**2. Problem Modeling**

The second grade nanofluid flow by stretchable rotating disk is assumed. The heat model of C-C is also taken in the nanofluid flow. The Hall current influence is considered in this nanofluid flow. Furthermore, EGM is considered with heat generation/absorption, Joule heating and non-linear thermal radiation. At *z* = 0 the disk rotates with angular velocity α1. The ambient and disk temperatures are *T*<sup>∞</sup> and *Tw* respectively. Similarly, the ambient and surface concentrations are *C*<sup>∞</sup> and *Cw*. Geometry of the fluid is displayed in Figure 1.

**Figure 1.** Fluid flow geometry.

The continuity, momentum, energy and concentration equations are taken as [40]:

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

$$u\frac{\partial\mu}{\partial r} - \frac{\mu^2}{r} + w\frac{\partial\mu}{\partial z} = \nu\_f \frac{\partial^2\mu}{\partial z^2} + \frac{\beta\_1}{\rho\_f} \left[\mu \frac{\partial^3\mu}{\partial r \partial z^2} - \frac{1}{r} \left(\frac{\partial\mu}{\partial z}\right)^2 + 2w\frac{\partial\mu}{\partial r}\frac{\partial^2\mu}{\partial z^2}\frac{\partial^3\mu}{\partial z^3} + \cdots\right] \tag{2}$$

 $\frac{\partial\eta}{\partial r}$  $\frac{\partial^2\upsilon}{\partial z^2} + \frac{\partial w}{\partial z}$  $\frac{\partial^2\mu}{\partial z^2} + \frac{\partial\upsilon}{\partial z}$  $\frac{\partial^2\upsilon}{\partial r dz} + 3$  $\frac{\partial\mu}{\partial r}$  $\frac{\partial^2\mu}{\partial r dz} - \frac{\partial^2\upsilon}{\partial z^2}$  $\frac{\mathbb{E}}{r}$  $\left[ -\frac{\sigma\_f B\_0^2}{\rho\_f (1 + m^2)} \left( \mu + m\upsilon \right) \right]$ 

$$\begin{split} \mu \frac{\partial \overline{v}}{\partial r} + \frac{u\overline{v}}{r} + w \frac{\partial \overline{v}}{\partial z} &= \nu\_f \frac{\partial \overline{v}\_{\overline{v}}}{\partial z^2} + \frac{\rho\_1}{\rho\_f} \Big[ \mu \frac{\partial^2 \underline{u}}{\partial z^2} - 2u \frac{\partial \overline{v}}{\partial z} \frac{\partial^2 \underline{u}}{\partial r \partial z} \frac{\partial^3 \overline{v}}{\partial r \partial z^2} \\ &+ w \frac{\partial^3 \underline{v}}{\partial z^3} - \frac{1}{r} \frac{\partial v}{\partial z} \frac{\partial \underline{u}}{\partial z} \Big] + \frac{\sigma\_f B\_0^2}{\rho\_f (1 + m^2)} (mu - v) \end{split} \tag{3}$$

$$\left(\rho c\_p\right)\left(u\frac{\partial T}{\partial r} + w\frac{\partial T}{\partial z}\right) + \gamma\left[u^2\frac{\partial^2 T}{\partial r^2} + w^2\frac{\partial^2 T}{\partial z^2} + 2uw\frac{\partial^2 T}{\partial r\partial z} + \frac{\partial T}{\partial r}\left(u\frac{\partial u}{\partial r} + w\frac{\partial u}{\partial z}\right)\right]$$

$$+ \frac{\partial T}{\partial z}\left(u\frac{\partial w}{\partial r} + w\frac{\partial w}{\partial z}\right)\right] = k\_f\frac{\partial^2 T}{\partial z^2} + Q(T - T\_{\infty}) + \frac{\sigma\_f B\_0^2}{(1 + m^2)}\left(u^2 + v^2\right) - \tag{4}$$

