Modelling and Analysis of Three-Dimensional Maxwell-Nanofluid Flow over a Bi-Directional Stretching Surface in the Presence of a Magnetic Field

1. Introduction

It is critical to include radiative heat flow while studying heat transfer processes, especially with nanofluids. Radiative heat transfer is made using electromagnetic waves throughout the energy transmission process. Heat transmission properties can be adjusted further with the presence of nanofluids, which are liquid suspensions containing microscopic particles. The Stefan-Boltzmann equation and other linear models that presuppose a linear relationship between the temperature difference and radiative heat flow have historically been used to characterize radiative heat transfer. When attempting to accurately depict the behaviour of heat transfer, a non-linear radiative heat flow model might be more suitable. To counteract the radiative characteristics of the nanofluid, such as the radiation absorption and scattering by the floating nanoparticles, a non-linear radiative heat flow is added. The fluid's effective thermal conductivity, absorption, and emission characteristics can all be changed by these factors, which can then affect radiative heat transfer. The non-linear radiative heat flow in the presence of a nanofluid can be described in a variety of ways. Nanoparticle concentration, size, and shape are often considered while estimating radiative heat transfer, using complex models that account for the interactions. The extinction coefficient and scattering phase function of the nanofluid, as well as other radiative properties, may be determined through experiments. In order to make more accurate forecasts, these observable characteristics can then be included in non-linear radiative heat transfer models. A fluid is a unique type of matter with the capacity to flow and undergo effortless distortion when an external force is applied. Non-Newtonian fluids have innumerable real-life applications in natural products, biomedical fields, agriculture, and food products. Exhaustive benefits in various fields have made budding researchers study and investigate the attributes of non-Newtonian fluids. Over the past decade, the non-Newtonian liquid stream has gained significant attention due to its diverse applications in engineering and industrial sectors. The Maxwell liquid model is a crucial rate-type fluid subclass, and recent studies have focused on modelling and exploring nanoliquid streams. Nanoparticles are used to increase heat transmission and overcome cooling issues in thermal frames. Researchers have studied nanoliquid streams as they pass through surfaces, as simple Navier-Stokes equations cannot represent some fluids due to their complicated rheology.

The radiative heat transport characteristics of nanofluids were studied in[1]. A comprehensive analysis of magnetohydrodynamic (MHD) Maxwell fluid flow incorporating nanoparticles over a stretching surface was conducted in[2], considering the effects of thermal radiation, convective boundary conditions, and an induced magnetic field. The thermal and diffusion influences on MHD Jeffrey fluid flow over a porous stretching sheet were explored in[3], factoring in activation energy. A model to analyze MHD micropolar nanofluid flow over a Darcian porous stretching surface was developed in[4], addressing thermophoretic and Brownian diffusion phenomena. A perturbative study on radiation absorption, Hall effect, and ion-slip current-driven MHD convection was performed in[5], focusing on spanwise cosinusoidally fluctuating temperature profiles. The role of a quadratic-linearly radiating heat source in conjunction with Carreau nanofluid and exponential space dependency was examined in[6], particularly in applications related to cones and wedges, with relevance to medical engineering and renewable energy contexts. Bio-convection phenomena involving slippery two-phase Maxwell nanofluid flow past a vertically induced magnetic stretching regime were investigated in[7], with implications for biotechnology and engineering. Insights into the flow dynamics of nanofluids were presented in[8], modelling various fluid flow challenges. A response surface methodology (RSM) analysis on slip effects in Casson nanofluid flow over a stretching sheet with activation energy considerations was conducted in[9]. A numerical study on magneto-hydro-convective nanofluid flow within a porous square enclosure was performed in[10]. The impact of nanofluids and porous structures on the thermal efficiency of wavy channel heat exchangers was investigated in[11]. Analytical solution of forced convective heat transfer in tubes partially filled with metallic foam using the two-equation model was found[12]. Articles[13-23] highlight potential applications. References[24-32] used the finite element method to address heat and mass transport in MHD flows effectively. Studies in[33-37] explored heat and mass transfer effects, significantly enhancing the understanding of the reported work.

When the magnetic field and non-radiative heat flux are present, relatively few studies have documented the three-dimensional analysis of nanofluid flow, according to a review of the works described above. There haven't yet been any reports in the literature on the combined effects of Soret and Dufour on steady, three-dimensional, incompressible, viscous, electrically conducting non-Newtonian upper convected Maxwell-Nanofluid flow over a bi-directional stretching sheet (surface) in the presence of non-linear radiative heat flux, Brownian motion, thermophoresis, thermal, and mass Biot numbers. This model is an enhanced version of previously published research. The outcomes of the current numerical simulations are also distinct.

2. Mathematical formulation

This section covers the effects on 3-D, steady, incompressible, viscous, non-Newtonian Maxwell fluid flow over a bidirectional stretching surface of thermophoresis, Brownian motion, Soret, and Dufour. Figure 1 depicts the fluid flow's shape.

