Brownian motion and thermophoresis influence in magnetized Maxwell upper-convected stagnation point fluid flow via a stretching porous surface

Abstract

This article focuses on the effects of Brownian motion and thermophoresis convection in the stagnation point flow of a Maxwell upper-convected fluid over a non-Darcian porous surface with slip conditions and a magnetic inclination effect. The Maxwell dissipative fluid accounts for Joule heating due to an imposed magnetic field and porous medium resistance. At the same time, the Cattaneo-Christov heat flux model represents thermal relaxation in contrast to the conventional Fourier law. The resulting nonlinear partial differential equations are transformed into ordinary differential equations (ODEs) using similarity variables. The analysis revealed the influence of dimensionless numbers, including Deborah, Eckert, Prandtl, chemical reaction, thermal relaxation, inclination and slip parameters. The findings were presented using graphs and tables. An increase in the thermophoresis parameter notably led to higher concentration and temperature profiles. In contrast, increasing Brownian motion (Nb) decreased the solutal boundary layer thickness but enhanced the thermal boundary layer.

KEYWORDS:

• Thermophoresis
• upper-convected Maxwell fluid
• thermal radiation
• Brownian motion
• stretching surface

1. Introduction

The significance of radiation in heat transfer processes is profound, with applications in electrical power generation, nuclear plants, space vehicles, and various engineering fields. A fundamental understanding of thermal radiation is crucial in thermodynamics and heat transfer, playing a central role in concepts such as blackbody radiation and the Stefan-Boltzmann law. This knowledge is essential for designing efficient heating and cooling systems and studying energy transfer in industrial processes. Extensive research and development in this field have led to numerous discussions on fluid dynamics, including recent studies such as Haroon et al.’s investigation of radiation’s impact on nanofluid flow toward a stagnation point subject to convective heating [1]. RamiReddy et al. [2] analyzed nonlinear radiation effects on Maxwell MHD fluid flow under the influence of a chemical reaction. Alam [3] examined biomagnetic fluid dynamics with thermal radiation effects, revealing that an increase in the ferromagnetic interaction number decreases the coefficient of heat transfer and the skin friction rate on the flowing surface. Zhang et al. [4] studied the nonlinear quadratic convection on magnetized viscous fluid. Additionally, Stanford and Gerald [5] studied the thermal radiation impacts on MHD mixed convection for Jeffrey fluid flow and demonstrated that an increase in the Deborah number results in a decrease in fluid viscosity, leading to an enhancement of fluid velocity in the considered medium.

The choice between no-slip and partial-slip boundary conditions depends on factors such as the nature of the flowing materials and the characteristics of the convection surface. Slip boundary conditions and chemical reactions are crucial in industrial applications and technological advancements related to heat transfer. Consequently, numerous studies have explored these aspects. Murtaza et al. [6] considered slip velocity about biomagnetic Maxwell fluids. In contrast, Reddy et al. [7] studied the effect of slip on Casson fluid with heat transfer coupled with the Cattaneo-Christov heat flux model. Their research highlighted that the velocity slip parameter positively enhanced the velocity field. Oyekunle and Agunbiade [8], Akolade et al. [9] and Akolade [10] investigated the influence of inclination, Soret effect, magnetic field and Dufour effect on slip-conditioned flow over-stretching and flat surfaces. Akolade [11] also considered quadratic convection and thermophysical influences on viscoelastic fluid with velocity slip over a convective sheet. Alali and Megahed [12] explored the impacts of radiation and velocity slip on nanofluid flow over an unsteady stretching surface. Ibrahim and Negera [13] presented a Maxwell nanofluid of upper-convected nature with a slip condition, finding that an increase in velocity slip enhances the Nusselt and Sherwood numbers. Sharada and Shankar [14] described the partial slip effect and convective MHD Williamson flow in a stretched exponential sheet, reporting that the slip condition decreases the velocity profile of the considered flow geometry. Reddy et al. [15] investigated the effects of thermophoresis and Brownian motion on MHD Jeffery nanofluid using NDSolver for numerical computations. In contrast, Arshad et al. [16] scrutinized the impacts of Brownian motion and thermophoresis on MHD nanofluid flow with chemical reaction, revealing that the magnetic field improved the velocity profile, among other findings in this extensive research area.

Non-Newtonian fluids can be classified into two main categories: the integral or differential type and the rate type. Maxwell fluid, known for its non-viscous behaviour, falls under the rate type category, with its characteristics influenced by the effects of relaxation time. In a study conducted by Alam et al. [17], the effect of a variable magnetic field on the three-dimensional squeezing of viscous fluid in a vertically rotating disc configuration was investigated. Their findings indicated that fluid velocity increases with higher squeezing Reynolds numbers. Shafique et al. [18] explored Maxwell boundary layer flow with a binary chemical reaction in a rotating frame, revealing that heat transfer decreases with larger rotation sizes. Researchers such as Shahid [19], Kashyap et al. [20], and Shafiq and Khalique [21] discussed the upper convected MHD Maxwell fluid over shrinking/stretching sheets. Farooq et al. [22] explained Maxwell MHD flow with nanomaterials over stretched exponential surfaces in a related study. Their collective results showed that increases in the Deborah (the fluid relaxation time) and Hartman numbers lead to an upsurge in the velocity profile.

Nanofluids are a class of engineering fluids that have garnered significant attention in recent years due to their remarkable thermal and transport properties. These nanofluids are essentially conventional liquids, such as water or oil, infused with nanoparticles at the nanoscale, typically in the range of 1–100 nanometres in size (Choi and Eastman [23], Akolade [24]). The addition of these nanoparticles can significantly alter the fluid’s behaviour, making it an excellent candidate for enhancing heat transfer and other thermal properties. Nanofluids have demonstrated superior heat conductivity compared to their base fluids, which can have profound implications in various industries, including electronics cooling, energy systems, and medical applications. Researchers continue to explore the potential of nanofluids for their unique properties, making them a promising area of study with the potential to revolutionize heat transfer and fluid dynamics, addressing critical challenges in energy efficiency and technological advancements (see Refs. [25–27]). However, it is important to note that nanofluids’ synthesis, stability, and safety aspects also remain areas of active research and development as their practical applications expand. Numerous studies on this concept include the works of Riaz et al. [28], Bhatti and R. Ellahi [29], and Awan et al. [30], to mention but a few.

