in the Eyring-Powell nanofluid system.

Fig. 4.

Effects of (a) velocity ratio parameter $A$, b activation energy parameter $E$, c curvature parameter $\gamma$, d chemical reaction parameter $\Gamma$, and (e) thermophoresis parameter $Nt$ on the concentration profile $\varphi \left(\eta \right)$

Figure 4(c) illustrates that increasing the curvature parameter $\gamma$ enhances the concentration profile within the boundary layer. Stronger curvature effects associated with the cylindrical geometry modify the radial diffusion of species and promote nanoparticle accumulation near the surface. Consequently, radial species diffusion is enhanced, leading to a thicker concentration boundary layer and a higher concentration profile. Figure 4(d) demonstrates that increasing the chemical reaction parameter $\Gamma$ suppresses the concentration profile and produces a thinner concentration boundary layer. Larger values of $\Gamma$ increase the rate of the destructive chemical reaction, accelerating nanoparticle consumption within the fluid layer. Consequently, the nanoparticle concentration profile decreases owing to enhanced reactive mass transfer. Figure 4(e) shows that the concentration profile increases, and the concentration boundary layer becomes thicker with increasing thermophoresis parameter $Nt$. Due to temperature gradients, thermophoretic forces transport nanoparticles from the heated cylinder surface toward the relatively cooler regions of the ambient fluid. The migration increases the accumulation of nanoparticles within the concentration boundary layer, thereby thickening the concentration boundary layer and enhancing the concentration profile. A similar inclination-angle behavior is observed in the concentration field, where larger inclination angles yield lower concentration profiles and thinner concentration boundary layers.

The influence of the velocity ratio parameter A , curvature parameter γ , microorganism parameter \varOmega , and bioconvection Peclet number Pe on the motile microorganism distribution is X (n)illustrated in Figure 5(a-d). Figure 5(a) shows that the motile microorganism profile decreases and the microorganism boundary layer becomes thinner with increasing velocity ratio parameter A. The stronger stretching motion promotes the axial flow of fluid particles along the cylindrical surface, thereby enhancing the convective transport of microorganisms away from the near-wall region. This results in reduced accumulation of motile microorganisms within the boundary layer, thereby decreasing the microorganism profile and thinning the microorganism boundary layer. Figure 5(b) illustrates that the motile microorganism profile increases, and the microorganism boundary layer becomes thicker with increasing curvature parameter . Higher values of increase the geometrical curvature effects of the cylindrical surface, which alter the radial transport characteristics of motile microorganisms. Increasing curvature causes microorganisms to accumulate toward the surface, thereby producing a thicker microorganism boundary layer.

Fig. 5.

Effects of (a) velocity ratio parameter A, b curvature parameter γ , c microorganism parameter Ω, and (d) bioconvection Peclet number Pe on the motile microorganism distribution x(η)

Figure 5(c) illustrates the influence of the microorganism parameter $\Omega$ on the motile microorganism profile. Increasing $\Omega$ suppresses the motile microorganism profile. Higher values of $\Omega$ correspond to a stronger ambient microorganism concentration relative to the wall microorganism concentration difference, which reduces the effective microorganism concentration gradient responsible for bioconvective transport within the boundary layer. This leads to reduced migration of motile microorganisms within the fluid region, thereby weakening bioconvective transport in the boundary layer. Figure 5(d) shows that the motile microorganism profile decreases, and the microorganism boundary layer becomes thinner with increasing bioconvection Peclet number $Pe$. The bioconvection Peclet number represents the ratio of advective to diffusive transport of motile microorganisms. Larger values of $Pe$ indicate that convective transport dominates over microorganism diffusion, thereby restricting the random spreading of motile microorganisms within the boundary layer. Consequently, the motile microorganism concentration near the surface decreases, leading to a thinner microorganism boundary layer. The slight overshoot behavior observed at larger $Pe$ values are attributed to the strong interaction between advective bioconvective transport and motile microorganism diffusion within the fluid layer. A similar inclination-angle behavior is observed in the motile microorganism field, where larger inclination angles yield lower motile microorganism profiles and thinner microorganism boundary layers.