In this study, the following premises are assumed:

The equation for continuity:

[TeX:] \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$ (1)

Momentum Equation:

[TeX:] u\left( \frac{\partial u}{\partial x} \right) + v\left( \frac{\partial u}{\partial y} \right) + w\left( \frac{\partial u}{\partial z} \right) + \lambda\begin{Bmatrix} \& u^{2}\left( \frac{\partial^{2}u}{\partial x^{2}} \right) + v^{2}\left( \frac{\partial^{2}u}{\partial y^{2}} \right) + w^{2}\left( \frac{\partial^{2}u}{\partial z^{2}} \right) + \\ \& 2uv\left( \frac{\partial^{2}u}{\partial x\partial y} \right) + 2vw\left( \frac{\partial^{2}u}{\partial y\partial z} \right) + 2uw\left( \frac{\partial^{2}u}{\partial x\partial z} \right) \\ \end{Bmatrix} = \frac{\partial^{2}u}{\partial z^{2}} - \left( \frac{\sigma B_{o}^{2}}{\rho} \right)u$u\left(\frac{\partial u}{\partial x}\right)+v\left(\frac{\partial u}{\partial y}\right)+w\left(\frac{\partial u}{\partial z}\right)+\lambda \left\{\begin{array}{c}\hfill u^2\left(\frac{{\partial }^{2}u}{\partial x^2}\right)+v^2\left(\frac{{\partial }^{2}u}{\partial y^2}\right)+w^2\left(\frac{{\partial }^{2}u}{\partial z^2}\right)+\hfill \\ \hfill 2uv\left(\frac{{\partial }^{2}u}{\partial x\partial y}\right)+2vw\left(\frac{{\partial }^{2}u}{\partial y\partial z}\right)+2uw\left(\frac{{\partial }^{2}u}{\partial x\partial z}\right)\hfill \end{array}\right\}=\frac{{\partial }^{2}u}{\partial z^2}-\left(\frac{\sigma B_o^2}{\rho }\right)u$ (2)

[TeX:] u\left( \frac{\partial v}{\partial x} \right) + v\left( \frac{\partial v}{\partial y} \right) + w\left( \frac{\partial v}{\partial z} \right) + \lambda\begin{Bmatrix} \& u^{2}\left( \frac{\partial^{2}v}{\partial x^{2}} \right) + v^{2}\left( \frac{\partial^{2}v}{\partial y^{2}} \right) + w^{2}\left( \frac{\partial^{2}v}{\partial z^{2}} \right) + \\ \& 2uv\left( \frac{\partial^{2}v}{\partial x\partial y} \right) + 2vw\left( \frac{\partial^{2}v}{\partial y\partial z} \right) + 2uw\left( \frac{\partial^{2}v}{\partial x\partial z} \right) \\ \end{Bmatrix} = \frac{\partial^{2}v}{\partial z^{2}} - \left( \frac{\sigma B_{o}^{2}}{\rho} \right)v$u\left(\frac{\partial v}{\partial x}\right)+v\left(\frac{\partial v}{\partial y}\right)+w\left(\frac{\partial v}{\partial z}\right)+\lambda \left\{\begin{array}{c}\hfill u^2\left(\frac{{\partial }^{2}v}{\partial x^2}\right)+v^2\left(\frac{{\partial }^{2}v}{\partial y^2}\right)+w^2\left(\frac{{\partial }^{2}v}{\partial z^2}\right)+\hfill \\ \hfill 2uv\left(\frac{{\partial }^{2}v}{\partial x\partial y}\right)+2vw\left(\frac{{\partial }^{2}v}{\partial y\partial z}\right)+2uw\left(\frac{{\partial }^{2}v}{\partial x\partial z}\right)\hfill \end{array}\right\}=\frac{{\partial }^{2}v}{\partial z^2}-\left(\frac{\sigma B_o^2}{\rho }\right)v$ (3)

Thermal energy Equation:

[TeX:] u\left( \frac{\partial T}{\partial x} \right) + v\left( \frac{\partial T}{\partial y} \right) + w\left( \frac{\partial T}{\partial z} \right) = \alpha\left( \frac{\partial^{2}T}{\partial z^{2}} \right) + \left( \frac{\left( \text{ρC} \right)_{p}}{\left( \text{ρC} \right)_{f}} \right)\left\{ D_{B}\left( \frac{\partial T}{\partial z} \right)\left( \frac{\partial\varphi}{\partial z} \right) + \frac{D_{T}}{T_{\infty}}\left( \frac{\partial T}{\partial z} \right)^{2} \right\}$u\left(\frac{\partial T}{\partial x}\right)+v\left(\frac{\partial T}{\partial y}\right)+w\left(\frac{\partial T}{\partial z}\right)=\alpha \left(\frac{{\partial }^{2}T}{\partial z^2}\right)+\left(\frac{{\left(\mathrm{\text{ρC}}\right)}_p}{{\left(\mathrm{\text{ρC}}\right)}_f}\right)\{D_B\left(\frac{\partial T}{\partial z}\right)\left(\frac{\partial \phi }{\partial z}\right)+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial z}\right)}^2\}$[TeX:] - \frac{1}{\rho C_{p}}\left( \frac{\partial q_{r}}{\partial z} \right) + \frac{D_{m}K_{T}}{C_{s}C_{p}}\left( \frac{\partial^{2}\varphi}{\partial z^{2}} \right)$-\frac{1}{\rho C_p}\left(\frac{\partial q_r}{\partial z}\right)+\frac{D_m{K}_T}{C_s{C}_p}\left(\frac{{\partial }^2\phi }{\partial z^2}\right)$ (4)