A stagnation point is a specific point in a flow field where the fluid velocity is zero. This point typically occurs when a fluid encounters an obstacle, such as a solid surface or another fluid, and its velocity decreases to zero due to the pressure increase caused by the obstruction. One classic real-life example of a stagnation point in fluid mechanics is the airflow over an aircraft wing. The significance of the stagnation point in stretching sheet fluid flow lies in its ability to help analyze flow behaviour and fluid separation from the surface. This knowledge has practical applications in various fields, including aerodynamics, heat transfer, and coating processes. As a result, Mehmood et al. [31] researched stagnation point flow of the oblique kind, considering the impact of diffusion-thermo and thermal diffusion. They reported that concentration profiles accelerated with a high Prandtl number. Makinde et al. [32] examined MHD stagnation point nanofluid flow over a convective stretching surface with slip conditions. Their results showed that the magnetic field tends to decrease the surface velocity. Meanwhile, Narayan et al. [33] investigated stagnation point MHD viscoelastic nanofluid flow over a heated stretching surface.

This study explores the influence of multiple slip, thermophoresis, and Brownian motion on magnetohydrodynamics (MHD) Maxwell fluid stagnation point flow. The analysis includes considerations of viscous dissipation, Joule heating due to an imposed magnetic field, the heating effect of a permeable medium, and a chemical reaction, all in the context of the Cattaneo-Christov heat flux model applied to a porous stretching sheet. To the best of the authors’ knowledge, the combined effects of these factors have not been previously studied, thereby creating a gap in knowledge part of which this study intends to fill.

2. Mathematical model

The researchers considered an incompressible boundary layer stagnation point flow of an upper-convected Maxwell fluid towards a stretching surface in a two-dimensional system, as shown in Figure 1. Due to two opposite and equal forces, the sheet is stretched along the $x$-axis with a velocity $U_w\left(x\right)=d_{0}x$. Along the flow in the $\xi$-axis, the velocity is denoted as $u$ and $C_w$ and $T_w$ represent the concentration and temperature at the plate, respectively. In contrast, $C_{\mathrm{\infty }}$, $T_{\mathrm{\infty }}$, and $U_e\left(x\right)=bx$ represent the concentration, temperature, and velocity at the free stream, respectively. Pressure is assumed to be constant in the flow field. The magnetic Reynolds number is considered significant, and the model incorporates dissipation effects, Joule heating, and Darcy dissipation heat. The analysis includes heat convection for Buongiorno’s nanofluid model with a fixed nanoparticle volume fraction. Additionally, it considers viscous dissipation effects, Joule heating, Brownian motion, Cattaneo-Christov heat flux, thermophoresis and radiation effects.

Figure 1. Physical model of the problem.

The vector form of the governing equations of continuity, momentum, energy and concentration in a steady state is as follows: (see Khan and Shaikh [34]; Ogunseye et al. [35]); (1) $\begin{array}{rl}\mathrm{\nabla }\cdot \overrightarrow{W}& =0,\end{array}$(1) (2) $\begin{array}{rl}{\rho }_f\left(\overrightarrow{W}\cdot \mathrm{\nabla }\overrightarrow{W}\right)& =\mathrm{\nabla }\cdot {\mathit{\tau }}_p+\text{si}{\text{n}}^2\left(\alpha \right)F_x-\varphi \left(K\right),\end{array}$(2) (3) $\begin{array}{rl}{\rho }_f{C}_p\left(\overrightarrow{W}\cdot \mathrm{\nabla }T\right)& ={\kappa }_f{\mathrm{\nabla }}^{2}T+{\tau }_f\left[D_B\mathrm{\nabla }T\mathrm{\nabla }C+\frac{D_T}{T_{\mathrm{\infty }}}{\left(\mathrm{\nabla }T\right)}^2\right]\\ & +\mathrm{\Phi }+G,\end{array}$(3) (4) $\begin{array}{rl}\left(\overrightarrow{W}\cdot \mathrm{\nabla }C\right)& =D_B{\mathrm{\nabla }}^{2}C+\frac{D_T}{T_{\mathrm{\infty }}}{\mathrm{\nabla }}^{2}T,\end{array}$(4) in the above equations, $\overrightarrow{W}$ is the velocity vector, ${\mathit{\tau }}_p$ is the Cauchy tensor stress of Maxwell fluid, $D_B$ represents the diffusion coefficient, $\varphi \left(K\right)$ is the Darcy effect, $\mathrm{\Phi }$ is the dissipation heat and $G$ is the additional energy into the system (Radiation, Joule heating, heat generation).