$$\frac{16\sigma^\*}{3k^\*}\left[T^3\frac{\partial^2 T}{\partial z^2} + 3T^2\left(\frac{\partial T}{\partial z}\right)^2\right] + \left(\rho c\_p\right)\_s\left[\frac{D\_T}{T\_{\infty}}\left(\frac{\partial T}{\partial z}\right)^2 + D\_B\left(\frac{\partial T}{\partial z}\frac{\partial C}{\partial z}\right)\right]$$

$$u\frac{\partial \mathbb{C}}{\partial r} + w\frac{\partial \mathbb{C}}{\partial z} = D\_B\frac{\partial^2 \mathbb{C}}{\partial z^2} + \frac{D\_T}{T\_{\infty}}\frac{\partial^2 T}{\partial z^2} \tag{5}$$

with

$$\begin{aligned} \mu &= ra, \ v = ra\_1, \ w = 0, \ T = T\_{\text{\textquotedblleft}}, \mathbb{C} = \mathbb{C}\_{\text{\textquotedblleft}} \text{ at } z = 0, \\\ u &= v = 0, \ T \to T\_{\text{\textquotedblleft}}, \mathbb{C} \to \mathbb{C}\_{\text{\textquotedblright}} \text{ when } z \to \infty \end{aligned} \tag{6}$$

where *u*, *v* and *w* are the components of velocity in *r*, θ and *z* directions, respectively, ν*<sup>f</sup>* is the kinematic viscosity, β<sup>1</sup> is the material parameter, *kf* is the thermal conductivity, *<sup>f</sup>* is the density, *cp* is the specific heat, *Q* is the heat absorption/generation, σ*<sup>f</sup>* is the electrical conductivity, (*cp*)*<sup>f</sup>* is the heat capacitance, *DT* is the thermophoretic diffusion coefficient and *DB* is the Brownian diffusion coefficient.

Similarity transformations are defined as [40]

$$u = r\alpha\_1 f'(\zeta), \; v = r\alpha\_1 g(\zeta), \; w = -2h\alpha\_1 f(\zeta), \; \Theta = \frac{T - T\_{\infty}}{T\_w - T\_{\infty}}, \; \phi = \frac{\mathbb{C} - \mathbb{C}\_{\infty}}{\mathbb{C}\_w - \mathbb{C}\_{\infty}}, \; \zeta = \frac{z}{h} \tag{7}$$

The dimensionless forms of the leading equations are

$$f^{\prime\prime} + \text{Re}\mathcal{W}\{2f^{\prime\prime} + \text{g}^{\prime} - 2ff^{\prime\prime\prime} + f^{\prime}f^{\prime\prime}\} - \text{Re}\{f^{\prime2} - 2ff^{\prime\prime} - \text{g}^2\} - \frac{M}{(1+m^2)}(f^{\prime} + mg) = 0 \tag{8}$$

$$\mathbf{g''} + \mathrm{Re}\mathcal{W}\varepsilon (2f'\mathbf{g''} - 2f\mathbf{g'''} - 3f''\mathbf{g'}) - \mathrm{Re}(2f'\mathbf{g} - 2f\mathbf{g'}) + \frac{M}{(1+m^2)}(mf'-\mathbf{g}) = 0\tag{9}$$