Species Concentration Equation:

[TeX:] u\left( \frac{\partial\varphi}{\partial x} \right) + v\left( \frac{\partial\varphi}{\partial y} \right) + w\left( \frac{\partial\varphi}{\partial z} \right) = D_{B}\left( \frac{\partial^{2}\varphi}{\partial z^{2}} \right) + \frac{D_{T}}{T_{\infty}}\left( \frac{\partial T}{\partial z} \right)^{2} + \frac{D_{m}K_{T}}{T_{m}}\left( \frac{\partial^{2}T}{\partial y^{2}} \right)$u\left(\frac{\partial \phi }{\partial x}\right)+v\left(\frac{\partial \phi }{\partial y}\right)+w\left(\frac{\partial \phi }{\partial z}\right)=D_B\left(\frac{{\partial }^2\phi }{\partial z^2}\right)+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial z}\right)}^2+\frac{D_m{K}_T}{T_m}\left(\frac{{\partial }^{2}T}{\partial y^2}\right)$ (5)

Fig. 1 Nano-Maxwell fluid flow graphical representation

The following are flow boundary conditions:

[TeX:] \left. \ \begin{matrix} \& u = U_{w} = ax,v = V_{w} = by,w = 0, - \kappa\frac{\partial T}{\partial z} = \beta_{1}\left( T_{f} - T \right), - D_{B}\frac{\partial C}{\partial z} = \beta_{2}\left( \varphi_{f} - \varphi \right)atz = 0 \\ \& u \rightarrow 0,v \rightarrow 0,T \rightarrow T_{\infty},\varphi \rightarrow \varphi_{\infty}\text{asz} \rightarrow \infty \\ \end{matrix} \right\}$\begin{array}{c}\hfill u=U_w=ax,v=V_w=by,w=0,-\kappa \frac{\partial T}{\partial z}={\beta }_1(T_f-T),-D_B\frac{\partial C}{\partial z}={\beta }_2({\phi }_f-\phi )atz=0\hfill \\ \hfill u\to 0,v\to 0,T\to T_{\infty },\phi \to {\phi }_{\infty }\mathrm{\text{asz}}\to \infty \hfill \end{array}\}$ (6)

With the Roseland approximation, the radiative heat flux is defined as:

[TeX:] \frac{\partial q}{\partial r} = - \frac{1.3\sigma^{*}}{K^{*}}T_{z}$\frac{\partial q}{\partial r}=-\frac{1.3{\sigma }^{*}}{K^{*}}{T}_z$ (7)

[TeX:] T^{4} = T_{\infty}^{4} + 4T_{\infty}^{3}\left( T - T_{\infty} \right) + 6T_{\infty}^{2}\left( T - T_{\infty} \right)^{2} + ............$T^4=T_{\infty }^4+4T_{\infty }^3(T-T_{\infty })+6T_{\infty }^2{(T-T_{\infty })}^2+............$ (8)

After disregarding higher-order terms after the first degree term [TeX:] \left( T - T_{\infty} \right)$(T-T_{\infty })$, we get:

[TeX:] T^{4} \cong 4T_{\infty}^{3}T - 3T_{\infty}^{4}$T^4\cong 4T_{\infty }^{3}T-3T_{\infty }^4$ (9)

From (7) and (9), we get:

[TeX:] q_{r} = - \frac{5.3T_{\infty}^{3}\sigma^{*}}{K^{*}}T_{z}$q_r=-\frac{5.3T_{\infty }^3{\sigma }^{*}}{K^{*}}{T}_z$ (10)

(8) allows Eq. (3) to be expressed as:

[TeX:] uT_{x} + vT_{y} + wT_{z} = \alpha T_{\text{zz}} + \left( \frac{\left( \text{ρC} \right)_{p}}{\left( \text{ρC} \right)_{f}} \right)\left( D_{B}T_{z}C_{z} + \frac{D_{T}}{T_{\infty}}\left( T_{z} \right)^{2} \right) + \frac{1}{\rho C_{p}}T_{\text{zz}}\left( \frac{16T^{3}\sigma_{\infty}^{*}}{3\kappa k^{*}} \right) + \frac{D_{m}K_{T}}{C_{p}C_{s}}C_{\text{zz}}$u{T}_x+v{T}_y+w{T}_z=\alpha T_{\mathrm{\text{zz}}}+\left(\frac{{\left(\mathrm{\text{ρC}}\right)}_p}{{\left(\mathrm{\text{ρC}}\right)}_f}\right)(D_B{T}_z{C}_z+\frac{D_T}{T_{\infty }}{\left(T_z\right)}^2)+\frac{1}{\rho C_p}{T}_{\mathrm{\text{zz}}}\left(\frac{16T^3{\sigma }_{\infty }^{*}}{3\kappa k^{*}}\right)+\frac{D_m{K}_T}{C_p{C}_s}{C}_{\mathrm{\text{zz}}}$ (11)