For the Rosseland approximation, the approximate radiative heat flux $q_r$ is given (Agunbiade et al. [36]); (5) $q_r=-\frac{4\sigma }{3k_1^{\ast }}\frac{\mathrm{\partial }{T}^4}{\mathrm{\partial }y},$(5) Where $\sigma$ and $k_1^{\ast }$ are constants of Stefan-Boltzmann and coefficient of volumetric absorption, respectively. According to Aliakbar et al. [37], $T^4$ is expanded with the help of Taylor series with ${\theta }_{\mathrm{\infty }}$, neglecting terms of higher-order gives; (6) $T^4\approx 4T_{\mathrm{\infty }}^{3}T-3T_{\mathrm{\infty }}^4,$(6) The vector $F$ is perpendicular to the velocity vector $u$ and the plane formed by both $J$ and $B$. $F_x$ represents the component of the magnetic force in the direction of the x-axis. We assume unidirectional flow, meaning that $\overline{v}$ and $\overline{w}$ are both equal to zero, and that $B_x$ and $B_y$ are also equal to zero. Since the magnetic field is oriented along the y-direction, we have $v=iu$, and the magnetic field is represented as $B=B_{0j}$, where $B_0$ is the component of magnetic field strength. Hence, (7) $F_x=\sigma \left[i\overline{u}\times j{B}_0\right]\times j{B}_0=\sigma {B_0}^{2}u.$(7) The equations governing the proposed flow model under the conditions stated above and the usual Boussinesq approximation are (Shahid [19], Khan et al. [38], Makinde et al. [32], Ur Rehman et al. [39]); (8) $\begin{array}{rl}& \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=0,\end{array}$(8) (9) $\begin{array}{rl}& u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+{\delta }_1\left(u^2\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+2uv\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }x\mathrm{\partial }y}+v^2\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}\right)\\ & ={\nu }_f\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+U_e\frac{d{U}_e}{dx}-\left[\frac{B_0^2\sigma \text{si}{\text{n}}^2\left(\alpha \right)}{{\rho }_f}+\frac{{\mu }_f}{{\rho }_{f}K}\right]\\ & \times \left(u-U_e+{\delta }_{1}v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right),\end{array}$(9) (10) $\begin{array}{rl}& u\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }T}{\mathrm{\partial }y}+{\delta }_2{\chi }_1\\ & =\frac{{\kappa }_f}{{\rho }_f{C}_p}\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}-\frac{1}{{\rho }_f{C}_p}\frac{\mathrm{\partial }{q}_r}{\mathrm{\partial }y}+\frac{{\mu }_f}{{\rho }_f{C}_p}{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^2\\ & +\tau \left[D_B\left(\frac{\mathrm{\partial }C}{\mathrm{\partial }y}\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\right)+\frac{D_T}{T_{\mathrm{\infty }}}{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\right)}^2\right]\\ & +\left[\frac{B_0^2\sigma \text{si}{\text{n}}^2\left(\alpha \right)}{{\rho }_f{C}_p}+\frac{{\mu }_f}{{\rho }_f{C}_{p}K}\right]{\left(U_e-u-{\delta }_{1}v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^2,\end{array}$(10) (11) $\begin{array}{rl}& u\frac{\mathrm{\partial }C}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }C}{\mathrm{\partial }y}=D_B\frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }{y}^2}-k_1\left(C-C_{\mathrm{\infty }}\right)+\frac{D_T}{T_{\mathrm{\infty }}}\left(\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}\right),\end{array}$(11)

The boundary conditions for the flow are as follows: (12) $\begin{array}{rl}v& =v_0,u=u_w+N_s{\nu }_f\frac{\mathrm{\partial }u}{\mathrm{\partial }y},T=T_w+R\frac{\mathrm{\partial }T}{\mathrm{\partial }y},\\ C& =C_w+E\frac{\mathrm{\partial }C}{\mathrm{\partial }y}aty=0,u=U_e,T=T_{\mathrm{\infty }},\\ C& =C_{\mathrm{\infty }}asy\to \mathrm{\infty }.\end{array}$(12) Where: (13) $\begin{array}{rl}{\chi }_1& =u^2\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{x}^2}+v^2\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}+2uv\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }y\mathrm{\partial }x}+u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\frac{\mathrm{\partial }T}{\mathrm{\partial }x}\\ & +u\frac{\mathrm{\partial }v}{\mathrm{\partial }x}\frac{\mathrm{\partial }T}{\mathrm{\partial }y}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }v}{\mathrm{\partial }y}\frac{\mathrm{\partial }T}{\mathrm{\partial }y},\end{array}$(13) Following the study of Khan et al. [38], Irfan and Farooq [40] and Khashi’ie et al. [41] the appropriate similarity transformations is employed to transform the PDEs to ODEs. Define (14) $\begin{array}{rl}v& =-\sqrt{d_0{\nu }_f}f\left(\eta \right),u=d_{0}x{f}^{\mathrm{\prime }}\left(\eta \right),\eta =\sqrt{\frac{d_0}{{\nu }_f}}y,\\ \varphi \left(\eta \right)& =\frac{C-C_{\mathrm{\infty }}}{C_w-C_{\mathrm{\infty }}},\vartheta \left(\eta \right)=\frac{T-T_{\mathrm{\infty }}}{T_w-T_{\mathrm{\infty }}}.\end{array}$(14) With the appropriate similarity Equation (14), Equations (8)–(12) are successfully transformed to non-linear and coupled ordinary differential equations given as; (15) $\begin{array}{rl}& f^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}+{\lambda }^2-\left(M_p\text{si}{\text{n}}^2\left(\alpha \right)+\frac{1}{Da}\right)\left[f^{\mathrm{\prime }}-\lambda +{\text{β}}_{n}f{f}^{\mathrm{\prime }\mathrm{\prime }}\right]\\ & +{\text{β}}_n\left(2f^{\mathrm{\prime }\mathrm{\prime }}{f}^{\mathrm{\prime }}f-f^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}{f}^2\right)+f{f}^{\mathrm{\prime }\mathrm{\prime }}-(f^{\mathrm{\prime }}{)}^2=0,\end{array}$(15) (16) $\begin{array}{rl}& \frac{1}{Pr}\left[1+\frac{4}{3R_d}\right]{\vartheta }^{\mathrm{\prime }\mathrm{\prime }}+f{\vartheta }^{\mathrm{\prime }}-\zeta \left[f{f}^{\mathrm{\prime }}{\vartheta }^{\mathrm{\prime }}+f^2{\vartheta }^{\mathrm{\prime }\mathrm{\prime }}\right]+E_c(f^{\mathrm{\prime }\mathrm{\prime }}{)}^2\\ & +E_c\left[M_p\text{si}{\text{n}}^2\left(\alpha \right)+\frac{1}{Da}\right](\lambda -f^{\mathrm{\prime }}+{\text{β}}_{n}f{f}^{\mathrm{\prime }\mathrm{\prime }}{)}^2\\ & +Nb{\varphi }^{\mathrm{\prime }}{\vartheta }^{\mathrm{\prime }}+Nt({\vartheta }^{\mathrm{\prime }}{)}^2=0,\end{array}$(16) (17) $\begin{array}{rl}& {\varphi }^{\mathrm{\prime }\mathrm{\prime }}+\frac{Nt}{Nb}{\vartheta }^{\mathrm{\prime }\mathrm{\prime }}+PrLef{\varphi }^{\mathrm{\prime }}-KnPrLe\varphi =0,\end{array}$(17) with boundary conditions: (18) $\begin{array}{rl}f\left(\eta \right)& =S,f^{\mathrm{\prime }}\left(\eta \right)=1+S_{\nu }{f}^{\mathrm{\prime }\mathrm{\prime }}\left(\eta \right),\vartheta \left(\eta \right)=1+S_t{\vartheta }^{\mathrm{\prime }}\left(\eta \right),\\ \varphi \left(\eta \right)& =1+S_o{\varphi }^{\mathrm{\prime }}\left(\eta \right)at\eta =0.\\ f^{\mathrm{\prime }}\left(\eta \right)& =\lambda ,\vartheta \left(\eta \right)=0,\varphi \left(\eta \right)=0as\eta \to \mathrm{\infty }.\end{array}$(18) Where ${\text{β}}_n={\delta }_1{d}_0$, $Le=\frac{\alpha }{D_B}$, $\lambda =\frac{b}{d_0}$, $Nb=\frac{\tau D_B\left(C_w-C_{\mathrm{\infty }}\right)}{\nu }$, $Da=\frac{\rho K{d}_0}{\mu }$, $M_p=\frac{{\alpha }_0^2\sigma }{\rho d_0}$, $R_d=\frac{k_1^{\ast }{\kappa }_f}{4\sigma T_{\mathrm{\infty }}^3}$, $\zeta ={\delta }_2{d}_0$, $Nt=\frac{\tau D_T\left(T_w-T_{\mathrm{\infty }}\right)}{{\nu }_f{T}_{\mathrm{\infty }}}$, $E_c=\frac{{\left(dx\right)}^2}{C_p\left(T_w-T_{\mathrm{\infty }}\right)}$, $Pr=\frac{\rho \nu C_p}{{\kappa }_f}=\frac{{\nu }_f}{\alpha }$, $S_{\nu }=N_s\sqrt{d_0{\nu }_f}$, $S_t=R\sqrt{\frac{d_0}{{\nu }_f}}$, $S_o=E\sqrt{\frac{d_0}{{\nu }_f}}$, $\tau =\frac{{\left(\rho C_p\right)}_p}{{\left(\rho C_p\right)}_f}$, and $Kn=\frac{k_1}{d_0}$.

