**3. Solution by HAM**

For the solution of three dimensional nanofluid thin layer flows through a steady rotating disk, the optimal approach is used. The obtained Equations (9)–(15) are solved by using HAM. The basic derivation and mechanism is explained below.

The operators *L f* , *L* <sup>θ</sup> and *<sup>L</sup>* <sup>φ</sup> are defined as [13,51]:

$$\begin{array}{l} L\_{\widehat{f}}(\widehat{f}) = \widehat{f}^{\prime\prime\prime}, L\_{\widehat{k}}(\widehat{k}) = \widehat{\mathbf{k}}^{\prime\prime}, L\_{\widehat{\mathbf{g}}}(\widehat{\mathbf{g}}) = \widehat{\mathbf{g}}^{\prime\prime}, \\ L\_{\widehat{s}}(\widehat{\mathbf{s}}) = \widehat{\mathbf{s}}^{\prime\prime}, L\_{\widehat{\mathbf{g}}}(\widehat{\boldsymbol{\theta}}) = \widehat{\boldsymbol{\theta}}^{\prime\prime}, L\_{\widehat{\boldsymbol{\phi}}}(\widehat{\boldsymbol{\phi}}) = \widehat{\boldsymbol{\Phi}}^{\prime\prime} \end{array} \tag{22}$$

where,

$$\begin{aligned} L\_{\widehat{f}}\left(\mathfrak{e}\_{1} + \mathfrak{e}\_{2}\mathfrak{n} + \mathfrak{e}\_{3}\mathfrak{n}^{2}\right) &= 0, L\_{\widehat{k}}\left(\mathfrak{e}\_{5} + \mathfrak{e}\_{6}\mathfrak{n}\right) = 0, L\_{\widehat{\mathfrak{k}}}\left(\mathfrak{e}\_{7} + \mathfrak{e}\_{8}\mathfrak{n}\right) = 0, \\ L\_{\widehat{\mathfrak{s}}}\left(\mathfrak{e}\_{9} + \mathfrak{e}\_{10}\mathfrak{n}\right) &= 0, L\_{\widehat{\mathfrak{g}}}\left(\mathfrak{e}\_{11} + \mathfrak{e}\_{12}\mathfrak{n}\right) = 0, L\_{\widehat{\mathfrak{g}}}\left(\mathfrak{e}\_{13} + \mathfrak{e}\_{14}\mathfrak{n}\right) = 0 \end{aligned} \tag{23}$$

The consistent non-linear operators are reasonably selected as N *f* , N *k* , N *<sup>g</sup>* , N *<sup>s</sup>* , N <sup>θ</sup> and N φ, and are recognized in the system [13,22]:

$$\mathrm{N}\_{\widehat{f}}\left[\widehat{f}\left(\mathfrak{n};\mathfrak{zeta}\right),\widehat{\mathfrak{g}}\left(\mathfrak{n};\mathfrak{zeta}\right)\right] = \widehat{f}\_{\eta\eta\eta\eta} - \widehat{f}\_{\eta}^{2} + \widehat{\mathfrak{g}}^{2} + 2\widehat{f}\widehat{f}\_{\eta\eta} - K\widehat{f}\_{\eta\eta\eta\eta\eta\eta\eta} - M\widehat{f}\_{\eta} \tag{24}$$

$$\mathrm{N}\_{\widehat{\mathscr{S}}}\left[\widehat{\mathscr{g}}\left(\mathfrak{n};\mathfrak{\zeta}\right),\widehat{f}\left(\mathfrak{n};\mathfrak{\zeta}\right)\right] = \widehat{\mathscr{g}}\_{\mathfrak{\eta}\mathfrak{\eta}} - 2\widehat{\mathscr{g}}\widehat{f}\_{\mathfrak{\eta}} + 2\widehat{\mathscr{g}}\_{\mathfrak{\eta}}\widehat{f} - \mathrm{K}\widehat{\mathscr{g}}\_{\mathfrak{\eta}\mathfrak{\eta}\mathfrak{\eta}\mathfrak{\eta}\mathfrak{\eta}\mathfrak{\eta}\mathfrak{\eta}} - \mathrm{M}\widehat{\mathscr{g}} \tag{25}$$

