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\small {\Large Finite Difference Analysis of MHD Bioconvective Nanofluid Flow with Radiation and Viscous Dissipation}
\end{center}
\begin{center}
\small
\author 1  Aimin Fatima*,  $^{a}$ \author 2 Muhammad Imran Asjad, $^{a}$  \author 3 Saraj Khan, $^{a}$  \author 3 Muhammad S. Alqarni, $^{b}$ 
%\author 3 xyz$^{b}$, 
\end{center}
\small
\begin{description}
\item[] $^a$ Department of Mathematics, University of Management and Technology Lahore, 54770 Pakistan.
%\item[] $^a$ Academy of Innovative Semiconductor and Sustainable Manufacturing, National Cheng Kung University, Tainan 701401, Taiwan
\item[] $^b$ Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia.
  %\item[] $^b$ Abdus Salam School of mathematical Sciences, Gc University, 68-B, New Muslim Town, Lahore 54600, Pakistan.
  %\item[] $^e$ Department of Mathematics, College of Science, King Khalid University, Abha, Saudi Arabia
%tabasam.rashid@umt.edu.pk}
\center \item[] {*Correspondence: aimanfatima377@gmail.com}
\end{description}
\begin{abstract}
This research investigates the combined impact of advanced features of flow properties on concerning a nanofluid infused with gyrotactic microorganisms over a nonlinear deforming sheet placed in a porous medium. Similarity transformations are employed to convert the governing partial differential equations into ordinary differential equations. These highly nonlinear equations are discretized using finite difference schemes and solved iteratively via the Successive Over-Relaxation (SOR) method. Although various authors have studied MHD flows of nanofluids, bioconvection, nonlinear stretching surfaces, and porous media effects separately, the combined influence of MHD, viscous and Joule heating, thermal radiation, and bioconvection in a porous medium over a nonlinear stretching sheet has not been comprehensively examined. The present study aims to fill this gap by formulating and solving a coupled system of nonlinear equations incorporating all these physical effects and solving them numerically using the finite difference method with SOR. The results presented through graphical profiles and tabulated data, illustrate the influence of key physical parameters on the velocity, temperature, concentration, and microorganism distributions. The findings reveal that increasing the Prandtl, Schmidt, Lewis, and Peclet numbers reduces the thickness of the thermal and concentration boundary layers, while higher values of radiation, Brownian motion, thermophoresis, and magnetic parameters enhance the temperature, nanoparticle concentration, and microorganism density within the flow. These results underscore the potential applications of bioconvective nanofluids in biomedical devices, enzyme-based sensors, microfluidics, microbe-assisted oil recovery, and biotechnological processes. 
\end{abstract}
\textbf{Key Terms:}
Nanofluid; MHD (magnetohydrodynamics); Thermal radiation; Viscous dissipation; Gyrotactic microorganisms; Finite difference method.
\section*{Nomenclature}
\begin{tabular}{>{\bfseries}c @{\hspace{1em}} l | >{\bfseries}c @{\hspace{1em}} l}
$u$ & Velocity in $x$ direction   & $v$ & Velocity in $y$ direction \\[4pt]
$S$ & Suction/Injection parameter & $\psi$ & Stream function \\[4pt]
$\Pr$ &  Dimensionless Prandtl number & $\mathrm{Sc}$ & Bioconvection Schmidt number \\[4pt]
$\theta$ & Dimensionless temperature & $\eta$ & Similarity independent variable \\[4pt]
$x$ & Coordinate along the plate  & $y$ & Coordinate normal to plate  \\[4pt]
$T_{wall}$ & Surface temperature  & $T_\infty$ & Ambient temperature  \\[4pt]
$C_{wall}$ & Wall concentration & $C_\infty$ & Ambient concentration \\[4pt]
$\mathrm{Nb}$ & Brownian motion parameter & $k^*$ & Absorption coefficient \\[4pt]
$\mu$ & Dynamic viscosity & $\nu$ & Kinematic viscosity \\[4pt]
$C_p$ & Specific heat  & $k$ & Thermal conductivity \\[4pt]
$\alpha$ & Thermal diffusivity & $\rho$ & Fluid density  \\[4pt]
$q_{wall}$ & Heat flux at wall  & $\mathrm{R_{c}}$ & dimensionless chemical reaction \\[4pt]
$\mathrm{M} $ & Magnetic field parameter & $\mathrm{Ec}$ & Eckert number (viscous dissipation) \\[4pt]
$\mathrm{R}$ & Radiation parameter & $\mathrm{Nt}$ & Thermophoresis parameter \\[4pt]
$\mathrm{Le}$ & Lewis number & $\mathrm{Pe}$ & Peclet number for motile microorganisms \\[4pt]
$\phi$ & Dimensionless nanoparticles concentration & $\mathrm{Nu_x}$ & Local Nusselt number (heat transfer rate) \\[4pt]
$\mathrm{ Sh_x}$ & Local Sherwood number (mass transfer rate) & $\mathrm {Nn_x }$ & Motile microorganism density number at wall \\[4pt]
\end{tabular}
\section{Introduction}\label{sec1}
\indent \indent Nanofluid flow influenced by motile microorganisms has emerged as a focal area of research due to its practical significance in fields like biomedical devices, bioenergy systems, and advanced thermal technologies. The presence of gyrotactic microorganisms induces bioconvective motion, which alters the transport dynamics of nanoparticles within the fluid. In addition, thermal radiation has a profound impact on heat transfer, especially in high-temperature applications such as solar thermal collectors, nuclear energy systems, and industrial heat exchangers. This study explores the interactive impact of thermal energy and microorganism motility on the viscous boundary layer of nanofluid past a stretching surface. The study provides insight into the Coupled influence of thermal radiation, thermophoresis, microbial, as well as Brownian motion activity on the temperature, velocity, and concentration distributions. Aziz et al. \cite{J1} investigates steady free convection motion of a nanofluid containing gyrotactic microorganisms across a horizontal surface embedded in a porous medium. Bioconvection parameters notably affect velocity, temperature, concentration, and microorganism distribution. The results, validated against existing data, show enhanced transport rates of heat, mass, and microbes. Mabood et al. \cite{J2} showed the delves into the influence of fusion dynamics and radiative heat on impinging flow toward deformable surfaces. The fourth-fifth order Runge-Kutta scheme is used to numerically solve the nonlinear system. Key physical metrics such as fluid velocity, thermal profile, species distribution, shear drag, and dimensionless heat and mass fluxes are scrutinized under diverse influences including surface deformation, primary and secondary slip effects, radiative flux, phase change, and diffusive parameters. A validation is performed through benchmarking against established findings in special limiting scenarios. \\
\indent Ahmad et al.\cite{J3} presented heat transfer enhancement the flow of nanofluid with suspended the gyrotactic microorganisms across a permeable layer near a stretching sheet. Chemical reactions are also considered in the model. The governing equations are simplified using similarity transformations and solved numerically via the SOR method. Results show that microorganism concentration and bioconvection parameters increase motile microorganism density, while chemical reactions enhance mass transfer. These findings have practical applications in biotechnology, microfluidics, and energy systems. Obalalu et al. \cite{J4} deliberated the bioconvection nanofluid flow based on the Oswald–DeWaele power-law model, incorporating nanoparticles and gyrotactic microorganisms. It explores microorganism swimming behavior and bioconvection structure formation. The Lie-group method is used to derive similarity solutions, and the Legendre-based collocation method (LBCM) is applied for numerical analysis. Results highlight the strong impact of bioconvective parameters on velocity, thermal distribution, and microbial concentration. The findings are accurate and efficient, with potential applications in nanofluid-based biofuels. Ahmad et al. \cite{J5} represented the novel scrutiny involving the circulation of a nanofluid enriched with gyrotactic motile organisms under radiative heat transfer through a permeable medium over a nonlinear stretching or shrinking surface. The system’s nonlinear equations are solved by employing the Successive Over-Relaxation technique. The outcomes indicate that thermal radiation elevates the temperature, while the bioconvection Peclet number and motility parameters enhance the concentration of microorganisms. Hamid et al. \cite{J6} scrutinized the unsteady, laminar mixed convection behavior of a magnetically influenced, electrically conducting hybrid nanofluid containing microorganisms showing gyrotactic behavior near a stagnation flow toward a porous deformable surface. By applying similarity transformations, the governing equations are simplified into a system of ordinary differential equations. The analysis highlights that higher values of the Brownian motion and Prandtl number coefficient lead to reductions in temperature and heat transfer rate, while increased magnetic diffusivity results in diminished flow velocity and magnetic intensity. kausar et al. \cite{J7} explored a computational model of steady bioconvective thermo-solutal Casson nanofluid motion along a vertical stretching surface immersed in a porous structure, incorporating thermal radiation, chemical reactions, and gyrotactic microorganisms. Using the Buongiorno model, the effect of parameters like Péclet number, and Schmidt number on Casson fluid properties on temperature, velocity, and microorganism density is examined. Shahid et al. \cite{J8} represented the effect of bioconvective activity due to gyrotactic cells in MHD nanofluid flow using Darcy’s law. A nonlinear mathematical model is resolve by the Successive Local Linearization Method combined with the Chebyshev spectral collocation technique. The precision and effectiveness of the suggested approach are validated through comparison with existing literature, and results are illustrated via tables and graphs. \\