2.1. Engineering quantity

Skin friction coefficient, Sherwood number and Nusselt number are the significance engineering quantities mathematically represented as follows: (19) $\begin{array}{rl}{C}_f& =\frac{{\tau }_w}{0.5{\rho }_f{u}_w^2},N{u}_x=\frac{\overline{x}{q}_w}{{\kappa }_0\left({\overline{T}}_w-{\overline{T}}_{\mathrm{\infty }}\right)},\\ S{h}_x& =\frac{\overline{x}{q}_m}{D_B\left({\overline{C}}_w-{\overline{C}}_{\mathrm{\infty }}\right)},\end{array}$(19) where the mass flux, surface heat flux and shear stress are: $q_m$, $q_w$, ${\tau }_w$ and are defined as: (20) $\begin{array}{rl}{\tau }_w& ={\rho }_f\sqrt{d_0^2{\nu }_f}\frac{\mathrm{\partial }u}{\mathrm{\partial }y},q_w=-{\kappa }_0\frac{\mathrm{\partial }\overline{T}}{\mathrm{\partial }\overline{y}}{|}_{\overline{y}=0}+q_r,\\ q_m& =-D_B\frac{\mathrm{\partial }\overline{C}}{\mathrm{\partial }\overline{y}}{|}_{\overline{y}=0}.\end{array}$(20) In dimensionless form, (19) using the definition defined in (20) becomes: (21) $\begin{array}{rl}\frac{1}{2}R{e}_{\overline{x}}^{\frac{1}{2}}{C}_f& =f^{\mathrm{\prime }\mathrm{\prime }}\left(0\right),R{e}_{\overline{x}}^{-\frac{1}{2}}N{u}_{\overline{x}}=-\left(1+\frac{4}{3R_d}\right){\vartheta }^{\mathrm{\prime }}\left(0\right),\\ R{e}_{\overline{x}}^{-\frac{1}{2}}S{h}_{\overline{x}}& =-{\varphi }^{\mathrm{\prime }}\left(0\right),\end{array}$(21) where: $R{e}_{\overline{x}}=\frac{u_w\overline{x}}{\nu }$.

