Insights for the Impacts of Inclined Magnetohydrodynamics, Multiple Slips, and the Weissenberg Number on Micro-Motile Organism Flow: Carreau Hybrid Nanofluid Model

Abstract

This study focuses on the analysis of the simultaneous impact of inclined magnetohydrodynamic Carreau hybrid nanofluid flow over a stretching sheet, including microorganisms with the effects of chemical reactions in the presence and absence of slip conditions for dilatant $(n>1.0)$ and quasi-elastic hybrid nanofluid $(n<1.0)$ limitations. Meanwhile, the transfer of energy is strengthened through the employment of heat sources and bioconvection. The analysis incorporates nonlinear thermal radiation, chemical reactions, and Arrhenius activation energy effects on different profiles. Numerical simulations are conducted using the efficient Bvp5c solver. Motile concentration profiles decrease as the density slip parameter of the motile microbe and Lb increase. The Weissenberg number exhibits a distinct nature depending on the hybrid nanofluid; the velocity profile, skin friction, and Nusselt number fall when $(n>1.0)$ and increase when $(n<1.0)$. For small values of inclination, the 3D surface plot is far the surface, while it is close to the surface for higher values of inclination but has the opposite behavior for the 3D plot of the Nusselt number. A detailed numerical investigation on the effects of important parameters on the thermal, concentration, and motile profiles and the Nusselt number reveals a symmetric pattern of boundary layers at various angles $\left(\alpha \right)$. Results are presented through tables, graphs, contour plots, and streamline and surface plots, covering both shear-thinning cases $(n<1.0)$ and shear-thickening cases $(n>1.0)$.

1. Introduction

Nano-fluids, first introduced in [1], are produced by suspending metallic or non-metallic nanoparticles in traditional heat transfer fluids to significantly enhance their thermal performance. The thermal conductivity of the nanofluid with copper nanophase materials was investigated, and the findings showed that one benefit of employing nanofluids is a significant decrease in heat exchanger pumping power. In recent years, extensive studies have examined slip effects in hybrid nanofluid flows over stretching or shrinking sheets. Thermal radiation, chemical reactions, and magnetohydrodynamics (MHD) were among the factors taken into consideration in these experiments, as explained by Sharma et al. [2].

Eid et al. [3] explored Carreau nanofluid’s boundary-layer flow across a nonlinearly stretched sheet, as well as chemical reactions and heat generation in porous media. Regarding the effects of chemical reaction against magnetohydrodynamics, Carreau–Yasudab investigated nanofluid flow caused by a nonlinear stretched surface under zero normal flux, slip, and convective boundary conditions [4]. The effect of a magnetic field and effects on a generalized Carreau fluid’s flow over a porous cylindrical surface without a slip requirement was investigated in the presence of thermal radiation, thermal stratification, and heat generation [5]. Reddy et al. [6] conducted a numerical investigation on Carreau nanofluid’s unsteady three-dimensional hydromagnetic flow over a sheet with the influences of velocity slip, thermal slip, solutal slip, Soret, and Dufour. The behavior of Newtonian, Maxwell, and Casson fluids in the existence of an induced MHD influence was compared using the slip effect and the bioconvective phenomenon in a study by Gautam et al. [7]. The decrease in the thicknesses of the momentum profile, thermal distribution, and concentration boundary layers when velocity, thermal, and mass slips are included was studied in hybrid nanofluid flow with various slips over stretching or shrinking sheets in recent research [8,9].

The role of microorganisms in bioconvective nanofluids has also received attention. Abdeen et al. [10] demonstrated that gyrotactic microorganisms enhance nanoparticle stability, and Rehman et al. [11] incorporated chemical reactions, MHD, and heat radiation into bioconvection models. These studies often use advanced numerical techniques such as the homotopy analysis method and the bvp4c function [12].

Bilal [13] identified the activation energy factors influencing the flow of Carreau fluid over a nonlinear stretching surface, including heat transfer and the Arrhenius relation.

Hayat et al. [14] looked at the melting heat transfer of nanofluid flow toward an extended surface in the presence of heat generation, a nonlinear thermal radiation parameter, and a magnetic field. The irreversibility of the Fe₃O₄-CO Kerosene hybrid nanofluid flow beyond a wedge with a heat source and radiation parameter was explained by Mabood et al. [15]. The Carreau fluid model was significantly increased by a high-volume-fraction parameter of hybrid particles more than in the Casson fluid model [16]. The theoretical and mathematical discussion of nanofluids, with the effects of the heat absorption parameter and Joule parameter of Carreau nanofluids in a magnetohydrodynamic flow with viscous dissipation, was provided by Rehman et al. [17]. It was found that the fluid characteristics and magnetic field have opposing effects on the flow [18].

Prasad et al. [19] investigated the influence of magnetic fields, radiation, heat sources, and porosity on hybrid nanofluid flow characteristics. Kathuroju et al. [20] observed that increasing the Jeffrey fluid parameter enhanced the Nusselt number and skin friction but reduced the Sherwood number. Vijaya et al. [21] reported that exponential heat source parameters increased temperature profiles, while variable viscosity decreased fluid velocity. These effects significantly influence heat and mass transfer in hybrid nanofluids, which consist of multiple nanoparticles dispersed in a base fluid [22,23]. For shear-thickening fluids, the radiation factor raised the fluid heat, while the chemical reaction factor lowered the flow concentration. In contrast, the local Weissenberg number had the opposite impact on the Sherwood and Nusselt numbers for shear-thinning fluids [24].

The magnetic parameter significantly increases shear stress and fluid velocity, while decelerating temperature and concentration curves [25]. The study of thermophoresis and Brownian motion effects on inclined MHD nanofluid flow with multiple slip conditions by Patel and Nanda [26] shows that while the inclination parameter and magnetic field parameter tend to reduce fluid motion, thermophoresis and Brownian motion improve the heat transfer process. The effects of an inclined magnetic field and a constant chemical reaction on the tri-hybrid Carreau nanofluid slip flow of blood across a porous inclined stenosed artery with viscous dissipation were studied by Kumar et al. [27]. Using a wedge-shaped artery and thermal radiation, the study examined the thermal transport properties of a tri-hybrid Carreau nanofluid (blood) containing copper (Cu), titanium dioxide (TiO₂), and aluminum oxide (Al₂O₃) nanoparticles with an inclined magnetic field [28].

Notwithstanding the expanding literature, considerable research gaps endure. A comprehensive analysis that integrates inclined MHD, multiple slip conditions, nonlinear thermal radiation, and Arrhenius activation energy within a Carreau hybrid nanofluid framework is notably absent. Furthermore, the confluence of nonlinear thermal conductivity and bioconvection phenomena in hybrid nanofluids comprising multiple nanoparticles has received scant attention. While the individual influences of parameters like slip, activation energy, and inclined magnetic fields have been investigated, their collective effect on heat and mass transfer dynamics is still poorly understood.

