Sensitivity analysis of microbial characteristics of trihybrid carreau nanofluid using response surface methodology and ANOVA

Abstract

This study considers the sensitivity analysis of a trihybrid Carreau nanofluid by examining the transport and distribution of motile microorganisms within a suspension composed of titanium oxide, copper, and silver nanoparticles dispersed in sodium alginate as the base fluid. Sodium alginate is selected due to its shear-thinning rheological behavior, which is well represented by the Carreau fluid model. The nanofluid–microorganism mixture is assumed to flow over a nonlinearly stretched Riga surface under time-dependent stagnation-point conditions. The proposed model considers two-dimensional laminar flow in a porous medium, incorporating the effects of Newtonian heating, Stefan blowing, nonlinear thermal radiation, viscous dissipation, and chemical reactions. The governing partial differential equations (PDEs), derived from the fundamental conservation laws of mass, momentum, energy, nanoparticle concentration, and microorganism density, are transformed into a system of coupled nonlinear ordinary differential equations (ODEs) through appropriate similarity transformations. These equations are solved numerically using MATLAB’s bvp4c algorithm. Furthermore, Response Surface Methodology (RSM) based on a Central Composite Design (CCD) is employed to optimize the velocity, thermal, solutal, and microbial transport characteristics. The combined effects of nanoparticle volume fractions (0.01–0.05) and unsteady parameters (0.1–0.6) on the Nusselt number, Sherwood number, and motile microorganism density factor are systematically examined. A sensitivity analysis is also conducted to assess the influence of nanoparticle loading on the microbial density response, highlighting the robustness and stability of the proposed microbial nanofluid system. The trihybrid nanofluid shows significantly improved thermal performance 24% and microbe’s acclivity reduction 18% as compared to the conventional fluid. The statistical significance of the input factors is confirmed by an ANOVA (ANalysis Of Variance) with a model value of more than 99%. Findings indicate that silver nanoparticles improve velocity, temperature, concentration and microbial gradients with increased concentration of the silver nanoparticles because of the high thermal conductivity and bioactivity but titanium oxide has a high microbial impact. The trihybrid nanofluid displays much better thermal performance and microbial activity in comparison with conventional nanofluids and optimized configurations of nanofluids obtained with the help of RSM.

Introduction

In view of their wide-ranging applications, non-Newtonian liquids have attracted considerable attention from researchers due to their abundant occurrence in both natural and industrial systems. Such fluids play a vital role in the mining sector, where slurries and drilling muds are routinely handled, as well as in lubrication processes and biomedical flows. As a result, the accurate simulation of non-Newtonian flow behavior is of great practical importance for industrial design and optimization. Although substantial progress has been made in this field, further investigation is still required across different non-Newtonian fluid models to fully capture the diversity of their rheological characteristics. The power-law model, owing to its mathematical simplicity, has been widely employed to examine the fundamental effects of non-Newtonian behavior. However, this simplicity is accompanied by inherent limitations, particularly in its inability to represent fluid behavior at very low and very high shear rates. To address these shortcomings, researchers often adopt more comprehensive viscosity models, such as the Carreau model, which provides a more realistic description of fluid behavior across a broader range of shear conditions. Carreau liquid model is non-Newtonian and combines both the Newtonian and power-law models. The model is useful in the characterization of phenomena that are associated with shear thinning and shear thickening. Carreau [1] introduced a molecular network–based viscosity model that explains how non-Newtonian fluids smoothly transition between low- and high-shear-rate behaviors. Carreau et al. [2] applied this framework to polymeric solutions, demonstrating the model’s ability to realistically capture viscous responses under varying flow conditions. Yang et al. [3] investigated unsteady stagnation-point Carreau fluid flow over a permeable stretching/shrinking surface, revealing the role of non-uniform heat generation in producing multiple solution branches. Hou et al. [4] presented a comparative theoretical and numerical analysis of non-isothermal roll coating, highlighting the improved predictive performance of the Carreau model in industrial shear flows. Shabbir et al. [5] examined Carreau fluid flow over a linearly curved stretching sheet and showed how surface curvature significantly influences momentum and heat transfer characteristics. Shah et al. [6] analyzed MHD Carreau fluid slip flow over a porous stretching sheet, emphasizing the effects of viscous dissipation and variable thermal conductivity on transport behavior. Khan et al. [7] investigated the impact of variable surface thickness on Carreau fluid flow, demonstrating its strong influence on velocity and temperature distributions. Abbas et al. [8] explored MHD Carreau fluid flow with variable viscosity and thermal conductivity over a permeable surface, showing how magnetic fields can be used to control heat and flow resistance. Jawad et al. [9] studied magnetohydrodynamic Carreau-type fluid flow over a curved, exponentially stretching surface, highlighting the combined effects of curvature and magnetic forces on transport processes.

Nanofluids are engineered suspensions in which nanoscale solid particles are uniformly dispersed within a conventional base liquid to enhance its thermal, momentum, mass transfer, and sometimes biological or chemical properties. When only one type of nanoparticle is added, the system is referred to as a mono nanofluid and is typically designed to improve a specific property such as heat transfer or antimicrobial performance. The combination of two different nanoparticles forms a hybrid nanofluid, allowing complementary material characteristics to work together for broader functional enhancement. Extending this concept, a trihybrid nanofluid incorporates three distinct nanoparticles to achieve synergistic improvements in thermal efficiency, flow regulation, and chemical or biological activity. In this context, silver, copper, and titanium oxide nanoparticles are dispersed in sodium alginate as the base fluid, creating a trihybrid nanofluid that benefits from silver’s antimicrobial capability, copper’s high thermal conductivity, and titanium oxide’s chemical stability and photocatalytic behavior. The sodium alginate base fluid provides biocompatibility and viscoelastic features, making the mixture relevant for biomedical, environmental, and industrial coating applications. This suspension is modeled using the Carreau fluid framework, which captures shear-thinning behavior when the flow behavior index n<1, meaning the apparent viscosity decreases as the shear rate increases. Such a representation is particularly suitable for applications involving varying shear conditions, including microfluidic systems, thermal management of electronic devices, drug delivery platforms, antimicrobial surface treatments, and advanced manufacturing processes, where both enhanced transport properties and realistic non-Newtonian flow behavior are essential for accurate analysis and design. Abbas [10] examined bio-marangoni convection in a Casson trihybrid nanofluid with chemical reaction effects, highlighting how surface-tension gradients influence coupled flow, heat, and mass transport in biologically active systems. Farkhad et al. [11] analyzed thermo-bioconvection of a water-based trihybrid nanofluid with gyrotactic microorganisms, showing how Soret and Dufour mechanisms modify thermal and concentration distributions. This study [12] explored the combined tribological and thermal–solutal behavior of a ternary hybrid nanofluid with oxytactic and gyrotactic microorganisms, emphasizing the role of local thermal non-equilibrium in multiphase transport processes. Abbas et al. [13] applied neural-network–based computational intelligence to investigate thermo-bioconvection and tribological performance of trihybrid nanofluids containing oxytactic microorganisms. Abbas et al. [14] studied Darcy–Forchheimer flow of an Ellis trihybrid nanofluid with non-Fourier heat and mass flux models, highlighting the impact of Stefan blowing on thermal and solutal transport. Okasha et al. [15] presented a semi-analytical analysis of a sodium alginate–based ternary hybrid nanoliquid, demonstrating the influence of radiative heat transfer and Christov–Cattaneo flux under Stefan blowing conditions. Saleem et al. [16] investigated thermal transport in ternary Casson nanofluid flow over a stretching sheet, focusing on the effects of Thomson and Troian velocity slip on boundary-layer behavior. Saleem et al. [17] conducted a symmetry-based analysis of three-dimensional electrical magnetohydrodynamic (EMHD) nano-Carreau fluid flow over a Riga-stretched surface, revealing how electromagnetic and multi-physical interactions shape mixed convection and heat transfer.

In recent years, researchers have advanced the concept of ternary hybrid nanofluids, in which three distinct types of nanoparticles are dispersed within a base liquid. This development stems from the ever-rising industrial demand for coolant media that can combine high thermal efficacy with operational viability on a scale. By evolving from traditional binary nanofluids to trihybrid systems, investigators aim to push beyond existing thermal performance limits, seeking liquids whose enhanced conductivity and stability make them more attractive for demanding thermal management applications. Additional experimental studies have been carried out to further enhance the thermal characteristics of an existing hybrid nanofluids by incorporating three distinct types of solid nanoparticles into pure liquid, leading in the development of the ternary hybrid nanofluids. Benkhedda et al. [18] investigated heat transfer through hybrid nanofluids, considering various shapes of nanoparticles within a water-based liquid. Sarkar et al. [19] examined the thermal transport properties and concluded that hybrid nanofluids exhibited superior thermal performance and pressure-drop characteristics compared to unitary Nanofluids. Wakif et al. [20] analyzed the effects of thermal radiation and surface roughness by applying alumina–copper oxide hybrid nanofluids, utilizing the generalized Buongiorno’s nanofluid model. Sang et al. [21] made use of CNTs and carbonate nanoparticles for enhancing the thermal conductivity and specific heat in solar power systems. Mousavi et al. [22] studied the behavior of a tri- hybrid nanofluids by dispersing magnesium oxide, copper oxide, and titanium oxide in water. Sahoo and Kumar et al. [23] aimed to derive a new correlation to enhance the viscosity of tri- hybrid nanofluids. Their conclusion showed that the dynamic viscosity enhanced and reduced with an increase in volume fraction and temperature, respectively. The density behaviors of different nanofluids were presented by equilibrium molecular dynamic simulations, as studied by Abbasi et al. [24], and it was emphasized in this research that the density of nanofluids is highly influenced by the nanoparticle's size and shape. Ahmad et al. [25] studied the effect of local magnetic forces on the flow regime of trihybrid nanofluids. Application of a magnetic field greatly altered the rotational mobility of the tri-hybrid nanoparticles. Rajesh et al. [26] also analyzed natural convection in an enclosure using ternary nanofluids. They studied how the nanomaterial composition and non-uniform heating affect energy transfer within the enclosure and observed that even though energy transfer is enhanced by a nanomaterial, heat conduction could be reduced because of the lowered thermal energy of the heated wall.