\indent Gangadhar et al. \cite{J9} introduced the behavior of gyrotactic microorganisms within a Casson nanofluid flow past a stretching cylinder influenced by a strong magnetic field and slip conditions. The mathematical model, expressed as a boundary value problem, is transformed into dimensionless form and solved numerically. Findings highlight the dominant role of the Prandtl number in heat transfer, while the magnetic field reduces flow velocity and boosts thermal radiation. Okasha et al. \cite{J10}  scrutinized the effects stagnation-point flow in a hybrid nanofluid embedded with gyrotactic microorganisms under local thermal non-equilibrium conditions. Findings show that greater Stefan blowing intensifies the flow but leads to a decline in temperature, heat, and mass transfer rates. khan et al. \cite{J11} established the bioconvective flow of a micropolar nanofluid with microorganisms over a stretchable inclined surface in a porous medium under magnetic, radiative, and heat absorption effects. The governing nonlinear system is derived using similarity transformations and solved using the finite difference method. Results offer insight into how various dimensionless parameters affect fluid flow, temperature, nanoparticle concentration, and microorganism distribution, with applications in environmental and biosciences engineering. Farooq et al. \cite{J12} established the bioconvective micropolar motion of nanofluid over a stretched surface using the Tiwari-Das model, treating blood as a micropolar fluid infused with nanoparticles. The governing dimensionless equations are solved via the local non-similarity method using the MATLAB. Findings reveal that increasing micropolar parameters boosts temperature, velocity and microrotation, while the microorganism and concentration distributions vary significantly with the Lewis, Schmidt, and Peclet numbers. Hussain et al. \cite{J13} explored the steady three-dimensional MHD flow of a tangent hyperbolic nanofluid infused with gyrotactic microorganisms over a stretching surface, considering the roles of activation energy, variable mass flux, and solar radiation. The findings reveal that increasing the Prandtl number and stretching rate lowers the temperature, while nanoparticle concentration and microorganism density are significantly affected by magnetic intensity, thermophoresis, power-law index, and bioconvective Lewis number. Zuhra et al. \cite{J14} developed in this study to examine the flow and thermal behavior of Casson and Williamson nanofluids with gyrotactic microorganisms that are affected by cubic autocatalytic processes in a porous media when buoyant forces are present. The effect of various dimensionless parameters on flow velocity, concentration, temperature distribution, and motile microorganism density is thoroughly examined. Li et al. \cite{J15} showed the flow near the stagnation region of a Maxwell-type nanofluid influenced by thermal radiation and containing gyrotactic microorganisms over a permeable, deformable surface, modeled using Buongiorno’s approach. The analysis incorporates key transport mechanisms such as Brownian diffusion, thermophoresis, and chemical reactivity. To simulate realistic boundary interactions, Robin-type conditions are applied for heat, mass, and microorganism transport processes. Kaswan et al. \cite{J16} scrutinized the LMA-based neural networks to solve magneto-bioconvective nanofluid flow with gyrotactic microorganisms and activation energy. Results show decreasing microorganism concentration with higher bioconvection Lewis number and rising temperature under stronger radiation effects. Hussain et al. \cite{J17} explored the unsteady 3D bioconvective Casson nanofluid along a extending surface, incorporating thermophoresis, Brownian motion, and thermal radiation. Results show that temperature increases with magnetic and Biot numbers, while concentration declines with a higher Lewis number, and microorganism density rises with stretching and Biot parameters. Muhammad et al. \cite{J18} numerically investigated bioconvective micropolar nanofluid flow over a stretching sheet influenced by motile microorganisms, velocity slip, micro-rotation, and thermal effects. Solving the transformed nonlinear equations via finite difference and relaxation methods reveals that velocity and temperature profiles are significantly affected by microrotation, slip parameters, and bioconvection intensity. \\
\indent Rehman et al. \cite{J19} the effect of Schmidt number, Peclet number including the concentration of microorganisms with distinct parameters on the concentration of motile gyrotactic cells, while under the influence of thermal stratification and magnetic field in nanofluid flow past a vertical stretching boundary. Varatharaj et al. \cite{J20} introduced MHD flow of Carreau nano-fluid with motile microorganisms through a porous medium, considering shear-dependent viscosity, activation energy, and thermal–mass transport phenomena. Solved via the Runge–Kutta shooting method, results reveal that flow, heat, and concentration profiles are sensitive to magnetic, Peclet, suction, and Schmidt numbers, with flat plate geometry showing minimal friction and transfer rates. Shahid et al. \cite{J21} scrutinized the nanofluid flow with gyrotactic particles and magnetic effects over a shrinking/stretching sheet, incorporating radiation effects and chemical reactions. By applying similarity transformations and solving with the Successive Taylor Series Linearization Method, the study reveals that magnetic intensity reduces the Sherwood number, Nusselt number, and microorganism density. Makkar et al. \cite{J22} explored MHD Casson nanofluid flow with gyrotactic microorganisms under convective boundary conditions, implemented using the Runge–Kutta–Fehlberg algorithm with shooting in MATLAB. Results show that increasing the thermal Biot number boosts temperature, while nanoparticle concentration decreases with stronger chemical reactions. 
\begin{figure}[H]
\setlength\abovecaptionskip{1\baselineskip}
\centering
\includegraphics[width=0.6\textwidth]{G:/Latex/fig1.png}
  \caption*{Fig. 1 Geometry of the problem}
\label{fig1}
\end{figure} 
\section{Description of Mathematical Model}\label{sec2}
Consider a time-independent, two-dimensional, laminar flow of an electrically conducting nanofluid containing both nanoparticles and gyrotactic microorganisms over a nonlinear stretching surface embedded in a porous medium. The nanofluid is assumed to be incompressible and confined to the boundary-layer region adjacent to the sheet. The model incorporates the effects of a transverse magnetic field, viscous dissipation, Joule heating, thermal radiation, thermophoresis, bioconvection, Brownian motion, and a homogeneous first-order chemical reaction to account for various transport phenomena occurring in realistic industrial and biological systems.\cite{J3}\\
The velocity of the external flow (free stream) far from the surface varies with $x$ as
\[
u_e(x) = a x^m,
\]
where $a>0$ is a constant and $m>0$, $m\neq1$ is the nonlinear stretching parameter. The stretching sheet at $y=0$ moves with velocity
\[
u_{wall}(x) = c x^m,
\]
where $c>0$ is a constant.
At the sheet surface ($y=0$), the temperature, nanoparticles concentration, and motile microorganism density are maintained at prescribed constant values $T_{wall} $, $C_{wall}$, and $N_{wall}$, respectively, while in the ambient fluid ($y\to\infty$), these quantities approach $T_\infty$, $C_\infty$, and $N_\infty$ shown in Fig 1.
\begin{eqnarray}
&& \frac{\partial v}{\partial y}+\frac{\partial u}{\partial x}=0,\\&&
v\frac{\partial u}{\partial y}+u\frac{\partial u}{\partial x} = u_e \frac{\partial u_e}{\partial x}+\frac{\mu}{\rho k^*} \big( u_e - u \big) -\frac{\sigma B_0^2}{\rho} u+\nu\frac{\partial^2 u}{\partial y^2},\\&&
u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \alpha \frac{\partial^2 T}{\partial y^2} + \frac{16 \sigma^* T^3}{3 K_0 \rho C_p} \frac{\partial^2 T}{\partial y^2} + \frac{\mu}{\rho C_p} \left( \frac{\partial u}{\partial y} \right)^2+\frac{\sigma_{p} B_0^2}{\rho C_p} u^2 \nonumber \\&&
+ \tau \bigg( \mathrm{D_B} \frac{\partial C}{\partial y} \frac{\partial T}{\partial y}+\frac{ \mathrm{D_T}}{T_\infty} \bigg( \frac{\partial T}{\partial y} \bigg)^2 \bigg),\\&&
u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y} =  \mathrm{D_{B}}\frac{\partial^2 C}{\partial y^2}+\left(\frac{ \mathrm{D_{T}}}{T_{\infty}}\right)\frac{\partial^2 T}{\partial y^2} -K_r (C - C_\infty),\\&&
u\frac{\partial N}{\partial x}+v\frac{\partial N}{\partial y} =  \mathrm{D_{N}}\frac{\partial^2 N}{\partial y^2}-\frac{bW_{c}}{C_{Wall}-C_{\infty}}\left[\frac{\partial}{\partial y}\left(N\frac{\partial C}{\partial y}\right)\right],
\end{eqnarray}