This study investigates the flow of a Carreau hybrid nanofluid, comprising Al₂O₃-Fe₂O₃ nanoparticles, over a nonlinear stretching sheet. The analysis incorporates the effects of inclined magnetohydrodynamics (MHD), multiple slip boundary conditions, nonlinear thermal radiation, Arrhenius activation energy, and non-uniform thermal conductivity for both dilatant and quasi-elastic fluid regimes. Furthermore, the model includes Brownian motion, thermophoresis, and bioconvection of motile microorganisms. The governing partial differential equations (PDEs) are converted into a system of ordinary differential equations (ODEs) through appropriate similarity transformations. This system is then solved numerically using the Bvp5c solver in MATLAB. The results are comprehensively presented through velocity, temperature, concentration, and microorganism profiles, alongside the variations in the skin friction coefficient, Nusselt number, and Sherwood number across a range of physical parameters.

2. Physical Pattern and Interpretation

This section presents a two-dimensional mathematical model for heat and mass transfer. The analysis encompasses the flow of a Carreau-type hybrid nanofluid containing microorganisms over a nonlinear stretching surface. The model incorporates inclined magnetohydrodynamics (MHD), chemically reactive species, thermal radiation, surface chemical reaction (kr), activation energy (Ea), and a multi-slip boundary condition. These elements are included to enhance the comprehensiveness and applicability of the research.

Assumption 1. 

Consider a laminar, two-dimensional, incompressible, water-based hybrid nanofluid (see [Figure 1]) containing ${\mathit{Al}}_2{O}_3$ and ${\mathit{Fe}}_3{O}_4$ nanoparticles. The flow is induced by a stretching surface with a velocity defined as $u=b{x}^r$, where b and r are positive constants. The model incorporates the effects of Brownian motion, thermophoresis, and a magnetic field. Heat transfer is governed by a convective boundary condition, where a hot working fluid at temperature $T_f$ with a heat transfer coefficient $h_f$ heats the nanofluidic medium. The ambient temperature and concentration, attained as y approaches infinity, are denoted by $T\infty$ and $C\infty$, respectively.

Governing equations [29,30,31]

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial v}{\partial y}}+{\displaystyle \frac{\partial u}{\partial x}}=0\hfill & \hfill & \\ \hfill & u.{\displaystyle \frac{\partial u}{\partial x}}+v.{\displaystyle \frac{\partial v}{\partial y}}={\nu }_{hnf}{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}{\left[1+{\Gamma }^2{\left({\displaystyle \frac{\partial u}{\partial y}}\right)}^2\right]}^{{\displaystyle \frac{-(1-n)}{2}}}-{\displaystyle \frac{{\sigma }_{hnf}}{{\rho }_{hnf}}}{B}^{2}usi{n}^2\left(\alpha \right)\hfill & \hfill & \end{array}$$

(1)

$$\begin{array}{cccc}\hfill & +{\nu }_{hnf}(n-1){\Gamma }^2{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}{\left({\displaystyle \frac{\partial u}{\partial y}}\right)}^2{\left[1+{\Gamma }^2{\left({\displaystyle \frac{\partial u}{\partial y}}\right)}^2\right]}^{{\displaystyle \frac{-(3-n)}{2}}}\hfill & \hfill & \\ \hfill & u{\displaystyle \frac{\partial T}{\partial x}}+v{\displaystyle \frac{\partial T}{\partial y}}={\displaystyle \frac{K_{hnf}}{{\left(\rho C_p\right)}_{hnf}}}{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}+{\displaystyle \frac{16{\sigma }^{*}}{3m^{*}{\left(\rho C_p\right)}_{hnf}}}{\displaystyle \frac{\partial }{\partial y}}\left(T^3{\displaystyle \frac{\partial T}{\partial y}}\right)+{\displaystyle \frac{Q_{*}}{{\left(\rho C_p\right)}_{hnf}}}(T-T_{\infty })\hfill & \hfill & \end{array}$$

(2)

$$\begin{array}{cccc}\hfill & +{\displaystyle \frac{{\left(\rho C_p\right)}_p}{{\left(\rho C_p\right)}_{hnf}}}\left[D_B{\displaystyle \frac{\partial C}{\partial y}}{\displaystyle \frac{\partial T}{\partial y}}+{\displaystyle \frac{D_T}{T_{\infty }}}{\left({\displaystyle \frac{\partial T}{\partial y}}\right)}^2\right]\hfill & \hfill & \end{array}$$

(3)

$$\begin{array}{cccc}\hfill & v.{\displaystyle \frac{\partial N}{\partial y}}+u.{\displaystyle \frac{\partial N}{\partial x}}=-{\displaystyle \frac{b{W}_c}{C_w-C_{\infty }}}\left[{\displaystyle \frac{\partial }{\partial y}}\left(N{\displaystyle \frac{\partial C}{\partial y}}\right)\right]+D_n\left({\displaystyle \frac{{\partial }^{2}N}{\partial y^2}}\right)\hfill & \hfill & \end{array}$$

(4)

$$\begin{array}{ccc}\hfill & v{\displaystyle \frac{\partial C}{\partial y}}+u{\displaystyle \frac{\partial C}{\partial x}}={\displaystyle \frac{D_T}{T_{\infty }}}{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}{D}_B{\displaystyle \frac{{\partial }^{2}C}{\partial y^2}}+kr(-1)(C-C_{\infty })exp\left({\displaystyle \frac{-E}{KT}}\right){\left({\displaystyle \frac{T}{T_{\infty }}}\right)}^n\hfill & \hfill \end{array}$$

(5)

Boundary conditions [32,33]:

$$\begin{array}{c}\hfill \left.\begin{array}{cccc}\hfill \left(a\right)& u=u=U_w={\displaystyle \frac{b{x}^r}{l}}+{\beta }_1{\displaystyle \frac{\partial u}{\partial y}}{\left(1+({\tau }_c{{\displaystyle \frac{\partial u}{\partial y}})}^2\right)}^{\frac{n-1}{2}},v=0,{\beta }_2{\displaystyle \frac{\partial T}{\partial y}}=\{T-T_w\},\hfill & \hfill & \\ \hfill & C=C_w+{\beta }_3{\displaystyle \frac{\partial C}{\partial y}},N=N_w+{\beta }_4{\displaystyle \frac{\partial N}{\partial y}}aty=0\hfill & \hfill & \\ \hfill \left(b\right)& u⟶\left(0\right),T⟶\left(T_{\infty }\right),N\to \left(N_{\infty }\right),C⟶\left(C_{\infty }\right)aty⟶\infty \hfill & \hfill & \end{array}\right\}\end{array}$$

(6)

Here, the velocity components along the x and y axes are denoted by u and v, respectively; $\nu$ is the kinematic viscosity; $\rho$ is the base fluid’s density; $Q_{*}$ is the coefficient of dimensional heat production/absorption; $D_B$ denotes the Brownian diffusion coefficient; and $D_T$ is the thermophoretic diffusion coefficient. Other parameters include the thermal relaxation time $\left(\delta \right)$, the chemical reaction parameter $\left(kr\right)$, the magnetic field strength $\left(B\right)=B_o{x}^{(r-1/2)}$ [34], the activation energy $\left(Ea\right)$, and the relaxation parameter $(\Gamma )$. The fluid properties are assumed constant except for the density variation, which induces a thermal buoyancy force.