$$\mathbf{N}\_{\widehat{\mathcal{S}}} \left[ \widehat{\mathcal{g}} (\mathfrak{n}; \boldsymbol{\zeta}), \widehat{f} (\mathfrak{n}; \boldsymbol{\zeta}) \right] = \widehat{\mathcal{g}}\_{\mathfrak{n} \| \boldsymbol{\eta}} - 2 \widehat{\mathcal{g}} \widehat{f}\_{\mathfrak{n}} + 2 \widehat{\mathcal{g}}\_{\mathfrak{n}} \widehat{f} - \mathsf{K} \widehat{\mathcal{g}}\_{\mathfrak{n} \| \boldsymbol{\eta} \boldsymbol{\eta} \boldsymbol{\eta} \boldsymbol{\eta} \boldsymbol{\eta} \boldsymbol{\eta} \boldsymbol{\eta}} - M \widehat{\mathcal{g}} \tag{26}$$

$$\mathrm{N}\_{\widehat{\mathbb{S}}}\left[\widehat{\boldsymbol{s}}\left(\boldsymbol{\eta};\boldsymbol{\zeta}\right),\widehat{\boldsymbol{g}}\left(\boldsymbol{\eta};\boldsymbol{\zeta}\right),\widehat{\boldsymbol{f}}\left(\boldsymbol{\eta};\boldsymbol{\zeta}\right),\widehat{\boldsymbol{k}}\left(\boldsymbol{\eta};\boldsymbol{\zeta}\right)\right] = \widehat{\boldsymbol{s}}\_{\boldsymbol{\eta}\boldsymbol{\eta}} - \widehat{\boldsymbol{k}}\,\widehat{\boldsymbol{g}} - \widehat{\boldsymbol{s}}\,\widehat{\boldsymbol{f}}\_{\boldsymbol{\eta}} + 2\widehat{\boldsymbol{s}}\_{\boldsymbol{\eta}\boldsymbol{\zeta}}\widehat{\boldsymbol{f}} - \boldsymbol{K}\widehat{\boldsymbol{s}}\_{\boldsymbol{\eta}\boldsymbol{\eta}\boldsymbol{\eta}\boldsymbol{\eta}\boldsymbol{\eta}\boldsymbol{\eta}} - \boldsymbol{M}\widehat{\boldsymbol{s}}\tag{27}$$

$$\mathrm{N}\_{\widehat{\boldsymbol{\Theta}}} \Big[ \widehat{\boldsymbol{\Theta}} \big( \boldsymbol{\eta}; \boldsymbol{\zeta} \big), \widehat{\boldsymbol{f}} \big( \boldsymbol{\eta}; \boldsymbol{\zeta} \big) \Big] = \widehat{\boldsymbol{\Theta}}\_{\boldsymbol{\eta} \boldsymbol{\eta}} + 2 \mathrm{Pr} \widehat{\boldsymbol{\Theta}} \widehat{\boldsymbol{f}} + \mathrm{Pr} \mathrm{N} \boldsymbol{\eta} \big\| \widehat{\boldsymbol{\Theta}}\_{\boldsymbol{\eta}} \widehat{\boldsymbol{\Theta}}\_{\boldsymbol{\eta}} + \mathrm{Pr} \boldsymbol{\Omega} \mathrm{N} \widehat{\boldsymbol{t}} \big\|\_{\boldsymbol{\eta}}^{2} \tag{28}$$

$$\mathrm{N}\_{\widehat{\Phi}} \Big[ \widehat{\Phi}(\eta;\zeta), \widehat{f}(\eta;\zeta), \widehat{\Phi}(\eta;\zeta) \Big] = \widehat{\Phi}\_{\eta\eta} + 2\mathrm{S}\boldsymbol{\varepsilon}\widehat{f}\widehat{\Phi}\_{\eta} + \frac{\mathrm{Nt}}{\mathrm{N}b} \widehat{\Theta}\_{\eta\eta} + \frac{\mathrm{S}}{2} \left( \eta \widehat{\Phi}\_{\eta} + \eta^{2} \widehat{\Phi}\_{\eta\eta} \right) \tag{29}$$