3. Numerical methodology: collocation method

Using an assumed Legendre polynomial, the collocation method is another efficient approach for solving linear and non-linear ordinary differential equations like other orthogonal polynomials (Chebeysev, etc.). The approach is flexible and gives an excellent convergence result. For the solution of Equations (15)–(18); the Legendre polynomial can be written as: (22) $f\left(y\right)=\sum _{j=0}^N{n}_j{Q}_j\left(y\right)\text{for}j=0,1,2,\dots ,N$(22) The Legendre polynomial is transformed from $\left[0,L\right]$ to $\left[-1,1\right]$ using: (23) $y=\frac{2\eta }{L}-1$(23) Equation (22) becomes; (24) $f\left(y\right)=n_0{Q}_0\left(y\right)+n_1{Q}_1\left(y\right)+n_2{Q}_2\left(y\right)+\dots$(24) But, (25) $Q_0\left(y\right)=1,Q_1\left(y\right)=y,Q_2\left(y\right)=\frac{1}{2}\left(3y^2-1\right)$(25) Equations (23), (24) and (25) result to: (26) $\begin{array}{rl}f\left(\eta \right)& =n_0+n_1\left(\frac{2\eta }{L}-1\right)+\frac{n_2}{2}\left(3{\left(2\frac{\eta }{L}-1\right)}^2-1\right)\\ & +\dots \end{array}$(26) Similarly, (27) $\begin{array}{rl}\vartheta \left(\eta \right)& =m_0+m_1\left(\frac{2\eta }{L}-1\right)+\frac{m_2}{2}\left(3{\left(2\frac{\eta }{L}-1\right)}^2-1\right)\\ & +\dots \end{array}$(27) (28) $\begin{array}{rl}\varphi \left(\eta \right)& =h_0+h_1\left(\frac{2\eta }{L}-1\right)+\frac{h_2}{2}\left(3{\left(2\frac{\eta }{L}-1\right)}^2-1\right)\\ & +\dots \end{array}$(28) where $L$ is the scaling parameter and the unknown coefficients $n_0$, $n_1$, $n_2,\dots n_N$, $m_0$, $m_1$, $m_2\dots m_N$, $h_0$, $h_1$, $h_2,\dots ,h_N$, are obtained using the collocation points and equations (15)–(17). With the following default values of fluid parameters; $\lambda =0.5$, $M_p=0.5$, $R_d=10$, $Pr=1.8$, $Kn=0.3$, $E_c=0.1$, $Nb=0.2$, $Le=1$, $S=-0.5$, $Nt=0.2$, $S_{\nu }=0.3$, $S_t=0.3$, $S_o=0.3$, ${\text{β}}_n=0.2$, $\zeta =0.01$, $Da=10$, solution of Equations (15)–(17) with the aid of MATHEMATICA Software is obtained as: (29) $\begin{array}{rl}{f}^{\mathrm{\prime }}\left(\eta \right)& =0.860997-0.463344\eta +0.255626{\eta }^2\\ & -0.0773976{\eta }^3+0.0138073{\eta }^4\\ & -0.00176396{\eta }^5+0.000670411{\eta }^6\\ & -0.00148969{\eta }^7+0.00397655{\eta }^8\\ & -0.00867939{\eta }^9+0.0152865{\eta }^{10}\\ & -0.0220815{\eta }^{11}+0.0265315{\eta }^{12}\\ & -0.0268199{\eta }^{13}+0.0230251{\eta }^{14}\\ & -0.0169198{\eta }^{15}+0.010712{\eta }^{16}\\ & -0.00587443{\eta }^{17}+0.00280288{\eta }^{18}\\ & -0.00116771{\eta }^{19}+0.000425951{\eta }^{20}\dots \end{array}$(29) (30) $\begin{array}{rl}\vartheta \left(\eta \right)& =0.920496-0.265015\eta -0.166217{\eta }^2\\ & +0.0184691{\eta }^3+0.0260388{\eta }^4\\ & +0.00339925{\eta }^5-0.00401595{\eta }^6\\ & -0.000946636{\eta }^7+0.000898848{\eta }^8\\ & -0.00112304{\eta }^9+0.00294137{\eta }^{10}\\ & -0.00561602{\eta }^{11}+0.00851816{\eta }^{12}\\ & -0.0107458{\eta }^{13}+0.0113873{\eta }^{14}\\ & -0.0102127{\eta }^{15}+0.00781191{\eta }^{16}\\ & -0.00512979{\eta }^{17}+0.00290663{\eta }^{18}\\ & -0.00142684{\eta }^{19}+0.000608791{\eta }^{20}\dots \end{array}$(30) (31) $\begin{array}{rl}\varphi \left(\eta \right)& =0.815527-0.61491\eta +0.1097{\eta }^2\\ & +0.11793{\eta }^3-0.0442722{\eta }^4\\ & -0.0263079{\eta }^5+0.0114962{\eta }^6\\ & +0.00418078{\eta }^7-0.00266259{\eta }^8\\ & +0.00123612{\eta }^9-0.00347493{\eta }^{10}\\ & +0.0070063{\eta }^{11}-0.0106887{\eta }^{12}\\ & +0.0135278{\eta }^{13}-0.0144151{\eta }^{14}\\ & +0.0130181{\eta }^{15}-0.0100353{\eta }^{16}\\ & +0.00664495{\eta }^{17}-0.00379813{\eta }^{18}\\ & +0.00188126{\eta }^{19}-0.00081001{\eta }^{20}\dots \end{array}$(31)

4. Results and discussion

The analytical solution of coupled nonlinear equations is generally unlikely. Therefore, numerical solutions are obtained for the velocity, temperature, and concentration equations, as shown in Equations (29)–(31). The validation of these results is presented in Table 2, which compares the present study with the studies conducted by Shahid [19] and Agunbiade et al. [36] under the conditions where $M_p=Pr=Le=1$, ${\text{β}}_n=\lambda =0.2$, $Da=Nt=\zeta =S\nu =0$, with variations in $Kn$. This study demonstrates good agreement with existing literature. Using the given data, including $\lambda =0.5$, $M_p=0.5$, $R_d=10$, $Pr=6.8$, $Kn=0.3$, $E_c=0.1$, $Nt=0.2$, $Nb=0.2$, $Le=1$, $S=-0.5$, $S_{\nu }=0.3$, $S_t=0.3$, $S_o=0.3$, ${\text{β}}_n=0.2$, $\zeta =0.01$, $Da=10$, and $\alpha =\pi /3$ (Shahid [19], Khan et al. [38], Makinde et al. [32], Ur Rehman et al. [39], and Agunbiade et al. [36]), The study explored the momentum, temperature and concentration fields.

