*3.1. Homotopy Analysis Method*

The solutions of Equations (13)–(16) with the related boundary conditions (17) and (18) are achieved using HAM. Consider that initial guesses on *f*(η), *g*(η), θ(η), and *φ*(η) satisfying the boundary conditions at η = 0 are as follows:

$$f\_0(\eta) = \frac{\eta^3}{2\mathfrak{P}^2} - \frac{3\mathfrak{\eta}^2}{2\mathfrak{\beta}} + \mathfrak{v}\_\prime \ g\_0(\mathfrak{\eta}) = 0,\\ \theta\_0(\mathfrak{\eta}) = 1, \,\phi\_0(\mathfrak{\eta}) = 1 \tag{20}$$

The linear operators for the given functions are the following:

$$L\_f(f) = f^{(iv)},\ L\_\emptyset(\mathfrak{g}) = \mathfrak{g}'',\ L\_\emptyset(\mathfrak{g}) = \mathfrak{g}'',\ L\_\emptyset(\mathfrak{g}) = \mathfrak{g}''.\tag{21}$$

satisfying the following properties:

$$L\_f\left(a\_1 + a\_2\eta + a\_3\eta^2 + a\_4\eta^3\right) = 0,\\ L\_g(a\_5 + a\_6\eta) = 0,\\ L\_\Phi(a\_7 + a\_8\eta) = 0,\\ L\_\Phi(a\_9 + a\_{10}\eta) = 0 \tag{22}$$

where *ai*(*i* = 1 − 10) are constants related to the general solution.

The corresponding nonlinear operators are as follows:

$$\begin{array}{ll} N\_f \left[ f \left( \mathfrak{n}; q \right), g \left( \mathfrak{n}; q \right) \right] = & f\_{\mathfrak{n} \| \mathfrak{n} \|} \left( \mathfrak{n}; q \right) + f \left( \mathfrak{n}; q \right) f\_{\mathfrak{n} \| \mathfrak{n}} \left( \mathfrak{n}; q \right) + \Delta g\_{\mathfrak{n}} \left( \mathfrak{n}; q \right) \\ & + \frac{1}{M \tau} \left( 1 - f\_{\mathfrak{n}} \left( \mathfrak{n}; q \right) \right) + N \tau \left( 1 - \left( f\_{\mathfrak{n}} \left( \mathfrak{n}; q \right) \right)^{2} \right), \end{array} \tag{23}$$

$$N\_{\mathfrak{g}}[f(\mathfrak{n};q),\mathfrak{g}(\mathfrak{n};q)] = Gr\_{\mathfrak{N}\mathfrak{\eta}\mathfrak{\eta}}(\mathfrak{n};q) - 2(2\mathfrak{g}(\mathfrak{n};q) + f\_{\mathfrak{\eta}\mathfrak{\eta}}(\mathfrak{n};q)) = 0,\tag{24}$$

$$N\_{\Theta}[f(\eta;q),\Theta(\eta;q)] = \left(1+\frac{4}{3}R\right)\Theta\_{\mathsf{T}\mathsf{T}}(\eta;q) - Pr(2\Theta(\eta;q)\,f\_{\mathsf{T}}(\eta;q) - f(\eta;q)\Theta\_{\mathsf{T}}(\eta;q)),\tag{25}$$

$$\begin{split} \mathcal{N}\phi\left[f(\mathfrak{n};q),\theta(\mathfrak{n};q),\phi(\mathfrak{n};q)\right] &= \phi\_{\mathfrak{N}\|\mathfrak{l}}(\mathfrak{n};q) + \mathcal{S}c(\mathcal{S}r - \mathfrak{r}\phi(\mathfrak{n};q))\,\theta\_{\mathfrak{l}\|\mathfrak{l}}(\mathfrak{n};q) + \\ &\mathcal{S}r(f - \mathfrak{r}\theta\_{\mathfrak{l}}(\mathfrak{n};q))\phi\_{\mathfrak{l}}(\mathfrak{n};q) - 2\mathcal{S}c\phi(\mathfrak{n};q)f\_{\mathfrak{l}}(\mathfrak{n};q) = 0. \end{split} \tag{26}$$