$$\begin{aligned} \boldsymbol{\Theta''} &+ 2\text{Re}\text{Pr}f\boldsymbol{\theta'} + \text{Re}\text{Pr}\boldsymbol{\eta}\boldsymbol{\theta} + \text{N}b\text{Pr}\boldsymbol{\theta'}\boldsymbol{\phi'} + \text{N}\text{Pr}\boldsymbol{\Theta'}^2 - 4\lambda \text{Pr}\text{Re}\left(f^2\boldsymbol{\theta''} + ff'\boldsymbol{\theta'}\right) \\ &+ \text{Rd}\left[3\left(\boldsymbol{\Theta}\_{\text{w}} - 1\right)\left(\boldsymbol{\theta'}^2 + \boldsymbol{\theta}^2\boldsymbol{\theta'}^2 \left(\boldsymbol{\Theta}\_{\text{w}} - 1\right)^2 + 2\boldsymbol{\theta}\boldsymbol{\theta'}^2 \left(\boldsymbol{\Theta}\_{\text{w}} - 1\right)\right) + \boldsymbol{\theta''} + \boldsymbol{\theta^3}\boldsymbol{\theta''}\left(\boldsymbol{\Theta}\_{\text{w}} - 1\right)^3 \\ &+ 3\left(\boldsymbol{\theta}\_{\text{w}} - 1\right)\boldsymbol{\theta}\boldsymbol{\theta''} + 3\left(\boldsymbol{\theta}\_{\text{w}} - 1\right)^2 \boldsymbol{\theta}^2 \boldsymbol{\theta''} \left[ + \frac{\text{MLP}\varepsilon}{\left(1 + \text{m}^2\right)} \left(f'^2 + \boldsymbol{\mathsf{g}}^2\right) = 0 \end{aligned} \tag{10}$$

$$\text{NCs}^{\prime} + \text{O}(\text{ew} - 1) \text{ or } \text{O}^{\prime} + \frac{1}{(1 + m^2)} \text{( $\uparrow$ } + \text{ $\chi$ )} - \text{O}$$

$$\text{\(\uparrow\text{''} + 2\text{Re}\text{Scf}\text{\'}\text{\'} + \frac{\text{Nt}}{\text{N}\text{\'}\text{\'}}\text{\textbf{6''}} = \text{0}\tag{11}$$

with

$$\begin{aligned} f(0) &= 0, \ f'(0) = A, \ f'(\infty) = 0, \ g(0) = 1, \\ g(\infty) &= 0, \ \theta(0) = 1, \ \theta(\infty) = 0, \ \phi(0) = 1, \ \phi(\infty) = 0 \end{aligned} \tag{12}$$

where *We* = β1/*h*<sup>2</sup> is the Weissenberg number, Re = α1*h*2/ν*<sup>f</sup>* is the Reynolds number, *A* = *a*/α<sup>1</sup> is the stretching parameter, Pr = (*cp*)ν*f*/*kf* is the Prandtl number, *q* = *Q*/*cp*α<sup>1</sup> is the heat absorption/generation parameter, *Nb* = τ*DB*(*Cw*−*C*∞)/ν Brownian motion parameter, *Nt* = τ*DB*(*Tw*−*T*∞)/*T*∞ν thermophoresis parameter, *M* = σ*B*<sup>0</sup> <sup>2</sup>/α1 magnetic parameter, *Sc* <sup>=</sup> <sup>ν</sup>/*DB* Schmidt number, <sup>θ</sup>*<sup>w</sup>* <sup>=</sup> *Tw*/*T*<sup>∞</sup> temperature difference, *Ec* = (*r*α1) <sup>2</sup>/*cp*(*Tw*−*T*∞) is the Eckert number, *Rd* <sup>=</sup> <sup>16</sup>σ\**T*∞3/3*kfk*\* radiation parameter and λ = *r*α<sup>1</sup> is the thermal relaxation parameter.

#### *2.1. Skin Friction and Nusselt Number*

Skin frictions along radial and tangential directions are

$$\mathcal{C}\_{fr} = \frac{\mathsf{\tau}\_{z\theta}}{\varrho (ra\_1)^2}, \ \mathcal{C}\_{f\Theta} = \frac{\mathsf{\tau}\_{zr}}{\varrho (ra\_1)^2} \tag{13}$$