Figure 2 displays the effects of $M_p$ on temperature and velocity profiles. The figure indicates that the thermal agitation of particles is more pronounced. As a result, there is greater random movement and mixing of particles, including any heat-carrying or mass-carrying particles. This enhanced mixing can lead to a more efficient distribution of heat or species in the fluid. A decline in velocity profiles is experienced with an increase in the $M_p$. Physically, in an electrically-conductive fluid, motion generates a force known as the Lorentz force due to the interaction between electric and magnetic fields. This force acts as resistance to the flow, causing a suppression of velocity in the direction perpendicular to the magnetic field lines, resulting in reduced flow rates.

Figure 2. $M_p$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\vartheta \left(\eta \right)$.

Figure 3 depicts the influence of $\alpha$ on velocity and temperature. The increase in inclination angle results in an increase in the effective gravitational force component acting opposite to the fluid flow direction. This additional resistance opposes the flow, and the fluid must overcome it to maintain the same velocity. As a result, the fluid’s velocity decreases because more energy is required to overcome the increased gravitational resistance. When $\alpha$ set to $\pi /2$ (90 degrees), the magnetic field is oriented completely perpendicular to the flow direction, leading to a higher magnitude of resistance force. Increase in the inclination angle promote better heat transfer within the fluid. The steeper angle encourages a more effective mixing of the fluid, which lead to improved thermal contact between the fluid and the solid surfaces or between different layers of the fluid. This, in turn, result in increased temperature profiles. The influence of the ratio parameter $\lambda \left(=b/d_0\right)$ on $f^{\mathrm{\prime }}\left(\eta \right)$, $\vartheta \left(\eta \right)$, and $\varphi \left(\eta \right)$ is portrayed. Mathematically, $d_0\ne 0$ and for $b=0$ signified a non-stagnation point flow. As depicted in Figure 4(a), a rise in the free stream velocity parameter results in an inverted boundary layer structure. In Figure 4, as the free stream velocity increases, it imparts additional kinetic energy to the fluid. Consequently, the fluid velocity throughout the flow field also increases. This result in higher flow rates and faster flow speeds in the entire fluid domain.

Figure 3. $\alpha$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\vartheta \left(\eta \right)$.

Figure 4. $\lambda$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\vartheta \left(\eta \right)$ and $\varphi \left(\eta \right)$.

Figure 5 presents the effects of the Darcy number ($Da$) on both $f^{\mathrm{\prime }}\left(\eta \right)$ and $\vartheta \left(\eta \right)$. The Darcy number characterizes the relative importance of inertial forces to viscous forces in a porous medium, influencing the flow rate within the porous medium. An increase in the Darcy number leads to a shift in the balance between viscous and inertial forces in porous media flows. With a stretching porous sheet, a higher Darcy number tends to flatten the velocity profile near the sheet’s surface. This can result in higher fluid velocity near the centre of the sheet and slower velocity near the surface. For low Darcy numbers, typical applications include groundwater flow, oil reservoir simulations, and filtration processes. However, with higher permeability (associated with a higher Darcy number), the thermal resistance within the porous medium is reduced. This leads to more efficient heat transfer because the fluid can more easily carry heat throughout the porous structure.

Figure 5. Darcy number $Da$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\vartheta \left(\eta \right)$.

Figure 6 illustrates the influence of $Kn$ on $\vartheta \left(\eta \right)$ and $\varphi \left(\eta \right)$. High values of $Kn$ retard both temperature and concentration profiles. This effect is due to destructive chemicals that reduce the thickness of the solutal boundary layer and improve mass transfer. A rise in the chemical reaction parameter can be associated with exothermic reactions, which release heat. As these reactions progress, they generate thermal energy that increases the fluid’s temperature, causing a decrease in temperature profiles. Figure 7 reveals the influence of $Nb$ on $\vartheta \left(\eta \right)$ and $\varphi \left(\eta \right)$. Brownian motion, which is the random motion of particles in a fluid due to thermal energy, contributes to the process of heat conduction. An increase in $Nb$ means that thermal energy is being more effectively distributed throughout the fluid. An increase in $Nb$ accelerates the $\varphi \left(\eta \right)$. However, a different trend is noticed for $\varphi \left(\eta \right)$ at $\eta =0.75$; it decelerates with an increase in $Nb$. In fluid systems with a high $Nb$, the decrease in concentration profiles indicates a more uniform and well-mixed distribution of particles or solutes. Brownian motion occurs due to the collision of fluid molecules with the suspended particles. The physical impact of Brownian motion on a fluid is notable in various scientific and engineering contexts.

Figure 6. Chemical reaction parameter $Kn$ on $\vartheta \left(\eta \right)$ and $\varphi \left(\eta \right)$.

Figure 7. $Nb$ on $\vartheta \left(\eta \right)$ and $\varphi \left(\eta \right)$.

Figure 8 portrays the influence of thermophoresis, represented as $Nt$, on $\vartheta \left(\eta \right)$ and $\varphi \left(\eta \right)$. The thermophoretic effect, which is the motion of particles in response to temperature gradients, becomes more significant as the thermophoresis constant increases. This effect enhances the transfer of heat within the fluid, leading to higher temperature profiles. Again, its implication on $\varphi \left(\eta \right)$ means that the movement of particles in response to temperature gradients becomes less significant, and particles tend to disperse or mix more uniformly within the fluid. Meanwhile, at $\eta =0.92$, there is a change in the trend of concentration profiles. Thermophoresis is noteworthy as a function of viscous force and thermal diffusion coefficient. $Nt$ is inversely proportional to viscous force; therefore, an increase in $Nt$ leads to a decrease in viscous force, which results in an improvement in the coefficient of thermal diffusion, thereby enhancing the temperature and concentration profiles. A positive thermophoresis parameter ($Nt>0$) indicates that particles will tend to move from lower temperature to higher temperature regions. Conversely, a negative parameter ($Nt<0$) would cause particles to move from hotter to cooler regions. This motion can affect the distribution and concentration of particles within the fluid.