#### (a) Zeroth-Order Deformation Problem

The main idea of HAM is explained in Equations (19)–(22). We formulate the zeroth-order problem from Equations (13)–(16) as follows:

$$(1 - q)L\_f\{f(\mathfrak{n}; q) - f\_0(\mathfrak{n})\} = qh\_f N\_f\{f(\mathfrak{n}; q), g(\mathfrak{n}; q)\},\tag{27}$$

$$\{(1-q)L\_{\mathcal{S}}\{\mathbf{g}(\mathfrak{n};q)-\mathfrak{g}\_{0}(\mathfrak{n})\}=q\hbar\_{\mathcal{S}}\mathcal{N}\_{\mathbb{S}}\{f(\mathfrak{n};q),\mathfrak{g}(\mathfrak{n};q)\},\tag{28}$$

$$\{(1-q)L\mathfrak{o}\{\theta(\mathfrak{n};q)-\theta\mathfrak{o}(\mathfrak{n})\} = qh\mathfrak{o}N\mathfrak{o}\{f(\mathfrak{n};q),\theta(\mathfrak{n};q)\},\tag{29}$$

$$(1 - q)L\_{\Phi} \{ \phi(\mathfrak{n}; q) - \phi\_0(\mathfrak{n}) \} = qh\_{\Phi}N\_{\Phi} \{ f(\mathfrak{n}; q), \mathfrak{g}(\mathfrak{n}; q), \mathfrak{f}(\mathfrak{n}; q), \phi(\mathfrak{n}; q) \},\tag{30}$$

Expanding the functions *f* , *g*, θ and *φ* by Taylor's series when *q* = 0, we have the following:

$$\begin{aligned} f(\mathfrak{n};q) &= f\_0(\mathfrak{n}) + \sum\_{\substack{\mathfrak{w}=1\\w(\mathfrak{n})=q}}^{\infty} f\_{\mathfrak{w}}(\mathfrak{n}) \, q^{\mathfrak{w}},\\ g(\mathfrak{n};q) &= g\_0(\mathfrak{n}) + \sum\_{\substack{\mathfrak{w}=1\\w=1}}^{\infty} \mathfrak{g}\_{\mathfrak{w}}(\mathfrak{n}) \, q^{\mathfrak{w}},\\ \Phi(\mathfrak{n};q) &= \Phi\_0(\mathfrak{n}) + \sum\_{\mathfrak{w}=1}^{\infty} \Phi\_{\mathfrak{w}}(\mathfrak{n}) \, q^{\mathfrak{w}}.\end{aligned} \tag{31}$$

where

$$\begin{array}{ll}f\_{\mathfrak{w}}(\mathfrak{n}) = \frac{1}{w!}f\_{\mathfrak{l}}^{w}(\mathfrak{n};q)|\_{\mathfrak{q}=0,\ \mathfrak{z}}\,\,\,\mathfrak{z}\_{\mathfrak{w}}(\mathfrak{n}) = \frac{1}{w!}g\_{\mathfrak{l}}^{w}(\mathfrak{n};q)|\_{\mathfrak{q}=0,\ \mathfrak{z}}\\ \Theta\_{\mathfrak{w}}(\mathfrak{n}) = \frac{1}{w!}\Phi\_{\mathfrak{l}}^{w}(\mathfrak{n};q)|\_{\mathfrak{q}=0,\ \mathfrak{z}}\,\,\phi\_{\mathfrak{w}}(\mathfrak{n}) = \frac{1}{w!}\Phi\_{\mathfrak{l}}^{w}(\mathfrak{n};q)|\_{\mathfrak{q}=0,\ \mathfrak{z}}.\end{array} \tag{32}$$

The supporting constraints *hf* , *hg*, *h*θ, and *h<sup>φ</sup>* are taken such that series (33) converges at *q* = 1. Substituting *q* = 1 in (33) we get the following:

$$f(\mathfrak{n}) = f\_0(\mathfrak{n}) + \sum\_{w=1}^{\infty} f\_w(\mathfrak{n})\_{\prime} \tag{33}$$

$$\mathcal{g}(\mathfrak{n}) = \mathcal{g}\_0(\mathfrak{n}) + \sum\_{w=1}^{\infty} \mathcal{g}\_w(\mathfrak{n}),\tag{34}$$

$$\theta(\mathfrak{u}) = \theta\_0(\mathfrak{u}) + \sum\_{w=1}^{\infty} \theta\_w(\mathfrak{u}),\tag{35}$$

$$
\phi(\eta) = \phi\_0(\eta) + \sum\_{w=1}^{\infty} \phi\_w(\eta). \tag{36}
$$
