**3. Numerical Procedure**

For the determination of the existing flow model, we used the RK shooting technique. The following substitution was required to begin the process:

$$w\_1 = f[\eta], \ w\_2 = f'[\eta], \ w\_3 = f''[\eta], \ w\_4 = f''[\eta], \ w\_5 = \theta[\eta], \ w\_6 = \theta'[\eta], \ w\_7 = \chi[\eta], \ w\_8 = \chi'[\eta] \tag{22}$$

First, in Equations (18)–(20), the model was changed in the following pattern:

$$f^{\prime\prime\prime}[\eta] = \frac{1}{G\_1} \left( (\operatorname{Ref}[\eta] - a\eta)f^{\prime\prime}[\eta] - (3a + \operatorname{Ref}'[\eta])f^{\prime}[\eta] + \mathcal{G}\_2 M f^{\prime\prime}[\eta] \right) \tag{23}$$

$$\theta''[\eta] = - \left( G\_5 G\_4 \left( Pr \{ a \eta - Ref[\eta] \} \theta'[\eta] \right) + (Pr \ast \sigma \ast \lambda) G\_4 \left( (1 + (\gamma \ast \kappa) \theta[\eta]) \right) (1 - E + (\gamma \ast E) \theta[\eta]) \chi[\eta] \right) \tag{24}$$

$$\chi''[\eta] = - \left( Sc(\eta a - Ref) \chi' + (Sc \ast \sigma) (1 + (n \ast \gamma) \theta[\eta]) (1 - E + (E \ast \gamma) \theta[\eta]) \chi[\eta] \right) \tag{25}$$

$$\chi''[\eta] = -\left(\mathrm{Sc}(\eta a - \mathrm{Re}f)\chi' + (\mathrm{Sc}\ast\sigma)(1 + (n\ast\gamma)\theta[\eta])(1 - E + (E\ast\gamma)\theta[\eta])\chi[\eta]\right) \tag{25}$$

The following system was obtained by using the substitution contained in Equation (22):

$$\begin{bmatrix} w\_1' \\ w\_2' \\ w\_3' \\ w\_4' \\ w\_5' \\ w\_6' \\ w\_7' \\ w\_8' \end{bmatrix} = \begin{bmatrix} w\_2 \\ w\_3 \\ w\_4 \\ \frac{1}{G\_1}((R\varepsilon w\_1 - a\eta)w\_4 - (3a + R\varepsilon w\_2)w\_3 + G\_2Mw\_3) \\ w\_6 \\ -(G\_3G\_4(Pr(u\eta - R\varepsilon w\_1)w\_6 + +(Pr\*\sigma\*\lambda)G\_4((1 + (\gamma\*\ast)w\_5)(1 - E + (\gamma\*E)w\_5)w\_7)) \\ w\tau \\ -(Sc(\eta\mathfrak{a} - R\varepsilon w\_1)w\_8 + (Sc\*\sigma)(1 + (\eta\*\ast)w\_5)(1 - E + (E\*\gamma)w\_5)w\_7) \end{bmatrix} \tag{26}$$

Consequently, the initial condition was as follows:

$$
\begin{bmatrix} w\_1' \\ w\_2' \\ w\_3' \\ w\_4' \\ w\_5' \\ w\_6' \\ w\_7' \\ w\_8' \end{bmatrix} = \begin{bmatrix} -1 \\ 0 \\ 1 \\ 0 \\ 1 \\ 0 \\ 1 \\ 0 \end{bmatrix} \tag{27}
$$

The above system was solved using mathematics and a suitable initial condition. Here, the accurate Runge–Kutta shooting technique was taken into consideration. The required dimensionless ODEs can easily be tackled with this method. We obtain the initial condition by using the shooting technique in such a way that boundary conditions are satisfied and achieve the desired level of accuracy.