where, $\alpha =\frac{\kappa}{\rho C_{p}}$ denotes the thermal diffusivity, A mass flux velocity is $v_{Wall}$ with injection $v_{Wall}>0$ as well as the fluid suction $v_{Wall}<0$, b is chemotaxis constant, T be the temperature of fluid, $W_{c}$ are the maximum speed of cell swimming, C symbol indicates the nanoparticles volume fraction, $\rho$ be density of the fluid, $B_0$ be the magnitude field, The symbol $\sigma^*$ stands for the Stefan–Boltzmann constant, $D_{n}$ stand for the Microorganism diffusion coefficient, $D_{B}$ indicates the Brownian diffusion, $K_r$ characterizes the intensity of the homogeneous chemical reaction in the fluid, $C_p$ represents the specific heat of the fluid at constant pressure, $D_{T}$ denotes the thermophoresis diffusion, and $N$ denotes the concentration of motile microorganisms.
The boundary constraints for velocity, temperature, and species concentration can be expressed as:\\
On the boundary surface $(y=0)$:
\begin{eqnarray}
v=v_{Wall},\ \ u= c x^m,\ \  N=N_{Wall},\ \ T=T_{Wall},\ \ C=C_{Wall},\ \
\end{eqnarray}
Away from the boundary layer, $ y \rightarrow \infty $:
\begin{eqnarray}
 C \rightarrow C_{\infty} ,\ T \rightarrow T_{\infty},\ \  N \rightarrow N_{\infty},\ \ u = ax^m,
\end{eqnarray}
By employing appropriate similarity transformations, the governing equations for mass conservation, momentum, heat transfer, nanoparticle diffusion, and microorganism density can be reduced to a system of ordinary differential equations (ODEs).
\begin{eqnarray}
\psi = \sqrt{a x^{m+1} \nu} \, G(\eta), \ \ \eta = y \sqrt{\frac{a}{\nu}} x^{\frac{m-1}{2}}, \chi (\eta)=-\frac{N_{\infty}-N}{N_{Wall}-N_{\infty}}, \\
\phi(\eta)=-\frac{C_{\infty}-C}{C_{Wall}-C_{\infty}},  \ \ \theta (\eta)=-\frac{T_{\infty}-T}{T_{Wall}-T_{\infty}}.
\end{eqnarray}
$\eta $ denotes the similarity variable, $\psi$ be the stream function defined by  $u=\frac{\partial \psi}{\partial y},  \ \ v=- \frac{\partial \psi}{\partial x},$. Upon substituting the above coordinates into equations (1)–(5), the following dimensionless equations are obtained:
\begin{eqnarray}
G''' + \mathrm{P_0} (1 - G') - \mathrm{M} G'-m(G'^2 -1)+\bigg(\frac{m+1}{2}\bigg) G G''&=&0,
\end{eqnarray}
The numerical algorithm for the present model is formulated using finite difference technique. To facilitate the numerical scheme, equation (10) is transformed into a second-order differential equation through appropriate substitutions.
by substituting 
\begin{eqnarray}
F=G'=\frac{d G}{d\eta},
\end{eqnarray}
The substitution yields the second-order equation:
\begin{eqnarray}
&&F''+(1-F)\mathrm{P_{0}}- m(F^2 -1)+(\frac{m+1}{2})GF'- \mathrm{M} F=0,\\&&
\bigg( \frac{1}{\mathrm{Pr}} + \frac{4}{3} \mathrm{R} \bigg) \theta'' + \mathrm{Nb}\, \phi'\, \theta' + \bigg(\frac{m+1}{2}\bigg) G \theta' + \mathrm{Nt}\, (\theta')^2 + \mathrm{Ec}\, (G'')^2 + \mathrm{M}\, (G')^2 \mathrm{Ec}= 0,\\&&
\phi'' - \mathrm{Le}\mathrm{R_c} \phi + \frac{\mathrm{Nt}}{\mathrm{Nb}} \theta'' + \bigg(\frac{1+m}{2}\bigg) \mathrm{Le}\, \mathrm{G}\, \phi'= 0,\\&&
\chi'' + \frac{1}{2}(1+m) \mathrm{Sc}\, G\, \chi'- \mathrm{Pe} \big[ \phi'' (\sigma + \chi) + \chi'\, \phi' \big]= 0.
\end{eqnarray}
Applying the similarity transformations, the boundary conditions (6) and (7) are rewritten in dimensionless terms.
The corresponding boundary conditions are defined below:
\begin{eqnarray} 
&& G(0)=S ,\ \ F(0) = \lambda ,\ \ \theta(0)=1,\ \phi(0)=1,\ \chi(0)=1,\\&&
F(\infty)= 1, \ \ \theta(\infty)=0,\ \phi(\infty)=0,\ \chi(\infty)=0,
\end{eqnarray}
$\mathrm{R_c}$ dimensionless parameter representing the chemical reaction, The parameter \( \lambda \) characterizes the sheet motion: the sheet stretches (expands) if \( \lambda > 0 \), and shrinks (contracts) if \( \lambda < 0 \), The symbol $K_{0}$ denotes the absorption coefficient, Pe signifies the Péclet number, $\mathrm{Le}$ represents the Lewis number, while $\mathrm{Pr}$ is denoted by the Prandtl number, $\mathrm{Sc}$ refers to the Schmidt number, $\mathrm{R}$ corresponds to the radiation parameter, $\mathrm{Ec}$ is the Eckert number, the parameters $\mathrm{Nb}$ and $\mathrm{Nt}$ represent the Brownian motion and thermophoresis effects, respectively. $\mathrm{M}$ is defined as the magnetic parameter, $\mathrm{P_0}$ be the dimensionless parameter quantifying the resistance due to the porous medium. The symbol $\sigma$ dimensionless constant, $\mathrm{S}$ is the mass flux parameter, where $\mathrm{S>0}$ indicates suction and $\mathrm{S<0}$ denotes injection. 
\begin{eqnarray} 
&&\mathrm{R_c} = \frac{xk_c}{u_e},\ \ \mathrm{Pe} = \frac{bW_{c}}{D_{n}},\ \ \mathrm{Sc} = \frac{\nu}{D_{n}},\ \ \mathrm{Pr} =\frac{\nu}{\alpha},\ \ \mathrm{Le}=\frac{\nu}{D_{B}},\ \ 
\mathrm{S} = -\frac{v_{Wall}}{\sqrt{a\nu}},\\&&
\sigma = \frac{N_{\infty}}{N_{Wall}-N_{\infty}},\ \ \mathrm{Nt} =\frac{\tau D_{T}(T_{Wall}-T_{\infty})}{\nu T_{\infty}}, \ \ \mathrm{Nb} = \frac{\tau D_{B}(C_{Wall}-C_{\infty})}{\nu}, \\&&
\mathrm{P_0} = \frac{\nu x}{k^* u_e}, \ \ \mathrm{M} = \frac{x\sigma_{p} B_0 ^2}{\rho u_e}, \ \ \mathrm{E_c} = \frac{u_w^2}{c_p (T_{Wall}-T_{\infty})} , \ \ \mathrm{R} = \frac{4 \sigma^* T^3_{\infty}}{K_{0}\mu C_{p}}. 
\end{eqnarray}
The following expressions represent the dimensionless forms of local Sherwood number, local Nusselt number, skin friction, and local density number of motile microorganisms are defined below \cite{J3}
\begin{eqnarray} 
&&\frac{1}{\text{Re}_x^{1/2}}  \mathrm{Sh_x} = -\phi'(0),\ \ \text{Re}_x^{-1/2}  \mathrm{Nu_x} = -\theta'(0),\\&&
\frac{1}{\text{Re}_x^{1/2}}  \mathrm{C_{fx}} = G''(0), \ \ \frac{1}{\text{Re}_x^{1/2}}  \mathrm{Nn_x} = -\chi'(0).
\end{eqnarray}
\section {Numerical Approach} 
\indent \indent The finite difference technique is applied to handle the nonlinear equations (12)-(15). Numerical solutions of complex dynamical models can encounter several difficulties, especially when the asymptotic boundary conditions are prescribed at infinity. Currently, no precise guideline exists for determining the suitable value of the independent variable that adequately approximates infinity. Selecting a value that is too small may reduce the desired accuracy, while an excessively large choice can cause the solution to diverge. At times, numerical methods for solving dynamical boundary value problems exhibit instability even when the boundary conditions are reasonably chosen. These instabilities may stem from the intrinsic unsteadiness of the governing equations or from the strong sensitivity of the system’s response to the initial states. These challenges can be addressed through the application of the Successive Over-Relaxation (SOR) method, which offers enhanced robustness and efficiency in achieving faster convergence. \\
\indent Other implicit and multi-step methods can also be applied to tackle nonlinear fluid flow problems. However, these approaches often require substantial computational time. Therefore, the Successive Over-Relaxation scheme is utilized to overcome such limitations. In the present study, the simulation procedure is formulated at $\eta = \eta _{i}$ (a chosen node of the grid) by approximating the derivatives using finite difference expressions. Afterward, the system of equations is solved iteratively using the SOR technique, incorporating the given boundary conditions. \cite{J5}\\
The numerical algorithm for the present model is formulated using finite difference technique. After applying finite difference discretization, equations (12)-(15) are written as:
\begin{eqnarray}
&&F_{k}=\frac{AF_{k+1}+BF_{k-1}+ C'}{D},\\&&
\theta_{k}=\frac{E\theta_{k+1}+G_{star}\theta_{k-1}+ H}{I},\\&&
\phi_{k}= \frac{J\phi_{k+1}+K'\phi_{k-1}+ L}{N},\\&&
\chi_{k}=\frac{M'\chi_{k+1}+O\chi_{k-1}+P}{Q}.
\end{eqnarray}
where,
\begin{eqnarray*} 
&& A = 1+\frac{hG_{k}(m+1)}{4},  \ B = 1-\frac{hG_{k}(m+1)}{4}, \nonumber \\&&
C'= h^2(\mathrm{P_{0}}-m(F_{k}^2-1), D = 2+h^2(\mathrm{P_{0}}+\mathrm{M}), \nonumber \\&& 
E = \left(\frac{1}{\mathrm{Pr}}+\frac{4\mathrm{R}}{3}\right)+\frac{\mathrm{Nb}}{4}\left(\phi_{k+1}-\phi_{k-1}\right) +\frac{(m+1)h}{4}G_{k}+\frac{\mathrm{Nt}}{4}\theta_{k+1}, \nonumber \\&& 
G_{star} = \left(\frac{1}{\mathrm{Pr}}+\frac{4\mathrm{R}}{3}\right)-\frac{\mathrm{Nb}}{4}\left(\phi_{k+1}-\phi_{k-1}\right) -\frac{(m+1)h}{4}G_{k}+\frac{\mathrm{Nt}}{4}\theta_{k-1},\nonumber \\&& 
H = -\frac{\mathrm{Nt}}{2}\left(\theta_{k+1}*\theta_{k-1}\right)+ \frac{\mathrm{M}\mathrm{Ec}}{4}\left(G_{k+1}-G_{k-1}\right)^2+\mathrm{Ec}\left(G_{k+1}-2G_{k}+G_{k-1}\right)^2, \nonumber \\&& 
I = 2\left(\frac{1}{\mathrm{Pr}}+\frac{4\mathrm{R}}{3}\right), J = 1+\frac{h(1+m)\mathrm{Le}G_{k}}{4},  \ K' = 1-\frac{h(1+m)\mathrm{Le}G_{k}}{4},  \nonumber \\&&
L = \frac{\mathrm{Nt}}{\mathrm{Nb}}(\theta_{k+1}-2\theta_{k}+\theta_{k-1}), N = 2+\mathrm{Le}\mathrm{R_{c}}h^2,\nonumber \\&&
M'= \left(1+\frac{h\mathrm{Sc}(1+m)G_{k}}{4}-\frac{\mathrm{Pe}}{4}*(\phi_{k+1}-\phi_{k-1})\right), \nonumber \\&&
O= \left(1-\frac{h\mathrm{Sc}(1+m)G_{k}}{4}+\frac{\mathrm{Pe}}{4}*(\phi_{k+1}-\phi_{k-1})\right), \nonumber \\&&
P = -\mathrm{Pe}*\sigma*\left(\phi_{k+1}-2\phi_{k}+\phi_{k-1})\right),  \nonumber \\&&
Q = [2+\mathrm{Pe}*\left(\phi_{k+1}-2\phi_{k}+\phi_{k-1}\right)].
\end{eqnarray*}
The differential equations~(23)-(26) are numerically solved using MATLAB software. Owing to their nonlinear characteristics, these equations are resolved by employing the boundary conditions specified in equations (16) and (17). To illustrate the outcomes, a range of parameters is examined and depicted through graphical representations and tabulated data.
\begin{itemize}
  \item An initial guess for \( F^{(0)} \), \( \theta^{(0)} \), \( \phi^{(0)} \), and \( \chi^{(0)} \) is chosen such that the boundary conditions in equations~(16)-(17) are identically satisfied.

