# Numerical Simulation of Williamson Nanofluid Flow over an Inclined Surface: Keller Box Analysis
## 2 Mathematical Formulation  
This section analyzes the flow behavior of Williamson nanofluid on an inclined surface. The angle \(\Omega\) is considered among stretching sheet and vertical direction. A further sheet stretched with the rate _t/c_ causes the flow of fluid. Heat generation or absorption impacts are under consideration. The concentration and wall temperature are taken as \(T_{w}\), \(C_{w}\), respectively, while \(u_{w}\) denotes the velocity on the wall. Moreover, at \(y\) approaches to infinity \(T_{\infty}\), \(C_{\infty}\) exhibits temperature and nanoparticles concentration, as shown in Figure 1. Additionally, Figure 1 presents the flow model and coordinate scheme. For Williamson fluid, the expression for Cauchy stress tensor is given as [34]:  
\[S=-pI+\tau \tag{1}\]  
\[\tau=\left[\mu_{\infty}+\frac{(\mu_{0}-\mu_{\infty})}{1-\Gamma\gamma^{*}} \right]A_{1}, \tag{2}\]where \(\tau\) denotes extra stress tensor, limiting viscosity at zero shear rate is \(\mu_{0}\) and at infinite shear rate is \(\mu_{\infty}\), \(\Gamma>0\) signifies time constant, and \(A_{1}\) denotes the first Rivlin-Erickson tensor. From [34], we obtain  
\[\tau=\mu_{0}[1+\Gamma\gamma^{*}]A_{1} \tag{3}\]  
The mathematical model for this flow is [35]  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \tag{4}\]  
\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=v\frac{ \partial^{2}u}{\partial y^{2}}+\sqrt{2}v\Gamma\frac{\partial u}{\partial y} \frac{\partial^{2}u}{\partial y^{2}}+g[\beta_{t}(T-T_{\infty})-\beta_{c}(C-C_ {\infty})]\cos\Omega-\frac{\sigma B_{0}^{2}}{\rho}u, \tag{5}\]  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha\frac{ \partial^{2}T}{\partial y^{2}}+\tau\left[D_{B}\frac{\partial C}{\partial y} \frac{\partial T}{\partial y}+\frac{D_{T}}{T_{\infty}}\left(\frac{\partial T }{\partial y}\right)^{2}\right]+\frac{Q_{0}}{\rho C_{p}}(T-T_{\infty}), \tag{6}\]  
\[u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_{B}\frac{ \partial^{2}C}{\partial y^{2}}+\frac{D_{T}}{T_{\infty}}\frac{\partial^{2}T}{ \partial y^{2}}-R^{*}(C-C_{\infty}). \tag{7}\]  
Let us consider the conditions as  
\[u=u_{w}(x)=cx,\,v=0,\,T=T_{w},\,C=C_{w}\,\,\text{at}\,\,y=0, \tag{8}\]  
\[u\to u_{\infty}(x)=0,\,\,\,v\to 0,\,\,\,T\to T_{\infty},\,\,\,C\to C_{\infty}\,\,\,\text{as}\,\,y\to\infty. \tag{9}\]  
The stream function is given as  
\[S=-pI+\tau \tag{1}\]0  
using appropriate similarity transformations such as  
\[S=-pI+\tau \tag{1}\]1  
\[S=-pI+\tau \tag{1}\]2  
On substituting Equation (11), the system of Equations (4)-(7) converted to  
\[S=-pI+\tau \tag{1}\]3  
\[S=-pI+\tau \tag{1}\]4  
\[S=-pI+\tau \tag{1}\]5  
\[S=-pI+\tau \tag{1}\]6 Physical structure with coordinates. where  
\[ \begin{array}{c}M=\frac{\sigma R_{\rm c}^{2}}{\rho\mu},Nt=\frac{\tau D_{\rm c}(T_ {\rm sp}-T_{\rm so})}{\
u T_{\rm so}},Gr_{x}=\frac{\delta\beta_{\rm c}(T_{\rm sp }-T_{\rm so})x^{-1}}{a^{2}},\text{Re}_{x}=\frac{u_{x}(x)x}{v},\\ Gc_{x}=\frac{\delta\beta_{\rm c}(C_{\rm sp}-C_{\rm so})x^{-1}}{a^{2}},Nt_{b}= \frac{Nt}{Nb},\lambda_{1}=\frac{Q_{0}}{\mu\rho c^{2}},\tau_{1}=\Gamma x\sqrt{ \frac{2c^{3}}{v}},\ Nb=\frac{\tau D_{\rm c}(C_{\rm sp}-C_{\rm so})}{v},\\ R=\frac{R_{x}}{a}.\end{array}  \tag{14}\]  
To find similarity solution, \(Gr_{x}\) and \(Gc_{x}\) should be independent of \(x\). Hence, suppose that [36; 37]  
\[\beta_{t}=nx^{1},\beta_{c}=n_{1}x^{1}, \tag{15}\]  
where \(n_{1}\) and \(n\) denote constants. Replacing Equation (15) into the parameters \(Gr_{x}\) and, \(Gc_{x}\) we obtain  
\[Gc=\frac{gn_{1}(C_{\rm sp}-C_{\rm so})}{a^{2}},\ Gr=\frac{gn(T_{\rm sp}-T_{ \rm so})}{a^{2}}. \tag{16}\]  
Equation (8) is converted to  
\[ \begin{array}{c}f(\eta)=0,f\prime(\eta)=1,\theta(\eta)=1,\phi(\eta)=1,\text {at }\eta=0,\\ f\prime(\eta)\to 0,\ \theta(\eta)\to 0,\ \phi(\eta)\to 0,\ \text{at }\eta\to\infty. \end{array}