Figure 8. $Nt$ on $\vartheta \left(\eta \right)$ and $\varphi \left(\eta \right)$.

In Figure 9, velocity slip at a solid boundary leads to a thinner boundary layer near that boundary. A thinner boundary layer signifies a decline in fluid velocity near the boundary. Likewise, an increase in velocity slip implies that the relative motion between the fluid and a solid boundary is more significant. This increased relative motion enhanced heat transfer, as it promotes greater mixing and fluid contact with the solid surface. This enhancement results from pulling the stretching sheet to transport the fluid. The thinner boundary layer allows for more efficient heat transfer from the solid surface to the fluid. This effect can be important in various engineering applications, such as cooling systems and microfluidics. In Figure 9(c), an increase in thermal slip implies that the thermal properties of the fluid near the boundary are affected by the slip condition. This result in a reduction in heat transfer from the solid boundary to the fluid and a decline in temperature profiles near the boundary which implies less efficient heat transfer. Figure 10 displays the effects of $S_o$ on $\vartheta \left(\eta \right)$ and $\varphi \left(\eta \right)$. Solutal slip refers to the relative motion of solute particles within the fluid. In Figure 10(a), an increase in solutal slip disrupt heat transfer mechanisms in the fluid. This disruption led to a reduction in temperature profiles as heat transfer from a solid boundary to the fluid becomes less efficient. The decline in concentration profiles, as depicted in Figure 10(b) is as a result of inability of solute particles in reaching the solid boundary efficiently due to $S_o$.

Figure 9. Velocity slip $S_{\nu }$ and thermal slip $S_t$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\vartheta \left(\eta \right)$.

Figure 10. Solutal slip $S_o$ on $\vartheta \left(\eta \right)$ and $\varphi \left(\eta \right)$.

Understanding and managing skin friction in fluid dynamics is essential for optimizing the performance of various engineering systems, from pipelines and heat exchangers to aircraft and ships. Engineers use techniques such as streamline shaping, surface treatments, and lubrication to minimize skin friction and enhance fluid flow efficiency. In Table 1, both Nusselt number and skin friction are enhanced with an increase in $\lambda$, $Da$ and $\alpha$. While an increase in $M_p$ and $S_v$ decelerates the skin friction. The physical impact of the Nusselt number on fluid flow is primarily related to heat transfer. A higher Nusselt number indicates more efficient convective heat transfer, meaning that heat is rapidly transferred from the sold surface to the fluid. Furthermore, Sherwood number improves with increase in $\lambda$, $Nt$, $Pr$, $Ec$, and $\alpha$, but is being retarded by $M_p$, $S_v$, $S_t$ and $R_d$. This could be beneficial in practical applications when trying to achieve fast reactions or efficient separation processes. To verify the convergence of the series solution obtained using the Legendre-based collocation method, Table 3 presents the residual analysis for the three distributions involved. Clearly, as we increase Np, the residue approaches zero. In contrast to the behaviour of some series solutions for larger Np values, where some had to introduce techniques like Pade approximation to control the series solution, the present method demonstrates its strength in handling nonlinear, coupled systems of equations. This strength is attributed to the properties of the polynomial (orthogonality) and the appropriately prescribed collocation technique.

Table 1. $f^{\mathrm{\prime }\mathrm{\prime }}\left(0\right)$, ${\vartheta }^{\mathrm{\prime }}\left(0\right)$ and ${\varphi }^{\mathrm{\prime }}\left(0\right)$ for constant values $Le=1$, $S=-0.5$, ${\text{β}}_n=0.01$, $\zeta =0.01$ and variation of other flow parameters.

$\lambda$	$Nt$	$Nb$	$Da$	$M_p$	$Pr$	$E_c$	$S_{\nu }$	$S_t$	$R_d$	$\alpha$	$f^{\mathrm{\prime }\mathrm{\prime }}\left(0\right)$	$-{\vartheta }^{\mathrm{\prime }}\left(0\right)$	$-{\varphi }^{\mathrm{\prime }}\left(0\right)$
0.5	0.2	0.2	5	0.5	1.8	0.1	0.3	0.3	10	$\pi /2$	−0.463344	0.265015	0.614910
0.8											−0.201600	0.340889	0.645896
	0.5										−0.463344	0.223149	0.726435
		0.3									−0.463344	0.244782	0.605082
			10								−0.454898	0.266954	0.615199
				1.0							−0.501774	0.256543	0.613629
					2.2						−0.463344	0.251899	0.658834
						0.2					−0.463344	0.252772	0.623277
							1.0				−0.276511	0.245801	0.595464
								1.0			−0.463344	0.225353	0.606913
									5		−0.463344	0.268798	0.606923
										$\pi /6$	−0.430097	0.307495	0.724908

Table 2. Comparison of $f^{\mathrm{\prime }\mathrm{\prime }}\left(0\right)$ and ${\varphi }^{\mathrm{\prime }}\left(0\right)$ with $Kn$ by setting $M_p=Pr=Le=1,$ ${\text{β}}_n=\lambda =0.2$ and $Da=Nt=\zeta =S_{\nu }=0$.

	Shahid [19]	Agunbiade et al. [36]	Present	Shahid [19]	Agunbiade et al. [36]	Present
$K_n$	$f^{\mathrm{\prime }\mathrm{\prime }}\left(0\right)$	$f^{\mathrm{\prime }\mathrm{\prime }}\left(0\right)$	$f^{\mathrm{\prime }\mathrm{\prime }}\left(0\right)$	$-{\varphi }^{\mathrm{\prime }}\left(0\right)$	$-{\varphi }^{\mathrm{\prime }}\left(0\right)$	$-{\varphi }^{\mathrm{\prime }}\left(0\right)$
1.0	1.272470	1.272470	1.272470	1.16786	1.16788	1.16788
1.2	–	–	–	1.25227	1.25228	1.25228
1.5	–	–	–	1.36885	1.36886	1.36886

Table 3. Residual error analysis of the respective momentum, energy and concentration distribution for the define data $\lambda =0.5$, $M_p=0.5$, $R_d=10$, $Pr=6.8$, $Kn=0.3$, $E_c=0.1$, $Nt=0.2$, $Nb=0.2$, $Le=1$, $S=-0.5$, $S_{\nu }=0.3$, $S_t=0.3$, $S_o=0.3$, ${\text{β}}_n=0.2$, $\zeta =0.01$, $Da=10$, and $\alpha =\pi /3$.

