*3.1. Analytic Solution*

In this section, we intend to find the analytical solutions by means of the homotopy analysis method [57]. We chose the initial guesses *u*0, θ0 and linear operators £1, £2, [58] as:

$$
\mu\_0(y) = \frac{1}{2}(1+y)\mathcal{U},\ \theta\_0(y) = \frac{1}{4}y(y-2). \tag{40}
$$

$$
\mathfrak{E}\_1 = \frac{d}{dy} \Big(\frac{du}{dy}\Big), \mathfrak{E}\_2 = \frac{d}{dy} \Big(\frac{d\theta}{dy}\Big). \tag{41}
$$

The deformation equations of homotopy for the zeroth-order are established as:

$$\begin{aligned} \left[ (1-\xi)\mathbb{E}\_1[\theta(y,\xi)-\theta\_0(y)] = \xi\mathbb{H}\_\mathsf{u}N\_1[\mathfrak{u}(y,\xi),\,\theta(y,\xi)]\_\mathsf{V} \right. \\ \left. (1-\xi)\mathbb{E}\_2[\theta(y,\xi)-\theta\_0(y)] = \xi\mathbb{H}\_\mathsf{N}N\_2[\mathfrak{u}(y,\xi),\,\theta(y,\xi)]\_\mathsf{V} \right. \end{aligned} \tag{42}$$

$$\begin{array}{lll}\text{For} & \xi = 0 & \xi = 1\\u(y,\xi): & u\_0(y) & u(y)\\\theta(y,\xi): & \theta\_0(y) & \theta(y)\end{array} \tag{43}$$

The nonlinear operators *N*1, *N*2 are can be written as:

$$\begin{aligned} N\_1[u(y,\xi),\theta(y,\xi)] &= \begin{pmatrix} 123\phi^2 + 7.3\phi + 1 \\ A\_3 \text{Gr}\theta(y,\xi) + \beta\_u \rho\_c - \text{Re}P\_\prime \end{pmatrix} \\ N\_2[u(y,\xi),\theta(y,\xi)] &= \begin{pmatrix} 4.97\phi^2 + 2.72\phi + 1 \end{pmatrix} \frac{\partial^2 \theta(y,\xi)}{\partial y^2} + A\_4 \text{Br}\mathcal{M}^2(u(y,\xi))^2 + \\ \text{Br}[123\phi^2 + 7.3\phi + 1] \left(\frac{\partial u(y,\xi)}{\partial y}\right)^{n+1} - B\_1 \gamma u(y,\xi) \end{pmatrix} \text{.} \tag{44}$$

The solution for velocity and temperature up to *mth*-order approximation can be expressed as:

$$
\mu(y) = \mu\_0(y) + \sum\_{l=1}^m \mu\_l(y), \theta(y) = \theta\_0(y) + \sum\_{l=1}^m \theta\_l(y). \tag{45}
$$

The best solutions for solving coupled differential equation can be presented as below at 20th order approximations:

$$u(y) = \mathbb{C}\_1 + \mathbb{C}\_2y + \mathbb{C}\_3y^2 + \mathbb{C}\_4y^3 + \mathbb{C}\_5y^4 + \mathbb{C}\_6y^5 + \mathbb{C}ry^6 + \mathbb{C}sy^7 \tag{46}$$

$$\theta(y) = D\_1 + D\_2 y + D\_3 y^2 + D\_4 y^3 + D\_5 y^4 + D\_6 y^5 \tag{47}$$

The constants *C*1 to *C*8 and *D*1 to *D*6 are specified in Appendix A.
