The system boundary restrictions are the following:

$$\begin{aligned} \mu = u\_{\text{in}}(\mathbf{x}) + L\_1 \frac{\partial u}{\partial y}, \upsilon = v\_{\text{in}} \frac{\partial T}{\partial y} = -\frac{h\_1}{k} (T\_f - T), \frac{\partial T}{\partial y} = -\frac{h\_2}{D\_m} (\mathbb{C}\_f - \mathbb{C}), \upsilon = 0 \quad \text{at } y = 0, \\ u \to 0, w \to 0, T \to T\_{\infty} \mathbb{C} \to \mathbb{C}\_{\infty} \quad \text{as } y \to \infty, \end{aligned} \tag{11}$$

where *B*<sup>0</sup> is the magnetic field magnitude, *T* (*ρ*) is the Carreau fluid temperature (density), *k* (*cs*) is the Carreau fluid thermal conductivity (susceptibility of concentration), *uw*(*x*) = *ax* (*v*) is the fluid velocity *x* (*y*) component, *KT* is the thermal diffusion ratio, *C*

is the concentration of the fluid, *L*<sup>1</sup> is the factor of the velocity slip, and *Dm* is the mass diffusivity. Furthermore, *vw* is the mass flow velocity, and *Cf* (*Tf*) is the convective fluid concentration (temperature).

The flux of the radiations *qr* is [75,78]:

$$\frac{\partial q\_r}{\partial y} = -\frac{16\sigma^s T\_0^3}{3k\_1} \frac{\partial^2 T}{\partial y^2} \, , \tag{12}$$

where *σ<sup>s</sup>* and *k*<sup>1</sup> are respectively the Stefan constant and average absorption coefficient. Applying Equation (12) to Equation (9), we will get the

$$
\mu \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} - \frac{D\_m K\_T}{c\_s c\_p} \frac{\partial^2 C}{\partial y^2} = \frac{\partial}{\partial y} \left[ \left( \mathfrak{a} + \frac{16 \sigma^s}{3k\_1} T\_\infty^3 \right) \frac{\partial T}{\partial y} \right]. \tag{13}
$$

Using the similarity variables as below [75]:

$$\begin{aligned} \psi &= \sqrt{a v} f(\eta) \mathbf{x}, \eta = \sqrt{\frac{a}{v}} y, \boldsymbol{T} - \boldsymbol{T}\_{\infty} = (\boldsymbol{T}\_f - \boldsymbol{T}\_{\infty}) \theta(\eta), \boldsymbol{w} = a \mathbf{x} \mathbf{g}(\eta), \\ \boldsymbol{\mathcal{C}} - \boldsymbol{\mathcal{C}}\_{\infty} &= (\boldsymbol{\mathcal{C}}\_f - \boldsymbol{\mathcal{C}}\_{\infty}) \boldsymbol{\phi}(\eta), \boldsymbol{T} - \boldsymbol{T}\_{\infty} = \boldsymbol{T}\_{\infty} (\theta\_w - 1) \theta\_r \theta\_w = \frac{\boldsymbol{T}\_f}{\boldsymbol{T}\_{\infty}}. \end{aligned} \tag{14}$$

Here, *a* is constant. The symbols *f* , *θ*, and *φ* represent the non-dimensional fluid velocity, temperature, and concentration, respectively. The symbol *ψ* denotes the stream function satisfying *u* = *∂ψ <sup>∂</sup><sup>y</sup>* and *<sup>v</sup>* <sup>=</sup> <sup>−</sup>*∂ψ ∂x* .

Applying these transformations in Equations (6), (7), (10), (13), and (14), we obtain

$$\begin{split} f^{\prime\prime\prime} \left[ 1 + \left( \frac{n-1}{2} \right) \mathsf{W} \mathrm{e} f^{\prime\prime 2} \right] + 2f^{\prime\prime\prime} \left[ \left( \frac{n-1}{2} \right) \mathsf{W} \mathrm{e} f^{\prime\prime 2} \right] \left[ 1 + \left( \frac{n-3}{2} \right) \mathsf{W} \mathrm{e} f^{\prime\prime 2} \right] \\ + f f^{\prime\prime} - f^{\prime 2} - \frac{M}{1 + m^2} (f^{\prime} + m \mathrm{g}) - r f^{\prime} = 0, \end{split} \tag{15}$$