Np	$\left|\text{Re}{\text{s}}_f\left(\eta \right)\right|$	$\left|\text{Re}{\text{s}}_{\vartheta }\left(\eta \right)\right|$	$\left|\text{Re}{\text{s}}_{\phi }\left(\eta \right)\right|$
$5$	$4.0518630417\times {10}^{-1}$	$1.6092651982\times {10}^{-2}$	$5.3660450658\times {10}^{-2}$
$8$	$1.5471094404\times {10}^{-2}$	$8.8612831793\times {10}^{-3}$	$6.1491649243\times {10}^{-3}$
$11$	$7.0553604843\times {10}^{-4}$	$7.8284204245\times {10}^{-4}$	$3.1576492279\times {10}^{-4}$
$14$	$3.1337779797\times {10}^{-5}$	$6.9267194511\times {10}^{-6}$	$3.1627430407\times {10}^{-5}$
$17$	$2.0307966870\times {10}^{-6}$	$3.8183103897\times {10}^{-6}$	$1.4977840037\times {10}^{-6}$
$20$	$1.6834982678\times {10}^{-8}$	$1.1788969374\times {10}^{-7}$	$1.1724493597\times {10}^{-8}$
$23$	$1.8282973616\times {10}^{-8}$	$3.4052651363\times {10}^{-8}$	$9.6155563438\times {10}^{-9}$
$26$	$8.4649453750\times {10}^{-9}$	$2.7721791457\times {10}^{-9}$	$2.0066340040\times {10}^{-9}$
$30$	$1.2158786195\times {10}^{-9}$	$5.0074415932\times {10}^{-9}$	$2.3271267156\times {10}^{-9}$

5. Conclusion

This study investigated the combined effects of Brownian motion and thermophoresis on magnetized Maxwell upper-convected stagnation point fluid flow over a stretching porous surface. The analysis considered a variety of parameters and their influence on the flow and heat transfer characteristics. The key findings and conclusions of the research follows:

1. that Brownian motion significantly affects the concentration profile of nanoparticles in the fluid. As the Brownian motion parameter increases, the concentration boundary layer becomes thinner, indicating enhanced dispersion of nanoparticles. This has important implications for applications involving nanoparticle transport.

2. that thermophoresis plays a crucial role in the heat transfer process. It induces temperature variations in the boundary layer, affecting the heat transfer rate. The direction of thermophoretic motion depends on the thermophoresis parameter, which can be harnessed to control and optimize heat transfer in various engineering applications.

3. the presence of a magnetic field in the flow significantly altered the velocity and temperature profiles. Magnetic forces have the potential to control and manipulate the flow and temperature distribution. Understanding this influence is vital for the design of magnetohydrodynamic systems.

4. the effect of velocity slip on skin friction shows that there is less frictional resistance acting on the fluid. This reduction in frictional resistance is often associated with lower drag forces, which is beneficial in various engineering application, such as in transportation (e.g cars, airplanes, ships) and fluid pipelines.

5. the increase in skin friction as a result of Darcy number indicates that the fluid experiences greater viscous resistance as it flows via the porous medium. This can lead to higher pressure drops, which can be important in applications like filtration or fluid flow in porous media. That is, it may indicate that the filter or porous medium is more effective at capturing particles or impurities from the fluid. This can be advantageous for achieving higher filtration efficiency.

Therefore, the study demonstrated that the interaction of Brownian motion, thermophoresis, Darcy Number, and a magnetic field in Maxwell upper-convected stagnation point fluid flow over a stretching porous surface is a complex and dynamic process. The results of the investigation provided valuable insights for researchers and engineers working in fields such as nanotechnology, magnetohydrodynamics, and porous media applications.

Nomenclature

$u$, $v$	=	velocity components in $x$, $y$ directions respectively [m]
${\delta }_1$	=	fluid relaxation time [s]
$\alpha$	=	inclination angle [degree]
${\kappa }_f$	=	fluid thermal conductivity [$s^2{m}^{-1}$]
${\nu }_f$	=	kinematic viscosity [$m^2{s}^{-1}$]
$T$	=	fluid temperature [$K$]
${\mu }_f$	=	viscosity of the base fluid [$Pa.s$]
${\delta }_2$	=	heat flux relaxation time [s]
${\rho }_f$	=	fluid density [$Kg/m^3$]
$C_p$	=	specific heat [$Jk{g}^{-1}{k}^{-1}$]
$D_B$	=	mass diffusion [$m^2/s$]
$C$	=	concentration field [$kmol/m^3$]
$k_1$	=	reaction rate [$mol/s{m}^3$]
$B_0$	=	magnetic field [$tesla$]
$\sigma$	=	the fluid electrical conductivity [$s{m}^{-1}$]
$K$	=	porous medium permeability [$m^2$]
${\text{β}}_n$	=	Deborah number
$Le$	=	Lewis numbers.
$\lambda$	=	positive constant.
$Nb$	=	Brownian motion number
$Da$	=	Darcy number.
$M_p$	=	magnetic parameter.
$R_d$	=	thermal radiation
$\zeta$	=	thermal relaxation.
$Nt$	=	thermophoresis constant.
$E_c$	=	Eckert number
$Pr$	=	Prandtl number.
$S_{\nu }$	=	velocity slip.
$S_t$	=	thermal slip
$S_o$	=	solutal slip.
$Kn$	=	chemical reaction constant.

Disclosure statement

No potential conflict of interest was reported by the author(s).