Figure 6(a-c) presents the contour distributions of the skin-friction coefficient under the combined influence of selected governing parameters. The contour plots provide a clear visualization of the interactive effects of these parameters on the wall shear stress characteristics of the Eyring-Powell nanofluid system. The combined influence of the velocity ratio parameter and the mixed convection parameter on the skin-friction coefficient is shown in Figure 6(a). The magnitude of the skin-friction coefficient increases with increasing velocity ratio parameter . A stronger stretching motion enhances axial momentum transport and increases the velocity gradient at the cylindrical surface, resulting in higher wall shear stress. On the other hand, the skin-friction coefficient decreases slightly with increasing mixed-convection parameter due to the opposing buoyancy contribution within the momentum boundary layer, which weakens the near-wall momentum gradient. Figure 6(b) depicts the combined effects of the velocity ratio parameter and the Eyring-Powell parameter on the skin-friction coefficient. A higher means a higher wall shear stress because more momentum is transported due to stretching. However, an increase in the Eyring-Powell parameter decreases the magnitude of the skin-friction coefficient. The parameter improves the nonlinear rheological behavior of the Eyring-Powell fluid and increases the fluid''s effective resistance to deformation. Consequently, axial momentum transport within the momentum boundary layer is weakened, leading to a smaller velocity gradient at the wall and, hence, a lower skin-friction coefficient.

Fig. 6.

Contour plots of the skin-friction coefficient for the combined effects of (a) the velocity ratio parameter A and mixed convection parameter Ri , (b) velocity ratio parameter A and the Eyring-Powell parameter ε, and (c) chemical reaction parameter Γ and activation energy parameter E .

The effect of the chemical reaction parameter $\Gamma$ and the activation energy parameter $E$ on the skin-friction coefficient is illustrated in Fig. 6(c). An increase in the chemical reaction parameter $\Gamma$ slightly increases the magnitude of the skin-friction coefficient, as higher reactive transport effects modify the near-wall momentum distribution. The magnitude of the skin-friction coefficient decreases with increasing activation energy parameter $E$. Increasing the activation energy parameter $E$ raises the energy barrier for chemical reactions, thereby reducing the reaction rate and weakening the coupled reactive transport mechanism within the boundary layer. Consequently, the axial momentum transport near the wall is slightly reduced, leading to a lower skin-friction coefficient.

The contour plots of local Nusselt number are presented in figure 7(a-d) for the influence of different thermophysical properties. The contour plots provide a clear visualization of the coupled thermal transport behavior within the Eyring-Powell nanofluid boundary layer. Figure 7(a) depicts the combined effect of the radiation parameter Rd and heat generation/absorption parameter Q on the local Nusselt number. The local Nusselt number increases with the radiation parameter Rd and decreases with the heat generation/absorption parameter Q . The higher values of Rd enhance the radiative heat transfer in the fluid layer, leading to increased thermal energy exchange between the cylinder surface and the surrounding fluid. The enhanced thermal radiation promotes thermal energy transport within the boundary layer, thereby increasing the wall temperature gradient and enhancing the local heat transfer rate. On the other hand, Increasing the heat generation parameter Q supplies additional thermal energy within the boundary layer, elevating the fluid temperature and reducing the temperature difference between the heated cylinder surface and the surrounding fluid. Consequently, the wall temperature gradient decreases, leading to a reduction in the local Nusselt number. Figure 7(b) displays the impact of the radiation parameter and thermophoresis parameter Rd on the local Nusselt number. Higher Rd somewhat increases the heat transfer rate due to the enhanced thermal radiation transport. On the other hand, larger Nt decreases the local Nusselt number. The thermophoretic force drives nanoparticles from the heated surface toward the cooler fluid region, which enhances thermal energy transport away from the wall and thickens the thermal boundary layer. Consequently, the wall temperature gradient decreases, leading to a reduction in the heat transfer rate from the cylindrical surface.

Fig. 7.

Contour plots of the local Nusselt number for the combined effects of (a) radiation parameter Rd and heat generation/absorption parameter Q, b radiation parameter Rd and Thermophoresis parameter Nt, (c) velocity ratio parameter A and Eyring-Powell parameter ε, and (d) chemical reaction parameter Γ and activation energy parameter E.