The following non-dimensional variables are introduced to simplify the problem:

$$\begin{array}{c}\left.\begin{array}{c}\hfill u={\displaystyle \frac{\partial \varphi }{\partial x}}=b{x}^r{f}^{\prime },v=-{\displaystyle \frac{\partial \varphi }{\partial y}}=-\sqrt{{\displaystyle \frac{(r+1)\nu }{2}}b{x}^{r-1}}\left[f^{\prime }\left({\displaystyle \frac{r-1}{r+1}}\right)\eta +f\right],\\ \hfill \varphi (x,y)=\sqrt{{\displaystyle \frac{2\nu b}{r+1}}}{x}^{{\displaystyle \frac{r+1}{2}}}f\left(\eta \right),\eta =y{x}^{{\displaystyle \frac{r-1}{2}}}\sqrt{{\displaystyle \frac{b(r+1)}{2\nu }}},\\ \hfill \theta ={\displaystyle \frac{T_{\infty }-T}{T_{\infty }-T_w}},\Psi ={\displaystyle \frac{C-C_{\infty }}{C_w-C_{\infty }}}\chi ={\displaystyle \frac{N-N_{\infty }}{N_w-N_{\infty }}},\end{array}\right\}\end{array}$$

(7)

Ordinary differential equations (ODEs) are derived by applying the similarity transformations and substituting Equation (6) into the governing partial differential Equations (2)–(7):

$$\begin{array}{c}{\left(1+W{e}^2{f}^{″2}\right)}^{{\displaystyle \frac{-(n-3)}{2}}}\left(1+nW{e}^2{f}^{″2}\right)f^{‴}+{\displaystyle \frac{{\nu }_f}{{\nu }_{hnf}}}f{f}^{″}-{\displaystyle \frac{2m}{m+1}}{\displaystyle \frac{{\nu }_f}{{\nu }_{hnf}}}{f}^{\prime 2}-{\displaystyle \frac{2}{m+1}}{\displaystyle \frac{{\sigma }_{hnf}}{{\sigma }_f}}{\displaystyle \frac{{\mu }_f}{{\mu }_{hnf}}}Msi{n}^2\left(\alpha \right)f^{\prime }=0\\ {\displaystyle \frac{k_{hnf}}{k_f}}{\theta }^{″}+{\displaystyle \frac{1}{3}}Rd4{(1+\theta ({\theta }_w-1))}^3{\theta }^{″}+4Rd{(1+({\theta }_w-1)\theta )}^2({\theta }_w-1){\theta }^{\prime 2}+{\displaystyle \frac{2Q_{e}Pr}{m+1}}\theta \end{array}$$

(8)

$$\begin{array}{c}+Pr{\displaystyle \frac{{\left(\rho C_p\right)}_{hnf}}{{\left(\rho C_p\right)}_f}}(Nb{\theta }^{\prime }{\psi }^{\prime }+Nt{\theta }^{\prime 2}+f{\theta }^{\prime })=0\end{array}$$

(9)

$$\begin{array}{c}{\chi }^{″}-Lb(\chi f^{\prime }-f{\chi }^{\prime })-Pe\left[(\chi +\omega ){\phi }^{″}+{\Psi }^{\prime }{\chi }^{\prime }\right]=0\end{array}$$

(10)

$$\begin{array}{c}{\Psi }^{″}+S_{c}f{\theta }^{\prime }+{\displaystyle \frac{Nt}{Nb}}{\theta }^{″}-{\displaystyle \frac{2}{m+1}}{S}_{c}Kr{(1+\delta \theta )}^{n}exp\left({\displaystyle \frac{-Ea}{1+\delta \theta }}\right)\Psi =0\end{array}$$

(11)

Moreover, the controlled limitations at the boundary are

$$\begin{array}{c}\left.\begin{array}{c}f\left(\eta \right)=0,f^{\prime }\left(\eta \right)=1+{(1+{W_e}^2{f^{″}}^2)}^{(n-1/2)}{\lambda }_1{f}^{″},\theta \left(\eta \right)=1+{\lambda }_2{\theta }^{\prime }\left(\eta \right),\chi \left(\eta \right)=1+{\lambda }_3{\chi }^{\prime },\Psi \left(\eta \right)=1+{\lambda }_4{\Psi }^{\prime }at\eta =0\hfill \\ f^{\prime }\left(\eta \right)⟶0,\theta \left(\eta \right)⟶0,{\chi }^{\prime }\left(\eta \right)⟶0,{\Psi }^{\prime }\left(\eta \right)⟶0at\eta ⟶\infty \hfill \end{array}\right\}\end{array}$$

(12)

3. Physical Quantities

The many contributing elements and involving factors are similarly described by Equations (7) through (10). The following parameters are considered: Péclet number $\left(Pe\right)$, Lewis number $\left(Lb\right)$, heat source $\left(Qe\right)$, microorganism difference parameter $\left(w\right)$, thermal slip factor $\left({\lambda }_2\right)$, concentration slip factor $\left({\lambda }_3\right)$, density slip factor $\left({\lambda }_4\right)$, Prandtl number $\left(Pr\right)$, magnetic parameter $\left(M\right)$, Weissenberg number $\left(We\right)$, and parameter $(\Gamma )$.

$$\begin{array}{c}\left.\begin{array}{c}\hfill W{e}^2={\displaystyle \frac{{\Gamma }^2{b}^3{x}^{3m-1}(m+1)}{2\nu }},Nb={\displaystyle \frac{\tau D_B{C}_{\infty }}{\nu }},Nt={\displaystyle \frac{\tau D_T(T_w-T_{\infty })}{\nu }},Rd={\displaystyle \frac{4{\sigma }^{*}{T}_{\infty }^3}{m^{*}\kappa }}\\ \hfill Qe={\displaystyle \frac{Q^{*}}{b{x}^{m-1}{\left(\rho C_p\right)}_f}},{\theta }_w={\displaystyle \frac{T_w}{T_{\infty }}},\delta ={\theta }_w-1,Sc={\displaystyle \frac{\nu }{D_B}},Ea={\displaystyle \frac{E}{K{T}_{\infty }}},Pr={\displaystyle \frac{\nu }{\alpha }},M={\displaystyle \frac{B_0^2{\sigma }_f}{{\rho }_{f}b{x}^{m-1}}}\\ \hfill \alpha ={\displaystyle \frac{\kappa }{{\left(\rho C_p\right)}_f}},Kr={\displaystyle \frac{kr}{b{x}^{m-1}}},\gamma ={\beta }_1\sqrt{{\displaystyle \frac{b(m+1)}{2\nu }}}{x}^{{\displaystyle \frac{m-1}{2}}},\\ \hfill Pr={\displaystyle \frac{{\mu }_f}{{\rho }_f{\alpha }_f}}Pe={\displaystyle \frac{b{W}_c}{D_m}},\omega ={\displaystyle \frac{N_{\infty }}{N_w-N_{\infty }}},Lb={\displaystyle \frac{{\mu }_f}{{\rho }_f{D}_m}},\end{array}\right\}\end{array}$$