Enhancing the rate of heat transfer in hybrid nanofluids is strongly facilitated by the presence of a porous medium combined with a Riga-stretched surface. The porous structure offers an interconnected network of microchannels that increases the effective surface area available for thermal interaction between the fluid and the heated surface. This promotes stronger fluid–solid contact and improves the dispersion and uniform distribution of nanoparticles throughout the flow field. As a result, the thermal boundary layer near the surface becomes thinner, allowing heat to be transferred more efficiently to the surrounding fluid. In addition, the electromagnetic forcing generated by the Riga surface induces secondary motion and flow acceleration within the porous matrix, which strengthens convective transport. The coupled action of porous resistance and Lorentz-driven flow therefore elevates the convective heat transfer coefficient and significantly enhances the overall thermal performance of the hybrid nanofluid system. Hayat et al. [27] investigated the flow of Carreau liquid under a nonlinear partial slip condition in porous media by utilizing the modified Darcy’s law concept. Kiyani et al. [28] studied the bidirectional unsteady flow of Williamson nanofluid saturation in porous media, applying modified Darcy’s law while considering heat sink/source factors and a zero-mass flux constraint. Waseem et al. [29] investigated how thermal radiation and internal heat generation influence entropy production and thermal efficiency in a three-dimensional magnetized hybrid nanofluid over a stretched surface. Sohail et al. [30] analyzed the combined effects of porous resistance (Darcy–Forchheimer model), viscous dissipation, and radiation on entropy generation in a squeezed hybrid nanofluid flow between rotating disks. Waseem et al. [31] applied the optimal homotopy analysis method (OHAM) technique to examine how temperature-dependent thermal conductivity and radiative heat transfer affect entropy behavior in hybrid nanofluids. Waseem et al. [32] numerically studied the role of heat generation and thermal radiation on thermal transport characteristics in water-based nanofluids using an OHAM framework. Abbas et al. [33] explored entropy production in three-dimensional stretched viscous fluid flow by considering the coupled influence of thermal radiation and double-diffusive transport. Sohail et al. [34] investigated the interaction of variable magnetic fields and radiative heat transfer on mass and thermal transport in bi-directional stretched Sisko fluid flows. Sohail et al. [35] examined the combined effects of rotation, Casson rheology, and bioconvection on three-dimensional nanofluid boundary layer behavior. Sohail et al. [36] analyzed the impact of magnetohydrodynamic forces and generalized heat and mass flux models on three-dimensional couple-stress non-Newtonian fluid flow using OHAM.

In the case of sodium alginate–based Carreau nanofluids, the movement of microbes is governed by a delicate balance between the shear-dependent flow behavior of the non-Newtonian liquid and the interactive forces introduced by the suspended nanoparticles. The Carreau character of sodium alginate creates regions of varying viscosity, which naturally guide microorganisms to migrate toward zones of lower resistance, while higher-shear regions tend to slow down and redistribute their motion. As a result, microbial swimming patterns become more structured, leading to controlled accumulation or dispersion within the flow domain. This bioconvective behavior is further shaped by external fields and chemical effects, as demonstrated in studies on cross nanofluid flows and electromagnetic regulation of motile microorganisms [37, 38]. Additionally, the coupling between thermal radiation, entropy generation, and non-Newtonian rheology plays a role in stabilizing microbial transport and clustering patterns [39]. The interaction of rotational and stagnation-point flows with nanoparticle-laden, micropolar and bioconvective systems also highlights how flow geometry and nanoparticle presence can redirect microbial pathways [40]. Zafar et al. [41] investigated how bioconvection in a Prandtl nanofluid around a stretched cylinder is shaped by Darcy–Forchheimer porous resistance and triple stratification, showing that combined thermal, solutal, and density gradients strongly regulate microbial accumulation and flow stability. Shah et al. [40] explored the bioconvective magneto-flow of a micropolar nanomaterial near an off-centered stagnation point over a rotating disk, demonstrating how rotation and magnetic forces can effectively redirect microbial pathways and nanoparticle transport. Falodun et al. [42] studied the antibacterial and antiviral effects of silver nanoparticles by conducting numerical simulation at Casson-Walters-B liquid with variable thermophysical characteristics. The work emphasized the biomedical significance of the liquid dynamics mediated by nanoparticles to make superior therapeutic and antimicrobial applications. A neural networking system was used by Ajmal et al. [43] to study the behavior of a tri-hybrid nanofluids on a lubricating flat plate under solar irradiation. Their results revealed the possibilities of the liquid as a good antimicrobial agent and identified the biomedical and engineering uses. Kumar et al. [44] modeled bio-magnetic blood flow of a Carreau hybrid nanofluid in stenotic arteries that contained active gyrotactic microorganisms. They determined the feasibility of the model in biomedical applications through the enhancement of targeted drug delivery and blood flow control by using the model through response surface optimization. Muntazir et al. [45] investigated numerically the MHD Carreau nanofluid flow of gyrotactic microorganisms on a plate, wedge, and stagnation point.

RSM is a statistical method for optimizing processes and product designs. It uses mathematical and statistical principles for developing and analyzing problems with multiple variables that affect the response of interest. RSM entails design experiments, data collection, and fitting a polynomial model to the response surface. The technique enables researchers to find the ideal set of factors to optimize or minimize the response of interest. Through contour plots or 3D surface plots, RSM gives a graphical illustration of the connection between the input variables and the response. It is specifically useful in manufacturing, chemical engineering, and biotechnology industries where process optimization is important to enhance efficiency and the quality of the product. Baithalu et al. [46] applied a statistical sensitivity and response surface framework to a micropolar hybrid nanofluid, revealing how key physical parameters jointly shape flow behavior and heat transfer efficiency, and identifying optimal conditions for thermal performance. Priyadharshini et al. [47] introduced a machine-learning–based regression approach to model magnetohydrodynamic flow in metallurgical processes, demonstrating how data-driven techniques can reliably predict complex thermal–fluid interactions. Sajid et al. [48] classified thermal performance in solar-powered vehicle systems using a binary tetra-hybrid nanofluid model, highlighting how nanoparticle geometry and effective thermal conductivity influence energy efficiency under realistic operating conditions. Wang et al. [49] numerically examined a Carreau fluid with melting and entropy generation over a variable-thickness Riga plate, showing how electromagnetic forcing and non-Newtonian viscosity strongly affect thermal irreversibility and flow stability. Fayyadh et al. [50] formulated a mathematical model of optimization of heat transfer of magnetized nanoparticles-containing Carreau liquid over an activating energy permeable surface. Using RSM, they showed improved thermal efficiency and transport characteristics of the liquid system. Tang et al. [51] studied bio-magnetic blood Carreau nanofluid flow in a stenotic artery with active gyrotactic microorganisms using response surface optimization and sensitivity analysis. Jindal and Sharma [52] optimized and modeled heat transfer in non-Darcy porous media with a non-Newtonian Ellis liquid. Applying response surface methodology, they proved enhanced thermal transport properties and process efficiency in intricate liquid systems. Takreem et al. [53] optimized the heat transfer characteristics of Casson hybrid nanofluid flow over a porous exponentially elongating surface. By applying the RSM approach, they revealed enhanced thermal performance and effective control of transport processes in hybrid nanofluid systems. Gull et al. [54] explored the slip flow of viscoelastic liquid under the influence of frictional heating effects. Using RSM with uncertainty analysis, they highlighted improved predictive accuracy and optimization of heat transfer performance in complex liquid systems.

Motivation

The motivation of this study is to advance the understanding of non-Newtonian Carreau-based nanofluid dynamics by addressing the limited availability of integrated models [57–61] that simultaneously describe thermal transport, mass diffusion, and microbial transport in complex electromagnetic and porous environments. Although earlier investigations have considered Carreau fluids and hybrid nanofluids independently, their combined influence on microorganism motion and spatial distribution under stagnation-point flow, nonlinear surface stretching, and Riga plate–induced magnetohydrodynamics remains insufficiently explored. This work is driven by the need for predictive and optimizable frameworks that can guide the design of practical systems where both heat management and controlled microbial behavior are important [39, 62–70].

Novelty

This study presents the first systematic integration of a trihybrid nanofluid system composed of TiO₂, Cu, and Ag nanoparticles suspended in sodium alginate with the Carreau rheological model under the combined effects of a Riga plate, Stefan blowing, stagnation-point flow, nonlinear surface stretching, and porous media. The work further introduces RSM and CCD to simultaneously optimize heat transfer, mass transfer, and microbial transport behavior within a defined parametric space. Statistical validation through analysis of variance, residual diagnostics, and percentile error analysis provides high-predictive empirical correlations suitable for design-level use. A mechanistic comparison of individual nanoparticle contributions demonstrates that silver, copper, and titanium oxide influence thermal, solutal, and microbial transports in distinct ways, showing that enhanced heat transfer does not necessarily coincide with stronger regulation of microbial distribution.

Industrial applications

In biomedical devices, the model supports the design of systems where precise thermal regulation and controlled microbial distribution along surfaces and flow channels are required. In water treatment and filtration systems, it provides guidance for managing heat transfer and microorganism transport within porous media and flow passages. The findings [71–75] are also relevant for functional surface coatings where long-term thermal performance and stable microbial interaction are desired. In geothermal and environmental engineering, the framework aids in predicting heat and mass transfer behavior in porous rock formations, subsurface reservoirs, and groundwater systems. Additionally, energy and filtration technologies can benefit from optimized porous heat exchangers and flow systems where thermal efficiency and microbial transport must be balanced.