Figure 7(c) displays the combined influences of the velocity ratio parameter A and the Eyring-Powell parameter ε on the local Nusselt number. The local Nusselt number increases with the increasing velocity ratio parameter A due to the stronger stretching motion which enhances the fluid motion along the cylindrical surface, producing a thinner thermal boundary layer and, consequently, a larger wall temperature gradient. However, the local Nusselt number decreases with an increase in the Eyring-Powell parameter ε . The larger values of increase the nonlinear rheological resistance of the Eyring-Powell fluid, which reduces axial momentum transport and convective heat transfer within the boundary layer. Consequently, convective heat transport is weakened relative to thermal diffusion, causing the thermal boundary layer to thicken, reducing the wall temperature gradient and lowering the local Nusselt number. Figure 7(d) illustrates the interactive effects of the chemical reaction parameter \varGamma and the activation energy parameter E on the local Nusselt number. Increasing the chemical reaction parameter \varGamma intensifies reactive species consumption, thereby altering the coupled heat and mass transfer processes within the boundary layer. Consequently, the coupled reactive transport weakens the wall temperature gradient, resulting in a decrease in the local Nusselt number. On the contrary, the local Nusselt number is marginally enhanced with the increase of the activation energy parameter E. Activation energy is the minimal energy necessary to start chemical reactions in the nanofluid system. As the reaction rate decreases, less thermal energy is absorbed through chemical reactions, preserving a steeper wall temperature gradient. Consequently, the local heat-transfer rate increases slightly, resulting in a marginal enhancement of the local Nusselt number.

The contour plot of the local Sherwood number $S{h}_x{\mathrm{Re}}_x^{-1/2}{X}^{-1}$ for the combined effects of the chemical reaction parameter $\Gamma$ and activation energy parameter $E$ is presented in Fig. 8. The contour plot clearly visualizes the coupled influences of reactive transport and activation energy on the mass transfer characteristics of the Eyring-Powell nanofluid system. The local Sherwood number increases with the chemical reaction parameter $\Gamma$. As the value of $\Gamma$ increases, the rate of the destructive chemical reaction in the concentration boundary layer increases, thereby increasing the rate of nanoparticle consumption and increasing the wall concentration gradient. This increases the mass transfer rate at the wall and gives larger Sherwood number values. On the other hand, an increase in the activation energy parameter $E$ leads to a decrease in the local Sherwood number. Activation energy represents the minimum energy barrier required for the Arrhenius chemical reaction to proceed. Higher values of $E$ raise the reaction energy barrier, suppressing the Arrhenius reaction rate and reducing nanoparticle consumption within the concentration boundary layer. Consequently, nanoparticle consumption decreases, resulting in a weaker wall concentration gradient and a lower mass transfer rate from the cylindrical surface. The contour distribution further indicates that the effect of the chemical reaction parameter is more pronounced at lower levels of activation energy, where reactive transport mechanisms dominate. As a result, weaker reactive mass transport reduces the wall concentration gradient, leading to lower mass transfer rates and smaller Sherwood number values throughout the computational domain.

Fig. 8.

Contour plot of the local Sherwood number $S{h}_x{\mathrm{Re}}_x^{-1/2}{X}^{-1}$ for the combined effects of the chemical reaction parameter $\Gamma$ and activation energy parameter $E$

Figure 9 presents the contour plot of the motile microorganism density number $N{n}_x{\mathrm{Re}}_x^{-1/2}{X}^{-1}$ for the combined effects of the chemical reaction parameter $\Gamma$ sand activation energy parameter $E$. The contour plot provides a clear visualization of the coupled influences of reactive transport and activation energy on the motile microorganism transport characteristics in the Eyring-Powell nanofluid system. The motile microorganism density number increases with the chemical reaction parameter $\Gamma$. Stronger chemical reactions modify the species concentration field, thereby enhancing the coupled bioconvective transport of motile microorganisms within the boundary layer. Consequently, the wall microorganism concentration gradient becomes steeper, leading to higher microorganism mass transfer rates and larger motile microorganism density numbers. On the other hand, increasing the activation energy parameter $E$ reduces the motile microorganism density number. Activation energy represents the minimum energy barrier required for the Arrhenius chemical reaction to proceed. Higher values of $E$ raise the reaction energy barrier, suppressing the Arrhenius reaction rate and weakening the coupled species transport responsible for bioconvection. Consequently, the wall microorganism concentration gradient decreases, resulting in lower microorganism mass transfer rates and smaller motile microorganism density numbers. The contour distribution further indicates that the influence of the chemical reaction parameter $\Gamma$ is more pronounced at lower activation energy levels, where reactive transport remains dominant. As the activation energy increases, chemical reactions are progressively suppressed, weakening the coupled bioconvective transport of microorganisms and reducing the motile microorganism density number throughout the computational domain.