For Equations (8)–(10) the 0th-order system is written as [21]:

$$(1 - \zeta)L\_{\widehat{f}}\left[\widehat{f}\left(\mathfrak{n};\zeta\right) - \widehat{f}\_{0}\left(\mathfrak{n}\right)\right] = p\hbar\_{\widehat{f}}\mathcal{N}\_{\widehat{f}}\left[\widehat{f}\left(\mathfrak{n};\zeta\right), \widehat{g}\left(\mathfrak{n};\zeta\right)\right] \tag{30}$$

$$\mathbb{E}\left[\widehat{\mathbf{1}}(1-\mathsf{L})L\_{\widehat{k}}\Big|\widehat{\widehat{k}}\left(\mathfrak{n};\mathsf{L}\right)-\widehat{k}\_{0}\big(\mathfrak{n}\Big)\right] = p\hbar\mathop{\mathsf{L}\_{\widehat{k}}}\mathrm{N}\_{\widehat{k}}\left[\widehat{\boldsymbol{k}}\left(\mathfrak{n};\mathsf{L}\right),\widehat{\boldsymbol{\mathcal{G}}}\left(\mathfrak{n};\mathsf{L}\right),\widehat{\boldsymbol{f}}\left(\mathfrak{n};\mathsf{L}\right),\widehat{\boldsymbol{s}}\left(\mathfrak{n};\mathsf{L}\right)\right] \tag{31}$$

$$\mathbb{P}\left(1-\zeta\right)L\_{\widehat{\mathbb{S}}}\left[\widehat{\mathbb{S}}\left(\mathfrak{n};\zeta\right)-\widehat{\mathbb{S}}\_{0}\left(\mathfrak{n}\right)\right]=p\hbar\_{\widehat{\mathbb{S}}}\mathrm{N}\_{\widehat{\mathbb{S}}}\left[\widehat{f}\left(\mathfrak{n};\zeta\right),\widehat{\mathbb{S}}\left(\mathfrak{n};\zeta\right)\right] \tag{32}$$

*Coatings* **2020**, *10*, 338

$$\mathbb{E}\left[ (1-\zeta)L\_{\widehat{s}}\Big|\widehat{\hat{s}}\left(\eta;\zeta\right)-\widehat{s}\_{0}(\eta)\right] = p\hbar\_{\widehat{s}}\mathrm{N}\_{\widehat{s}}\left[ \widehat{\hat{s}}\left(\eta;\zeta\right),\widehat{\hat{g}}\left(\eta;\zeta\right),\widehat{f}\left(\eta;\zeta\right),\widehat{k}\left(\eta;\zeta\right) \right] \tag{33}$$

$$\mathbb{P}\left(1-\zeta\right)L\,\widehat{\,}\_{\widehat{\theta}}\left[\widehat{\theta}\left(\eta;\zeta\right)-\widehat{\theta}\_{0}\left(\eta\right)\right]=p\hbar\,\widehat{\,}\_{\widehat{\theta}}\mathrm{N}\_{\widehat{\theta}}\left[\widehat{\theta}\left(\eta;\zeta\right),\widehat{f}\left(\eta;\zeta\right),\widehat{\Phi}\left(\eta;\zeta\right)\right]\tag{34}$$

$$\left[ (1 - \zeta) \, L\_{\widehat{\phi}} \widehat{\left[ \phi(\eta; \zeta) - \widehat{\phi}\_0(\eta) \right]} \right] = p \hbar \widehat{\,}\_{\widehat{\phi}} \mathcal{N}\_{\widehat{\phi}} \left[ \widehat{\phi}(\eta; \zeta), \widehat{f}(\eta; \zeta), \widehat{\theta} \left( \eta; \zeta \right) \right] \tag{35}$$

where the boundary conditions are [22]:

 *f* (η; ζ) η=0 = 0, <sup>∂</sup> *f* (η;ζ) ∂η η=0 <sup>=</sup> 0, <sup>∂</sup><sup>2</sup> *f* (η;ζ) ∂η2 η=δ <sup>=</sup> 0, *g*(η; ζ) <sup>η</sup>=<sup>0</sup> <sup>=</sup> <sup>0</sup> ∂ *g* (η;ζ) ∂η η=δ <sup>=</sup> 0, *k* (η; ζ) η=0 = 0, <sup>∂</sup> *k* (η;ζ) ∂η η=δ <sup>=</sup> 0, *s* (η; ζ) <sup>η</sup>=<sup>0</sup> <sup>=</sup> <sup>0</sup> ∂ *s* (η;ζ) ∂η η=δ <sup>=</sup> 0, θ(η; ζ) η=0 = 0, <sup>∂</sup> θ(η;ζ) ∂η η=δ <sup>=</sup> 1, φ(η; ζ) η=0 <sup>=</sup> 0, φ(η; ζ) η=δ = 1 (36)

while the embedding constraint is <sup>ζ</sup> <sup>∈</sup> [0, 1], to regulate for the solution convergence - *f* , - *k* , - *g* , - *s* , - <sup>θ</sup> and - φ are used. When ζ = 0 and ζ = 1 we have:

$$\begin{array}{l} \widehat{f} \left( \mathfrak{n}; 1 \right) = \widehat{f} \left( \mathfrak{n} \right), \widehat{\mathfrak{k}} \left( \mathfrak{n}; 1 \right) \right) = \widehat{\mathfrak{k}} \left( \mathfrak{n} \right), \widehat{\mathfrak{g}} \left( \mathfrak{n}; 1 \right) = \widehat{\mathfrak{g}} \left( \mathfrak{n} \right) \\\widehat{\mathfrak{s}} \left( \mathfrak{n}; 1 \right) = \widehat{\mathfrak{s}} \left( \mathfrak{n} \right), \widehat{\mathfrak{g}} \left( \mathfrak{n}; 1 \right) = \widehat{\mathfrak{g}} \left( \mathfrak{n} \right), \widehat{\mathfrak{g}} \left( \mathfrak{n}; 1 \right) = \widehat{\mathfrak{g}} \left( \mathfrak{n} \right) \end{array} \tag{37}$$

expand the *f* (η; ζ), *k* (η; ζ), *g* (η; ζ), *s* (η; ζ), <sup>φ</sup> (η; <sup>ζ</sup>) and φ (η; ζ) through Taylor's series for ζ = 0:

 *<sup>f</sup>* (η; <sup>ζ</sup>) <sup>=</sup> *<sup>f</sup>* <sup>0</sup> (η) + <sup>∞</sup> *n*=1 *<sup>f</sup> <sup>n</sup>* (η)ζ*n*, *<sup>k</sup>* (η; <sup>ζ</sup>) <sup>=</sup> *k* <sup>0</sup> (η) + <sup>∞</sup> *n*=1 *k <sup>n</sup>* (η)ζ*<sup>n</sup> <sup>g</sup>* (η; <sup>ζ</sup>) <sup>=</sup> *<sup>g</sup>*<sup>0</sup> (η) + <sup>∞</sup> *n*=1 *<sup>g</sup> <sup>n</sup>* (η)ζ*n*, *<sup>s</sup>* (η; <sup>ζ</sup>) <sup>=</sup> *s* <sup>0</sup> (η) + <sup>∞</sup> *n*=1 *s <sup>n</sup>* (η)ζ*<sup>n</sup>* <sup>θ</sup> (η; <sup>ζ</sup>) <sup>=</sup> θ<sup>0</sup> (η) + <sup>∞</sup> *n*=1 θ *<sup>n</sup>* (η)ζ*n*, <sup>φ</sup> (η; <sup>ζ</sup>) <sup>=</sup> φ<sup>0</sup> (η) + <sup>∞</sup> *n*=1 φ*<sup>n</sup>* (η)ζ*<sup>n</sup>* (38)

$$\begin{aligned} \widehat{\boldsymbol{f}}\_{n}(\mathfrak{n}) &= \frac{1}{n!} \frac{\widehat{\partial}\boldsymbol{f}(\mathfrak{n};\boldsymbol{\zeta})}{\partial\boldsymbol{\eta}}\bigg|\_{p=0}, \widehat{\boldsymbol{k}}\_{n}(\mathfrak{n}) = \frac{1}{n!} \frac{\widehat{\partial}\boldsymbol{k}(\mathfrak{n};\boldsymbol{\zeta})}{\partial\boldsymbol{\eta}}\bigg|\_{p=0}, \widehat{\boldsymbol{g}}\_{n}(\mathfrak{n}) = \frac{1}{n!} \frac{\widehat{\partial}\widehat{\boldsymbol{\zeta}}\_{n}(\mathfrak{n};\boldsymbol{\zeta})}{\partial\boldsymbol{\eta}}\bigg|\_{p=0} \\ \widehat{\boldsymbol{s}}\_{n}(\mathfrak{n}) &= \frac{1}{n!} \frac{\widehat{\partial}\boldsymbol{\tilde{s}}(\mathfrak{n};\boldsymbol{\zeta})}{\partial\boldsymbol{\eta}}\bigg|\_{p=0}, \widehat{\boldsymbol{\Theta}}\_{n}(\mathfrak{n}) = \frac{1}{n!} \frac{\widehat{\partial}\boldsymbol{\tilde{s}}(\mathfrak{n};\boldsymbol{\zeta})}{\partial\boldsymbol{\eta}}\bigg|\_{p=0}, \widehat{\boldsymbol{\Phi}}\_{n}(\mathfrak{n}) = \frac{1}{n!} \frac{\widehat{\partial}\boldsymbol{\tilde{\Phi}}(\mathfrak{n};\boldsymbol{\zeta})}{\partial\boldsymbol{\eta}}\bigg|\_{p=0} \end{aligned} \tag{39}$$

where the boundary restrictions are:

$$\begin{aligned} \widehat{f}\left(0\right) = 0, \widehat{f}'\left(0\right) = 0, \widehat{f}''\left(\delta\right) = 0, \widehat{g}\left(0\right) = 0, \widehat{g}'\left(\delta\right) = 0, \widehat{k}\left(0\right) = 0, \widehat{k}'\left(\delta\right) = 0, \\\widehat{s}\left(0\right) = 0, \widehat{s}'\left(\delta\right) = 0, \widehat{\theta}\left(0\right) = 0, \widehat{\theta}\left(\delta\right) = 1, \widehat{\phi}\left(0\right) = 0, \widehat{\phi}\left(\delta\right) = 1. \end{aligned} \tag{40}$$

now define

$$\mathfrak{R}\_{n}^{\hat{f}}(\mathfrak{q}) = \hat{\boldsymbol{f}}^{\prime\prime\prime}\_{\boldsymbol{n}-1} - \hat{\boldsymbol{f}}^{\prime\prime 2}\_{\boldsymbol{n}-1} + \hat{\boldsymbol{g}}^{2}\_{\boldsymbol{n}-1} + 2\sum\_{j=0}^{w-1} \hat{\boldsymbol{f}}\_{w-1-j} \hat{\boldsymbol{f}}\_{j}^{\prime\prime} - \sum\_{j=0}^{w-1} \hat{\boldsymbol{k}}\_{w-1-j} \hat{\boldsymbol{f}}\_{j}^{\prime\prime i} - \hat{\boldsymbol{M}} \hat{\boldsymbol{f}}\_{n-1}^{\prime} \tag{41}$$

$$\mathfrak{R}\_{n}^{\widehat{k}}(\eta) = \widehat{\mathbf{k}}\_{n-1}^{\prime\prime} + \sum\_{j=0}^{w-1} \widehat{g}\_{w-1-j} \widehat{s}\_{j} - \sum\_{j=0}^{w-1} \widehat{k}\_{w-1-j} \widehat{f}\_{j}^{\prime} + 2 \sum\_{j=0}^{w-1} \widehat{k}\_{w-1-j} \widehat{f}\_{j} - \widehat{k}\_{k} \widehat{k}\_{n-1} - M \widehat{k}\_{n-1} \tag{42}$$

$$\mathfrak{R}\_{n}^{\widehat{\mathscr{S}}}(\boldsymbol{\eta}) = \widehat{\mathscr{S}}\_{n-1}^{\prime\prime} - 2\sum\_{j=0}^{w-1} \widehat{\mathscr{g}}\_{w-1-j} \widehat{\boldsymbol{f}}\_{j}^{\prime} + 2\sum\_{j=0}^{w-1} \widehat{\mathscr{g}}\_{w-1-j}^{\prime} \widehat{\boldsymbol{f}}\_{j} - \boldsymbol{K}\_{\mathscr{S},n-1}^{\prime\overline{w}i} - \boldsymbol{M} \widehat{\boldsymbol{g}}\_{n-1},\tag{43}$$

*Coatings* **2020**, *10*, 338

$$\mathfrak{R}\_{n}^{\widehat{\,^{s}}}(\mathfrak{q}) = \widehat{\boldsymbol{s}}\_{n-1}^{\prime\prime} - \sum\_{j=0}^{w-1} \widehat{\boldsymbol{k}}\_{w-1-j} \widehat{\boldsymbol{\mathfrak{z}}\_{j}} - \sum\_{j=0}^{w-1} \widehat{\boldsymbol{s}}\_{w-1-j} \widehat{\boldsymbol{f}}\_{j}^{\prime} + 2 \sum\_{j=0}^{w-1} \widehat{\boldsymbol{s}}\_{w-1-j}^{\prime} \widehat{\boldsymbol{f}}\_{j} - \boldsymbol{K} \widehat{\boldsymbol{s}}^{\prime i} - \boldsymbol{M} \widehat{\boldsymbol{s}}\_{n-1} \tag{44}$$

$$\hat{\mathfrak{R}}\_{n}^{\hat{\mathfrak{G}}}(\boldsymbol{\eta}) = \left(\hat{\boldsymbol{\Theta}}\_{n-1}^{\prime\prime}\right) + 2\text{Pr}\sum\_{j=0}^{w-1} \hat{\boldsymbol{\Theta}}\_{w-1-j} \boldsymbol{f}\_{j} + \text{PrN}l\boldsymbol{\eta}\boldsymbol{\Pi}\sum\_{j=0}^{w-1} \hat{\boldsymbol{\Phi}}\_{w-1-j}^{\prime} \hat{\boldsymbol{\Theta}}\_{j}^{\prime} + \text{Pr}\boldsymbol{\Omega}Nt\boldsymbol{\hat{\Theta}}\_{n-1}^{\prime\prime2} \tag{45}$$

$$\hat{\mathfrak{R}}\_{n}^{\hat{\Phi}}(\eta) = \hat{\Phi}\_{n-1}^{\prime\prime} + 2\text{Sc} \sum\_{j=0}^{w-1} \hat{f}\_{w-1-j} \hat{\Phi}\_{j}^{\prime} + \frac{\text{Nt}}{\text{Nb}} \hat{\mathfrak{\theta}}\_{n-1}^{\prime\prime} + \frac{\text{S}}{2} \left( \boldsymbol{\eta} \hat{\boldsymbol{\phi}}\_{n-1}^{\prime} + \boldsymbol{\eta}^{2} \hat{\boldsymbol{\phi}}\_{n-1}^{\prime\prime} \right) \tag{46}$$

where,

$$\chi\_n = \begin{cases} 0, \text{ if } \zeta \le 1 \\ 1, \text{ if } \zeta > 1 \end{cases} \tag{47}$$