Formation of the problem

Consider a two-dimensional, incompressible Carreau flow of an electrically conducting trihybrid nanofluid composed of sodium alginate as the base fluid with suspended nanoparticles, flowing over a nonlinearly stretched Riga surface subject to porosity (Stefan blowing) and viscous dissipation effects. The Riga surface generates a spatially varying electromagnetic field that interacts with the electrically conducting nanofluid, thereby influencing the near-wall momentum transport. The wall velocity is included in the sheet, where b is stretching-shrinking constant and m is the power-law index. The stagnation point velocity is as: . Here, a>0, which governs the free stream boundary conditions. The effects of external forces, gravity and pressure gradients are ignored. The velocity components (U, V) in the streamwise and transverse directions are represented by the fluid motion parallel and normal to the surface, respectively (Fig. 1).

Fig. 1.

Fluid flow over the nonlinear Riga stretched surface

The current investigation has the following objectives:

1. To develop and solve a two-dimensional, unsteady, laminar stagnation system flow model of a triple hybrid Carreau nanofluid along a nonlinearly stretched Riga surface, with Stefan blowing and suction forces on a porous material.

2. To describe the shear-thinning rheology of the base fluid by a Carreau fluid model, having a power-law index that is less than one, and to determine its effect on the velocity field, the thermal distribution, solutal concentration, and the motile microorganism density.

3. To explore the joint impact of nonlinear thermal radiation, viscous dissipation and Newtonian heating on the characteristics of heat and mass transfers under time-dependent conditions of stagnation point.

4. To investigate the importance of linear chemical reactions in controlling the nanoparticle transport and microbial density in the boundary layer.

5. To measure how nanoparticle volume fraction and unsteady flow parameters influence important engineering performance indicators, such as Nusselt number, Sherwood number and motile microorganism density factor.

6. To apply similarity transformations to reduce the governing PDEs to a coupled system of nonlinear ODEs, and to find the correct numerical solution by applying the bvp4c solver of MATLAB.

7. To use RSM, based on CCD, to do systematic optimization of the velocity, thermal, solutal and microbial responses to transport.

8. To conduct ANOVA to determine statistically significant parameters and interactions on the microbial performance and the transport efficiency.

9. To perform a sensitivity analysis to determine the strength of the motile microorganism density response to changes in the nanoparticle loading and flow unsteadiness.

Mathematical modelling

Based on the problem formulations and objectives to make the mathematical model, we have the following constitutive equations [3–5]:

1

2

3

4

5

Here, is called the Cauchy stress tensor. represent as velocity and body force of the fluid particles respectively. Moreover, T, C, N represent temperature, concentration and microbial characteristics for this flow phenomena. Equations (1)–(5) represent the laws of conservation of mass, temperature, concentration and motile movement. Carreau fluids belong to the class of generalized Newtonian fluids, and the constitutive model was originally proposed by Carreau [1, 2] to describe the rheological behavior of fluids whose viscosity varies with the applied shear rate. Unlike the classical power-law model, which is unable to accurately predict viscosity in the limiting cases of very low and very high shear rates, the Carreau model provides a smooth transition between the zero-shear-rate viscosity and the infinite-shear-rate viscosity. This makes it particularly suitable for modeling a wide range of non-Newtonian fluids, including blood, plasma, polymer solutions, and viscoelastic materials.

The Carreau constitutive relation is expressed as [1, 2]:

6

where denotes the extra stress tensor and represents the generalized shear rate. The effective viscosity is defined in terms of the zero-shear-rate viscosity , the infinite-shear-rate viscosity , the material time constant λ , and the power-law index n, which characterizes the shear-thinning or shear-thickening behavior of the fluid. In terms of the second invariant of the rate-of-deformation tensor, the generalized shear rate is given by:

7

where A₁ is the first Rivlin–Ericksen tensor. In the present study, the special case is considered, which simplifies the constitutive relation. Accordingly, the extra stress tensor can be rewritten as:

8

Model approximations, assumptions with limitations

To achieve the above-mentioned objectives, we have the following assumptions:

1. The flow is assumed to be two-dimensional, unsteady, and laminar, with negligible three-dimensional and turbulent effects throughout the boundary layer.

2. The base fluid (sodium alginate) is modeled as a shear-thinning non-Newtonian fluid, governed by the Carreau constitutive relation with a power-law index less than unity.

3. The nanofluid is treated as a single-phase homogeneous mixture, where titanium oxide, copper, and silver nanoparticles are uniformly dispersed, and thermal equilibrium and no slip are assumed between the base fluid and nanoparticles.

4. The suspension of motile microorganisms is assumed to be dilute, such that microorganism–microorganism interactions and volume exclusion effects are negligible.

5. The flow occurs over a nonlinearly, time-dependent stretched Riga surface, and the electromagnetic forcing is confined to the near-wall region and decays exponentially normal to the surface.

6. The external flow is characterized by a time-dependent stagnation-point velocity field, which governs the far-field boundary conditions.

7. The porous medium is considered homogeneous and isotropic, and the resistance offered by the medium follows Darcy’s law, neglecting inertial (Forchheimer) effects.

8. Stefan blowing/suction is prescribed at the wall, and the mass flux is assumed to be uniform along the surface.

9. Heat transfer is modeled using Newtonian heating conditions, where the wall heat flux is proportional to the local surface temperature difference.

10. Thermal radiation is represented using the Rosseland diffusion approximation, assuming the medium is optically thick and radiation heat flux occurs primarily in the normal direction.

11. Viscous dissipation effects are included, while pressure work and Joule heating are assumed negligible.

12. The chemical reaction is assumed to be homogeneous, first-order, and temperature-dependent, and its influence is restricted to the nanoparticle concentration field.

13. The thermophysical properties of the nanofluid are assumed to be constant, except for the density variations appearing in the buoyancy terms, if applicable.

14. Brownian motion and thermophoresis effects are neglected, as the Buongiorno model is not adopted in the present formulation.

15. Microorganism transport is governed by a concentration coupled Fokker–Planck-type bioconvection model, assuming gyrotactic behavior and negligible microorganism inertia.

16. Wedge and leading-edge effects are neglected, consistent with the boundary-layer approximation and semi-infinite surface assumption.

After using Eqs. (1–8) and the above-mentioned approximations, we have the following boundary layer equations for the momentum, heat, solutal concentration and microbes’ movement as [6–9, 11–16]:

9

10

11

12

13

After applying the boundary-layer assumptions, the governing model is formulated from the fundamental conservation laws of mass, momentum, energy, nanoparticle concentration, and microorganism density. The energy balance accounts for the combined effects of thermal radiation and viscous dissipation, which are enhanced by the presence of trihybrid nanoparticles which are chosen as spherical type shapes. The subjected boundary conditions are [18–21, 63]:

14

The involved terms in Eqs. (9–14) have been mentioned in the nomenclature with their SI units. The governing mathematical model describes the unsteady stagnation-point flow of a Carreau fluid over a permeable Riga surface embedded in a porous medium_. The unsteady term represents the temporal evolution of the flow field, capturing how the boundary layer develops and adjusts with time rather than remaining in a steady state, while the stagnation-point component accounts for the deceleration and redirection of the fluid as it impinges on the surface. The Carreau fluid model characterizes the shear-dependent viscosity, allowing the fluid to behave as Newtonian at low shear rates and shear-thinning or thickening at higher shear levels. The Darcy term models the resistance imposed by the porous medium, which opposes fluid motion and reduces the velocity through drag effects. The Riga term represents the electromagnetic body force generated by the Riga surface, which actively accelerates the fluid along the plate and enhances momentum transport. In the energy equation, thermal radiation accounts for radiative heat exchange within the fluid, becoming significant at elevated temperatures, while viscous dissipation converts mechanical energy into internal thermal energy due to fluid friction. The chemical reaction term in the concentration equation describes the rate of species consumption or generation within the flow, influencing mass transfer characteristics. Finally, the motile microorganism reflects the relative contribution of swimming microorganisms to bioconvective transport, indicating how microbial motion alters the distribution and stability of the boundary layer. At the nonlinearly stretched Riga surface, the fluid is assumed to adhere to the surface, meaning that the tangential velocity of the fluid matches the surface velocity. The normal motion at the wall accounts for mass transfer through the porous surface, which represents either injection or suction of the fluid caused by concentration-driven effects associated with the nanoparticle suspension (Stefan blowing) The thermal condition at the surface is modeled using a convective heating mechanism, in which heat is exchanged between the surface and the adjacent fluid. The rate of heat transfer is governed by the temperature difference between the surface and the surrounding fluid, reflecting a realistic representation of Newtonian heating commonly encountered in engineering systems. At the wall, the nanoparticle concentration and the motile microorganism density are prescribed, indicating a controlled supply of both species at the surface. Far from the surface, the flow gradually approaches the free-stream stagnation-point conditions, where the velocity, temperature, nanoparticle concentration, and microorganism density attain their ambient values.

Similarity transformations

The application of similarity transformations is a widely used mathematical approach for simplifying complex governing equations in fluid dynamics [11–20]. This method introduces appropriately scaled dimensionless variables, which enable the identification of characteristic physical scales and the emergence of key dimensionless parameters that control the system behavior. By recasting the governing equations into a dimensionless form, the explicit dependence on physical units is removed, thereby facilitating both physical interpretation and numerical implementation. In the present study, the governing equations, represented by Eqs. (9)– (13), form a strongly nonlinear system of PDEs corresponding to the velocity, temperature, nanoparticle concentration, and motile microorganism density fields, together with the associated boundary conditions in Eq. (14). Although these equations can be solved directly using advanced numerical techniques, such an approach is computationally demanding. To reduce this complexity, a transformation-based framework is adopted to convert the coupled system of PDEs into a corresponding set of ODEs. This reduction significantly lowers the computational cost while preserving the essential physical structure and accuracy of the underlying mathematical model. Accordingly, the similarity transformations reported in [63] are employed to reformulate the governing equations into the following system of nonlinear ODEs:

15

By applying (15) to (10)–(14), the dimensionless form of the ODEs which are obtained is as follows:

16

17

18

19

The dimensionless boundary conditions reduce to

20

where non- dimensional factors like Peclet number, microorganism difference factor, Bio-convection Lewis number, Weissenberg number, unsteady factor, Biot number, magnetic factor, Prandtl number, Eckert number, index constant, Stefan blowing parameter, Schmidt number, chemical reaction parameter, modified Hartmann number, Darcy number, thermal radiation, and stretching-shrinking constants are mathematically defined as:

The value of m is defined as: . So, the value of β can be grow from 1 to <2.

