**2. Formulation**

Let *V* = [*u*(*t*,*r*, *θ*, *z*) *v*(*t*,*r*, *θ*, *z*) *w*(*t*,*r*, *θ*, *z*)] be the velocity of unsteady, incompressible and viscous multi nanofluids axi-symmetrically rotating disk having an angular velocity Ψ, as shown in Figure 1.

**Figure 1.** Physical configuration of the rotating disk.

The layer of nanofluid across the surface is evenly spread out; thus, appropriate assumptions can be enlisted as:


The governing equations in components form are:

$$\frac{\partial \overline{u}}{\partial \overline{t}} + \frac{\overline{u}}{\overline{r}} + \frac{\partial \overline{w}}{\partial \overline{z}} = 0 \tag{1}$$

$$\overline{\rho}\_{nf} \left( \frac{\partial \overline{u}}{\partial \overline{t}} + \overline{\mu} \frac{\partial \overline{u}}{\partial \overline{\tau}} + \overline{w} \frac{\partial \overline{u}}{\partial \overline{z}} - \frac{\overline{v}^2}{\overline{r}} \right) = -\frac{\partial \overline{p}}{\partial \overline{\tau}} + \overline{\mu}\_{nf} \left\{ \frac{\partial^2 \overline{u}}{\partial \overline{\tau}^2} + \frac{\partial}{\partial \overline{\tau}} \left( \frac{\overline{u}}{\overline{r}} \right) + \frac{\partial^2 \overline{u}}{\partial \overline{z}^2} \right\} \tag{2}$$

$$
\overline{\rho}\_{nf} \left( \frac{\partial \overline{\upsilon}}{\partial \overline{t}} + \overline{u} \frac{\partial \overline{\upsilon}}{\partial \overline{\tau}} + \overline{w} \frac{\partial \overline{\upsilon}}{\partial \overline{z}} + \frac{\overline{u} \overline{\upsilon}}{\overline{\tau}} \right) = -\frac{\partial \overline{p}}{\partial \overline{\theta}} + \overline{\mu}\_{nf} \left\{ \frac{\partial^2 \overline{\upsilon}}{\partial \overline{\tau}^2} + \frac{\partial}{\partial \overline{\tau}} \left( \frac{\overline{\upsilon}}{\overline{\tau}} \right) + \frac{\partial^2 \overline{\upsilon}}{\partial \overline{z}^2} \right\} \tag{3}
$$

$$\left(\overline{\rho}\_{nf}\left(\frac{\partial\overline{\overline{w}}}{\partial\overline{t}} + \overline{\pi}\frac{\partial\overline{\overline{w}}}{\partial\overline{\overline{r}}} + \overline{w}\frac{\partial\overline{w}}{\partial\overline{z}}\right) = -\frac{\partial\overline{p}}{\partial\overline{z}} + \overline{\mu}\_{nf}\left\{\frac{\partial^2\overline{w}}{\partial\overline{r}^2} + \frac{\overline{w}}{\overline{r}} + \frac{\partial^2\overline{w}}{\partial\overline{z}^2}\right\} \tag{4}$$

$$\left(\left(\overline{\rho}\mathbb{C}\_{\mathcal{P}}\right)\_{nf}\left(\frac{\partial\overline{T}}{\partial\overline{t}}+\overline{u}\frac{\partial\overline{T}}{\partial\overline{\tau}}+\overline{w}\frac{\partial\overline{T}}{\partial\overline{z}}\right)=\overline{k}\_{nf}\left\{\frac{\partial^{2}\overline{T}}{\partial\overline{\tau}^{2}}+\frac{1}{\overline{\tau}}\frac{\partial\overline{T}}{\partial\overline{\tau}}+\frac{\partial^{2}\overline{T}}{\partial\overline{z}^{2}}\right\}\tag{5}$$

Initial and boundary conditions associated with Equations (1)–(5) are defined in the following sub sections:
