## 2 Problem Formulation  
Let us take the steady two-dimensional boundary layer flow of an incompressible viscous nanofluid past an exponentially stretching sheet flooded in a permeable medium. Utilizing the boundary layer estimates the governing equations for the flow problem are obtained as  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \tag{1}\]  
\[\rho_{nf}\bigg{(}u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y }\bigg{)}+\frac{\partial p}{\partial x}-\mu_{nf}\frac{\partial^{2}u}{\partial y ^{2}}=-\frac{\mu_{nf}\varepsilon}{K}u. \tag{2}\]  
In the aforementioned equations, \(u\) and \(v\) represent the velocity components along with the \(x\) and \(y\) directions respectively, \(\varepsilon\) indicates the porosity of the medium and \(K\) is regarded as the permeability of the fluidic medium. The parameters \(\mu_{nf}\) (nanofluid viscosity) and \(\rho_{nf}\) (nanofluid density) are defined below by  
\[\mu_{nf}=\frac{\mu_{f}}{\left(1-\phi\right)^{2.5}},\ \rho_{nf}=(1-\phi)\rho_{f}+ \phi\rho_{s}. \tag{3}\]  
Here \(\phi\) indicates the nanoparticles volume fraction of nanofluid, \(\rho_{f}\) denotes the fluid density of the base fluid, \(\rho_{s}\) specifies nanoparticles density and \(\mu_{f}\) represents base fluid viscosity. For the steady boundary layer flow, the equation of energy is  
\[-\mathbf{V}.\
abla T=\frac{\
abla.\mathbf{q}}{\left(\rho c_{p}\right)_{nf}}, \tag{4}\]  
where \(\left(\rho c_{p}\right)_{nf}\) signifies specific heat capacity of nanofluid, \(T\) demonstrates local fluid temperature and \(q\) represents heat flux satisfying Cattaneo-Christov heat flux law given below  
\[\mathbf{q}+\lambda_{2}\bigg{[}\frac{\partial\mathbf{q}}{\partial t}+\ \left( \
abla.\mathbf{V}\right)\mathbf{q}-\mathbf{q}.\
abla\mathbf{V}+\mathbf{V}. \
abla\mathbf{q}\bigg{]}=-k_{nf}\
abla T, \tag{5}\]in which \(\lambda_{2}\) indicates thermal relaxation time, \(\mathbf{V}\) denotes the velocity vector and \(k_{nf}\) represents the thermal conductivity of the nanofluid. Using the fact that fluid is incompressible and utilizing Equation (5) in Equation (4), the temperature equation takes the form  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+\lambda_{2}\left( \begin{array}{c}\left\{u\frac{\partial u}{\partial x}\frac{\partial T}{ \partial x}\right\}+\left\{v\frac{\partial v}{\partial y}\frac{\partial T}{ \partial y}\right\}+\left\{u\frac{\partial v}{\partial x}\frac{\partial T}{ \partial y}\right\}+\left\{v\frac{\partial u}{\partial y}\frac{\partial T}{ \partial x}\right\}+\\ \left\{2 uv\frac{\partial^{2}T}{\partial x\partial y}\right\}+\left\{u^{2} \frac{\partial^{2}T}{\partial x^{2}}\right\}+\left\{v^{2}\frac{\partial^{2}T }{\partial y^{2}}\right\}\end{array} \right)=\alpha_{nf}\frac{\partial^{2}T}{ \partial y^{2}}. \tag{6}\]  
Here, the parameter \(\alpha_{nf}\) denotes the thermal diffusivity of nanofluid defined as follow  
\[ \begin{array}{c}\alpha_{nf}\big{(}\rho c_{p}\big{)}_{nf}=k_{nf},\ \ \big{(}\rho c_{p}\big{)}_{nf}-\left(1-\phi\right)\big{(}\rho c_{p}\big{)}_{f}= \phi\big{(}\rho c_{p}\big{)}_{s},\\ \frac{k_{nf}}{k_{f}}=\frac{\left(2k_{f}+k_{*}\right)-2\phi\big{(}k_{f}-k_{*} \big{)}}{\left(2k_{f}+k_{*}\right)+\phi\big{(}k_{f}-k_{*}\big{)}},\ \ v_{nf}\rho_{nf}=\mu_{nf}.\end{array}  \tag{7}\]  
Here \(k_{f}\) indicates the thermal conductivity of base fluid whereas \(k_{*}\) indicates solid thermal conductivity. The associated boundary conditions are imposed as below  
\[ \begin{array}{c}v|_{y=0}=0,\,u|_{y=0}=U_{w}(x)=U_{0}e^{(x/l)},\ T_{y=0}=T_{w }(x)=T_{0}+ae^{(x/l)},\\ u|_{y\rightarrow\infty}\to 0,\ \ T_{y\rightarrow\infty}\to T_{\infty}=T_{0}+ be^{(x/l)}.\end{array}  \tag{8}\]  
Here \(U_{w}(x)\) denotes the velocity of the stretching surface, \(U_{0}\) identifies the reference velocity, \(l\) signifies a characteristic length, \((a,b)\) are dimensional constants, \(T_{w}(x)\) and \(T_{\infty}\) represent the wall and ambient temperature or far away temperature, respectively.