Initiating the following similarity transformations:

[TeX:] \left. \ \eta = \left( \sqrt{\frac{a}{\nu}} \right)z,u = axf^{'}\left( \eta \right),v = ayg^{'}\left( \eta \right),\theta = \frac{T - T_{\infty}}{T_{f} - T_{\infty}},\phi = \frac{\varphi - \varphi_{\infty}}{\varphi_{f} - \varphi_{\infty}},w = \nu\sqrt{\text{νa}}\left\{ f\left( \eta \right) + g\left( \eta \right) \right\}, \right\}$\eta =\left(\sqrt{\frac{a}{\nu }}\right)z,u=ax{f}^{\prime }\left(\eta \right),v=ay{g}^{\prime }\left(\eta \right),\theta =\frac{T-T_{\infty }}{T_f-T_{\infty }},\varphi =\frac{\phi -{\phi }_{\infty }}{{\phi }_f-{\phi }_{\infty }},w=\nu \sqrt{\mathrm{\text{νa}}}\{f\left(\eta \right)+g\left(\eta \right)\},\}$ (12)

Via 12 Eqs. (2), (3), (4), and (11) take the following form:

The associated boundary conditions (6) become:

If the relevant physical parameters are described as:

[TeX:] \left. \ \begin{matrix} \& M = \frac{\sigma B_{o}^{2}}{\rho U_{o}},K = \lambda_{1}a,C = \frac{b}{a},\theta_{w} = \frac{T_{w}}{T_{\infty}},Nb = \frac{\tau_{B}D_{B}\left( \varphi_{f} - \varphi_{\infty} \right)}{\nu},\Pr = \frac{\nu}{\alpha},\delta = \frac{\beta_{1}}{\kappa}\sqrt{\frac{\nu}{a}},\zeta = \frac{\beta_{2}}{D_{B}}\sqrt{\frac{\nu}{a}}, \\ \& Nt = \frac{\tau_{B}D_{T}\left( T_{f} - T_{\infty} \right)}{\nu T_{\infty}},Sr = \frac{D_{m}K_{T}(T_{f} - T_{\infty})}{T_{m}\nu\left( \varphi_{f} - C_{\infty} \right)},Du = \frac{D_{m}K_{T}\left( \varphi_{f} - \varphi_{\infty} \right)}{C_{s}C_{p}\nu\left( T_{f} - T_{\infty} \right)},Sc = \frac{\nu}{D_{B}},R = \frac{16\sigma^{*}T_{\infty}^{3}}{3\kappa k^{*}} \\ \end{matrix} \right\}$\begin{array}{c}\hfill M=\frac{\sigma B_o^2}{\rho U_o},K={\lambda }_{1}a,C=\frac{b}{a},{\theta }_w=\frac{T_w}{T_{\infty }},Nb=\frac{{\tau }_B{D}_B({\phi }_f-{\phi }_{\infty })}{\nu },Pr=\frac{\nu }{\alpha },\delta =\frac{{\beta }_1}{\kappa }\sqrt{\frac{\nu }{a}},\zeta =\frac{{\beta }_2}{D_B}\sqrt{\frac{\nu }{a}},\hfill \\ \hfill Nt=\frac{{\tau }_B{D}_T(T_f-T_{\infty })}{\nu T_{\infty }},Sr=\frac{D_m{K}_T(T_f-T_{\infty })}{T_m\nu ({\phi }_f-C_{\infty })},Du=\frac{D_m{K}_T({\phi }_f-{\phi }_{\infty })}{C_s{C}_p\nu (T_f-T_{\infty })},Sc=\frac{\nu }{D_B},R=\frac{16{\sigma }^{*}{T}_{\infty }^3}{3\kappa k^{*}}\hfill \end{array}\}$ (13)

Quantities of physical interest, the physical parameters, are presented as follows:

[TeX:] C_{\text{fx}} = \frac{\tau_{\text{wx}}}{\rho U_{w}^{2}} = \frac{1}{\rho U_{w}^{2}}\mu\left( \frac{\partial u}{\partial y} \right)_{y = 0} \Rightarrow Cfx = C_{\text{fx}}\left( \sqrt{\text{Re}_{x}} \right) = f^{''}\left( 0 \right)$C_{\mathrm{\text{fx}}}=\frac{{\tau }_{\mathrm{\text{wx}}}}{\rho U_w^2}=\frac{1}{\rho U_w^2}\mu {\left(\frac{\partial u}{\partial y}\right)}_{y=0}\Rightarrow Cfx=C_{\mathrm{\text{fx}}}\left(\sqrt{{\mathrm{\text{Re}}}_x}\right)=f^{″}\left(0\right)$ (14)