4. Engineering Quantities

Key engineering parameters of interest include the skin friction factor $C{f}_{x}R{e}_x^{1/2}$, the local Sherwood number $S{h}_{x}R{e}_x^{-1/2}$, and the local Nusselt number $N{u}_{x}R{e}_x^{-1/2}$.

$$\begin{array}{cccc}\hfill & {\tau }_w={\left[{\mu }_0{\displaystyle \frac{\partial u}{\partial y}}{\left(1+{\Gamma }^2{\left({\displaystyle \frac{\partial u}{\partial y}}\right)}^2\right)}^{n-1/2}\right]}_{y=0}{q}_w={\left[\kappa {\displaystyle \frac{\partial T}{\partial y}}-{\displaystyle \frac{16{\sigma }^{*}{T}_{\infty }^3}{3\kappa }}{\displaystyle \frac{\partial T}{\partial y}}\right]}_{y=0},\hfill & \hfill & \\ \hfill & j_m=-D_B{\left[{\displaystyle \frac{\partial C}{\partial y}}\right]}_{y=0}S{h}_x={\displaystyle \frac{-x}{C_w-C_{\infty }}}\left({\displaystyle \frac{\partial C}{\partial y}}\right)N_{nx}={\displaystyle \frac{-x{D}_n}{D_n(N_w-N_{\infty })}}\left({\displaystyle \frac{\partial N}{\partial y}}\right)\hfill & \hfill \end{array}$$

Applying the aforementioned similarity transformations allows for the computation of the wall shear stress ${\tau }_w$, the wall heat flux $q_w$, and the wall mass flux $j_m$.

$$\begin{array}{cccc}\hfill & C{f}_{x}R{e}_x^{{\displaystyle \frac{-1}{2}}}=\sqrt{{\displaystyle \frac{r+1}{2}}}{f}^{″}\left(0\right){\left(1+W{e}^2{\left(f^{″}\left(0\right)\right)}^2\right)}^{{\displaystyle \frac{n-1}{2}}}\hfill & \hfill & \end{array}$$

(13)

$$\begin{array}{cccc}\hfill & N{u}_{x}R{e}_x^{{\displaystyle \frac{-1}{2}}}=-{\displaystyle \frac{4}{3}}Rd\sqrt{{\displaystyle \frac{r+1}{2}}}\left(1+{(1-\theta \left(0\right)(1-{\theta }_w))}^3\right){\theta }^{\prime }\left(0\right)\hfill & \hfill & \end{array}$$

(14)

$$\begin{array}{ccc}\hfill & S{h}_{x}R{e}_x^{{\displaystyle \frac{-1}{2}}}=-\sqrt{{\displaystyle \frac{r+1}{2}}}{\displaystyle \frac{{\Psi }^{\prime }\left(0\right)}{\Psi \left(0\right)}}{\displaystyle \frac{N_{nx}}{\sqrt{R{e}_x}}}={\chi }^{\prime }\left(0\right)\hfill & \hfill \end{array}$$

(15)

Here, $R{e}_x=x{u}_w/{\nu }_f$ is the local Reynold’s number.

5. Execution of Methodology

By considering the effects of significant active physical factors, this work employs the Bvp5c method to solve the fundamental flow equations for MHD Carreau hybrid nanofluid flow across a stretched surface. Computational simulations were performed using MATLAB-2021. The highly nonlinear coupled ordinary differential equation (ODE) system, represented by Equations (8)–(11) along with boundary conditions (12), was solved using the built-in bvp5c solver in MATLAB. This technique is particularly well-suited for solving ODEs when the problem’s precise behavior is not known a priori, as it provides a flexible approach that effectively balances computational accuracy and efficiency. The solution process involves continually refining the initial guess for the unknown boundary limits based on the model’s residual. The initial guess strategy and solver setup are as follows:

For all dependent variables—$f\left(\eta \right)$, $\theta \left(\eta \right)$, $\Psi \left(\eta \right)$, and $\chi \left(\eta \right)$—an initial mesh of 200 uniformly spaced points is generated across the computational domain $0<\eta <{\eta }_{\infty }$. These initial guesses are iteratively refined by the Bvp5c solver until the solution residuals fall within a predefined tolerance. The initial profiles are constructed according to physically expected trends, such as a linear variation for velocity and an exponential decay for temperature, concentration, and motile microorganism density.

Tolerance and convergence criteria:

High numerical precision is achieved by setting both the relative and absolute error tolerances to ${10}^{-7}$. The solver iteratively updates the solution until the maximum residual norm, $\left|R\right(\eta \left)\right|$, is reduced below ${10}^{-7}$ throughout the entire domain. Additionally, it was verified that further mesh refinement (up to 200 points) resulted in changes of less than 0.01.

Choice of ${\eta }_{\infty }$ = 7 and asymptotic verification:

The far-field boundary, ${\eta }_{\infty }=7$, was selected based on a series of sensitivity tests where ${\eta }_{\infty }$ was varied from 6 to 10. The results showed that for values beyond $\eta =7$, the velocity, temperature, concentration, and motile microorganism profiles all asymptotically approached their ambient values ($T_{\infty }$, $C_{\infty }$), with subsequent computed variations being negligible (in the order of ${10}^{-7}$). This value ensures computational efficiency without compromising accuracy. The asymptotic convergence of the profiles was confirmed by visually comparing plots for progressively larger ${\eta }_{\infty }$ values until no discernible difference was observed in the curves.

$${\displaystyle \begin{array}{l}f={\zeta }_1, f^{\prime }={\zeta }_2, f^{″}={\zeta }_3, \theta ={\zeta }_4, {\theta }^{\prime }={\zeta }_5, \psi ={\zeta }_6, {\psi }^{\prime }={\zeta }_7, \chi ={\zeta }_8, {\chi }^{\prime }={\zeta }_9\\ {\zeta }_2^{\prime }={\zeta }_3, \\ {\zeta }_3^{\prime }=-{\displaystyle \frac{1}{{\left(1+W{e}^2{\zeta }_3^2\right)}^{{\displaystyle \frac{n-3}{2}}}\left(1+nW{e}^2{\zeta }_3^2\right)}}\left({\displaystyle \frac{{\nu }_f}{{\nu }_{hnf}}}{\zeta }_1{\zeta }_3-{\displaystyle \frac{2m}{m+1}}{\displaystyle \frac{{\nu }_f}{{\nu }_{hnf}}}{\zeta }_2^2-{\displaystyle \frac{2}{m+1}}{\displaystyle \frac{{\sigma }_{hnf}}{{\sigma }_f}}{\displaystyle \frac{{\mu }_f}{{\mu }_{hnf}}}Msi{n}^2\left(\alpha \right)f{\zeta }_2\right)=0\\ {\zeta }_4^{\prime }={\zeta }_5,\\ {\zeta }_5^{\prime }=-{\displaystyle \frac{1}{{\displaystyle \frac{k_{hnf}}{k_f}}+{\displaystyle \frac{4}{3}}Rd{(1+({\theta }_w-1){\zeta }_4)}^3}}\left(4Rd{(1+({\theta }_w-1){\zeta }_4)}^2({\theta }_w-1){\zeta }_5^2+{\displaystyle \frac{2Q_{e}Pr}{m+1}}{\zeta }_4\right)\\ {\zeta }_6^{\prime }={\zeta }_7,\\ {\zeta }_7^{\prime }=-\left(S_{c}f{\zeta }_5+{\displaystyle \frac{Nt}{Nb}}{\zeta }_5^{\prime }-{\displaystyle \frac{2}{m+1}}{S}_{c}Kr{(1+\delta {\zeta }_4)}^{n}exp\left({\displaystyle \frac{-E}{1+\delta {\zeta }_4}}\right){\zeta }_6\right)\\ {\zeta }_8^{\prime }={\zeta }_9\\ {\zeta }_9^{\prime }=[L_b({\zeta }_2{\zeta }_8+{\zeta }_1{\zeta }_9)+Pe(({\zeta }_7^{\prime }+w)+{\zeta }_7{\zeta }_9)]\\ \mathrm{boundary\; conditions}\\ {\zeta }_2\left(0\right)=1+{\lambda }_1{\zeta }_3\left(0\right){(1+{W_e}^2{{\zeta }_3}^2)}^{((n-1)/2)}, {\zeta }_4\left(0\right)=1+{\lambda }_2{\zeta }_5, {\zeta }_6\left(0\right)=1+{\lambda }_3{\zeta }_7, {\lambda }_8\left(0\right)=1+{\lambda }_4{\zeta }_9,\\ {\zeta }_2\left(\eta \right)⟶0, {\zeta }_4\left(\eta \right)⟶0, {\zeta }_6\left(\eta \right)⟶0{\zeta }_8\left(\eta \right)⟶0 at \eta ⟶\infty \end{array}}$$

The rescaled infinity condition is now $\eta$ = 7. The thermal and physical properties of the hybrid nanofluid are provided in

Table 1. The physical properties [35,36].

Physical Properties	$\mathit{\rho }(\mathbf{kg}·{\mathbf{m}}^{-3})$	${\mathit{C}}_{\mathit{p}}(\mathbf{J}/\mathbf{kg}·\mathbf{K})$	$\mathit{\kappa }(\mathbf{W}/\mathbf{m}·\mathbf{K})$	$\mathit{\sigma }\times {10}^5{\mathit{k}}^{-1}$
${\mathrm{Al}}_2{\mathrm{O}}_3$	3970	765	040	$03.69\times {10}^7$
${\mathrm{Fe}}_3{\mathrm{O}}_4$	05180	0670	$09.7$	$02.5\times {10}^4$
${\mathrm{H}}_2\mathrm{O}$	$0997.1$	04179	$00.613$	$00.05$

and

Table 2. Physical properties [37,38].

$\mathbf{Physical}\mathbf{Properties}$	$\mathbf{In}\mathbf{Terms}\mathbf{of}\mathbf{Hybrid}\mathbf{Nanofluid}$
$\mathrm{Effective}\mathrm{Dynamic}\mathrm{Viscousity}$	${\displaystyle \frac{{\mu }_{hnf}}{{\mu }_f}}={\displaystyle \frac{1}{{(1-{\varphi }_1)}^{2.5}{(1-{\varphi }_2)}^{2.5}}}$
$\mathrm{Effective}\mathrm{Density}$	${\displaystyle \frac{{\rho }_{hnf}}{{\rho }_f}}=(1-{\varphi }_2)\left[(1-{\varphi }_1)+\left({\displaystyle \frac{{\rho }_{s1}}{{\rho }_f}}\right){\varphi }_1\right]+\left({\displaystyle \frac{{\rho }_{s2}}{{\rho }_f}}\right){\varphi }_2$
$\mathrm{Effective}\mathrm{Electrical}\mathrm{Conductivity}$	${\displaystyle \frac{{\sigma }_{hnf}}{{\sigma }_f}}=1+{\displaystyle \frac{3\left[{\displaystyle \frac{{\varphi }_1{\sigma }_{s1}+{\varphi }_2{\sigma }_{s2}}{{\sigma }_f}}-({\varphi }_1+{\varphi }_2)\right]}{2+\left[{\displaystyle \frac{{\varphi }_1{\sigma }_{s1}+{\varphi }_2{\sigma }_{s2}}{({\varphi }_1+{\varphi }_2){\sigma }_f}}\right]-\left[{\displaystyle \frac{{\varphi }_1{\sigma }_{s1}+{\varphi }_2{\sigma }_{s2}}{{\sigma }_f}}-({\varphi }_1+{\varphi }_2)\right]}}$
$\mathrm{Effective}\mathrm{Thermal}\mathrm{Conductivity}$	${\displaystyle \frac{{\kappa }_{hnf}}{{\kappa }_f}}=\left({\displaystyle \frac{{\kappa }_{hnf}}{{\kappa }_{nf}}}\right)\left({\displaystyle \frac{{\kappa }_{nf}}{{\kappa }_f}}\right)$
	where${\displaystyle \frac{{\kappa }_{hnf}}{{\kappa }_{nf}}}={\displaystyle \frac{{\kappa }_{s2}+2{\kappa }_{nf}-2{\varphi }_2({\kappa }_{nf}-{\kappa }_{s2})}{{\kappa }_{s2}+2{\kappa }_{nf}+{\varphi }_2({\kappa }_{nf}-k_{s2})}}$
	${\displaystyle \frac{{\kappa }_{hnf}}{{\kappa }_f}}={\displaystyle \frac{{\kappa }_{s1}+2{\kappa }_f-2{\varphi }_1({\kappa }_f-{\kappa }_{s1})}{{\kappa }_{s1}+2{\kappa }_f+{\varphi }_1({\kappa }_f-{\kappa }_{s1})}}$
$\mathrm{Specific}\mathrm{Heat}\mathrm{Capacity}$	${\displaystyle \frac{{\left(\rho C_p\right)}_{hnf}}{{\left(\rho C_p\right)}_f}}=(1-{\varphi }_2)\left[(1-{\varphi }_1)+\left({\displaystyle \frac{{\left(\rho C_p\right)}_{s1}}{{\left(\rho C_p\right)}_f}}\right){\varphi }_1\right]+\left({\displaystyle \frac{{\left(\rho C_p\right)}_{s2}}{{\left(\rho C_p\right)}_f}}\right){\varphi }_2$

. Numerical results obtained using the Bvp5c and Rk45 methods are compared with each other, and and results obtained using the Bvp5c and Rk45 methods is shown in

Table 3. Assessment of $-f^{\prime \prime }\left(0\right)$ for different values of n, where We and other parameters are constant.

n	${\mathit{W}}_{\mathit{e}}$	Bvp5c [39]	RKF45 [39]	RK4 [39]	Current Result
00.5	1.0	00.989702	00.989709	00.989709	00.989705
	2.0	00.901331	00.901343	00.901343	00.901340
	3.0	00.830948	00.830972	00.830972	00.830960
01.5	1.0	01.145708	01.145708	01.145708	01.145708
	2.0	01.367059	01.367061	01.367061	01.367058
	3.0	01.611580	01.611585	01.611585	01.611583

and

Table 4. Validation of $-{\theta }^{\prime }\left(\eta \right)$ at $\eta =0$ for different values of $Pr$.