Thermophysical properties of the nanoparticles

Thermophysical properties describe how a fluid or material responds to heat, energy, and motion under thermal effects. These properties include density, specific heat capacity, thermal conductivity, viscosity, and electrical conductivity, which collectively govern the transport of momentum and heat within the system. Density reflects the compactness of the fluid, while specific heat capacity indicates its ability to store thermal energy. Thermal conductivity determines how efficiently heat is conducted through the medium, whereas viscosity controls the resistance to flow. In electrically influenced flows, electrical conductivity plays a crucial role by coupling the fluid motion with applied electric and magnetic fields. In nanofluid and trihybrid nanofluid systems, the inclusion of nanoparticles significantly alters these thermophysical properties, leading to enhanced heat transfer performance, modified flow resistance, and improved thermal energy distribution near the surface. The detailed description of the Tables 1 and 2 can be explained as: thermophysical properties of the base fluids, nanofluids and trihybrid nanoparticles.

Table 1.

Physical properties of micro-sized particles and conventional fluid

Characteristics
SA	989	4175	0.6376	23	5.5*10−6
TiO2,	4250	686.2	8.953	0.9	2.38*10−6
Ag,	10,500	235	429	1.89	2.8*105
Cu,	8933	385	401	1.67	29*105

Table 2.

Correspondences for THNF thermophysical properties [22–26]

Density
Viscosity
Thermal conductivity
the nanoparticles shapes are spherical
Brownian diffusion constant for nanoparticles
Motile diffusion constant for nanoparticles

The viscosity of a hybrid nanofluid is significantly influenced by factors such as temperature, nanoparticle shape, size, and concentration. Typically, increasing the temperature, utilizing smaller spherical structures, and reducing nanoparticle concentration directed to a decrease in the nanofluid's viscosity. Hence, we express the viscosity model for hybrid spherical nanoparticles as follows:

Physical quantities

The intersecting physical quantities are the skin friction coefficient , local Nustle number and microorganism number are shown below [63]:

21

The skin friction coefficient (), local Nusstle number () and Sherwood number ( ) and Microorganism number () in the simplified form as:

22

where .

Numerical simulation

The coupled nonlinear ODEs (16)–(19), together with the boundary conditions (20), form a two-point boundary value problem that cannot be solved analytically due to the strong nonlinearity introduced by the Carreau fluid behavior and the interacting thermal, solutal, and bioconvective effects (Fig. 2). Therefore, a numerical approach is adopted. The system is solved using MATLAB’s bvp4c solver, which is well suited for nonlinear boundary value problems. To ensure convergence, the semi-infinite domain is truncated by taking the far-field boundary at a sufficiently large finite value, set as 10, where all dependent variables asymptotically satisfy the boundary conditions. Convergence of the solution is confirmed by monitoring successive mesh refinements and residual errors, with the numerical results showing stable and rapidly converging profiles within the prescribed tolerance limits across the entire parameter range.

23

24

25

26

Fig. 2.

Graphical abstract of the modelled problem

The dimensionless boundary conditions reduce to.

.

27

Numerical results

Tables 3 and 4 present a comparative analysis of the skin friction coefficient and Nusselt number for both nanofluid and hybrid nanofluid cases. The computed values of the surface drag force show excellent agreement, matching up to four decimal places with the results reported in references [61–63]. Similarly, the predicted heat transfer rates closely align, up to four decimal places, with the numerical values available in references [57–60], as shown in Table 4. Table 5 validates the present model by comparing the values of and at a Prandtl number of 6.2 while setting all other parameters to zero, for different values of the copper nanoparticle volume fraction . The obtained results are consistent with those reported in reference [60]. In addition, Tables 3, 4, 5 examine the limiting cases by considering different values of the Prandtl number and the stretching–shrinking parameter. A very strong agreement is observed for both the Nusselt number and the skin friction coefficient.

Table 3.

Model endorsement of when Pr=6.2 with numerous values of λ and all others are zero

λ
	Wang [61]	Bachok et al. [62]	Saleem et al. [63]	Current findings
2	− 1.887310	− 1.887307	− 1.887307	− 1.887306492
1	0	0	0	0
0.5	0.713300	0.713295	0.713295	0.713294347
0	1.232588	1.232588	1.232588	1.232586594
− 0.5	1.495670	1.495670	1.495670	1.495667445
− 1	1.328820	1.328817	1.328817	1.328815513
− 1.15	1.082230	1.082231	1.082231	1.082230661
− 1.2		0.932473	0.932473	0.932473046
− 1.2465	0.554300	0.584281	0.584281	0.584028247

Table 4.

Model endorsement of with numerous values of Pr and all others are zero

Pr	Khan and Pop [59]	Wang [57]	Reddy and Sidawi [58]	Devi and Devi [60]	Current values
2.0	0.9113	0.9114	0.9114	0.91135	0.911362125554452
6.13	–	–	–	1.75968	1.759699217508779
7	1.8954	1.8954	1.8954	1.89540	1.895423658180989
20	3.3530	3.3530	3.3539	3.35390	3.353955082789159

Table 5.

Model endorsement of , when Pr=6.2 and all other are zero with numerous values of

	Skin friction coefficient [60]	Nusselt number [60]	Current values Cu/water	Current values Cu/water
0.005	− 1.026272	1.768118	− 1.026188760	1.768171647
0.02	− 1.104327	1.791406	− 1.104481659	1.791408385
0.04	− 1.208432	1.822984	− 1.208874794	1.822923410
0.06	− 1.313402	1.855179	− 1.314110645	1.855060954

Discussion

The overall focus of this part is to provide a detailed insight into the flow regime, that is, the variations in velocity of the trihybrid nanofluid, temperature, concentration, and microorganism velocity of three types of nanoparticles depending on the varying regulatory flow parameters. MATLAB uses the in-built solver function bvp4c to produce both numerical and graphical outputs. The `bvp4c` method, which is incorporated into MATLAB, is a landmark development in scientific computation. It solves complex boundary value problems, including complicated systems of ODEs, addressing a wide range of real-world problems. Through its flexibility, precision, adaptive mesh refinement, and incorporation of MATLAB's functionality, `bvp4c` provides an easy-to-use platform for scientists and engineers to model and solve numerous physical phenomena efficiently. While it has its merits, users must keep an eye out for its limitations and adjust factors appropriately for best performance, especially in instances of highly nonlinear or complex problems where additional methods might prove helpful. The impact of various factors like unsteady factors, volume fraction of nanoparticles, Weissenberg number, Biot number, Eckert number, Schmidt number, bio-convection Lewis number, microorganism difference factor and slip factor are studied in detail.

The observed variations in the current profile are examined by utilizing the governing equations and related boundary conditions of the current continuum model. In addition to that, it should be noted that the current discussion is confined to the analysis of the transport processes predicted by the current mathematical model without including any complex microscale interactions among nanoparticles and microorganisms, which include aggregation, adhesion, toxicity, sedimentation, or biodegradation mechanisms. Thus, the current findings provide important information about the transport behavior of trihybrid nanofluids with the inclusion of the considered physical effects.

Figure 3a–d portrays the outcomes of unsteady factor on velocity, temperature, concentration and microbial outlines respectively [38–40]. The velocity, temperature, concentration and microbial gradients upturn for higher estimations of . The variations in the nanoparticle volume fraction on the velocity, temperature, concentration and microbial outlines are illustrated in Fig. 4a–d respectively. It can be observed that all the gradients are augmented by growing values of . When the amount of nanoparticle in the liquid increases, the velocity increases more efficiently. The velocity increases because the tiny particles help the liquid flow more smoothly by reducing friction between liquid layers, making it easier for the liquid to move faster. The temperature goes up because is very good at conducting heat, so when adding more of these particles, they create better pathways for heat to travel through the liquid, making the whole liquid warmer and transferring heat more effectively. However, the concentration and microbial gradients decrease because the higher amount of particles create more obstacles and resistance for other substances trying to move through the liquid, making it harder for chemicals or nutrients to spread evenly, while the increased particle density also makes it more difficult for microorganisms like bacteria to swim freely through the thicker, more crowded nanofluid.

Fig. 3.

Influence of unsteady factor on a , b , c and d

Fig. 4.

Influence of volume fraction on a , b , c and d

Figure 5a–d depicts the behavior of nanoparticle (Cu) volume fraction on velocity , temperature , Concentration , and microbial gradient respectively. When the volume fraction of (Cu) nanoparticles increase, the velocity gradient , improves because of its excellent electrical conductivity that create favorable electromagnetic effects that enhance liquid movement. The concentration gradient increases because (Cu) nanoparticles are biocompatible and don't interfere as much with the transport of other dissolved substances, helping to create better mixing patterns that distribute chemicals and nutrients more evenly throughout the liquid. The microbial gradient also increases because (Cu) nanoparticles, unlike some other metals, can provide beneficial trace nutrients. However, the temperature gradient decreases because (Cu) has such extremely high thermal conductivity that it acts like a heat sink, rapidly conducting heat away from hot areas and spreading it throughout the entire liquid so quickly that it prevents the buildup of high temperatures in any region.

Fig. 5.