  \item Simpson’s rule provides an approximation for the solution of the linear system shown in equation~(11). Subsequently, \( F^{(1)} \) in equation~(23) is determined based on the known value of \( G^{(1)} \).

  \item In equations~(12)-(15), the derivatives are replaced by finite differences, and the resulting system~(23)-(26) is obtained by employing the SOR technique.

  \item Starting from the initial sequences \( F^{(1)} \), \( \theta^{(1)} \), \( \phi^{(1)} \), and \( \chi^{(1)} \), successive series of functions are generated until the sequences \( \{F^{(k)}\} \), \( \{\theta^{(k)}\} \), \( \{\phi^{(k)}\} \), and \( \{\chi^{(k)}\} \) converge to \( F \), \( \theta \), \( \phi \), and \( \chi \), respectively.

  \item The iterative procedure continues until the following convergence condition is met:
  \[
  \max \left\{
  \frac{\| \chi^{(k+1)} - \chi^{(k)} \|_{L^2}}{\| \chi^{(k)} \|_{L^2}}, \,
  \frac{\| \phi^{(k+1)} - \phi^{(k)} \|_{L^2}}{\| \phi^{(k)} \|_{L^2}}, \,
  \frac{\| F^{(k+1)} - F^{(k)} \|_{L^2}}{\| F^{(k)} \|_{L^2}}, \ \\
  \frac{\| \theta^{(k+1)} - \theta^{(k)} \|_{L^2}}{\| \theta^{(k)} \|_{L^2}}, \,
  \right\} 
  < \text{TOL}_{\text{iter}}.
  \]

  \item The tolerance \( \text{TOL}_{\text{iter}} \) is fixed at \( 10^{-11} \) throughout the computations.
\end{itemize}