Fig. 9.

Contour plot of the motile microorganism density number $N{n}_x{\mathrm{Re}}_x^{-1/2}{X}^{-1}$ for the combined effects of the chemical reaction parameter $\Gamma$ and activation energy parameter $E$

The present results demonstrate that the transport characteristics of the Eyring-Powell nanofluid are strongly influenced by the combined effects of non-Newtonian rheology, nonlinear thermal radiation, activation energy, gyrotactic microorganisms, Darcy porous medium resistance, and the dual Cattaneo-Christov heat and mass flux models. The velocity ratio parameter enhances axial momentum transport, whereas the Eyring-Powell parameter suppresses the velocity profile by increasing the effective rheological resistance within the momentum boundary layer. Thermal radiation, Brownian motion, thermophoresis, and curvature significantly modify the thermal transport characteristics by altering the temperature profile and thermal boundary layer thickness. Species transport is predominantly governed by the activation energy and chemical reaction parameters, which regulate nanoparticle consumption and concentration boundary layer thickness. Likewise, motile microorganism transport is strongly influenced by the bioconvection Peclet number and microorganism parameter, which control microorganism migration and the associated bioconvective boundary layer. Furthermore, increasing the cylinder inclination weakens buoyancy-assisted transport because the gravitational component proportional to $\cos \alpha$ decreases. The contour analyses further reveal the coupled influences of the governing parameters on the skin-friction coefficient, heat transfer rate, mass transfer rate, and motile microorganism transport, thereby providing a comprehensive understanding of the interactive transport mechanisms in the present Eyring-Powell nanofluid system.

Conclusions

The present study investigated the stagnation point flow of an Eyring-Powell nanofluid over an inclined cylindrical surface embedded in a Darcy porous medium under the combined influences of nonlinear thermal radiation, Arrhenius activation energy, gyrotactic microorganisms, and dual Cattaneo-Christov heat and mass fluxes. The resulting nonlinear similarity equations were transformed into a system of ordinary differential equations and solved numerically using the MATLAB bvp4c solver. The present investigation provides a comprehensive numerical assessment of the coupled effects of non-Newtonian rheology, porous medium resistance, nonlinear thermal radiation, activation energy, bioconvection, and non-Fourier heat and mass transport on the momentum, thermal, concentration, and microorganism fields over an inclined cylindrical surface. The principal findings are summarised as follows:

• An increase in the velocity ratio parameter enhances axial momentum transport and thickens the momentum boundary layer, whereas the Eyring-Powell parameter suppresses the velocity profile by increasing the effective nonlinear rheological resistance within the momentum boundary layer.

• Larger values of the mixed convection parameter, buoyancy ratio parameter, and bioconvection Rayleigh number reduce the velocity profile because the associated buoyancy forces oppose the primary external stagnation flow along the inclined cylindrical surface.

• Increasing the curvature, thermal radiation, Brownian motion, and thermophoresis parameters enhance the temperature profile and thicken the thermal boundary layer, whereas the thermal relaxation parameter suppresses thermal transport owing to the finite-speed heat propagation described by the dual Cattaneo-Christov heat flux model.

• The activation energy parameter weakens the Arrhenius reaction rate, thereby preserving nanoparticle species concentration within the concentration boundary layer and enhancing the concentration profile, whereas the chemical reaction parameter accelerates nanoparticle consumption and consequently reduces the concentration distribution.

• The motile microorganism density decreases with increasing bioconvection Peclet number and microorganism parameter because advective transport becomes dominant over microorganism diffusion, leading to thinner microorganism boundary layers.

• Increasing the inclination angle weakens buoyancy assisted transport because the effective gravitational component proportional to $\cos \alpha$ decreases, resulting in reductions in the momentum, thermal, concentration, and microorganism boundary layer thicknesses.

• The contour analyses revealed the coupled influences of the governing parameters on the skin-friction coefficient, local Nusselt number, local Sherwood number, and motile microorganism density number, thereby providing a comprehensive understanding of the interactive momentum, heat, mass, and bioconvective transport mechanisms in the Eyring-Powell nanofluid system.

• Comparison of the present limiting-case results with previously published studies showed excellent agreement, confirming the accuracy and reliability of the proposed numerical procedure.