$\mathit{Pr}$	Khan [40]	Wang [41]	Current Results
00.7	00.4539	00.453932	00.453946
02.0	00.9113	00.911359	00.911364
07.0	01.8954	01.895428	01.895437
020	03.3539	03.354174	03.354172

.

6. Results and Discussion

The effects of the physical parameters—M, $Ea$, $Rd$, ${\theta }_w$, ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, ${\lambda }_4$, and $Sc$—on the velocity profile $f^{\prime }\left(\eta \right)$, temperature distribution $\theta \left(\eta \right)$, hybrid nanofluid concentration profile $\Psi \left(\eta \right)$, and motile microorganism profile $\chi \left(\eta \right)$, along with the Nusselt number $N{u}_{x}R{e}_x^{-1/2}$, skin friction coefficient $C{f}_{x}R{e}_x^{1/2}$, and non-dimensional Sherwood number $S{h}_{x}R{e}_x^{-1/2}$, are illustrated using various graphical profiles and condition tables for both dilatant and pseudo-plastic fluids.

The variation in the velocity profile with the local Weissenberg number ($We$), under the combined influence of velocity slip (${\lambda }_1$) and the magnetic parameter ($\alpha$), is depicted in Figure 2a,b for both shear-thinning ($n<1.0$) and shear-thickening ($n>1.0$) fluids. It is observed that the fluid velocity responds differently to changes in $We$ depending on the fluid behavior: it decreases for shear-thickening fluids ($n>1.0$) but increases for shear-thinning fluids ($n<1.0$). Physically, the Weissenberg number represents the ratio of the fluid’s relaxation time to a characteristic process time. An increase in $We$ signifies a dominant elastic response, which enhances the effective viscosity and consequently reduces the fluid velocity. Furthermore, the figure shows that the dimensionless velocity decreases as $\alpha$ increases. This occurs because the magnetic parameter $\alpha$ is directly related to the Lorentz force, which opposes flow. A higher $\alpha$ strengthens this resistive magnetic effect, thereby decelerating the flow. The velocity slip parameter (${\lambda }_1$) quantifies the interaction between the fluid and a solid boundary. An increase in ${\lambda }_1$ implies greater slip, creating a larger effective gap between the fluid and the wall. This reduces momentum transfer from the boundary, resulting in a lower fluid velocity. Thus, the velocity profile declines as the slip parameter increases. This behavior of the velocity profile with the Weissenberg number is consistent across both shear-thinning and shear-thickening cases, which aligns with the findings of Sabu et al. [39].

Figure 3a,b demonstrate the influence of the Weissenberg number on the temperature profile for dilatant and pseudo-plastic fluids, respectively. The presence of thermal slip (${\lambda }_2$) at the boundary, alongside parameter $\alpha$, exhibits a concurrent effect on the profile. It is observed that the temperature profile for dilatant fluids intensifies with an increasing Weissenberg number ($We$), whereas pseudo-plastic fluids display the opposite trend. This temperature rise is attributed to extended particle relaxation times and a higher frequency of inter-particle collisions. Furthermore, the results indicate that the dimensionless temperature profile increases with higher values of $\alpha$. This correlation is due to an enhanced magnetic effect, which grows with $\alpha$. The transverse magnetic field acts to decelerate the fluid’s velocity, thereby significantly diminishing the rate of energy transmission. This increased flow resistance is a result of the strengthening Lorentz force associated with a higher magnetic parameter (M). Consequently, the thermal boundary layer thickens for both fluid types. Conversely, an increase in the thermal slip parameter (${\lambda }_2$) reduces the size of the thermal boundary layer and decreases the susceptibility of the fluid flow within it.

The radiation factor $\left(Rd\right)$ is influenced by the simultaneous effects of thermal slip (${\lambda }_2$) and $\alpha$ on the temperature profile of the Carreau fluid for both dilatant $(n<1.0)$ and pseudo-plastic $(n>1.0)$ hybrid nanofluids, as shown in Figure 4a,b. In both cases, an increase in the radiation factor $\left(Rd\right)$ corresponds to a rise in the nanofluid’s temperature profile. From a physical perspective, an increase in the radiation parameter enhances the emission of heat energy from the flow region. This process elevates the temperature of the nanofluid, thereby cooling the system. A higher radiation parameter introduces more heat into the system, and a thicker thermal boundary layer further contributes to this heating effect. The thermal field is also enhanced as K1 increases, due to a reduction in the mean absorption coefficient. Consequently, to achieve a more efficient cooling process, radiative heat transfer should be minimized. Furthermore, an increase in the thermal slip parameter (${\lambda }_2$) raises the temperature at the surface by inhibiting heat exchange between the fluid and the wall. This effect causes the fluid to retain more thermal energy, reduces the rate of heat transfer, and diminishes the thermal gradient. The overall temperature profile is consequently elevated due to the thickening of the thermal boundary layer.

Figure 5a,b show trends of $Qe$ over the temperature field for $(1.0>n)$ and $(n>1.0)$. The angle of inclination $\alpha$ and the exponentially space-dependent heat source $Qe$ affect the temperature profile, improving the thermal boundary layer profiles and corresponding thickness by raising $Qe$ for both scenarios. In practical systems, Qe models effects such as volumetric heating from chemical reactions, nuclear or electrical heating, and radiative absorption. It directly influences the energy balance by altering temperature gradients, modifying convection strength, and impacting overall heat transfer characteristics between the surface and the surrounding fluid. Figure 6a,b demonstrate the positive change in the temperature ratio parameter $\theta$ and temperature profile for dilatant and quasi-elastic liquids, respectively, with the parallel effect of $\alpha$ in the presence of thermal slip ${\lambda }_2$ at the boundary. Figure 7a,b illustrate the effects of the parameter $Kr$ and concentration slip ${\lambda }_3$ on the concentration profile for both dilatant and pseudo-plastic fluids with the angle of inclination $\alpha$. Increasing $Kr$ and $\alpha$ leads to a decline in the profile in both scenarios.

Figure 8a,b illustrate the characteristics of the concentration field under the influence of the Eckart number ($Ea$), the inclination angle ($\alpha$), and the parallel effects of the concentration slip parameter ${\lambda }_3$. The concentration profile is shown to increase with both the Eckart number and $\alpha$. Physically, activation energy ($Ea$) critically influences reaction rates in reactive flows according to the Arrhenius law; a lower $Ea$ or a higher temperature accelerates reactions. A high $Ea$ slows species consumption, resulting in elevated concentration levels, whereas a lower $Ea$ enhances reaction rates, causing sharper declines in species concentration. This makes its inclusion vital for accurately modeling mass transfer and conversion in reactive systems. Consequently, higher-dimensional activation energy ($Ea$) leads to an increase in the concentration profile for both dilatant ($1.0>n$) and pseudo-plastic ($n>1.0$) hybrid nanofluids.