[TeX:] C_{\text{fy}} = \frac{\tau_{\text{wy}}}{\rho V_{w}^{2}} = \frac{\mu}{\rho V_{w}^{2}}\left( \frac{\partial v}{\partial y} \right)_{y = 0} \Rightarrow Cfy = C_{\text{fy}}\left( \sqrt{\text{Re}_{y}} \right) = g^{''}\left( 0 \right)$C_{\mathrm{\text{fy}}}=\frac{{\tau }_{\mathrm{\text{wy}}}}{\rho V_w^2}=\frac{\mu }{\rho V_w^2}{\left(\frac{\partial v}{\partial y}\right)}_{y=0}\Rightarrow Cfy=C_{\mathrm{\text{fy}}}\left(\sqrt{{\mathrm{\text{Re}}}_y}\right)=g^{″}\left(0\right)$ (15)

[TeX:] Nu = - \left( 1 + R\theta_{w}^{3} \right)\left( \sqrt{\text{Re}_{x}} \right)\theta'\left( 0 \right)$Nu=-(1+R{\theta }_w^3)\left(\sqrt{{\mathrm{\text{Re}}}_x}\right)\theta \prime \left(0\right)$ (16)

[TeX:] Sh = \frac{xq_{m}}{D_{B}\left( C_{f} - C_{\infty} \right)} = - \frac{x\left( \frac{\partial C}{\partial y} \right)_{y = 0}}{D_{B}\left( C_{f} - C_{\infty} \right)} \Rightarrow Sh = - \left( \sqrt{\text{Re}_{x}} \right)\phi^{'}\left( 0 \right)$Sh=\frac{x{q}_m}{D_B(C_f-C_{\infty })}=-\frac{x{\left(\frac{\partial C}{\partial y}\right)}_{y=0}}{D_B(C_f-C_{\infty })}\Rightarrow Sh=-\left(\sqrt{{\mathrm{\text{Re}}}_x}\right){\varphi }^{\prime }\left(0\right)$ (17)

[TeX:] \text{Re}_{x} = \frac{u_{x}\left( x \right)x}{\nu}${\mathrm{\text{Re}}}_x=\frac{u_x\left(x\right)x}{\nu }$ (18)

[TeX:] \text{Re}_{y} = \frac{v_{w}\left( y \right)y}{\nu}${\mathrm{\text{Re}}}_y=\frac{v_w\left(y\right)y}{\nu }$ (19)

Based on the stretching velocities above, local Reynolds numbers (18) and (19) have been shown.

3. Solution of the mathematical problem using the Finite Element Method

This study employs the finite element method, a powerful technique for solving linear and nonlinear partial and ordinary differential equations prevalent in physics and mechanical engineering. Its utility extends to future research endeavours. A primary application of this method involves solving constructed equations, where the accurate approximation of real functions using form functions is crucial when dealing with real numbers. Rigorous adherence to this methodology ensures computational precision. The steps involved in FEM are shown in Figure 2.

The inflow sphere model comprises 10,000 identical quadratic factors, resulting in 20,001 nodes. Each factor possesses a uniform shape and magnitude, mirroring its counterparts. Following the formulation of element equations, 80,004 nonlinear equations remained for analysis. Subsequently, these equations were processed. Gaussian elimination, a well-established technique, was applied to resolve the equations, incorporating boundary conditions. This method yielded a solution with accuracy within 1/100,000 degrees. Gaussian quadrature facilitated the necessary integration simplifications. The computational implementation was achieved using the Mathematica programming language.

Fig. 2 Fundamental steps in FEM

Program code validation

Table 1 Comparison between the current numerical results and the published results for various values of C from Ammar Mushtaq et al.[38] at Pr = 1.0 when M = 0, Nb = 0, Nt = 0, Sr = 0, Du = 0, ζ → ∞ and δ → ∞

Table 2 Comparison of present numerical results with the published results of Liu and Anderson[39] for different values of C at Pr = 1.0 when M = 0, K = 0, R = 0, Nb = 0, Nt = 0, Sr = 0, Du = 0, ζ → ∞ and δ → ∞

Tables 1 and 2 compare the authors' most recent numerical findings for various values of C at Pr = 1.0 with those previously reported by Ammar Mushtaq et al.[38] and Liu and Anderson[39]. The aforementioned findings show how closely our numerical results agree with the results of the previous research.

4. Results and discussion

The authors solved the governing flow ordinary differential equations (8), (9), (10), and (11) using the finite element method and boundary equations (12). In this part, the authors use several figures to demonstrate how various factors might affect equations for temperature, concentration, and dimensionless velocity (velocity components along the x and y axes). The parameter ranges for the graphical representations are built on the following critical values: M (Magnetic field parameter) = 0.5, K (Maxwell fluid parameter or Deborah number) = 0.5, C (Velocity ratio parameter) = 0.5, Pr (Prandtl number) = 0.22, R (Thermal radiation parameter) = 0.5, Nb (Brownian motion parameter) = 0.5, Nt (Thermophoresis parameter) = 0.5, Du (Dufour number) = 0.5, Sr (Soret number) = 0.5, Sc (Schmidt number) = 0.22, δ (Thermal Biot number) = 0.1, ζ (Mass Biot number) = 0.1, and θ_w (Temperature ratio parameter) = 0.5 (see 3 – Figure 20).