*Coatings* **2020**, *10*, 610

in which τ*zr* and τ*z*<sup>θ</sup> are called shear stresses in radial and tangential directions respectively, and are defined as

$$\begin{split} \mathsf{T}\_{zr} &= \mu \Big( \frac{\partial \underline{w}}{\partial r} + \frac{\partial \underline{u}}{\partial z} \Big) + \beta\_1 \Big[ 2 \frac{\partial \underline{w}}{\partial z} \Big( \frac{\partial \underline{v}}{\partial r} - \frac{\underline{v}}{r} \Big) + \left( \frac{\partial \underline{w}}{\partial r} + \frac{\partial \underline{u}}{\partial z} \Big) \Big( \frac{\partial \underline{u}}{\partial r} + \frac{\partial \underline{w}}{\partial z} \Big) + \frac{\partial \underline{w}}{\partial r} \frac{\partial \underline{u}}{\partial z} + \\ & \frac{\partial \underline{u}}{\partial r} \frac{\partial \underline{w}}{\partial r} + 3 \Big( \frac{\partial \underline{w}}{\partial r} \frac{\partial \underline{w}}{\partial z} + \frac{\partial \underline{u}}{\partial z} \frac{\partial \underline{u}}{\partial r} \Big) \Big] - \beta\_1 \Big[ \frac{\partial \underline{w}}{\partial z} \Big( \frac{\partial \underline{w}}{\partial r} - \frac{\underline{v}}{r} \Big) + \left( \frac{\partial \underline{u}}{\partial r} + \frac{\partial \underline{w}}{\partial z} \Big) \Big( \frac{\partial \underline{w}}{\partial r} + \frac{\partial \underline{u}}{\partial z} \Big) \Big] \end{split} \tag{14}$$

$$\begin{split} \pi\_{z0} &= \mu \frac{\partial \upsilon}{\partial z} + \beta\_1 \Big[ w \frac{\partial^2 \upsilon}{\partial z^2} - \frac{\upsilon}{r} \frac{\partial \mu}{\partial z} + \frac{\partial \upsilon}{\partial z} \frac{\partial \upsilon}{\partial z} + u \frac{\partial^2 \upsilon}{\partial r \partial z} + \frac{\partial u}{\partial z} \frac{\partial \upsilon}{\partial r} + 3 \frac{u}{r} \frac{\partial \upsilon}{\partial z} \Big] \\ &- \beta\_1 \Big[ \frac{\partial u}{\partial z} \frac{\partial \upsilon}{\partial r} + 2 \frac{u}{r} \frac{\partial \upsilon}{\partial z} + 2 \frac{\partial v}{\partial z} \frac{\partial \upsilon}{\partial z} - \frac{\upsilon}{r} \frac{\partial \upsilon}{\partial r} - \frac{\upsilon}{r} \frac{\partial u}{\partial z} + \frac{\partial u}{\partial r} \frac{\partial \upsilon}{\partial r} \Big] \end{split} \tag{15}$$

Dimensionless forms are

$$\operatorname{Re}\_r \mathbb{C}\_{fr} = f''(\zeta) + \operatorname{Re} \mathcal{We}[\mathfrak{H}f'(\zeta)f''(\zeta) - 2f(\zeta)f''''(\zeta)]\Big|\_{\zeta=0} \tag{16}$$

$$\operatorname{Re}\_r \mathbb{C}\_{f\Phi} = g'(\zeta) + \operatorname{Re} \mathbb{W}e[4f'(\zeta)g'(\zeta) - 2f(\zeta)g''(\zeta)]\Big|\_{\zeta=0} \tag{17}$$

Heat transfer rate is defined as

$$N\mu\_{\rm X} = \frac{hq\_w}{k(T\_{\rm av} - T\_{\rm os})}\tag{18}$$

where *qw* is the wall heat flux, which is defined as

$$\left.q\_{\mu\nu}\right|\_{z=0} = \left.-\left(k + \frac{16\sigma^\*T^3}{3k^\*}\right)\frac{\partial T}{\partial z}\right|\_{z=0} \tag{19}$$

The dimensionless form is

$$Nu\_x = \left. - \left( 1 + R d\Theta\_w^3 \right) \Theta'(\zeta) \right|\_{\zeta=0} \tag{20}$$