\section{Results and Interpretation}
\indent Fig. 2 represent the significance of the stretching rate $\lambda$ on the velocity distribution. As $\lambda$ enhanced from 0.1 to 0.5, the velocity gradient near the wall Increases. The results align with classical boundary layer theory for stretching sheets, where larger stretching parameter values enhance momentum diffusion. Fig. 3 represents the impact of the mass flux parameter $\mathrm{S}$ on the normal velocity. It is evident from the figure that the velocity expand with rising values of $\mathrm{S}$. Fig. 4 depicts the impact of the magnetic parameter on the velocity distribution. As $\mathrm{M}$ increases, the velocity decreases, indicating boundary layer thinning due to the opposing Lorentz force. Fig. 5 presents the effect of the porosity parameter $\mathrm{P_0}$ on the streamwise velocity $G'(\eta)$. A rise in both $\mathrm{S}$ tends to amplify the velocity $G'(\eta)$. Fig. 6 represents the streamwise velocity for distinct values of the nonlinearity parameter $m$. An increase in the positive constant $m$ enhances $G'(\eta)$. In fig. 7 indicates that the temperature profile of the radiation number increases as enhanced the values of $\theta(\eta)$. A rises in the Eckert number $\mathrm{Ec}$ shows an enhancement in the fluid temperature. This is because the Eckert number accounts for the velocity term in the energy equation, which significantly contributes to the elevation of temperature as shown in fig. 8. The fig. 9 demonstrates the effect of the suction parameter $\mathrm{S}$ on the temperature field $\theta(\eta)$. With increasing $\mathrm{S}$, the temperature profile becomes higher, indicating that enhanced suction augments the thermal boundary layer and raises the fluid temperature.\\
\indent Fig. 10 establish the impact of the Prandtl number $\mathrm{Pr}$ on the temperature distribution $\theta(\eta)$ across the boundary layer. It is observed that as the Prandtl number increases, the temperature profile consistently declines. Higher values of $\mathrm{Pr}$ correspond to a fluid with lower thermal diffusivity, which leads to a thinner thermal boundary layer. Fig.~11 represents the temperature profiles for distinct values of the nonlinearity parameter $m$. As reducing $\theta(\eta)$. This indicates that the temperature profile accelerates in the case of nonlinear shrinking/stretching of the sheet, as shown in the figure. From fig.~12, it is evident that increasing the mass flux parameter $\mathrm{S}$ reduces the temperature profile.  Higher $\mathrm{M}$ values fig.~13 exhibit the graph increase through the boundary layer. Fig. 14 highlight the influence of increasing the thermophoresis parameter $\mathrm{Nt}$. It's noticed that the increased values of $\mathrm{Nt}$ lead to an enhanced the temperature profile. Fig. 15 portray the variation of nanoparticles concentration profile with the chemical reaction parameter $\mathrm{Rc}$. The presence of this parameter produces to a decline in the concentration profile. Fig. 16 shows the profile of nanoparticles concentration in the parameter of $\mathrm{S}$, as S enhanced the nanoparticles decline. Fig. 17 illustrates the behavior of the thermophoresis parameter $\mathrm{Nt}$, the increased values of $\mathrm{Nt}$ lead to an enhanced the motile microorganism density profile. The fig. 18 demonstrates the effect of the Lewis number $\mathrm{le}$, its shows decreases behaviour along the boundary layer. Fig. 19 indicates the increased values of $\mathrm{Nb}$, represent a declines in the nano-particle concentration. Fig. 20 showing a decreasing trend as the microorganism concentration $\sigma$ increases. Fig. 21  establish the impact of the motile microorganism density profile enhanced  as the Peclet number show decay behaviour. The fig. 22 demonstrates the effect of the Schmidt number decreases in a nanoparticles concentration profiles. \\
\textbf{Explanations for the tables are presented as follows:}\\
Table 2 presents the influence of the power-law index \(m\) and porosity parameter $\mathrm{P_{0}}$ on the skin friction coefficient \(C_{fx}\), Sherwood number $\mathrm{Sh_{x}}$, Nusselt number \(\mathrm{Nu_{x}}\), and motile microorganism density number \(Nn_{x}\). With increasing m from 2 to 6, the wall shear stress \(G''(0)\) and heat transfer rate \(-\theta'(0)\) exhibit a steady increase, reflecting enhanced momentum and thermal transport at the surface. The mass transfer rate \(-\phi'(0)\) and microorganism flux \(-\chi'(0)\) also increase in magnitude while remaining negative, which indicates that these fluxes are directed away from the surface. Overall, higher m values correspond to stronger transport phenomena due to the intensified stretching or flow associated with larger m. Whereas, with an increase in $\mathrm{P_0}$, the value of $G''(0)$ rises noticeably, reflecting an elevated wall shear stress. Similarly, $\theta'(0)$ increases, indicating improved heat transfer at the surface. On the other hand, $\phi'(0)$ and $\chi'(0)$ decrease as $\mathrm{P_0}$ grows, which implies enhanced transport of solute and microorganisms away from the boundary as shown in table 2.\\
\indent Table 3 highlights the distinct impact of various parameters on heat transfer and the behavior of motile microorganisms. An increase in the Prandtl number $\mathrm{Pr}$ increases the Nusselt number \(\mathrm{Nu_{x}}\), signifying improved heat transfer due to the thinning of the thermal boundary layer. In contrast, the radiation parameter reduces \(\mathrm{Nu_{x}}\) because thermal radiation weakens the temperature gradient at the surface. Similarly, higher values of the Eckert number $\mathrm{Ec}$ and the magnetic parameter \(\mathrm{M}\) also lead to a decrease in \(\mathrm{Nu_{x}}\), as the additional heat generated by viscous dissipation and Lorentz forces thickens the thermal boundary layer. On the other hand, the Péclet number \(\mathrm{Pe}\) positively influences the motile microorganism density number $\mathrm{Nn_x}$, as stronger advection sharpens the microorganism gradient at the wall, thereby enhancing bioconvective transport. Table~3 illustrates the individual impacts of different parameters on the Sherwood number $\mathrm{Sh_{x}}$ and the motile microorganism density number $\mathrm{Nn_x}$. An increase in the Lewis number $\mathrm{Le}$ leads to a rise in $\mathrm{Sh_{x}}$, indicating enhanced mass transfer due to reduced solute diffusivity. Similarly, a higher chemical reaction parameter $\mathrm{Rc}$ increases $\mathrm{Sh_{x}}$, as stronger reactions steepen the concentration gradient at the wall. Conversely, both the Brownian motion parameter $\mathrm{Nb}$ and the thermophoresis parameter $\mathrm{Nt}$ result in a decrease in $\mathrm{Sh_x}$, since enhanced nanoparticle diffusion and thermophoretic effects thicken the concentration boundary layer. Meanwhile, the Schmidt number $\mathrm{Sc}$ positively influences $\mathrm{Nn_x}$, as lower microorganism diffusivity sharpens the microorganism density gradient at the surface, promoting bioconvective transport.
\begin{table}[htbp]
\centering
\caption{Impact of m and $\mathrm{P_0}$ on \(\mathrm{C_{fx}}\), \(\mathrm{Nu_x}\), \(\mathrm{Sh_x}\), and \(\mathrm{Nn_x}\)}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\(m\) & $\mathrm{C_{fx}}$ & $ \mathrm{Nu_x} $ & $ \mathrm{Sh_x} $ & $ \mathrm{Nn_x}$ & $\mathrm{P_0}$ & $\mathrm{C_{fx}}$ & $ \mathrm{Nu_x} $ & $\mathrm{Sh_x} $ & $ \mathrm{Nn_x}$ \\
\hline
2 & 1.6200 & 1.0320 & -0.4137 &  0.0277 & 11 & 2.9093 & 0.8566 & -0.3052 & 0.0608 \\ 
3 & 2.0110 & 1.2604 & -0.5615 & -0.0238 & 22 & 3.9167 & 0.8778 & -0.3220 & 0.0531 \\
4 & 2.3512 & 1.4738 & -0.7016 & -0.0769 & 33 & 4.6848 & 0.8892 & -0.3313 & 0.0484 \\
5 & 2.6589 & 1.6772 & -0.8370 & -0.1315 & 44 & 5.3225 & 0.8967 & -0.3375 & 0.0450 \\
6 & 2.9436 & 1.8734 & -0.9693 & -0.1875 & 55 & 5.8745 & 0.9022 & -0.3420 & 0.0425 \\
\hline
\end{tabular}
\label{tab:m_effect}
\end{table}

\begin{table}[h!]
\centering
\caption{Impact of $\mathrm{Pr}$, $\mathrm{R}$, $\mathrm{Ec}$, and $\mathrm{M}$ on $\mathrm{Nu_x}$ and $\mathrm{Pe}$ on $\mathrm{Nn_x}$}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
$\mathrm{Pr}$ & $ \mathrm{Nu_x} $ & $\mathrm{R}$ & $ \mathrm{Nu_x} $ & $\mathrm{Ec}$ & $\mathrm{ Nu_x} $ & $\mathrm{M}$ & $ \mathrm{Nu_x}$ & $\mathrm{Pe}$ & $\mathrm{Nn_x}$\\ \hline
0.2 & 0.3884 & 1 & 0.4245 & 0.1 & 0.8062 & 0.1 & 0.8175& 0.1  &	0.7202\\ 
0.3 & 0.4552 & 2 & 0.3797 & 0.2 & 0.7936 & 0.5 & 0.7927& 0.3  & 1.0181\\
0.4 & 0.5050 & 3 & 0.3461 & 0.3 & 0.7810 & 1.1 & 0.7607& 0.5  & 1.3217\\ 
0.5 & 0.5441 & 4 & 0.3198 & 0.4 & 0.7684 & 1.5 & 0.7425& 0.7  & 1.6302\\ 
0.6 & 0.5760 & 5 & 0.2986 & 0.5 & 0.7558 & 2.4 & 0.7094& 0.9 &1.9430 \\
\hline
\end{tabular}
\label{tab:m_effect}
\end{table}

\begin{table}[h!]
\centering
\caption{Impact of $\mathrm{Le}$, $\mathrm{Rc}$, $\mathrm{Nb}$, and $\mathrm{Nt}$ on $\mathrm{Sh_x}$ and $\mathrm{Sc}$ on $Nn_x$}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
$\mathrm{Le}$ & $\mathrm{Sh_x}$ & $\mathrm{Rc}$ & $\mathrm{Sh_x}$ & $\mathrm{Nb}$ & $\mathrm{Sh_x}$ & $\mathrm{Nt}$ & $\mathrm{Sh_x}$ & $\mathrm{Sc}$ & $\mathrm{Nu_x}$  \\ \hline
0.9 & 0.9663 & 1.5 & 0.3274 & 0.05 & 0.6555 & 0.2 & 1.0553 &0.1 & 1.1438   \\ 
1.1 & 1.1418 & 2.5 & 0.4982  & 0.09 & 0.7936 & 0.4 & 1.0553&0.2 & 1.3216  \\
1.6 & 1.5506 & 3.5 & 0.6442 & 0.15 & 1.0042 & 0.6 & 0.8389&0.3 &  1.4845 \\ 
2.1 & 1.9346 & 4.5 & 0.7727 & 0.21 & 1.0541 & 0.8 & 0.7395&0.4 &  1.6394 \\ 
2.6 & 2.3043 & 5.5 & 0.8882 & 0.29 & 1.0884 & 1.0 & 0.6457&0.5 & 1.7893 \\
\hline
\end{tabular}
\label{tab:m_effect}
\end{table}

