## 2 Problem Formulation
\begin{array}{ll}\mu_{nf}=\frac{\mu_{f}}{\left(1-\phi\right)^{25}},\quad\alpha _{nf}=\frac{k_{nf}}{\left(\rho C_{p}\right)_{nf}},\quad\rho_{nf}=(1-\phi)\rho_ {f}+\phi\rho_{s},\\ \left(\rho\beta\right)_{nf}=(1-\phi)(\rho\beta)_{f}+\phi(\rho\beta)_{s^{\prime }},\quad\left(\rho C_{p}\right)_{nf}=(1-\phi)\big{(}\rho C_{p}\big{)}_{f}+\phi \big{(}\rho C_{p}\big{)}_{s^{\prime}},\\ \frac{\sigma_{nf}}{\sigma_{f}^{\prime}}=1+\frac{3\left(\sigma_{s}/\sigma_{f} -1\right)\phi}{\sigma_{s}/\sigma_{f}+2-\big{(}\sigma_{s}/\sigma_{f}-1\big{)} \phi},\quad\frac{k_{nf}}{k_{f}}=\frac{k_{s}+2k_{f}-2\phi\big{(}k_{f}-k_{s} \big{)}}{k_{s}+2k_{f}+\phi\big{(}k_{f}-k_{s}\big{)}},\end{array}  \tag{6}\]  
where \(\mu_{nf}\) is the dynamic viscosity of the nanofluid, as described by the Brinkman model, \(\mu_{f}\) denotes the dynamic viscosity of the base fluid, \(\phi\) denotes the solid volume fraction of nanoparticles, \(\rho_{f}\) and \(\rho_{s}\) are the density of the base fluid and the solid nanoparticle, respectively, \(\big{(}\rho C_{p}\big{)}_{nf}\) is the heat capacitance of the nanofluid, \(k_{nf}\) is the thermal conductivity of the nanofluid, as approximated by Maxwell-Garnett model, in which the subscript '\(f\)' refers to the base fluid while '\(s\)' refers to the solid nanoparticle. Furthermore, \(N_{1}\) and \(K_{1}\) are the velocity slip factor and the thermal slip factor, respectively (see [PERSON] and [PERSON] [37]), and it has been assumed that \(u_{w}\), \(v_{w}\), \(w_{w}\), \(T_{w}\), \(B\), \(N_{1}\), and \(K_{1}\) take the following forms:  
\[ \begin{array}{ll}u_{w}=v_{w}=a(x+y)^{n},&w_{w}=-\frac{n+1}{2}\sqrt{\alpha v_ {f}}(x+y)^{(n-1)/2}s,\quad T_{w}=T_{\infty}+T_{0}\ (x+y)^{2n-1},\\ B=B_{0}(x+y)^{(n-1)/2},&N_{1}=N_{0}(x+y)^{(1-n)/2},\quad K_{1}=K_{0}(x+y)^{(1- n)/2}.\end{array}  \tag{7}\]  
Here, \(s\) refers to the suction/injection parameter with \(s<0\) for injection and \(s>0\) for suction, \(B_{0}\), \(N_{0}\) and \(K_{0}\) are the constants, \(T_{0}\) is the constant characteristic temperature with \(T_{0}<0\) that implies cooled surface (opposing flow) while \(T_{0}>0\) corresponds to heated surface (assisting flow), \(N\) and \(K\) are the dimensionless slip parameters, and lastly, \(a\) and \(n\) are the positive constants. Moreover, the governing Equations (1)-(4) along with boundary conditions (5), can be converted into ordinary differential equations by the dimensionless functions \(u\), \(v\), \(w\) and \(\theta\), which are related to the similarity variable \(\eta\), as follows:  
\[ \begin{array}{ll}u=a(x+y)^{n}f^{\prime}(\eta),&v=a(x+y)^{n}h^{\prime}(\eta ),\\ w=-\sqrt{\alpha v_{f}}\ (x+y)^{(n-1)/2}\Big{[}\frac{n+1}{2}(f+h)+\frac{n-1}{2}\eta(f ^{\prime}+h^{\prime})\Big{]},\\ \theta(\eta)=\frac{T-T_{\infty}}{T_{w}-T_{\infty}},&\eta=\sqrt{\frac{a}{v_{f} }}(x+y)^{(n-1)/2}z.\end{array}  \tag{8}\]  
Note that \(f\) and \(h\) are the dimensionless quantities, \(f^{\prime}\) and \(h^{\prime}\) are the dimensionless velocity, \(\theta\) is the dimensionless temperature and prime denotes differentiation with respect to \(\eta\). Substituting similarity transformation (8) into governing Equations (1)-(4), the following equations are obtained:  
\[