Overall, the present numerical results demonstrate that the coupled interaction among non-Newtonian rheology, porous medium resistance, nonlinear thermal radiation, activation energy, bioconvection, and non-Fourier heat and mass transport governs the momentum, heat, mass, and microorganism transport characteristics over the inclined porous cylindrical surface.

Potential applications

The proposed Eyring-Powell bioconvective nanofluid model is applicable to a wide range of advanced thermal and reactive transport systems involving non-Newtonian fluids, porous structures, and chemically reacting flows. The combined influences of Darcy porous medium resistance, nonlinear thermal radiation, Arrhenius activation energy, dual Cattaneo-Christov heat and mass fluxes, and gyrotactic microorganisms make the present model relevant to microbial fuel cells, catalytic chemical reactors, geothermal energy extraction, polymer-processing operations, porous heat exchangers, solar thermal energy systems, and advanced thermal management technologies employing nanofluids in cylindrical configurations.

Furthermore, the inclusion of gyrotactic microorganisms and activation energy effects provides a useful theoretical basis for designing advanced bioconvective transport systems encountered in biomedical engineering, bio-microfluidic devices, tissue engineering, targeted drug delivery, biosensors, and microorganism assisted thermal management technologies. The present findings provide useful guidance for the design and optimization of porous cylindrical systems requiring enhanced heat and mass transfer under non-Newtonian nanofluid flow conditions, while also serving as valuable benchmark solutions for future numerical, experimental, and machine-learning-assisted investigations.

Limitations and future scope

The present study is limited to a steady, laminar, two-dimensional boundary layer flow of an Eyring-Powell nanofluid over an inclined porous cylinder under the boundary layer approximations. The thermophysical properties of the nanofluid are assumed to remain constant, while nanoparticle aggregation, sedimentation, variable viscosity, temperature-dependent thermal conductivity, and the induced magnetic field are neglected. Furthermore, the analysis is restricted to the Darcy porous medium model and is limited to numerical simulations without experimental validation. These assumptions simplify the mathematical formulation while preserving the essential transport mechanisms governing the present problem.

The present framework can be further extended by considering the unsteady and three-dimensional transport phenomena, variable thermophysical properties, Joule heating, viscous dissipation, velocity and thermal slip conditions, entropy generation, and temperature-dependent thermal conductivity. Additional developments may also include ternary and multi-component hybrid nanofluids, alternative non-Newtonian constitutive models, non-uniform magnetic fields, and more realistic porous media. Experimental validation, stability analysis, uncertainty quantification, and machine-learning-assisted optimization may further improve the predictive capability of the model. Such extensions would provide a more comprehensive understanding of coupled momentum, heat, mass, and bioconvective transport processes in porous cylindrical systems relevant to advanced thermal and biomedical engineering applications.

Nomenclature

$A$

Velocity ratio parameter

$B_0$

Applied uniform magnetic field strength (T)

$C$

Nanoparticle concentration (kg m^-3)

$C_f$

Skin-friction coefficient

$C_p$

Specific heat at constant pressure (J kg^-1 K^-1)

$C_w$

Nanoparticle concentration at the cylinder surface (kg m^-3)

$C_{\infty }$

Ambient nanoparticle concentration (kg m^-3)

$D_B$

Brownian diffusion coefficient (m² s^-1)

$D_T$

Thermophoretic diffusion coefficient (m² s^-1)

$D_n$

Microorganism diffusivity (m² s^-1)

$E_a$

Activation energy (J mol^-1)

$f(\eta )$

Dimensionless stream function

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

Dimensionless velocity profile

$g$

Gravitational acceleration (m s^-2)

$i_w$

Motile microorganism flux at the cylinder surface (m^-2 s^-1)

$J_w$

Mass flux at the cylinder surface (kg m^-2 s^-1)

$k$

Thermal conductivity (W m^-1 K^-1)

$k^{⁎}$

Mean absorption coefficient (m^-1)

$K_0$

Permeability of the porous medium (m²)

$K_r$

Chemical reaction rate constant (s^-1/2)

$l$

Characteristic length (m)

$N$

Density of motile microorganisms (m^-3)

$N{n}_x$

Local motile microorganism density number

$N{u}_x$

Local Nusselt number

$N_w$

Motile microorganism density at the cylinder surface (m^-3)

$N_{\infty }$

Ambient motile microorganism density (m^-3)