\begin{figure}[H]
\setlength\abovecaptionskip{0.5\baselineskip}
    \centering
     
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig7.png}
       \caption*{Fig. 2 \(G'(\eta) \) for various $\lambda$ and  $\mathrm{M}$ = 0.2, $\mathrm{Pe}$ = 0.9, $\mathrm{Nt}$ = 0.3, \(\sigma = 0.6\), $\mathrm{Pr}$ = 0.5, $\mathrm{Le}$ = 0.2, $\mathrm{P_{0}}$ = 0.2, $\mathrm{Ec}$ = 0.01, $\mathrm{S}$ = 0.2, $\mathrm{R}$ = 1.7,  $\mathrm{Nb}$ = 0.4, m = 2, $\mathrm{Rc}$ = 0.8, $\mathrm{Sc}$ = 0.8. }
        \label{fig:sub1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig8.png}
        \caption*{Fig. 3 \(G'(\eta) \) for various $\mathrm{S}$ and $\mathrm{M}$ = 0.2, \(\lambda = 0.6\), $\mathrm{Nt}$ = 0.3, \(\sigma = 0.5\), $\mathrm{Sc}$ = 0.2, $\mathrm{Pe}$ = 0.9, $\mathrm{P_{0}}$ = 0.2, $\mathrm{Ec}$ = 0.01, $\mathrm{Nb}$ = 0.4, $\mathrm{Le}$ = 0.5,  $\mathrm{Pr}$ = 0.5, m = 3, $\mathrm{Rc}$ = 0.8, $\mathrm{R}$ = 0.4.}
        \label{fig:sub2}
    \end{subfigure} 
    \label{fig:sidebyside}
\end{figure}

\begin{figure}[H]
\setlength\abovecaptionskip{0.5\baselineskip}
    \centering
     
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig15.png}
       \caption*{Fig. 4 \(G'(\eta) \) for various $\mathrm{M}$ and $\mathrm{Rc}$ = 0.4, \(\lambda = 0.2 \), \(\sigma = 0.1 \), $\mathrm{Sc}$ = 0.7, $\mathrm{Pe}$ = 0.4, $\mathrm{P_{0}}$ = 1.2, $\mathrm{Pr}$ = 0.3, $\mathrm{S}$ = 0.3, $\mathrm{Le}$ = 0.2,  $\mathrm{Nb}$ = 0.1, $\mathrm{Nt}$ = 0.1, m = 0.2, $\mathrm{Ec}$ = 0.01, $\mathrm{R}$ = 1.9.}
        \label{fig:sub1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig17.png}
        \caption*{Fig. 5 \(G'(\eta) \) for various $\mathrm{P_{0}}$ and $\mathrm{Rc}$ = 0.4, \(\lambda = 0.2 \), $\mathrm{Nt}$ = 0.4, \(\sigma = 0.5 \), $\mathrm{Sc}$ = 0.6, $\mathrm{Pe}$ = 0.4, $\mathrm{M}$ = 0.9, $\mathrm{Pr}$ = 0.2, $\mathrm{S}$ = 0.3, $\mathrm{Le}$ = 0.3,  $\mathrm{Nb}$ = 0.4, m = 0.2, $\mathrm{Ec}$ = 0.01, $\mathrm{R}$ = 0.1.}
        \label{fig:sub2}
    \end{subfigure} 
    \label{fig:sidebyside}
\end{figure}

\begin{figure}[H]
\setlength\abovecaptionskip{0.5\baselineskip}
    \centering
     
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig21.png}
       \caption*{Fig. 6 \(G'(\eta) \) for various m and $\mathrm{Rc}$ = 0.4, \(\lambda = 0.2 \), $\mathrm{Nt}$ = 0.1, \(\sigma = 1.3 \), $\mathrm{Sc}$ = 0.2, $\mathrm{Pe}$ = 0.4, $\mathrm{M}$ = 0.2, $\mathrm{Pr}$ = 2.2, $\mathrm{S}$ = 0.3, $\mathrm{Le}$ = 0.1, $\mathrm{Nb}$ = 0.1, $\mathrm{Ec}$ = 0.01, $\mathrm{P_{0}}$ = 0.2, $\mathrm{R}$ = 0.2.}
        \label{fig:sub1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig4.png}
        \caption*{Fig. 7 \(\theta(\eta) \) for various $\mathrm{R}$ and  \(\lambda = 0.2\), \(\sigma = 0.6\), $\mathrm{Pr}$ = 0.2, $\mathrm{Pe}$ = 0.4, $\mathrm{P_{0}}$ = 0.2, $\mathrm{Ec}$ = 0.01, $\mathrm{S}$ = 0.9, $\mathrm{Le}$ = 1.0,  $\mathrm{Nb}$ = 0.1, m = 1.5, $\mathrm{Rc}$ = 0.4, $\mathrm{Nt}$ = 0.2, $\mathrm{M}$ = 0.1, $\mathrm{Sc}$ = 0.7.}
        \label{fig:sub2}
    \end{subfigure} 
    \label{fig:sidebyside}
\end{figure}

\begin{figure}[H]
\setlength\abovecaptionskip{0.5\baselineskip}
    \centering
     
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig19.png}
       \caption*{Fig. 8 \(\theta(\eta) \) for various $\mathrm{Ec}$ and $\mathrm{Rc}$ = 0.4, \(\lambda = 0.2 \), $\mathrm{Nt}$ = 0.2, \(\sigma = 0.5 \), $\mathrm{Sc}$ = 0.2, $\mathrm{Pe}$ = 0.4, $\mathrm{M}$ = 0.1, $\mathrm{Pr}$ = 1.0, $\mathrm{S}$ = 0.3, $\mathrm{Le}$ = 0.2,  $\mathrm{Nb}$ = 0.4, m = 0.3, $\mathrm{P_{0}}$ = 0.2, $\mathrm{R}$ = 0.8.}
        \label{fig:sub1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig20.png}
        \caption*{Fig. 9 \(\theta(\eta) \) for various $\mathrm{S<0}$ and $\mathrm{M}$ = 0.2, \(\lambda = 0.6\), $\mathrm{Nt}$ = 0.3, \(\sigma = 0.5\), $\mathrm{Sc}$ = 0.2, $\mathrm{Pe}$ = 0.9, $\mathrm{P_{0}}$ = 0.2, $\mathrm{Ec}$ = 0.01, $\mathrm{Pr}$ = 0.5, $\mathrm{Le}$ = 0.5, $\mathrm{Nb}$ = 0.4, m = 3, $\mathrm{Rc}$ = 0.8, $\mathrm{R}$ = 0.4.}
        \label{fig:sub2}
    \end{subfigure} 
    \label{fig:sidebyside}
\end{figure}

\begin{figure}[H]
\setlength\abovecaptionskip{0.5\baselineskip}
    \centering
     
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig23.png}
       \caption*{Fig. 10 \(\theta(\eta) \) for various $\mathrm{Pr}$ and $\mathrm{M}$ = 0.1, \(\lambda = 0.2\), $\mathrm{Nt}$ = 0.2, \(\sigma = 0.6\), $\mathrm{Sc}$ = 0.7, $\mathrm{Pe}$ = 0.4, $\mathrm{P_{0}}$ = 0.2, $\mathrm{Ec}$ = 0.01, $\mathrm{S}$ = 0.3, $\mathrm{Le}$ = 1.0,  $\mathrm{Nb}$ = 0.1, m = 2, $\mathrm{Rc}$ = 0.4, $\mathrm{R}$ = 0.8.}
        \label{fig:sub1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig22.png}
        \caption*{Fig. 11 \(\theta(\eta) \) for various for various m and $\mathrm{R{c}}$ = 0.4, \(\lambda = 0.2 \), $\mathrm{Nt}$ = 0.1, \(\sigma = 1.3 \), $\mathrm{Sc}$ = 0.2, $\mathrm{Pe}$ = 0.4, $\mathrm{M}$ = 0.2, $\mathrm{Pr}$ = 2.2, $\mathrm{S}$ = 0.3, $\mathrm{Le}$ = 0.1, $\mathrm{Nb}$ = 0.1, $\mathrm{Ec}$ = 0.01, $\mathrm{P_{0}}$ = 0.2, $\mathrm{R}$ = 0.2.}
        \label{fig:sub2}
    \end{subfigure} 
    \label{fig:sidebyside}
\end{figure}