Influence of Cu volume fraction on a , b , c and d

Figure 6a–d explains that higher values of nanoparticles (Ag) volume fraction develop the velocity , temperature , concentration , and microbial gradients correspondingly due to silver's unique blend of exceptional properties and bioactive properties. The velocity gradient surges because (Ag) nanoparticles have exceptional dispersion stability and even surface properties that lessen internal friction within the liquid, letting it flow more easily and with less fighting, while their small size and spherical shape create negligible flow disruption. The temperature outline increases because (Ag) has one of the highest thermal conductivities among all metals, creating very efficient heat transfer pathways that help spread thermal energy throughout the liquid and keep higher overall temperatures by discontinuing heat loss and improving thermal diffusion. The concentration outline increases because silver nanoparticles enhance mixing through their ability to create electrochemical environments that promote better dissolution and transport of other substances, while their presence improves the liquid's ability to carry dissolved materials without causing precipitation or separation. Most surprisingly, the microbial profile also increases despite (Ag) well-known antimicrobial properties, because at low concentrations, silver nanoparticles can stimulate certain beneficial microorganisms [40–42].

Fig. 6.

Influence of Ag volume fraction on a , b , c and d

Figure 7a–b depicts the effect of Weissenberg number on velocity , and temperature distributions respectively. The Fig. 7a illustrates that as the Weissenberg number increases, the velocity curve decreases. This occurs because higher values of lead to an increase in the relaxation time of the liquid particles, resulting in greater viscosity. This heightened viscosity creates resistance to liquid flow, ultimately leading to a reduction in the velocity curve. Figure 7b demonstrates that increasing values of decline the thermal distribution . The variation in temperature scattering with respect to the Biot number and the Eckert number is illustrated in Fig. 8a, b. Figure 8a clearly shows a one-to-one relationship between and at every point within the boundary layer region. Due to the increase in , it can be inferred that the conduction heat transfer rate is significantly greater than that of convection close to the surface of the sheet. Consequently, the temperature curve will exhibit much more pronounced extremes. Figure 8b illustrates the effect of the Eckert number on θ(η), indicating that an increase in leads to enhanced temperature distributions for trihybrid nano-liquid. A higher signifies that heat is generated within the flow region due to friction between two neighboring liquid layers, a phenomenon directly attributed to viscous dissipation. Consequently, the temperature increases, resulting in a thicker thermal boundary layer area.

Fig. 7.

Influence of on a , and b   

Fig. 8.

Influence of   

Figure 9a–d illustrates the impact of the Schmidt number on the velocity , temperature , Concentration , and microbial gradients, respectively. The Schmidt number is a dimensionless quantity defined as the ratio of liquid viscosity to mass diffusivity. Therefore, if the viscosity of the liquid exceeds the mass diffusivity, a higher Schmidt number will be present throughout the entire boundary layer. As the rises, the concentration diminishes. This reduction leads to a decrease in the buoyancy effects of concentration, resulting in a lower liquid velocity. The declines in velocity, temperature, and concentration gradients are accompanied by a simultaneous thickening of the momentum and concentration boundary layers, which is similar to the effect of an increasing Prandtl number on the thickness of a thermal boundary layer. Figure 9d illustrates that with an increase in , there is a corresponding decrease in the microbial gradient . This indicates that the microorganisms are diffusing at a slower rate through the liquid. In the flow's nodal region, the diminished mass diffusivity restricts the dispersion of microorganisms, resulting in a more concentrated distribution and a reduction in the overall microbial gradient. Figure 10a, b signifies the impact of bio-convection Lewis number and micro-organism difference factor via microbial gradient respectively. It can be clearly noticed that the increasing values of and diminished the microbial gradient

Fig. 9.

Influence of on a , b , c and d

Fig. 10.

Influence of

Figure 11a–d illustrates how the velocity slip factor λ affect the velocity temperature , Concentration , and microbial gradients, respectively. It is observed that an increase in the velocity slip factor λ leads to a higher velocity gradient but decline the thermal boundary layer , Concentration , and microbial gradients. By the stronger slip λ at the boundary, the liquid near the wall doesn’t stick as tightly, so it moves more freely. This raises the overall velocity gradient because less momentum is wasted overcoming friction at the wall. Meanwhile, the layers through which heat, solute, or microbes diffuse toward the wall become thinner, the faster moving liquid carries away energy, species, or microbes more quickly, giving them less time to accumulate near the boundary. As a result, the temperature , concentration , and microbial density near the surface drop compared to the no-slip case.

Fig. 11.

Influence of a , b , c and d

Optimizing heat, solute and microbes transfer rates using RSM via CCD model

For the specified numerical values of the factors influencing flow phenomena, the rate coefficients, such as the heat transfer rate represented by the Nusselt number, the mass transfer rate indicated by the Sherwood number, and the microorganism transfer rate denoted by the motile density factor, are calculated. To achieve an optimized heat and mass transfer rate for the predicted values, a statistical method known as RSM with CCD is beneficial for simulating variations in different factors. Factors like the volume fraction of nanoparticles (A, B, C) and the unsteady factor (S) are utilized to determine the responses of the Nusselt number, Sherwood number, and motile density factor, and these are then briefly explained. The current research provides an illustration of how the incorporated the independent factors influence the response function , , and when using the statistical RSM methodology. The simulation is presented for their specific range, that is, , , , and , and these values are deployed in Table 6. The simulation process is conducted using a quadratic order model, which incorporates an in-built CCD utilizing a face-centered technique that encompasses various levels, including low, middle, and high values. Therefore, Table 7 exhibits the computed values of responses of the Nusselt number, Sherwood number and motile density factor results for the factors A, B, C, and S. The allocations of the responses are presented for 30 usual runs. These standard CCD experimental runs represent center point replicates where all factors are maintained at mid-levels (A=0.03, B=0.03, C=0.03, S=0.35). These replicates are essential for estimating pure experimental error, assessing numerical reproducibility, and validating model adequacy in RSM analysis. Table 8 depicts the standard deviation for the quadratic model with a mean and the actual values of the R adjusted and predicted values are also displayed. In accordance with the stated, the structure of the quadratic polynomial model is established using Minitab Statistical software as follows:

28

Table 6.

Design arguments with their levels for and experiment

Design arguments	Levels
	Low (− 1)	Mid (0)	High (+ 1)
	0.01	0.03	0.05
	0.01	0.03	0.05
	0.01	0.03	0.05
S	0.1	0.35	0.6

Table 7.

Experimental design and measured responses using CCD model

Exp. No	Real values				Responses
	A	B	C	S
1	0.01	0.01	0.01	0.1	0.042849411	0.972139143	1.555139705
2	0.05	0.01	0.01	0.1	0.042137898	1.115210206	1.8826829
3	0.01	0.05	0.01	0.1	0.040396975	0.853971064	1.292452495
4	0.05	0.05	0.01	0.1	0.039777057	0.978675982	1.55932156
5	0.01	0.01	0.05	0.1	0.03954812	0.851766048	1.291193719
6	0.05	0.01	0.05	0.1	0.038958548	0.976373741	1.557852907
7	0.01	0.05	0.05	0.1	0.037444914	0.745164391	1.074426632
8	0.05	0.05	0.05	0.1	0.036933306	0.853846583	1.292385204
9	0.01	0.01	0.01	0.6	0.041381353	0.383312409	1.223272511
10	0.05	0.01	0.01	0.6	0.040694519	0.425147235	1.445323805
11	0.01	0.05	0.01	0.6	0.039156052	0.351485312	1.039972196
12	0.05	0.05	0.01	0.6	0.03855285	0.389233704	1.227359628
13	0.01	0.01	0.05	0.6	0.038374229	0.349488478	1.038743715
14	0.05	0.01	0.05	0.6	0.037799677	0.387166048	1.225933432
15	0.01	0.05	0.05	0.6	0.03644742	0.317213572	0.881897619
16	0.05	0.05	0.05	0.6	0.035945978	0.35137769	1.039905734
17	0.01	0.03	0.03	0.35	0.039521851	0.573265172	1.158233608
18	0.05	0.03	0.03	0.35	0.03892207	0.645292701	1.380899216
19	0.03	0.01	0.03	0.35	0.040330265	0.642295501	1.376898152
20	0.03	0.05	0.03	0.35	0.038175484	0.573960942	1.156981299
21	0.03	0.03	0.01	0.35	0.040747297	0.643524641	1.377724319
22	0.03	0.03	0.05	0.35	0.037784365	0.573079094	1.156454168
23	0.03	0.03	0.03	0.1	0.039651943	0.913872981	1.419876372
24	0.03	0.03	0.03	0.6	0.038454008	0.369222569	1.130066942
25	0.03	0.03	0.03	0.35	0.039214047	0.608042058	1.263350278
26	0.03	0.03	0.03	0.35	0.039214047	0.608042058	1.263350278
27	0.03	0.03	0.03	0.35	0.039214047	0.608042058	1.263350278
28	0.03	0.03	0.03	0.35	0.039214047	0.608042058	1.263350278
29	0.03	0.03	0.03	0.35	0.039214047	0.608042058	1.263350278
30	0.03	0.03	0.03	0.35	0.039214047	0.608042058	1.263350278

Table 8.

ANOVA analysis

Source	DF	Adj SS	Adj MS	F-value	P-value
Model	14	6.8		64,027.04	0.000
Linear	4	6.8		223,062.03	0.000
Square	4	0.000000	0.000000	275.12	0.000
2-way interaction	6	0.000000	0.000000	504.98	0.000
Error	16	0.000000	0.000000	*	*
Lack-of-fit	10	0.000000	0.000000	*	*
Pure error	6	0.000000	0.000000
Total	30	6.8
Adj. Pred
Model	14	1.47006	0.10500	6577.34	0.000
Linear	4	1.43756	0.35939	22,511.72	0.000
Square	4	9.4	2.36	147.54	0.000
2-way interaction	6	2.308	3.85	240.96	0.000
Error	16	2.6	2.0	*	*
Lack-of-fit	10	2.6	3.0	*	*
Pure error	6	0.00000	0.00000
Total	30	1.47032
Adj.Pred
Model	14	1.12279	8.02	2581.25	0.000
Linear	4	1.09349	2.734	8798.68	0.000
Square	6	0.02513	4.188	134.80	0.000
2-way interaction	16	5.0	3.1
Error	10	5.0	5.0	*	*
Lack-of-fit	6	0.00000	0.000000
Pure error	30	1.12328
Total	14	1.12279	8.02	2581.25	0.000
Adj.Pred

where is intercept term, , and are regression coefficients of the linear, quadratic and interaction factors respectively.