$n_1$

Reaction order

$Q_0$

Volumetric heat generation/absorption coefficient (W m^-3)

$q_r$

Radiative heat flux (W m^-2)

$q_w$

Heat flux at the wall (W m^-2)

$r$

Radial coordinate normal to the cylindrical surface (m)

$R$

Radius of the cylindrical surface (m)

${\mathrm{Re}}_x$

Local Reynolds number

$S{h}_x$

Local Sherwood number

$T$

Temperature of the Eyring-Powell nanofluid (K)

$T_w$

Temperature at the surface of the cylinder (K)

$T_{\infty }$

Ambient temperature (K)

$u_1$

Axial velocity component (m s^-1)

$u_e$

External stagnation point velocity (m s^-1)

$U_{\infty }$

Free-stream stagnation velocity constant (m s^-1)

$U_w$

Stretching velocity (m s^-1)

$U_0$

Characteristic stagnation point velocity (m s^-1)

$v_1$

Radial velocity component (m s^-1)

$W_c$

Maximum swimming speed of motile microorganisms (m s^-1)

$x$

Axial coordinate along the cylindrical surface (m)

$\alpha$

Inclination angle (rad)

${\beta }_f$

Thermal expansion coefficient of base fluid (K^-1)

${\beta }_p$

Solutal expansion coefficient (m³ kg^-1)

${\beta }_c$

Eyring-Powell material parameter (Pa s)

$\chi \left(\eta \right)$

Dimensionless microorganism density profile

$\delta ,\epsilon$

Dimensionless Eyring-Powell model parameters

$\eta$

Similarity variable

$\gamma$

Curvature parameter

${\gamma }_1$

Microorganism density difference parameter (m³ kg^-1)

$\theta \left(\eta \right)$

Dimensionless temperature profile

${\lambda }_1$

Thermal relaxation parameter (s)

${\lambda }_2$

Concentration relaxation parameter (s)

${\mu }_f$

Dynamic viscosity of the base fluid (Pa s)

$v_f$

Kinematic viscosity of the base fluid (m² s^-1)

${\rho }_f$

Density of base fluid (kg m^-3)

${\rho }_p$

Density of nanoparticles (kg m^-3)

${\rho }_m$

Density of motile microorganisms (kg m^-3)

$\sigma$

Electrical conductivity (S m^-1)

${\sigma }^{⁎}$

Stefan-Boltzmann constant (W m^-2 K^-4)

$\tau$

Ratio of nanoparticle heat capacity to the heat capacity of the base fluid

$\varphi \left(\eta \right)$

Dimensionless concentration profile

$\psi$

Stream function (m² s^-1)

$\Omega$

Microorganism parameter

$E$

Dimensionless activation energy parameter

$K_p$

Permeability parameter

$Lb$

Bioconvection Lewis number

$M$

Magnetic parameter

$Nb$

Brownian motion parameter

$Nc$

Bioconvection Rayleigh number

$Nr$

Buoyancy ratio parameter

$Nt$

Thermophoresis parameter

$Pe$

Bioconvection Peclet number

$\mathrm{Pr}$

Prandtl number

$Q$

Heat generation/absorption parameter

$Rd$

Radiation parameter

$Ri$

Mixed convection parameter

${\Lambda }_1$

Thermal relaxation parameter

${\Lambda }_2$

Concentration relaxation parameter

$Sc$

Schmidt number

$\Gamma$

Chemical reaction parameter

${\theta }_w$

Temperature ratio parameter

$f$

Fluid

$m$

Microorganism

$p$

Nanoparticle phase

$w$

Wall

$\infty$

Ambient (free-stream) condition

$\text{'}\text{'}\text{'}\text{'}\text{'}\text{'}\text{'}\text{'}$

First derivative with respect to $\eta$

$"$

Second derivative with respect to $\eta$

$\partial$

Partial derivative operator

MHD

Magnetohydrodynamics

ODE

Ordinary Differential Equations

PDE

Partial Differential Equations

Author contributions

P. Jayasri: Conceptualization, Investigation, Writing-Original Draft, Validation, Visualization. P. Durgaprasad: Validation, Writing-Review & Editing, Supervision.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Data availability

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

Declarations

Ethical approval

Not applicable. This study did not involve human participants or animals.

Consent to participate

Not applicable.

Consent to publish

Not applicable.

Competing interests

The authors declare no competing interests.