## 2 Mathematical Formulation  
The system of two-dimensional laminar incompressible unsteady stagnation-point flow (density is constant), permeable surface, stretching/shrinking sheet in the water as the base fluid with different types of nanoparticles, i.e., Copper, Alumina, and Titania, is studied, as the Tiwari and Das model is considered. Let the unsteady free stream velocity be \(U_{\infty}=(a/(1-ct))e^{x/l}\), \(a\) is the stagnation length, and \(c\) is the flow unsteadiness parameter. Assume that the plate velocity is \(u_{w}=\varepsilon(a/(1-ct))e^{x/l}\) where \(\varepsilon>0\) for stretching velocity rate, \(\varepsilon<0\) for shrinking velocity rate, \(l\) is the characteristic length, and \(v_{w}=v_{0}e^{x/2l}/(1-ct)\) represents mass flux velocity with \(v_{0}\) as a constant. Let the slip occur between the fluid and the surface, which is represented as \(u_{s}=L\partial u/\partial y\), where \(L\) is the slip length (see [PERSON] et al. [8]). The reference temperature, ambient temperature, and temperature distribution near the surface are \(T_{0}\), \(T_{\infty}\) and \(T_{w}=T_{\infty}+(T_{0}/(1-ct))e^{x/2l}\), where both \(T_{0}\) and \(T_{\infty}\) are constants. Apart from that, the \(x\)-axis is parallel to stream direction while the \(y\)-axis is perpendicular to it. A schematic diagram of the problem is shown in Figure 1.  
Based on the above assumptions, the governing equations can be written as follows (see [PERSON] et al. [4] and [PERSON] et al. [7]):  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \tag{1}\]  
\[\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial v }{\partial y}=\frac{\partial U_{\infty}}{\partial t}+U_{\infty}\frac{ \partial U_{\infty}}{\partial x}+\frac{\mu_{nf}}{\rho_{nf}}\frac{\partial^{2} u}{\partial y^{2}}, \tag{2}\]  
\[\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T }{\partial y}=\alpha_{nf}\frac{\partial^{2}T}{\partial y^{2}}, \tag{3}\]  
subject to boundary conditions  
Figure 1: Schematic diagram of the problem. \[v=-v_{w},u=\varepsilon+L\frac{\partial u}{\partial y},\ T=T_{w}\quad\text{at}\ y=0, \tag{4}\]  
\[u\to U_{\infty},\ T\to T_{\infty}\quad\text{as}\ y\to\infty.\]  
where \(u\) and \(v\) are velocity component in \(x\) and \(y\) axes respectively, \(t\) denotes time \(v_{w}=v_{0}e^{x/2l}/(1-ct)^{1/2}\) is the variable suction velocity with \(v_{0}>0\) being a constant, \(T\) is the nanofluid temperature, \(\mu_{nf}\), \(\alpha_{nf}\), \(k_{nf}\) and \(\left(\rho c_{p}\right)_{nf}\) represent the viscosity, thermal diffusivity, thermal conductivity and heat capacity of the nanofluid, respectively. Equation (5) is given by [PERSON] et al. [42]:  
\[\mu_{nf}=\frac{\mu_{f}}{(1-\varphi)^{2.5}}, \tag{5}\]  
\[\alpha_{nf}=\frac{k_{nf}}{\left(\rho c_{p}\right)_{nf}}, \tag{6}\]  
\[\frac{k_{nf}}{k_{f}}=\frac{k_{s}+2k_{f}-2\varphi(k_{f}-k_{s})}{k_{s}+2k_{f}+ \varphi(k_{f}-k_{s})}, \tag{7}\]  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \tag{1}\]0  
where \(\varphi\) denotes the volume of the nanoparticle, \(\mu_{f}\) is the viscosity of the fluid, \(\left(\rho C_{p}\right)_{nf}\), \(\left(\rho C_{p}\right)_{s}\) and \(\left(\rho C_{p}\right)_{f}\) is the heat capacity of the nanofluid, nanoparticle and fluid \(k_{s}\) and \(k_{f}\) are the thermal conductivities of the nanoparticle and fluid, respectively. The densities of the fluid and nanoparticle are denoted as \(\rho_{f}\) and \(\rho_{s}\), respectively. The system of ordinary differential Equations (1)-(3) bounded by (4) is transformed to the simpler form by using the similarity transformation variables, which are given by:  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \tag{1}\]1  
The stream function \(\psi\) is defined as \(u=\partial\psi/\partial y\) and \(v=-\partial\psi/\partial x\), where  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \tag{1}\]2  
satisfies the continuity Equation (1).