Accuracy of the model

The model's accuracy is utilized for testing, thus an effective statistical method, namely "ANalysis Of Variance" (ANOVA), is recommended, with results illustrated in Table 8. The model F-value indicates that the proposed model holds significance. However, it is important to note that there is a 0.01 probability that a high F-value may arise from noise. The simulated P-values demonstrate numeric values below 0.0500, signifying that the model terms are indeed significant. Therefore, it is suggested that the model terms are not significant if the values exceed 0.1000. If multiple model terms are deemed insignificant, the proposed model may represent the design space. The ANOVA results presented in Table 8 demonstrate exceptional model validity for all three responses. For , the model F-value of 64,027.04 with p < 0.000 indicates the quadratic model is highly significant, with linear terms contributing most strongly (F = 223,062.03), followed by interaction effects (F = 504.98) and square terms (F = 275.12). Similarly, for , the model F-value of 6577.34 confirms significance, with linear effects dominating (F = 22,511.72), while square and interaction terms (F = 147.54 and 240.96) capture important curvature and interactive effects. For , the F-value of 2,581.25 validates the model, with linear terms again being most influential (F = 8,798.68). The negligible lack-of-fit and pure error terms confirm model adequacy, while values exceeding 99.9% for all responses indicate that the quadratic models account for virtually all variability in the data. The high adjusted and predicted values (differing by < 0.2%) further confirm excellent model fit without overfitting, validating the use of Eqs. (14–16) for predictive optimization in practical applications.

Table 9 shows the estimated regression coefficients for the responses with its corresponding P-value set for the non-linear polynomial model in Eq. (28). It is worthy of attention that a large P-value indicates that the change in input has no noticeable impact on the output, making it statistically insignificant. On the other hand, the term with a small P-value (łe 0.05) that is statistically significant elsewhere, can be omitted. Consequently, and AS are insignificant terms for , and BC are insignificant terms for and are insignificant terms for . The asterisk signs in these tables indicate that no values are available. Therefore, Eq. (28) for , , and , can be written as:

29

30

31

Table 9.

Estimated regression coefficients for , and

Terms	Coefficients	P-value
Constant	0.039214	0.0000
A	- 0.000300	0.0000
B	- 0.001069	0.0000
C	- 0.001470	0.0000
S	- 0.274299	0.0000
	0.000007	0.186
	0.000038	0.0000
	0.000051	0.0000
	-0.000162	0.0000
AB	0.000020	0.0000
AC	0.000028	0.0000
AS	0.000004	0.074
BC	0.000079	0.0000
BS	0.000050	0.0000
CS	0.000066	0.0000
Constant	0.60795	0.0000
A	0.040251	0.0000
B	- 0.038221	0.0000
C	- 0.039290	0.0000
S	- 0.274299	0.0000
	0.00144	0.571
	0.00029	0.910
	0.00046	0.856
	0.03370	0.0000
AB	-0.002618	0.0000
AC	-0.002639	0.0000
AS	-0.021853	0.0000
BC	0.001578	0.134
BS	0.021752	0.0000
CS	0.021807	0.0000
Constant	1.26326	0.0000
A	0.11424	0.0000
B	- 0.11291	0.0000
C	- 0.11358	0.0000
S	- 0.14849	0.0000
	0.00642	0.082
	0.00379	0.290
	0.00394	0.272
	0.01182	0.0000
AB	-0.01083	0.0000
AC	-0.01088	0.0000
AS	-0.02027	0.0000
BC	0.01014	0.0000
BS	0.02026	0.0000
CS	0.02027	0.0000

where A, B, C represent the volume fractions of TiO₂, Cu, and Ag nanoparticles respectively, and S denotes the unsteady factor. All variables in Eqs. (29–31) are expressed in coded units ranging from -1 to +1, corresponding to the actual ranges specified in Table 6.

Graphical analysis of RSM

RSM relies heavily on graphical analysis through contour and surface plots to visualize and interpret the relationship between response variables and experimental factors. Contour plots are a two-dimensional representation, showing lines of equal response on a plane by two factors. Contour plots are handy in finding optimal regions and in understanding variable interactions. Surface plots give a three-dimensional representation of the response surface, making it possible to inteapproach is especially beneficial forrpret intuitively the interaction between factors and the response. This visualization facilitates the detection of maximum or minimum points, the recognition of curvature of the response surface, and evaluation of the type of critical points (whether they are maxima, minima, or saddle points). The vertical axis height represents the predicted response value, making it simple for researchers to detect peaks, valleys, and slopes in the response surface. Together, the surface and contour plots provide complementary information and assist in the optimization of process factors and in the understanding of intricate systems in engineering, chemistry, and biotechnology. Figures 12a, b, 13a, b, and 14a, b illustrate the variations of Nusselt Number , Sherwood number , and Motile density factor using 2-D contours and three- dimensional (3-D) surface plots for various volume fraction of nanoparticles and unsteady factor (S) respectively. Figure 12a, b presents contour plots and surface plots of the Nusselt number as functions of pairwise interactions among nanoparticle volume fractions (A, B, C) and the unsteady factor (S).The contour fill indicates magnitude ranges of , with lighter tones corresponding to lower values (< 0.037) and progressively darker tones to higher values (up to > 0.041). In the B× A and C× A planes, increases with increasing volume fractions of both nanoparticles, the darkest contour regions are found where both A and B, or A and C, are at or near their maximum normalized values. This suggests strong main‐effects of nanoparticle concentration. In the S× A interaction plot, the unsteady factor S also exhibits a positive effect on ​, especially when A is high, the contours curve indicating that the incremental benefit of increasing S is greater at higher A. Similarly, in the C× B subplot, the contour lines are approximately diagonal and more uniformly spaced, implying that B and C contribute additively to ​ with modest interaction. In contrast, the S× B and S× C plots show contours with pronounced non‐linearity, increases in S yield greater enhancements in ​ when B or C are already large. These interaction effects suggest that optimizing nanoparticle concentrations together with unsteady flow conditions will yield more substantial augmentation of heat transfer than varying each factor independently.

Fig. 12.

Interaction plots for a contour plots, b surface plots

Fig. 13.

Interaction plots for a contour plots, b surface plots

Fig. 14.

Interaction plots for a contour plots, b surface plots

Figure 13a, b presents contour and surface plots of the local Sherwood number ​ across two-variable planes formed among nanoparticle volume fractions (A, B, C) and the unsteady factor (S) respectively. The contour shading corresponds to ranges of ​, from deep blue (< 0.4) through lighter blues and greens toward higher values (> 0.9). In the B× A and C× A subplots, there is a clear dichotomy in the shading, as either A or B(or C) increases, the contours shift from blue to green, indicating an increase in . This suggests that increasing nanoparticle volume fractions enhances mass transfer (via higher Sherwood number). The boundary between blue and green regions tends to lie near the midline of the normalized axes, implying that moderate increases in these fractions yield appreciable change in ​. In the S× A plot, the vertical progression from blue at low S to green at high S(especially when A is also large) highlights that S positively influences . The isolines are nearly horizontal indicating that variation in S has a stronger effect than variation in A on ​. Similar behavior is seen in the S× B and S× C plots as S increases, ​ increases markedly, with the darkest (green) zones concentrated toward the upper right or upper regions of the plots. Notably, the S× B and S× C contours are more evenly spaced (gradual shading transitions), implying a relatively steady incremental effect of S across the range of B and C. In contrast, the C× B subplot exhibits a roughly diagonal transition from blue in the lower right to green in the upper left. This orientation suggests that increasing C while decreasing B, or vice versa, may yield compensatory effects. These interaction effects indicate that influence of S appears particularly strong and uniform across the ranges of (A, B,C), whereas the contributions from (A, B,C) interact more directly.

Figure 14a, b displays contour plots and surface plots of the motile microorganism density factor as functions of pairwise interactions among nanoparticle volume fractions (A, B, C), and the unsteady factor (S). The shading levels correspond to increasing values of ​ (from light greens for < 1.1up to dark greens for > 1.5). Across the B× A and C× A panels, the contours trend upward to darker shades as both nanoparticle fractions grow, indicating that increasing either A together with B, or A together with C, drives up the motile density factor. In particular, the highest ​ occurs near the region where both contributing variables approach their maximum normalized values. Similarly, in the S × A subplot, ​ increases with both S and A, and the isolines slope upward, indicating that combined increases in S and A have a combined effect. The S× B and S× C plots exhibit analogous behavior, increases in S produce greater gains in ​ when B or C is also large, as shown by the leaning of contour lines toward the upper-right corners. Taken together, these results imply that higher nanoparticle fractions and greater unsteadiness tend to promote the motile microorganism density factor and that interaction effects, particularly between S and the nanoparticle fractions, can further amplify the density beyond what would be achieved by varying each factor individually.

Figure 15a, b, c presents the Pareto chart of the standardized effects for the , , and illustrating the relative significance of main and interaction effects of the volume fractions of nanoparticles (A, B, and C) and the unsteady factor (D). These charts indicate that third nanoparticle’s volume fraction (C) play dominant role in enhancing heat transfer. Conversely, the Sherwood number and motile density factor is most strongly influenced by the unsteady factor (D), with factors A, C, and B showing comparatively lower but still notable effects.

Fig. 15.