Figures 3 and 4 are affected by the Magnetic field parameter (M) on velocity profiles along the x - and the y – directions, respectively. The velocity fields are shown to diminish as the value of M rises. The fluid flow tends to slow down in the presence of a magnetic field, which reduces the thickness of the velocity and momentum boundary layers. The influence of the Maxwell fluid parameter or Deborah number (K) on the x - and y - components of velocity is sketched in Figures 5 and 6, respectively. The deviation of the fluid relaxation period from its basic time scale is quantified by the Deborah number. The relaxation period is the amount of time that fluid requires to reach equilibrium following the application of the shear force. It is anticipated that fluids with higher viscosity may take longer to relax. It is possible to interpret an increase in K in this way because fluid viscosity restricts fluid motion and lowers velocity. As a result, the hydrodynamic boundary layer thins when K is increased. It is also noted that the change in the velocity fields f ʹ and g ʹ is larger in the three-dimensional flow when compared with the two-dimensional and axi-symmetric flows.

Figures 7 and 8 show the behavior of the velocity ratio parameter (C) on primary and secondary velocity profiles, respectively. From these figures, it is observed that both the primary and secondary velocity profiles are increasing with rising values of the velocity ratio parameter. The variation of the Prandtl number on temperature outlines is shown in Figure 9. It is concluded that increasing values of the Prandtl number result in a thinner temperature boundary layer thickness. Fluids with larger Prandtl numbers have lower thermal diffusivity, and hence the temperature decreases. Figure 10 shows the variations in temperature profiles due to an increase in the values of (R). Because the conduction effect of the nanofluid increases in the presence of R, it is noticed that the fluid temperature rises as R increases. Higher values of R thus imply higher surface heat flow, which raises the temperature in the boundary layer area.

The influence of the Brownian motion parameter (Nb) on temperature and concentration profiles is depicted through Figure 11 and Figure 12, respectively. These graphs show that when the values of the Brownian motion parameter increase, the thickness of the thermal boundary layer grows and the temperature differential at the surface decreases. When the Brownian motion parameter is raised, however, the concentration profiles and concentration boundary layer thickness show the reverse trend. Figs. 13 and 14 serve to demonstrate the impact of the thermophoresis parameter (Nt) on temperature and concentration curves. The findings demonstrate that the temperature and concentration boundary layer thickness rise as the thermophoresis parameter grows. By contrasting Figures 15 and 16 one after the other, you can see how the Dufour number (Du) and the Soret number (Sr) change in relation to the temperature and concentration profiles. Looking at this graph, we can see that increasing Du's value led to a rise in the temperature and thickness of the thermal layer. Physically, Du has to do with how a concentration gradient affects the thermal energy of a liquid. Growing Sr values also enhance concentration profiles, much like growing Sr levels do. This is because when there is a temperature gradient, the mass may move more readily from a location of lower to higher solute concentration. How the Schmidt number (Sc) impacts the concentration profiles is seen in Figure 17. The Sc value is a representation of the mass diffusivity to momentum ratio. It is possible to evaluate the relative importance of momentum and mass transfer using diffusion in the concentration (species) boundary layer. Lower concentration profiles will arise from a decrease in the fluid's mass diffusivity brought on by higher Sc. Due to the inverse relationship between mass diffusivity and Sc, weaker concentration boundary layers are linked to higher Sc concentrations. The temperature field's impact on the thermal Biot number (δ) is shown in Figure 18. With an increase, convection intensifies, raising the temperature field.

Figure 19 illustrates the effect of the mass Biot number (ζ) on the concentration profiles. Concentration profiles are an increasing function of the mass Biot number. Figure 20 includes the behavior of the temperature ratio parameter θ_w on the temperature distribution. Higher values of θ_w correspond to a higher temperature and thicker thermal boundary layer. This is explained as follows. It is clear from energy Eq. (10) that effective thermal diffusivity is the sum of classical thermal diffusivity (α) and thermal diffusivity due to the radiation effect. Thus, one anticipates that parameter θ_w, being the coefficient of the later term, would support the thermal boundary layer thickness. It can be noticed that the profiles attain a special S-shaped form when θ_w enlarges, which dictates the existence of an adiabatic case. In other words, the wall temperature gradient approaches zero value when the wall-to-ambient temperature ratio is sufficiently large.

