3. Solution by homotopy analysis method

Initially guessed values for f , θ, and ϕ at η ¼ 1 are

$$\begin{split} f\_{0}(\eta) &= \frac{\beta}{2\left(\beta - 1\right)^{3}} \left[ \eta^{3} - 3\beta\eta^{2} - \left( 3 - 6\beta \right) \eta + \left( 2 - 3\beta \right) \right] + \eta, \\ \theta\_{0}(\eta) &= \frac{-\eta^{2}}{2} + \beta(\eta - 1) + \frac{3}{2}, \phi\_{0}(\eta) = \frac{-\eta^{2}}{2} + \beta(\eta - 1) + \frac{3}{2}. \end{split} \tag{23}$$

The linear operators for the given functions Lf , L<sup>θ</sup> and L<sup>ϕ</sup> are selected as

$$L\_f = \frac{\partial^4 f}{\partial \eta^4}, \quad L\_\theta = \frac{\partial^2 \theta}{\partial \eta^2} \quad \text{and} \quad L\_\phi = \frac{\partial^2 \phi}{\partial \eta^2}, \tag{24}$$

which satisfies the following general solution:

$$L\_f \left( A\_1 + A\_2 \eta + A\_3 \eta^2 + A\_4 \eta^3 \right) = 0, \quad L\_\theta (A\_5 + A\_6 \eta) = 0 \quad \text{and} \quad L\_\phi (A\_7 + A\_8 \eta) = 0,\tag{25}$$

where Aiði ¼ 1 � 8Þ are constants of general solution.

The corresponding nonlinear operators Nf , Nθ, and N<sup>ϕ</sup> are defined as

$$\begin{split} N\_f[f(\xi;p), \theta(\xi;p)] &= \left(1 + \frac{1}{\beta\_1}\right) \left(\eta \frac{\partial^3 f(\eta;p)}{\partial \eta^3} + \frac{\partial^2 f(\eta;p)}{\partial \eta^2}\right) \\ &+ \text{Re } \ f(\eta;p) \frac{\partial^2 f(\eta;p)}{\partial \eta^2} - \left(\frac{\partial f(\eta;p)}{\partial \eta}\right)\bigg) \bigg( + \left(\lambda \theta(\eta;p) + Nr\phi(\eta;p)\right) = 0, \end{split} \tag{26}$$

$$\begin{split} N\_{\theta}[f(\xi;p),\theta(\xi;p)] &= \eta \frac{\partial^{2}\theta(\eta;p)}{\partial\eta^{2}} + \frac{\partial\theta(\eta;p)}{\partial\eta} + \text{Pr.Re}\left(f(\eta;p)\frac{\partial\theta(\eta;p)}{\partial\eta} - 2\frac{\partial f(\eta;p)}{\partial\eta}\theta(\eta;p)\right) \\ &+ \eta \frac{\partial\theta(\eta;p)}{\partial\eta} \left(N\_{t}\frac{\partial\theta(\eta;p)}{\partial\eta} + N\_{b}\frac{\partial\phi(\eta;p)}{\partial\eta}\right) = 0, \end{split} \tag{27}$$

$$\begin{split} N\_{\phi} \left[ f(\eta; p), \phi(\eta; p) \right] &= \eta \frac{\partial^{2} \phi(\eta; p)}{\partial \eta^{2}} + \frac{\partial \phi(\eta; p)}{\partial \eta} + Le. \text{Re} \left( f(\eta; p) \frac{\partial \phi(\eta; p)}{\partial \eta} - 2 \frac{\partial f(\eta; p)}{\partial \eta} \phi(\eta; p) \right) \\ &+ \frac{N\_{t}}{N\_{b}} \left( \eta \frac{\partial^{2} \theta(\eta; p)}{\partial \eta^{2}} + \frac{\partial \theta(\eta; p)}{\partial \eta} \right) = 0, \end{split} \tag{28}$$

where p∈ ½0; 1] is embedded parameter.
