## 2 Background  
Consider 2-D steady, and incompressible boundary layer flow of Casson ternary hybrid nanofluid over a porous lessening sheet with uniform surface temperature. The ternary hybrid nanofluid model is utilized to model flow equations with an assumption of magnetic (\(M\)) and thermal radiation (\(Rd\)) effect on a contracting sheet. There are three different nanoparticles are used. The red nanoparticles represent Aluminium oxide (\(Al_{2}O_{3}\)), green particles show Titanium oxide (\(TiO_{2}\)) and yellow particles show the copper (\(Cu\)). \(u_{w}\) represents wall velocity and \(u_{e}\) is ambient velocity. Figure 1 depicts the schematic diagram of a contracting sheet. \(T_{w}\) are the temperature at the surface of a sheet where \(T_{\infty}\) is ambient temperature. Additionally, a constant magnitude of a magnetic field is implemented in \(y\)-aixs. The under-consideration flow equations of Casson nanofluid [49]. \[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \tag{1}\]  
\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=u_{\infty} \frac{du_{\infty}}{dx}+\left(\frac{u_{thmf}}{\rho_{thmf}}\right)\left(1+ \frac{1}{\beta}\right)v\frac{\partial^{2}u}{\partial y^{2}}+\frac{\sigma_{ thmf}}{\rho_{thmf}}B^{2}(u_{\infty}-u), \tag{2}\]  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\left(\frac{1}{ \left(\rho cp\right)_{thmf}}\left(k_{thmf}+\frac{16\sigma T_{\infty}{}^{3}}{ 3K}\right)\frac{\partial^{2}T}{\partial y^{2}}\right). \tag{3}\]Boundary conditions of governing flow problems are;  
\[ \begin{split} u&=u_{w}(x),\ v=v_{w}(x),\ T=T_{w}\ \text{at}\ y=0,\\ &\ u\to u_{e}(x)\,\ T\to T_{\infty}\ \text{as}\ y\to\infty.\end{split}  \tag{4}\]  
where \(u\) and \(v\) are velocity components along \(x\) and \(y\)-direction. The temperature of the fluid is represented by \(T\). Here, \(v\) is kinematics viscosity, \(B(x)=B_{0}x^{(m-1)/2}\), \(B_{0}\) is constant (see [PERSON] et al. [50]), and \(q_{r}=-\frac{4e^{+}}{3k^{2}}\frac{3T^{4}}{dy}\) (for detail see reference [51]). The flow Equations (1)-(3) and boundary condition (4) are PDEs, to convert these equations into ODEs we exhibit an efficient similarity transformation as follows;  
\[ \begin{split}\psi(x,y)=f(\eta)\sqrt{\frac{2v}{a(\eta+1)}x^{n-1}}, \ \eta=y\sqrt{\frac{a(\eta+1)}{2v}x^{n-1}y},\ \theta=\frac{T-T_{\infty}}{T_{w}-T_{\infty}},\ u_{e}=U_{e}x^{n},\\ u_{w}=U_{w}x^{n}.\end{split}  \tag{5}\]  
The dimensionless variables/ transformation is presented in Equation (5). These dimensionless variables satisfied the continuity equation identically. However, the Equations (2) and (3) and boundary conditions Equation (4) are transformed into a system of ODEs and dimensionless boundary conditions by applying similarity variables and velocity components \(u\) and \(v\) as follows;  
\[ \begin{split}\left(\frac{u_{thf}/\mu_{f}}{\rho_{thnf}/\rho_{f}} \right)\left(1+\frac{1}{\beta}\right)f^{\prime\prime}(\eta)+f(\eta)f^{\prime \prime}(\eta)+\frac{2n}{n+1}-\frac{2n}{n+1}f^{\prime 2}(\eta)-\frac{\sigma_{ thnf}/\sigma_{f}}{\rho_{thnf}/\rho_{f}}M^{2}f^{\prime}(\eta)=0,\end{split}  \tag{6}\]  
\[ \begin{split}\frac{1}{Pr}\frac{1}{\left(\frac{\langle\rho c_{T} \rangle_{thnf}}{\langle\rho c\rangle_{f}}\right)}\left(\frac{k_{thnf}}{k_{f}}+ \frac{4}{3}Rd\right)\theta^{\prime\prime}(\eta)+f(\eta)\theta^{\prime}(\eta)=0.\end{split}  \tag{7}\]  
The dimensionless boundary conditions are;  
\[ \begin{split} f(\eta)=S,\ f^{\prime}(\eta)=\lambda,\ \theta(\eta)=1,\ \text{at}\ \eta=0,\\ f^{\prime}(\eta)\to 1,\ \theta(\eta)\to 0\,\ \text{at}\ \eta\to\infty.\end{split}  \tag{8}\]  
Figure 1: Flow Geometry. Here prime denotes differentiation w.r.t \(\eta\), \(\lambda=\frac{U_{w}}{U_{x}}\) are the elongating and lessening factors with \(\lambda>0\) for extending case and \(\lambda<0\) for the lessening case and \(S\) is a suction and injection parameters with \(S<0\) for injection and \(S>0\) for suction. The Prandtl number is defined as \(Pr=\frac{(\mu cp)_{f}}{k_{f}}\).