The numerical values of various parameters, namely, M (magnetic field parameter), K (Maxwell fluid parameter or Deborah number), C (velocity ratio parameter), Pr (Prandtl number), R (thermal radiation parameter), Nb (Brownian motion parameter), Nt (thermophoresis parameter), Du (Dufour number), Sr (Soret number), Sc (Schmidt number), δ (thermal Biot number), ζ (mass Biot number), and θ_w (temperature ratio parameter) on skin-friction coefficients along the x - (Cfx) and y - (Cfy) directions are discussed in tables 3 and 4, respectively. From these tables, it is observed that the skin-friction coefficients along the x - (Cfx) and y - (Cfy) directions increase with increasing values of C (velocity ratio parameter), R (thermal radiation parameter), Nb (Brownian motion parameter), Nt (thermophoresis parameter), Du (Dufour number), Sr (Soret number), δ (thermal Biot number), ζ (mass Biot number), and θ_w (temperature ratio parameter) and decrease with increasing values of M (magnetic field parameter), K (Maxwell fluid parameter or Deborah number), Pr (Prandtl number), and Sc (Schmidt number). The tabular values of the rate of heat transfer coefficient or Nusselt number (Nu_x) for different values of Pr (Prandtl number), R (thermal radiation parameter), Nb (Brownian motion parameter), Nt (thermophoresis parameter), Du (Dufour number), δ (thermal Biot number), and θ_w (temperature ratio parameter) are shown in Table 5. From this table, it is observed that the rate of heat transfer coefficient increases with rising values of R (thermal radiation parameter), Nb (Brownian motion parameter), Nt (thermophoresis parameter), Du (Dufour number), δ (thermal Biot number), and θ_w (temperature ratio parameter), and the opposite effect is observed with the effect of Prandtl number (Pr). The effects of Nb (Brownian motion parameter), Nt (thermophoresis parameter), Sr (Soret number), Sc (Schmidt number), and ζ (mass Biot number) on the rate of mass transfer coefficient or Sherwood number (Sh_x) are studied through tabular forms in Table 6. From this table, it is observed that the rate of mass transfer coefficient increases with rising values of Nt (thermophoresis parameter), Sr (Soret number), and ζ (mass Biot number) and decreases with increasing values of Nb (Brownian motion parameter), and Sc (Schmidt number).

Fig. 2 Velocity profiles along x - direction for different values of M	Fig. 3 Velocity profiles along y - direction for different values of M
Fig. 4 Velocity profiles along x - direction for different values of K	Fig. 5 Velocity profiles along y - direction for different values of K
Fig. 6 Velocity profiles along x - direction for different values of C	Fig. 7 Velocity profiles along y - direction for different values of K
Fig. 8 Temperature for different values of Pr	Fig. 9 Temperature profiles for different values of R
Fig. 10 Temperature profiles for different values of Nb	Fig. 11 Concentration profiles for different values of Nb
Fig. 12 Temperature profiles for different values of Nt	Fig. 13 Concentration profiles for different values of Nt
Fig. 14 Temperature profiles for different values of Du	Fig. 15 Concentration profiles for different values of Sr
Fig. 16 Concentration profiles for different values of Sc	Fig. 17 Temperature profiles for different values of δ
Fig. 18 Concentration profiles for different values of ζ	Fig. 19 Temperature profiles for different values of θw

Table 3 Skin-friction coefficient along x-direction results

Table 4 Skin-friction coefficient along y-direction results

Table 5 Rate of heat transfer coefficient results

Table 6 Rate of mass transfer coefficient results

5. Conclusion

This work investigates the impact of Thermo-diffusion and Diffusion-thermo on a non-Newtonian Maxwell fluid with nanofluid particles, thermal and mass Biot numbers, Brownian motion, thermophoresis, and magnetic field. It models the problem using partial differential equations and numerically solves them, allowing visualization of velocity, temperature, and nanofluid concentration profiles. The conclusions of this study project are:

Applications

The present study addressed in this paper has several scientific and engineering applications, including the following:

Nomenclature:

List of symbols:

u, v, w : Velocity components in x, y and z Axes, respectively (m/s)

x, y, z : Cartesian coordinates measured along the stretching sheet (m)

[TeX:] f$f$: Dimensionless stream function along the x - direction (kg/m.s)

[TeX:] f^{'}$f^{\prime }$: Fluid velocity along the x - direction(m/s)

[TeX:] g$g$: Dimensionless stream function along the y - direction (kg/m.s)

[TeX:] g^{'}$g^{\prime }$: Fluid velocity along the y - direction(m/s)

Pr: Prandtl number

T: Fluid temperature (K)

[TeX:] T_{f}$T_f$: Temperature of hot fluid (K)

[TeX:] T_{\infty}$T_{\infty }$: Temperature of the fluid far away from the stretching sheet (K)

[TeX:] Cf_{x}$C{f}_x$: Skin-friction coefficient along x - direction (s^-1)

[TeX:] \text{Cfx}$\mathrm{\text{Cfx}}$: Non-dimensional skin-friction coefficient along x - direction (s^-1)

[TeX:] M$M$: Magnetic field parameter

[TeX:] B_{o}$B_o$: Uniform magnetic field (Tesla)

a, b: Constants

[TeX:] T_{w}$T_w$: Temperature at the surface (K)