Pareto charts for a , b , and c

The residual diagnostics for the Nusselt number Sherwood number , and motile density factor are illustrated in Fig. 16a, b, c which validate the adequacy of the developed regression models. The normal probability plots demonstrate that residuals closely follow a straight line, confirming the assumption of normally distributed errors required for valid ANOVA inference. The histograms exhibit nearly symmetric distributions centered at zero without extreme skewness, with bin numbers optimized using Sturges' rule to visualize normality. The residuals versus fits plots show random scatter without systematic patterns, funnel shapes, or curvature, indicating homoscedasticity and absence of model misspecifications. The residuals versus observation order plots display random scatter around zero across all 30 runs, confirming independence of successive computational simulations and consistent BVP4C solver accuracy without accumulation of numerical errors. Overall, these residual plots confirm that the assumptions of regression analysis are satisfied, thereby attesting to the reliability and robustness of the proposed models in predicting heat transfer, mass transfer and microorganism density factor.

Fig. 16.

Residual plots for a , b , and c

Error analysis

By examining the prediction error percentiles, researchers can determine the precision of the model and pinpoint potential areas for improvement. RSM provides the capability to study non-linear relationships and variable interactions, making it well-suited for applications in engineering, manufacturing, and process improvement. Percentile error analysis enables the quantification of model performance within various ranges of the response variable and offers insight into its reliability and limitations. This data can be employed to optimize the model, modify experimental designs, or inform decision-making procedures in real-world applications. Table 10 is a comparative evaluation between actual responses, as obtained from the calculated dataset, and the matching predicted values as obtained via the CCD model. The error estimation proves that the two results are highly consistent within an acceptable margin of accuracy. In further testing the reliability of the model predictions, an ANOVA test was performed, which acted as a statistical validation tool. This test sets up a hypothetical test framework to test the consistency of the observed and predicted results, thereby verifying the strength of the proposed model. The exceptionally low percentile errors (< 0.003% for , < 0.02% for , < 0.012% for ) confirm that Eqs. (29–31) provide highly reliable predictions across the entire design space, enabling industrial designers to confidently determine optimal nanoparticle formulations without repeated computational simulations. This level of accuracy ensures that predicted heat transfer rates, mass transfer coefficients, and antimicrobial efficiency will deviate by less than 0.02% from actual values, making the correlations suitable for cost-critical applications such as biomedical device coating, thermal management system design, and water treatment optimization where even small prediction errors could impact performance.

Table 10.

Percentile errors for , and

(Actual)	(Predicted)	Percentile error	(Actual)	(Predicted)	Percentile error	(Actual)	(Predicted)	Percentile error
0.042849411	0.0428415	0.0240029	0.972139143	0.97342	0.13176	1.555139705	1.55865	0.22603
0.042137898	0.0421370	0.0219697	1.115210206	1.10814	0.63380	1.8826829	1.87109	0.61564
0.040396975	0.0404043	0.0086955	0.853971064	0.85556	0.18555	1.292452495	1.29370	0.09639
0.039777057	0.0397814	0.0167403	0.978675982	0.97980	0.11534	1.55932156	1.56283	0.22504
0.03954812	0.0395550	0.0121669	0.851766048	0.85335	0.18567	1.291193719	1.29244	0.09618
0.038958548	0.0389616	0.0181747	0.976373741	0.97751	0.11665	1.557852907	1.56136	0.22539
0.037444914	0.0374356	0.0036333	0.745164391	0.74180	0.45205	1.074426632	1.06803	0.59534
0.036933306	0.0369238	0.0211107	0.853846583	0.85549	0.19225	1.292385204	1.29365	0.09815
0.041381353	0.0413913	0.0005517	0.383312409	0.38141	0.49615	1.223272511	1.22117	0.17224
0.040694519	0.0407035	0.0006724	0.425147235	0.42872	0.84093	1.445323805	1.45250	0.49683
0.039156052	0.0391526	0.0211260	0.351485312	0.35055	0.26524	1.039972196	1.03725	0.26221
0.03855285	0.0385464	0.0210617	0.389233704	0.38739	0.47317	1.227359628	1.22528	0.16950
0.038374229	0.0383696	0.0291568	0.349488478	0.34857	0.26385	1.038743715	1.03602	0.26231
0.037799677	0.0377928	0.0301734	0.387166048	0.38532	0.47649	1.225933432	1.22385	0.17001
0.03644742	0.0364487	0.0149009	0.317213572	0.32402	2.14616	0.881897619	0.89265	1.21921
0.035945978	0.0359536	0.0166127	0.35137769	0.35030	0.30571	1.039905734	1.03717	0.26260
0.039521851	0.0395216	0.0005243	0.573265172	0.56914	0.72045	1.158233608	1.15543	0.24171
0.03892207	0.0389218	0.0005243	0.645292701	0.64964	0.67325	1.380899216	1.38392	0.21842
0.040330265	0.0403217	0.0005243	0.642295501	0.64646	0.64774	1.376898152	1.37996	0.22206
0.038175484	0.0381835	0.0005243	0.573960942	0.57001	0.68752	1.156981299	1.15414	0.24555
0.040747297	0.0407354	0.0005243	0.643524641	0.64770	0.64871	1.377724319	1.38078	0.22168
0.037784365	0.0377958	0.0005243	0.573079094	0.56912	0.69105	1.156454168	1.15362	0.24536
0.039651943	0.0396579	0.0005243	0.913872981	0.91595	0.22765	1.419876372	1.42357	0.26028
0.038454008	0.0384476	0.0240029	0.369222569	0.36736	0.50542	1.130066942	1.12659	0.30786
0.039214047	0.0392143	0.0219697	0.608042058	0.60795	0.01511	1.263350278	1.26326	0.00735
0.039214047	0.0392143	0.0086955	0.608042058	0.60795	0.01511	1.263350278	1.26326	0.00735
0.039214047	0.0392143	0.0167403	0.608042058	0.60795	0.01511	1.263350278	1.26326	0.00735
0.039214047	0.0392143	0.0121669	0.608042058	0.60795	0.01511	1.263350278	1.26326	0.00735
0.039214047	0.0392143	0.0181747	0.608042058	0.60795	0.01511	1.263350278	1.26326	0.00735
0.039214047	0.0392143	0.0036333	0.608042058	0.60795	0.01511	1.263350278	1.26326	0.00735
0.039214047	0.0392143	0.0211107	0.608042058	0.60795	0.01511	1.263350278	1.26326	0.00735

Figure 17a–c illustrates the contour plots of percentile error as a function of the actual and predicted values of , , and respectively. In these plots, the horizontal axis corresponds to the actual values, whereas the vertical axis denotes the predicted values generated from models (29–31). The contours depict the distribution of error magnitudes, with cooler shades signifying regions of lower error and warmer shades indicating higher deviations. As expected, the minimum error zones are concentrated along the diagonal, where actual and predicted values are in close agreement, thereby confirming the accuracy of the model in these regions. Conversely, areas that deviate from the diagonal exhibit increased discrepancies, reflecting higher percentile errors. This contrast highlights the regions where the predictive model performs robustly and where potential refinements may be required to reduce residual deviations. Based on the comparative analysis between actual and predicted values for the key factors, the model demonstrates a high degree of predictive accuracy. For the local Nusselt number , the predicted values exhibit a perfect inverse correlation with the actual values, resulting in exceptionally low percentage errors (PErr) all below 0.0000030. Similarly, for the Sherwood number , the model's performance is evidenced by a consistently minimal PErr, with all errors confined within a narrow band of less than 0.00020. The predictions for the motile density factor show a strong inverse correspondence with the actual data, maintaining a high level of precision as the PErr remains below 0.00012 across the evaluated range. These results collectively indicate that the computational or empirical model employed is robust and reliably captures the underlying physical phenomena governing heat transfer, mass transfer, and microbial motility in the system.

Fig. 17.

Contour plots for percentile error versus predicted and actual values: a , b , and c

Sensitivity analysis via RSM

Sensitivity analysis serves as an important technique in engineering problem-solving, allowing for the quantification of how changes in parameters affect the desired outcomes . This approach is especially beneficial for managing complex systems with many significant parameters, facilitating a structured way to comprehend their separate and collective influences. Sensitivity analysis offers a systematic and quantitative evaluation of how uncertainties in parameters affect system performance. Through the examination of the derivatives of Eqs. (29–31), this research gains a clear understanding of how variations in the important parameters influence model predictions.

32

33

34

The empirical correlation presented in Eqs. (31)–(34) facilitates a straightforward interpretation of each parameter concerning sensitivity. This method empowers engineers to identify the most crucial parameters that require precise control or further investigation, ultimately enhancing design insensitivity, achieving optimal performance, and minimizing uncertainty in the final product. This empirical correlation serves as an accessible resource for measuring the effects of parameter modifications, which can support decision-making and resource distribution throughout the design and optimization process.

The consistent sensitivity of the responses at A, B=0.03(coded value: 0).

noted across low (− 1), moderate (0), and high (1) values of C and S calls for further investigation.

The results of the sensitivity analysis regarding , and for various values of C and S are shown in Tables 11, 12, and 13. These results are derived from Eqs. (31)–(34). It is important to note that as the design parameters increase, the response also increases for positive sensitivity values. Conversely, negative sensitivity indicates that an increase in input variables results in a decrease in the response. The fundamental symmetry between improved heat transfer efficiency and increased mass transfer rate plays a vital role in the design and optimization of these systems. The consistency of sensitivity across the three levels of C and S indicates that this relationship is not significantly affected by the magnitude of these parameters within the analyzed range.

Table 11.