\begin{figure}[H]
\setlength\abovecaptionskip{0.5\baselineskip}
    \centering
     
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig9.png}
       \caption*{Fig. 12 \(\theta(\eta) \) for various $\mathrm{S>0}$ and $\mathrm{M}$ = 0.2, \(\lambda = 0.6\), $\mathrm{Nt}$ = 0.3, \(\sigma = 0.6\), $\mathrm{Sc}$ = 0.2, $\mathrm{Pe}$ = 0.9, $\mathrm{P_{0}}$ = 0.2, $\mathrm{Ec}$ = 0.01, $\mathrm{Nb}$ = 0.4, $\mathrm{Le}$ = 0.5,  $\mathrm{Pr}$ = 0.5, m = 0.5, $\mathrm{Rc}$ = 0.8, $\mathrm{R}$ = 0.1.}
        \label{fig:sub1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig16.png}
        \caption*{Fig. 13 \(\theta(\eta) \) for various $\mathrm{M}$ and $\mathrm{Rc}$ = 0.4, \(\lambda = 0.2 \), $\mathrm{Nt}$ = 0.4, \(\sigma = 1.5 \), $\mathrm{Sc}$ = 0.7, $\mathrm{Pe}$ = 0.4, $\mathrm{P_{0}}$ = 1.2, $\mathrm{Pr}$ = 1.5, $\mathrm{S}$ = 0.3, $\mathrm{Le}$ = 0.2,  $\mathrm{Nb}$ = 0.4, m = 0.2, $\mathrm{Ec}$ = 0.01, $\mathrm{R}$ = 2.}
        \label{fig:sub2}
    \end{subfigure} 
    \label{fig:sidebyside}
\end{figure}
\begin{figure}[H]
\setlength\abovecaptionskip{0.5\baselineskip}
    \centering
     
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig13.png}
       \caption*{Fig. 14 \(\theta(\eta) \) for various $\mathrm{Nt}$ and $\mathrm{M}$ = 0.1, \(\lambda = 0.2\), \(\sigma = 0.6\), $\mathrm{Sc}$ = 0.2, $\mathrm{P_{0}}$ = 0.2, $\mathrm{Ec}$ = 0.1, $\mathrm{S}$ = 0.1, $\mathrm{}$Pe = 0.5, $\mathrm{Le}$ = 0.5,  $\mathrm{Pr}$ = 0.2, m = 2, $\mathrm{Rc}$ = 0.4, $\mathrm{Nb}$ = 0.1, $\mathrm{R}$ = 0.3.}
        \label{fig:sub1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig18.png}
        \caption*{Fig. 15 \(\phi(\eta) \) for various $\mathrm{Rc}$ and $\mathrm{M}$ = 0.1, \(\lambda = 0.2\), $\mathrm{Nt}$ = 0.2, \(\sigma = 1.5\), $\mathrm{Sc}$ = 0.2, $\mathrm{Pe}$ = 0.4, $\mathrm{P_{0}}$ = 0.2, $\mathrm{Pr}$ = 1.0, $\mathrm{S}$ = 0.3, $\mathrm{Le}$ = 0.2,  $\mathrm{Nb}$ = 0.4, m = 2, $\mathrm{Ec}$ = 0.01, $\mathrm{R}$ = 0.1.}
        \label{fig:sub2}
    \end{subfigure} 
    \label{fig:sidebyside}
\end{figure}

\begin{figure}[H]
\setlength\abovecaptionskip{0.5\baselineskip}
    \centering
     
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig10.png}
       \caption*{Fig. 16 \(\phi(\eta) \) for various $\mathrm{S}$ and $M$ = 0.2, \(\lambda = 0.6\), $\mathrm{Nt}$ = 0.3, \(\sigma = 0.6\), $\mathrm{Sc}$ = 0.2, $\mathrm{Pe}$ = 0.9, $P_{0}$ = 0.2, $\mathrm{Ec}$ = 0.01, $\mathrm{Nb}$ = 0.4, $\mathrm{Le}$ = 0.5,  $\mathrm{Pr}$ = 0.5, m = 0.5, $\mathrm{Rc}$ = 0.8, $\mathrm{R}$ = 0.1.}
        \label{fig:sub1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig14.png}
        \caption*{Fig. 17 \(\phi(\eta) \) for various $\mathrm{Nt}$ and \(\lambda = 0.4\), $\mathrm{Nb}$ = 0.2, \(\sigma = 0.9\), $\mathrm{Sc}$ = 0.2, $\mathrm{Pe}$ = 0.5, $\mathrm{P_{0}}$ = 0.2, $\mathrm{Ec}$ = 0.01, $\mathrm{S}$ = 1.2, $\mathrm{Le}$ = 0.9, $\mathrm{M}$ = 0.1, $\mathrm{Pr}$ = 0.2, m = 3, $\mathrm{Rc}$ = 0.7, $\mathrm{R}$ = 0.3.}
        \label{fig:sub2}
    \end{subfigure} 
    \label{fig:sidebyside}
\end{figure}

\begin{figure}[H]
\setlength\abovecaptionskip{0.5\baselineskip}
    \centering
     
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig5.png}
       \caption*{Fig. 18 \(\phi(\eta) \) for various $\mathrm{Le}$ and  $\mathrm{M}$ = 0.1, \(\lambda = 0.2\), $\mathrm{Nt}$ = 0.2, \(\sigma = 0.6\), $\mathrm{Pr}$ = 0.2, $\mathrm{Pe}$ = 0.4, $\mathrm{P_{0}}$ = 0.2, $\mathrm{Ec}$ = 0.01, $\mathrm{S}$ = 0.9, $\mathrm{R}$ = 0.8,  $\mathrm{Nb}$ = 0.1, m = 0.5, $Rc$ = 0.4, Sc = 0.7.}
        \label{fig:sub1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig12.png}
        \caption*{Fig.19 \(\phi(\eta) \) for various $\mathrm{Nb}$ and $\mathrm{M}$ = 0.1, \(\lambda = 0.2\), $\mathrm{Nt}$ = 0.1, \(\sigma = 0.6\), $\mathrm{Sc}$ = 0.2, $\mathrm{Pe}$ = 0.5, $\mathrm{P_{0}}$ = 0.2, $\mathrm{Ec}$ = 0.1, $\mathrm{S}$ = 0.7, $\mathrm{Le}$ = 0.5,  $\mathrm{Pr}$ = 0.2, m = 2, $\mathrm{Rc}$ = 0.4, $\mathrm{R}$ = 0.3.}
        \label{fig:sub2}
    \end{subfigure} 
    \label{fig:sidebyside}
\end{figure}
\begin{figure}[H]
\setlength\abovecaptionskip{0.5\baselineskip}
    \centering
     
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig11.png}
       \caption*{Fig.20 \(\chi(\eta) \) for various \(\sigma \) and $\mathrm{M}$ = 0.5, \(\lambda = 2.2\), $\mathrm{Nt}$ = 0.5, $\mathrm{Nb}$ = 0.2, $\mathrm{Sc}$ = 0.2, $\mathrm{Pe}$ = 0.5, $\mathrm{P_{0}}$ = 2.2, $\mathrm{Ec}$ = 0.01, $\mathrm{S}$ = 2.2, $\mathrm{Le}$ = 0.9, $\mathrm{Pr}$ = 0.2, m = 0.5, $\mathrm{R}$ = 0.4, $\mathrm{R}$ = 0.7.}
        \label{fig:sub1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig6.png}
        \caption*{Fig. 21 \(\chi(\eta) \) for various $\mathrm{Pe}$ and $\mathrm{M}$ = 0.1, \(\lambda = 0.2\), $\mathrm{Nt}$ = 0.2, \(\sigma = 0.6\), $\mathrm{Pr}$ = 0.2, $\mathrm{Le}$ = 1.0, $\mathrm{P_{0}}$ = 0.2, $\mathrm{Ec}$ = 0.01, $\mathrm{S}$ = 0.9, $\mathrm{R}$ = 0.8,  $\mathrm{Nb}$ = 0.1, m = 2, $\mathrm{Rc}$ = 0.4, $\mathrm{Sc}$ = 0.7}
        \label{fig:sub2}
    \end{subfigure} 
    \label{fig:sidebyside}
\end{figure}

\begin{figure}[H]
\setlength\abovecaptionskip{1\baselineskip}
\centering
\includegraphics[width=0.6\textwidth]{G:/Latex/fig3.png}
  \caption*{ Fig. 22 \(\chi(\eta)\) for various $\mathrm{Sc}$ and \(\lambda = 0.2\), \(\sigma = 0.6\), $\mathrm{Pr}$ = 0.2, $\mathrm{Pe}$ = 0.4, $\mathrm{P_{0}}$ = 0.2, $\mathrm{Nt}$ = 0.2, $\mathrm{Ec}$ = 0.01, $\mathrm{S}$ = 0.9, $\mathrm{Le}$ = 1.0, $\mathrm{Nb}$ = 0.1, m = 0.3, $\mathrm{Rc}$ = 0.4, $\mathrm{M}$ = 0.1, $\mathrm{R}$ = 0.8.}
\label{fig1}
\end{figure} 