The influence of the bioconvective Lewis parameter ($Lb$) on the profile of gyrotactic motile microorganisms is shown in Figure 9a,b for both thinning ($1.0>n$) and thickening ($n>1.0$) cases, under the simultaneous effects of the microorganism slip parameter (${\lambda }_4$) and inclined MHD ($\alpha$). It is observed that an increase in the bioconvective Lewis number reduces the fluid motility in both scenarios. The parameter $Lb$, which represents the ratio of thermal to microorganism diffusivity, determines the relative rates of heat and microbe diffusion. A higher $Lb$ value sharpens the concentration gradients and thins the microorganism boundary layer, whereas a lower $Lb$ promotes smoother profiles through enhanced microbial dispersion. Furthermore, Lorentz forces generated by the inclined magnetic field impede fluid motion, thereby reducing convection and mitigating excessive microbial accumulation. The interplay between $Lb$ and the magnetic field collaboratively governs the stability and behavior of the bioconvective flow by shaping the microorganism distribution, altering the boundary layer thickness, and modulating the associated transport of heat, mass, and microbes.

The influence of $Pe$ on the motile profiles are displayed in Figure 10a,b for both dilatant $(1.0>n)$ and pseudo-plastic hybrid nanofluids $(n>1.0)$, with a parallel effect of $\alpha$ as the microbe slip parameter ${\lambda }_4$ is present at the boundary. It has been noted that for both dilatant $(1.0>n)$ and pseudo-plastic hybrid nanofluids $(n>1.0)$, the interpretation of the profile decreases as the parameter $Pe$ increases.

The streamlined graphs illustrate how fluid flow is simultaneously influenced by the magnetic parameter M and an angled magnetic field, as depicted in Figure 10a,b and Figure 11a,b. Asymmetric streamline patterns emerge when the flow direction is skewed by the magnetic field’s inclination. This inclination alters the balance between inertial and magnetic forces, redirecting the flow and causing the boundary layer to thicken on one side. Higher values of M enhance damping effects, resulting in a smoother flow field and reduced velocity. In MHD flow management and heat transfer optimization, the combined influence of magnetic strength and angle dictates the flow behavior. In contrast, a higher $Kr$ depletes the solute, thickens the concentration boundary layer, and diminishes the diffusive driving force at the wall, collectively causing a decline in $S{h}_x$. As illustrated in Figure 12b, a larger $\alpha$ suppresses convective transport. This effect lowers the absolute $Sh$ levels and flattens the concentration surface.

This analysis suggests that MHD (magnetohydrodynamics) dampens the sensitivity of mass transfer to parameters ${\lambda }_3$ and $Kr$. These pictorial results validate that the same trend holds for different parameter values involving an inclined magnetic field and slip effect [42,43]. Furthermore, the response of the microorganism slip parameter ${\lambda }_4$ and the Peclet number $Pe$ to the local motile–microorganism transfer measure is mapped for various inclined MHD angles $\alpha$ in Figure 13a.

The Peclet number ($Pe$) decrease corresponds to enhanced advective cell sweeping. This weakens wall-normal diffusion, resulting in flattened near-wall number-density gradients. Consistent with the observed reduction at high $Pe$, a lower ${\lambda }_4$ further diminishes the relative mobility of microorganisms near the surface. This reduces the wall flux and pushes the surface concentration lower. Conversely, a higher ${\lambda }_4$ combined with a smaller $Pe$ elevates $N{n}_x$ and steepens the near-wall slopes. Figure 13b plots the local Sherwood number against the reaction parameter $Kr$ and the solutal slip parameter ${\lambda }_3$. An increase in ${\lambda }_3$ raises $S{h}_x$ by intensifying wall-normal concentration gradients and relaxing the wall’s constraint on species transport. Figure 14 presents a contour map demonstrating how the radiation parameter (Rd), thermophoresis (Nt), Brownian motion (Nb), and inclined MHD interact collectively to influence the local Nusselt number. This number serves as a measure of heat transport at the surface for both dilatant $(n<1.0)$ and pseudo-plastic $(n>1.0)$ hybrid nanofluids. Maximum heat transfer is achieved in both cases with higher values of Rd and Nt coupled with a lower value of Nb. Although the introduction of inclined MHD slightly diminishes convection and the overall Nusselt number due to Lorentz forces that oppose fluid motion, the overarching trends of the parameter interactions confirm that Rd and Nt play a dominant role in enhancing thermal transport.

The combined effects of the radiation parameter (Rd), Brownian motion parameter (Nb), and thermophoresis parameter (Nt) on heat transfer in both dilatant $(n<1.0)$ and pseudo-plastic $(n>1.0)$ hybrid nanofluids are illustrated in the 3D surface plots in Figure 15a,b. The Nusselt number increases with higher Nt values. This occurs because the thermophoretic force, which drives nanoparticles from hotter to cooler regions, intensifies, thereby enhancing energy transmission and steepening the wall temperature gradient. Similarly, a larger Rd enhances radiative heat transfer, which reduces thermal resistance and further accelerates the rate of heat transfer. Conversely, a higher Nb increases the random motion of nanoparticles. This random Brownian diffusion disrupts the directed transport caused by thermophoresis, thickens the thermal boundary layer, and ultimately reduces the Nusselt number, despite also promoting local thermal mixing. The surface plots effectively capture this complex, nonlinear interplay among the parameters, revealing optimal ranges for heat transfer optimization. Prominent peaks occur at high combinations of Nt and Rd with low Nb, while troughs are observed where Nb is high or Rd is low.

The residual versus fitted plot in Figure 16 is used to assess the reliability of the estimated model. This plot, which indicates a maximum error of 0.01001, further confirms the model’s accuracy. This section presents the numerical results for the skin friction and Nusselt number with respect to various parameters and the inclination angle ($\alpha$).

Table 5. The variation in the skin friction coefficient $\left\{C_{f}xR{e}_x^{-1/2}\right\}$ and the Nusselt number $\left\{N{u}_{x}R{e}_x^{-1/2}\right\}$ for different values of $n,Rd,We,Nt,Nb,$ and ${\lambda }_1$ while other parameters remain fixed in active and passive cases.