[TeX:] O$O$: Origin

[TeX:] \text{Sc}$\mathrm{\text{Sc}}$: Schmidt number

[TeX:] K$K$: Maxwell fluid parameter

[TeX:] R$R$: Thermal radiation parameter

[TeX:] q_{r}$q_r$: Radiative heat flux

[TeX:] Cf_{y}$C{f}_y$: Skin-friction coefficient along y - direction (s^-1)

[TeX:] \text{Cfy}$\mathrm{\text{Cfy}}$: Non-dimensional Skin-friction coefficient along the y - direction (s^-1)

[TeX:] U_{w}\left( x \right)$U_w\left(x\right)$: Stretching velocity of the fluid along the x - direction (m/s)

[TeX:] V_{w}\left( y \right)$V_w\left(y\right)$ : Stretching velocity of the fluid along the y - direction (m/s)

[TeX:] q_{w}$q_w$: Heat flux coefficient

[TeX:] q_{m}$q_m$: Mass flux coefficient

[TeX:] \text{Nt}$\mathrm{\text{Nt}}$: Thermophoresis parameter

[TeX:] \text{Nb}$\mathrm{\text{Nb}}$: Brownian motion parameter

[TeX:] \text{Nu}$\mathrm{\text{Nu}}$: Rate of heat transfer coefficient (or) Nusselt number

[TeX:] Sh$Sh$: Rate of mass transfer coefficient (or) Sherwood number

[TeX:] C_{p}$C_p$: Specific heat capacity of nano particles (J/kg/K)

[TeX:] C$C$: Stretching sheet parameter

[TeX:] \text{Re}_{x}${\mathrm{\text{Re}}}_x$: Reynolds number along the x - direction

[TeX:] \text{Re}_{y}${\mathrm{\text{Re}}}_y$: Reynolds number along the y - direction

[TeX:] D_{B}$D_B$: Brownian diffusion coefficient (m²/s)

[TeX:] D_{T}$D_T$: Thermophoresis diffusion coefficient

[TeX:] \text{Sr}$\mathrm{\text{Sr}}$: Soret number

[TeX:] \text{Du}$\mathrm{\text{Du}}$: Dufour number

[TeX:] K^{*}$K^{*}$: Mean absorption coefficient

[TeX:] C_{w}$C_w$: Dimensional nanoparticle volume Concentration at the stretching surface (mol/m³)

[TeX:] C_{\infty}$C_{\infty }$: Dimensional ambient volume fraction (mol/m³)

[TeX:] C_{s}$C_s$: Concentration susceptibility

[TeX:] K_{T}$K_T$: Thermal diffusion ratio

[TeX:] D_{m}$D_m$: Solutal diffusivity of the medium

[TeX:] T_{m}$T_m$: Fluid Mean temperature

[TeX:] C_{f}$C_f$: Dimensional Concentration of hot fluid (mol/m³)

Greek symbols:

[TeX:] \eta$\eta$: Dimensionless similarity variable (m)

[TeX:] \theta$\theta$: Dimensionless temperature (K)

[TeX:] \nu$\nu$: Kinematic viscosity (m²/s)

[TeX:] \sigma$\sigma$: Electrical Conductivity

[TeX:] \rho$\rho$: Fluid density (kg/m³)

[TeX:] \kappa$\kappa$: Thermal conductivity of the fluid

[TeX:] \tau_{\text{wx}}${\tau }_{\mathrm{\text{wx}}}$: Wall shear stress along the x - direction

[TeX:] \tau_{\text{wy}}${\tau }_{\mathrm{\text{wy}}}$: Wall shear stress along the y - direction

[TeX:] \phi$\varphi$: Dimensionless nano-fluid Concentration (mol/m³)

[TeX:] \delta$\delta$: Thermal Biot number

[TeX:] \zeta$\zeta$: Mass Biot number

[TeX:] \beta_{1}${\beta }_1$: Non-uniform heat transfer coefficient

[TeX:] \beta_{2}${\beta }_2$: Non-uniform mass transfer coefficient

[TeX:] \alpha$\alpha$: Thermal diffusivity, (m²/s)

[TeX:] \theta_{w}${\theta }_w$: Temperature ratio parameter

[TeX:] \varphi$\phi$: Fluid nanoparticle volume Concentration (mol/m³)

[TeX:] \varphi_{\infty}${\phi }_{\infty }$: Dimensional ambient volume fraction (mol/m³)

[TeX:] \sigma^{*}${\sigma }^{*}$: Stefan-Boltzmann constant

[TeX:] \lambda$\lambda$: The fluid relaxation time

[TeX:] \mu$\mu$: Dynamic viscosity of the fluid

[TeX:] \rho_{f}${\rho }_f$: Density of the fluid (kg/m³)

Superscript:

[TeX:] /$/$: Differentiation w.r.t η

Subscripts:

[TeX:] f$f$: Fluid

[TeX:] w$w$: Condition on the sheet

[TeX:] \infty$\infty$: Ambient Conditions