\subsection{Comparative Analysis and Code Validation}
The results obtained from the numerical results are contrasted with previously published data, as illustrated in Table 1. The analysis demonstrates excellent agreement with the expected outcomes. This evaluation confirms the accuracy of our simulation procedure. The data in Table 1 show that escalating the injection/suction parameter (S) enhances all measured physical quantities.\\
\indent We further validated our graphical results by comparing them with existing simulation data, as shown in Fig 23(a) and Fig 23(b). The comparison demonstrates reliable results. Fig 23(a) portrays the results obtained in the literature \cite{J5}, while Fig 23(b) shows the present work on the temperature trend escalates due to the Brownian motion parameter in the absence of magnetic effects, chemical reaction, and Eckert number with these parameters \(\sigma = 0.6\), $\mathrm{M}$ = 0, \(\lambda = 0.5\), $\mathrm{Sc}$ = 0.7, $P_{0}$ = 0, $\mathrm{S}$ = 1.0, $\mathrm{Le}$ = 1.0, $\mathrm{Pe}$ = 0.1, $\mathrm{Pr}$ = 1, m = 1, $\mathrm{Rc}$ = 0, $\mathrm{Nt}$ = 0.1, $\mathrm{R}$ = 0, $\mathrm{Ec}$ = 0. A comparable increasing pattern in \( \theta(\eta) \) was scrutinized by Ahmad et al. [5].

\begin{table}[htbp]
\centering
\caption{Comparison of numerical results for \( Nu_x \) and \( Nn_x \) over a range of \( \mathrm{S} \), under the condition \( \mathrm{P_0} = \mathrm{Rc} = \mathrm{M} = \mathrm{R} = 0 \).}
\scalebox{0.85}{ % scale factor < 1 shrinks the table
\begin{tabular}{c|ccc|ccc}
\hline
\multirow{2}{*}{\( \mathrm{S} \)} & \multicolumn{3}{c|}{\( Nu_x \)} & \multicolumn{3}{c}{\( Nn_x \)} \\ 
\cline{2-7}
 & Zaimi et al. \cite{J23} & Ahmad et al. \cite{J5} & Obtained results 
 & Zaimi et al. \cite{J23} & Ahmad et al. \cite{J5} & Obtained results \\ 
\hline
2.5 & 3.38557 & 3.3859 & 3.3861 & 5.48217 & 5.4748 & 5.4921 \\
3.0 & 4.13260 & 4.1336 & 4.1342 & 6.49841 & 6.4862 & 6.4984 \\
3.5 & 4.86811 & 4.8698 & 4.8701 & 7.54285 & 7.5237 & 7.5432 \\
4.0 & 5.59672 & 5.5994 & 5.5999 & 8.60180 & 8.5736 & 8.6019 \\
4.5 & 6.32096 & 6.3248 & 6.3250 & 9.66858 & 9.6289 & 9.6684 \\
\hline
\end{tabular}}
\label{tab:comparison}
\end{table}
\begin{figure}[H]
\setlength\abovecaptionskip{0.5\baselineskip}
    \centering
     
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig2(a).png}
       \caption*{Fig.23(a) \(\theta(\eta)\) for various $\mathrm{Nb}$ and \(\sigma = 0.6\), $\mathrm{M}$ = 0, \(\lambda = 0.5\), $\mathrm{Sc}$ = 1, $P_{0}$ = 0, $\mathrm{S}$ = 1.0, $\mathrm{Le}$ = 1.0, $\mathrm{Pe}$ = 0.1, $\mathrm{Pr}$ = 1, m = 1, $\mathrm{Rc}$ = 0, $\mathrm{Nt}$ = 0.5, $\mathrm{R}$ = 0, $\mathrm{Ec}$ = 0.}
        \label{fig:sub1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[]{0.45\textwidth}
        \includegraphics[width=\linewidth]{G:/Latex/fig2(b).png}
        \caption*{Fig.23(b) \(\theta(\eta)\)for various $\mathrm{Nb}$ and \(\sigma = 0.6\), $\mathrm{M}$ = 0, \(\lambda = 0.5\), $\mathrm{Sc}$ = 0.7, $P_{0}$ = 0, $\mathrm{S}$ = 1.0, $\mathrm{Le}$ = 1.0, $\mathrm{Pe}$ = 0.1, $\mathrm{Pr}$ = 1, m = 1, $\mathrm{Rc}$ = 0, $\mathrm{Nt}$ = 0.1, $\mathrm{R}$ = 0, $\mathrm{Ec}$ = 0.}
        \label{fig:sub2}
    \end{subfigure} 
    \label{fig:sidebyside}
\end{figure}
\section{Conclusions}
This study presents a novel investigation of an electrically conducting nanofluid containing nanoparticles as well as gyrotactic microorganisms over a nonlinear stretching or shrinking surface within a porous medium. Thermal radiation is incorporated into the energy equation, and the combined influences of Joule heating, magnetohydrodynamics, viscous dissipation, thermal radiation, and bioconvection are examined. To obtain the numerical solutions, the problem is mathematically formulated and solved using the SOR scheme. The key outcomes of this study are summarized as described here:\\
\begin{itemize}
\item The velocity $G'(\eta)$ increases with higher porosity $P_0$, injection $\mathrm{S}$, and nonlinearity parameter $m$, indicating enhanced fluid motion and reduced resistance.
\item The temperature profile $\theta(\eta)$ rises with increasing Eckert number $\mathrm{Ec}$,  magnetic parameter $\mathrm{M}$, radiation parameter $\mathrm{R}$, thermophoresis $\mathrm{Nt}$, and suction $\mathrm{S}$, reflecting stronger thermal effects and a thicker thermal boundary layer.
\item The nanoparticles concentration profile $\phi(\eta)$ diminishes with decreasing chemical reaction $\mathrm{Rc}$, brownaian motion $\mathrm{Nb}$, injection $S$, and Lewis number $\mathrm{Le}$. 
\item The density motile microorganism $\chi(\eta)$ decreases as the Schmidt number $\mathrm{Sc}$, Peclet number $\mathrm{Pe}$, and motile microorganism density $\sigma$, showing reduced mass diffusion and thinner concentration layers. 
\item The magnetic parameter suppresses the velocity $G'(\eta)$ and reduces the thickness of the velocity boundary layer due to the opposing Lorentz force, confirming the damping effect of magnetic fields on fluid motion. As the Prandtl number $\mathrm{Pr}$  enhanced the values, the thermal boundary layer thickness decay.\\
\end{itemize}
\indent Now, we outline some limitations and advantages of the present study. The assumption of self-similarity, adopted here to reduce the governing partial differential equations (PDEs) to ordinary differential equations (ODEs), significantly simplifies the mathematical analysis. However, it is essential to ensure that the obtained solutions correspond to physically realistic scenarios. In future work, these findings could be validated by solving the full PDEs numerically without imposing similarity transformations. Furthermore, the study assumes steady, laminar, two-dimensional flow and does not account for possible three-dimensional or unsteady effects, which might occur in practical applications.
\section*{Declarations}
\subsection*{Ethical approval}
Not applicable.
\subsection*{Consent to participate}
Not applicable.
\subsection*{Consent to publish}
Not applicable.
\section*{Funding statement}
No specific assistance was provided for this research by government, private, or nonprofit organizations.
\section*{Ethical approval}
This study does not involve the use of human subjects or animals, therefore ethical approval is not required. The research relies exclusively on publicly available data, and no personally identifiable information is being collected or analyzed. All procedures and methodologies strictly adhere to established ethical guidelines and regulations.
\section*{Credit authorship contribution statement}
\textbf{Aimin Fatima Muhammad Imran Asjad:}\\
Investigation, Writing – original draft, Methodology, Visualization.\\
\textbf{Muhammad Imran Asjad, Saraj Khan:}\\
Conceptualization, Formal analysis, Investigation, Supervision, Edit and Review.\\
\textbf{M. S. Alqarni, Muhammad Imran Asjad, Saraj Khan:}\\
Funding acquisition, Project administration, Resources, Validation, Edit and Review.
%\textbf{Aimin Fatima}: Investigation,
%Writing – original draft, Methodology.\\
%\textbf{Muhammad Imran Asjad}: Conceptualization, Formal analysis, Investigation, Supervision.\\
%\textbf{Magda Abd El-Rahman}: Funding acquisition, Project administration, Resources, Validation.\\
%\textbf{Acknowledgement}\\
%Magda Abd El-Rahman extends their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP2/39 /46. \\
\section*{Declaration of Competing Interest}
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
\section*{Data availability}
The datasets used or analysed during the current study available from the corresponding author on reasonable request.
\bibliographystyle{ieeetr}
%\bibliography{finalrefer}
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