n	$\mathit{We}$	${\mathit{\lambda }}_1$	$\mathit{Rd}$	$\mathit{Nt}$	$\mathit{Nb}$	${\mathit{Cf}}_{\mathit{x}}{\mathit{Re}}_{\mathit{x}}^{-1/2}$			${\mathit{Nu}}_{\mathit{x}}{\mathit{Re}}_{\mathit{x}}^{-1/2}$
						$\mathit{\alpha }=\mathbf{0}$	$\mathit{\alpha }=\mathit{\pi }/\mathbf{4}$	$\mathit{\alpha }=\mathit{\pi }/\mathbf{2}$	$\mathit{\alpha }=\mathbf{0}$	$\mathit{\alpha }=\mathit{\pi }/\mathbf{4}$	$\mathit{\alpha }=\mathit{\pi }/\mathbf{2}$
0.5	2.0	0.5	0.3	0.5	0.03	−0.527111	−0.601869	−0.659435	1.0446460	0.939929	0.830034
-	4.0	-	-	-	-	−0.476473	−0.539980	−0.588676	1.014796	0.888463	0.749531
-	6.0	-	-	-	-	−0.436829	−0.494217	−0.538261	0.985187	0.838113	0.668262
-	-	0.1	-	-	-	−0.703466	−0.806995	−0.890672	1.162428	1.094829	1.026973
-	-	0.5	-	-	-	−0.527111	−0.601869	−0.659435	1.044646	0.939929	0.830034
-	-	0.7	-	-	-	−0.468163	−0.533047	−0.581870	0.994626	0.86976285	0.733840
-	-	-	0.3	-	-	−0.527111	−0.601869	−0.659435	1.044646	0.939929	0.830034
-	-	-	0.5	-	-	−0.527111	−0.601869	−0.659435	1.197162	1.054621	0.900435
-	-	-	0.7	-	-	−0.527111	−0.601869	−0.659435	1.324502	1.140270	0.937680
-	-	-	-	0.02	-	−0.527111	−0.601869	−0.659435	1.027009	0.920977	0.809490
-	-	-	-	0.04	-	−0.527111	−0.601869	−0.659435	1.009391	0.902054	0.788964
-	-	-	-	0.06	-	−0.527111	−0.601869	−0.659435	0.991797	0.883165	0.768461
-	-	-	-	-	0.03	−0.527111	−0.6018699	−0.659435	1.055302	0.950957	0.8413789
-	-	-	-	-	0.05	−0.527111	−0.6018699	−0.659435	1.033974	0.928902	0.8187097
-	-	-	-	-	0.07	−0.527111	−0.6018699	−0.659435	1.012584	0.906852	0.796126
1.5	2.0	0.5	0.3	0.5	0.03	−0.577454	−0.66744630	−0.7383706	1.0663477	0.9788646	0.89124861
-	4.0	-	-	-	-	−0.604035	−0.6980943	−0.7720928	1.0721157	0.9885325	0.9057679
-	6.0	-	-	-	-	−0.6250351	−0.7215622	−0.7973422	1.0744292	0.9928002	0.9126137
-	-	0.1	-	-	-	−0.8693453	−1.0319911	−1.1725263	1.216047194	1.178152592	1.1427349440
-	-	0.5	-	-	-	−0.57745424	−0.6674463	−0.7383706	1.0663477	0.9788646	0.89124861
-	-	0.7	-	-	-	−0.44236496	−0.5048088	−0.5511111	0.9603352	0.82694211	0.6808565
-	-	-	0.3	-	-	−0.5774542	−0.6674463	−0.738370	1.066347	0.97886468	0.8912486
-	-	-	0.5	-	-	−0.5774542	−0.6674463	−0.738370	1.2290514	1.1115718	0.9908486
-	-	-	0.7	-	-	−0.5774542	−0.6674463	−0.738370	1.367745	1.2171891	1.0596390
-	-	-	-	0.02	-	−0.5774542	−0.6674463	−0.738370	1.048988	0.9604717	0.8717179
-	-	-	-	0.04	-	−0.5774542	−0.6674463	−0.738370	1.031643	0.942104	0.8522147
-	-	-	-	0.06	-	−0.5774542	−0.6674463	−0.738370	1.014317	0.923767	0.8327426
-	-	-	-	-	0.03	−0.5774542	−0.6674463	−0.738370	1.0767765	0.989572	0.9021788
-	-	-	-	-	0.05	−0.5774542	−0.6674463	−0.738370	1.0559031	0.968155	0.8803323
-	-	-	-	-	0.07	−0.5774542	−0.6674463	−0.738370	1.0349712	0.946738	0.858548

displays the corresponding calculated numerical values. An increase in the radiation parameter (Rd) within the range $0.3<\mathrm{Rd}<0.7$ leads to a decrease in skin friction. Higher radiation enhances energy transfer via thermal radiation, which thickens the thermal boundary layer. This reduces the momentum boundary layer gradient, resulting in lower wall shear stress and consequently lower skin friction. Conversely, an increase in the Weissenberg number within the range $3<\mathrm{We}<5$ raises the skin friction values. Furthermore, as the inclination angle increases from $0^{°}<\alpha <{90}^{°}$, the buoyant driving force diminishes. This slows the fluid motion near the surface, reducing wall shear stress and thus skin friction.

As the inclination angle $\alpha$ increases within the range $0<$$\alpha$$<90$, the component of surface gravity tangential to the slope decreases. This reduction diminishes the buoyant driving force, weakens the flow velocity, and consequently lowers the Nusselt number, impairing convective heat transfer in both active and passive scenarios. A higher Biot number signifies that surface convection is more dominant relative to internal conduction within the fluid, which enhances overall heat transfer and raises the Nusselt number. The Nusselt number increases for both cases as the radiation parameter Rd grows from $0.3<$$Rd$$<0.7$. Conversely, the Nusselt number declines for all cases with rising values of the Brownian motion parameter Nb $0.03<$$Nb$$<0.07$, the thermophoresis parameter Nt $0.02<$$Nt$$<0.6$, and the Weissenberg number We $3<$$We$$<5$. The decrease associated with a higher Weissenberg number is attributed to heightened elastic effects, which reduce the thermal efficiency of the flow.

7. Concluding Remarks

This study numerically investigates the magnetohydrodynamic flow of a two-dimensional, incompressible Carreau hybrid nanofluid over a nonlinear stretching sheet, incorporating microorganisms. The model accounts for the simultaneous effects of nonlinear thermal radiation, chemical reactions, and multiple slip boundary conditions for both dilatant ($n>1.0$) and pseudo-plastic ($n<1.0$) fluid behaviors. The governing partial differential equations (PDEs) were transformed into ordinary differential equations (ODEs) via similarity transformations and solved numerically using the Bvp5c solver in MATLAB. The major findings of this research are the following:

• A magnetic field promotes higher temperature and elevates the motile concentration profiles. However, it induces a reversing character in the velocity and concentration profiles
• The physical parameters of heat sources, nonlinear radiation, and the Weissenberg number intensify the temperature field, and the structure of the boundary layers exhibits symmetry as a result of parameters acting at various angles $\left(\alpha \right)$.
• The Weissenberg number affects heat transfer, skin friction, and the velocity profile differently at various angles $\alpha$, with effects intensifying when $(n>1.0)$ and weakening when $(1.0>n)$.
• The hybrid nanofluid concentration is effectively affected by activation energy, whereas a significant decrease is noted upon increasing the chemical reaction parameter with the simultaneous effect of concentration slip and inclined magnetohydrodynamic for shear thinning $(1.0>n)$ and shear thickening $(n>1.0)$.
• The motile profile shrinks with increasing $Lb$ and $Pe$ in dilatant and quasi-elastic liquids, showing a parallel effect from $\alpha$, under the presence of the boundary slip parameter ${\lambda }_4$ for microorganisms.
• Increasing the inclination angle ($\alpha$) causes the skin friction surface to shift toward the base plane; conversely, the Nusselt number surface shows an opposite trend.
• The maximum heat transfer rate occurs at larger radiation parameter values. An increase in the inclination angle within the range of $0^{°}$ to ${90}^{°}$ ($0<\alpha <{90}^{°}$) results in a reduction in the Nusselt number.
• The hybrid nanofluid model enhances cooling efficiency in applications like solar panels and electronic equipment. It is also highly effective in industries such as coating and polymer extrusion, where precise control of mass and heat transfer under magnetic fields is essential.