*Skin Friction and Nusselt Number*

The following are the definitions of the skin friction and the local Nusselt number:

$$\mathbf{c}\_{f} = \frac{\mathbf{r}\_{\omega}}{\rho\_{f} \mathbf{u}\_{\omega}^{2}}, \mathbf{N}\_{\mu} = \frac{zq\_{\omega}}{k\_{f} \left(T\_{f} - T\_{\infty}\right)}\tag{13}$$

where *τω* and *qω* are the surface shear stress and heat flux, respectively. These are defined as:

$$\pi\_{\omega} = \left[ \left( \mu\_{nf} + \frac{1}{\beta c} \right) \left( \frac{\partial u}{\partial r} \right) - \frac{1}{6\beta c^3} \left( \frac{\partial u}{\partial r} \right)^3 \right]\_{r=R}, \\ q\_{\omega} = -\left( k\_{nf} + \frac{16\sigma^\* T\_{\infty}^{-3}}{3k^\*} \right) \left( \frac{\partial T}{\partial r} \right)\_{r=R} \tag{14}$$

In dimensionless form, the skin friction and Nusselt number are:

$$C\_f Re\_z^{1/2} = \left(\frac{1}{\left(1-\phi\_1\right)^{2.5}} + a\right) f''(0) - \frac{\lambda}{3} a \left(f''(0)\right)^3,\\ N\_u Re\_z^{-1/2} = -\left(\frac{k\_{nf}}{k\_f} + Rd\right) \theta'(0) \tag{15}$$