$$\begin{aligned} \text{g}^{\prime\prime} \left[ 1 + \left( \frac{n-1}{2} \right) \text{W} \text{cg}^{\prime 2} \right] + 2 \text{g}^{\prime\prime} \left[ \left( \frac{n-1}{2} \right) \text{W} \text{cg}^{\prime 2} \right] \left[ 1 + \left( \frac{n-3}{2} \right) \text{W} \text{cg}^{\prime 2} \right] \\ - \text{g}^{\prime} + \text{g}^{\prime} f + \frac{M}{1 + m^2} (m \text{f}^{\prime} - \text{g}) + r \text{g} = 0, \end{aligned} \tag{16}$$

$$\theta^{\prime\prime} \left( 1 + R d (1 + (\theta\_w - 1)\theta)^3 \right) + \left( 3(\theta\_w - 1)\theta^{\prime 2} (1 + \theta\_w - 1)\theta \right)^2 + Pr f \theta^{\prime} + Pr Du \phi^{\prime\prime} = 0,\tag{17}$$

$$
\phi'' + \mathcal{S}cf\phi' + \mathcal{S}cSr\theta'' = 0.\tag{18}
$$

The boundary restrictions are transformed as:

$$f = \varrho, f' = 1 + \chi\_1 f(0)'', \varrho = 0, \theta' = -\chi\_2 (1 - \theta(0)), \mathbf{g} = 0, \mathbf{0}' = -\chi\_3 (1 - \phi(0)) \text{ at } \eta = 0,$$

$$f' \to 0, \theta \to 0, \phi \to 0, \mathbf{g} \to 0 \quad \text{as} \quad \eta \to \infty.$$

Here, the symbol *We* represents the Weissenberg number, shows the mass transfer parameter which describes suction ( > 0) and injection ( < 0). The symbol *Rd* denotes the radiation parameter, whereas *Pr*, *Sc*, and *Du* are respectively the Prandtl, Soret, and Dufour numbers. The symbols *χ*2, *χ*<sup>3</sup> are the thermal and concentration profiles slip parameters, respectively. These parameters have the following definitions:

$$\begin{aligned} Sc &= \frac{\nu}{D\_m}, Pr = \frac{\nu}{a}, Rd = \frac{16\sigma \ast T\_{\infty}^3}{3kk\_{\varepsilon}}, Du = \frac{D\_m K\_T (\mathbb{C}\_f - \mathbb{C}\_{\infty})}{c\_s c\_p \nu (T\_f - T\_{\infty})}, r = -\frac{-\nu}{ak}, \\ \mathcal{W}c &= \frac{\lambda^2 l^2 a^3}{\nu}, \chi\_1 = L\sqrt{\frac{a}{\nu}}, \chi\_2 = \frac{h\_1}{k}\sqrt{\frac{a}{\nu}}, \chi\_3 = \frac{h\_2}{D\_m}\sqrt{\frac{a}{\nu}}, \varrho = -\frac{v\_w}{\sqrt{a\nu}}, M = \frac{\sigma B\_0^2}{\rho a} \end{aligned} \tag{20}$$

The basic physical quantities of engineering interest (Sherwood and Nusselt numbers, and the skin frictions (along *x* and *z* axis) are defined by [79]:

> *Sh*(*Rex*)−1/2 <sup>=</sup> <sup>−</sup>*φ*(0) , (21)

$$N\mu (Re\_x)^{-1/2} = -\left(1 + \frac{1}{Rd}\theta\_{(w)}(0)^3\right)\theta(0)',\tag{22}$$

$$\mathcal{C}\_{fx}(Re\_x)^{1/2} = \frac{f''(0)}{2} \left( 2 + \mathcal{Wc}(n-1) (f''(0))^2 \right),\tag{23}$$

$$\mathcal{C}\_{fz}(Re\_x)^{1/2} = \frac{g'(0)}{2} \left( 2 + (n-1)We(g'(0))^2 \right),\tag{24}$$

where *Rex* = *xuw <sup>ν</sup>* represents the Reynolds number.