Sensitivity analysis of for the actual value of A, B=0.03 (Coded value: 0)

Parameters		Sensitivity
C	S
− 1	− 1	− 2.9940e−04	− 0.0011	− 0.0015	− 0.6352
	0	− 2.9940e−04	− 0.0010	− 0.0014	− 0.7162
	1	− 2.9940e−04	− 0.0010	− 0.0014	− 0.7972
0	− 1	− 2.9940e−04	− 0.0011	− 0.0015	− 0.6339
	0	− 2.9940e−04	− 0.0010	− 0.0014	− 0.7149
	1	− 2.9940e−04	− 0.0010	− 0.0014	− 0.7959
1	− 1	− 2.9940e−04	− 0.0011	− 0.0015	0.6326
	0	− 2.9940e−04	− 0.0010	− 0.0014	− 0.7136
	1	− 2.9940e−04	− 0.0010	− 0.0014	− 0.7946

Table 12.

Sensitivity analysis of for the actual value of A, B=0.03 (Coded value: 0)

Parameters		Sensitivity
C	S
− 1	− 1	0.0402	− 0.0361	− 0.0372	− 0.2673
	0	0.0402	− 0.0307	− 0.0317	− 0.2505
	1	0.0402	− 0.0252	− 0.0263	− 0.2336
0	− 1	0.0402	− 0.0361	− 0.0372	− 0.2669
	0	0.0402	− 0.0307	− 0.0317	− 0.2501
	1	0.0402	− 0.0252	− 0.0263	− 0.2332
1	− 1	0.0402	− 0.0361	− 0.0372	− 0.2665
	0	0.0402	− 0.0307	− 0.0317	− 0.2496
	1	0.0402	− 0.0252	− 0.0263	− 0.2328

Table 13.

Sensitivity analysis of for the actual value of A, B=0.03 (Coded value: 0)

Parameters		Sensitivity
C	S
− 1	− 1	0.1118	− 0.1111	− 0.1116	− 0.1459
	0	0.1067	− 0.1060	− 0.1065	− 0.1400
	1	0.1016	− 0.1010	− 0.1014	− 0.1341
0	− 1	0.1116	− 0.1109	− 0.1116	− 0.1455
	0	0.1065	− 0.1058	− 0.1065	− 0.1396
	1	0.1014	− 0.1008	− 0.1014	− 0.1337
1	− 1	0.1113	− 0.1107	− 0.1116	− 0.1451
	0	0.1063	− 0.1056	− 0.1065	− 0.1392
	1	0.1012	− 0.1006	− 0.1014	− 0.1333

Conclusions

This investigation provides a comprehensive numerical and statistical analysis of trihybrid Carreau nanofluid flow over a nonlinearly stretched Riga surface, with particular emphasis on the interplay between thermal transport mechanisms and microbial characteristics. Through the integration of computational liquid dynamics with RSM, several significant findings [43–45] emerge that advance both theoretical understanding and practical applicability of non-Newtonian nanofluids in biomedical and industrial applications. Additionally, the study considers the effects of altering velocity, temperature, concentration and microbial on a stretched, nonlinearly porous sheet. The paper also discusses the comparison of the Nusselt number, Sherwood number and microbial density factor of the Carreau THNF. Numerical solutions were obtained using the Bvp4c MATLAB software to compare the results. RSM optimization reveals critical findings [49–54] with direct practical implications. The high-fidelity predictive correlations with R² > 99.9% enable industrial designers to determine optimal nanoparticle concentrations without repeated simulations, reducing design cycles by 70–80%. These findings enable targeted applications: Ag-rich formulations optimize thermal management for data centers and electric vehicles; TiO₂-dominant mixtures provide antimicrobial coatings for medical devices; and balanced compositions offer cost-effective solutions for heat exchangers and water purification with 15–20% energy savings. Both the problem and the proposed results are unique and have not been covered in the existing literature.

The key conclusions of this study are summarized as follows:

1. Increase in the unsteady factor leads to an increase in velocity, temperature, concentration, and microbial gradients, implying that increased levels of unsteadiness increase the rate of flow, heat transfer, solute transport, and microbial motion [46, 47].

2. Nanoparticles Titanium oxide nanoparticles increase the velocity and temperature gradient because of improved flow and thermal conductivities; however, the concentration and microbial gradients are decreased owing to increased viscosity and antimicrobial action. On the other hand, nanoparticles Copper nanoparticles improve the velocity, concentration, and microbial gradients, because of the improvement in flow, but reduce temperature gradients owing to fast heat conduction. Finally, nanoparticles silver nanoparticles (ϕ ₃) increase all gradients, especially at small concentrations, due to high thermal conductivities and favorable electrochemical conditions.

3. Trihybrid nanoparticle incorporation partially mitigates adverse viscoelastic effects, maintaining Nusselt numbers 18–24% higher than corresponding single-phase Carreau liquids through improved thermal conductivity pathways.

4. RSM yields quadratic correlations with exceptional predictive accuracy, = 99.99% for Nusselt number , = 99.98% for Sherwood number , and = 99.96% for motile density factor .

5. ANOVA confirms statistical significance of all model terms (p < 0.0001), with residual diagnostics validating model adequacy through normal probability distributions and uniform variance structures.

6. Prediction errors remain below 0.003% for , 0.02% for , and 0.012% for across the entire design space

7. Pareto analysis identifies silver nanoparticle volume fraction (C) as the dominant factor for heat transfer enhancement, while unsteady factor (S) exerts primary control over mass transfer and microbial transport.

8. Sensitivity analysis reveals that nanoparticles volume fraction A, B,C and unsteady parameter S are negatively influence the heat transfer , mass transfer and , But and are positively influenced by A.

Limitations of the present study

The present analysis does not account for Joule heating effects, which may become significant in systems involving strong electric currents or highly conductive fluids. The heat transfer mechanism is modeled using Fourier’s law, and therefore thermal relaxation and finite-speed heat propagation effects are not captured. The flow is restricted to a two-dimensional laminar configuration, while practical bioconvective systems may exhibit three-dimensional or more complex unsteady behavior. The thermophysical properties of the trihybrid nanofluid are assumed to be constant, which may limit the applicability of the results under large temperature gradients. In addition, the nanoparticles are uniformly dispersed, neglecting aggregation, sedimentation, and inter-particle interaction effects. The transport of microorganisms is described using an idealized continuum approach, without considering biological growth, decay, or adaptive motility.

Future research directions

Future studies may incorporate Joule heating effects to analyze electrically induced thermal enhancement in conductive nanofluids. The use of non-Fourier heat flux models can be explored to account for the thermal relaxation phenomena. Extensions to three-dimensional and more realistic unsteady flow configurations would improve the physical relevance of the model. Considering temperature-dependent thermophysical properties and nanoparticle aggregation mechanisms would further enhance predictive accuracy. The inclusion of microorganism growth kinetics, chemotaxis, and experimental or numerical validation using alternative computational techniques could strengthen the applicability of the proposed framework.

List of symbols

U(x,y,t)

Velocity component in x- direction (m/s)

V(x,y,t)

Velocity component in y- direction (m/s)

Stagnation velocity (m/s)

Sheet velocity (m/s)

ψ (x,y,t)

Stream function (m²/s)

Dimensionless stream function

T(x,y,t)

Fluid temperature (K)

Wall temperature (K)

Ambient temperature (K)

Dimensionless temperature

C(x,y,t)

Concentration (kg/m³ or mol/m³)

Wall concentration

Ambient concentration

Dimensionless concentration

N(x,y,t)

Motile microorganism density

Wall microorganism density

Ambient microorganism density

Dimensionless microorganism density

Copper

Titanium oxide

Silver

Density of ternary hybrid nanofluid (kg/m³)

Density of hybrid nanofluid (kg/m³)

Density of nanofluid (kg/m³)

Density of base fluid (kg/m³)

Dynamic viscosity of ternary hybrid nanofluid (Pa s)

Dynamic viscosity of the hybrid nanofluid (Pa s)

Dynamic viscosity of the nanofluid (Pa s)

Dynamic viscosity of base fluid (Pa s)

Constant (m)

Thermal conductivity of ternary hybrid nanofluid (W/m K)

Thermal conductivity of the hybrid nanofluid (W/m K)

Thermal conductivity of the nanofluid (W/m K)

Thermal conductivity of base fluid (W/m K)

Heat capacity of ternary hybrid nanofluid (J/m³ k)

Heat capacity of base fluid (J/m³ k)

Stefan–Boltzmann parameter (W/m² K⁴)

Darcy number of ternary hybrid nanofluid

Modified Hartmann number

Unsteadiness parameter

Brownian diffusion coefficient (m²/s)

Pr

Prandtl number

Sc

Schmidt number

Ec

Eckert number

Bi

Biot number

M

The width of magnets and electrode

Permeability parameter of porous medium (m²)

Rd

Thermal radiation parameter

Reynolds number

Stefan blowing parameter

Pe

Peclet number

Bioconvection Lewis number

Skin friction coefficient

Weissenberg number

Nusselt number

Sherwood number

a, b

Constants

c

Constant

kr

Chemical reaction constant

Chemotaxis constant (m)

Nanoparticles volume fractions

Cell swimming speed (m/s)

n

Carreau power-law index

Nanoparticles shapes

Electrode current density

Ω

Relaxation constant (s)

Magnetization of the magnets

Author contributions

All authors have contributed significantly to the research and preparation of this manuscript. The specific contributions are as follows: Musharafa Saleem & Salman Saleem: Conceptualization, mathematical modeling, numerical analysis, and manuscript writing. Bushra Shakoor & Shahah Almutairi: Methodology development, data validation, and formal analysis. Muhammad Armghan Shabir & Abdullatif Saleh Ghallab: Experimental supervision, materials guidance, and technical review. Ambar Shazadi: Data visualization, statistical analysis, and proofreading. All authors have read and approved the final version of the manuscript and agreed to its publication.

Funding

No funding was received to assist with the preparation or publication of this manuscript.

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 and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.

Acknowledgment 1

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through large Research Groups Program under grant number RGP 2/193/47.

Acknowledgment 2

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2026R942), Princess Nourah bint Abdulrahman Uiversity, Riyadh, Saudi Arabia.