**2. Mathematical Formulation**

Consider the thin film micropolar fluid flow on a stretched plate, which is being stretched with a linear velocity *Uw* = *ax*. Here, *a* > 0 is a constant and shows the stretching rate and *x* displays the direction of the flow. The thickness δ of the thin film is chosen uniform and the medium is considered porous, as displayed in Figure 1. The stretching plate is kept at temperature *Tw* and concentration *Cw*. The temperature *Tw* <sup>=</sup> *<sup>T</sup>*<sup>0</sup> <sup>−</sup> *Tref* - *Uwx* 2*υ* and concentration *Cw* <sup>=</sup> *<sup>C</sup>*<sup>0</sup> <sup>−</sup> *Cref* - *Uwx* 2*υ* on the surface are assumed to vary with distance *x* from the plate. *T*<sup>0</sup> and *C*<sup>0</sup> are the temperature and concentration at the plate, while *Tref* and *Cref* are the constant reference temperature and concentration. Further, it is assumed that the liquid film is gripping and releasing radiation. The radiate heat flux is considered along the *x*-axis, while neglecting along the *y*-axis.

**Figure 1.** Physical geometry of the problem.

The basic flow equations of our proposed model are as follows:

$$u\_x + v\_y = 0\tag{1}$$

$$
tau\_x + \upsilon u\_y = \upsilon u\_{xx} + k\_c \sigma\_y + \frac{\upsilon \varphi}{K} (\mathcal{U} - \mu) + \mathbb{C}\_r \varphi (\mathcal{U}^2 - u^2) \tag{2}$$

$$G\_1 \sigma\_{yy} - 2\sigma - u\_{\mathcal{Y}} = 0 \tag{3}$$

$$
\rho c\_p \left( \mu T\_\mathbf{x} + \nu T\_\mathbf{y} \right) = k T\_{\mathbf{y}\mathbf{y}} - \left( q\_r \right)\_\mathbf{y} \tag{4}
$$

$$u\mathbb{C}\_{x} + v\mathbb{C}\_{y} = D\_{m}\mathbb{C}\_{yy} + \frac{D\_{m}k\_{T}}{T\_{m}}T\_{yy} - \left(V\_{T}\mathbb{C}\right)\_{y} \tag{5}$$

The modeled boundary conditions for the two-dimensional liquid film are as follows:

$$\mu = lL\_{\overline{w}} = a\mathbf{x}, \; \mathbf{v} = 0, \; \sigma = 0, \; T = T\_{\overline{w}}, \; \mathbf{C} = \mathbf{C}\_{\overline{w}}\mathbf{a}\mathbf{t}\mathbf{y} = \mathbf{0},\tag{6}$$

$$\mathfrak{a}\_y = T\_y = \sigma\_y = \mathbb{C}\_y = 0, \; \mathfrak{v} = \delta\_{\mathfrak{x}}, \mathsf{at}\mathfrak{y} = \delta. \tag{7}$$

The Rosseland approximation is defined as follows:

$$q\_r = -\frac{4\sigma^\*}{3k^\*} \partial\_y T^4 \tag{8}$$

where *qr* is radiative heat flux, σ∗ is Stefan–Boltzman constant, and *k*∗ represents the mean absorption coefficient. The flux is assumed to be small, such that *T*<sup>5</sup> <sup>1</sup> and higher terms are ignored, as in the existing literature. After expanding by Taylor's series, *T*<sup>4</sup> is reduced to the following form:

$$T^4 = 4T\_1^3T - 3T\_1^4 \tag{9}$$

*T*<sup>1</sup> is used as the temperature at the free surface. Using Equations (8) and (9), Equation (4) is reduced as follows:

$$
\Delta u T\_x + v T\_y = \frac{k}{\rho c\_p} T\_{yy} + \frac{16 \sigma^\* T\_1^3}{3 \rho c\_p k^\*} T\_{yy} \tag{10}
$$

Abo-Eldahab and Ghonaim [17], Rashidi et al. [18,19] and Heydari et al. [20] introduced the following transformations:

$$\Psi(\mathbf{x}, \mathbf{y}) = (2v\mathcal{U}\_{\mathbf{w}}\mathbf{x})^{\frac{1}{2}} f(\eta), \; \sigma = (\frac{\mathcal{U}\_{\mathbf{w}}}{2v\mathbf{x}})^{\frac{1}{2}} \mathcal{U}\_{\mathbf{w}} g(\eta), \; \eta = (\frac{\mathcal{U}\_{\mathbf{w}}}{2v\mathbf{x}})^{\frac{1}{2}} y, \; u\_{\mathbf{x}} = \psi\_{y} \text{ and } u\_{\mathbf{y}} = -\psi\_{x} \tag{11}$$

In the recent research of Khan [27] and Qasim et al. [29], the thin film flows are modeled using reference temperature and concentration for steady and unsteady problems, respectively.

$$T = T\_0 - T\_{ref}(\frac{\mathcal{U}\_w \mathbf{x}}{2\upsilon})\theta(\eta),\\\mathbb{C} = \mathbb{C}\_0 - \mathbb{C}\_{r\varepsilon f}(\frac{\mathcal{U}\_w \mathbf{x}}{2\upsilon})\theta(\eta) \tag{12}$$

where *T*<sup>0</sup> is temperature at the stretched surface and *Tref* is used as a constant reference temperature, such that 0 ≤ *Tref* ≤ *T*0. Similarly, *C*<sup>0</sup> is the concentration at the stretched surface and *Cref* is used as a constant reference concentration, such that 0 ≤ *Cref* ≤ *C*0. Substituting Equations (11) and (12) into Equations (1)–(7), the basic governing equations of velocity, velocity rotation, and temperature with boundary conditions yield the following forms:

$$f'''' + ff'' + \Delta \mathbf{g'} + \frac{1}{Mr} \left(1 - f'\right) + Nr \left(1 - \left(f'\right)^2\right) = 0\tag{13}$$

$$\mathcal{G}\,\mathcal{g}'' - \mathcal{2}(\mathcal{2}\mathcal{g} + f'') = 0 \tag{14}$$

$$\left(1 + \frac{4}{3}R\right)\theta'' - Pr\left(2\Theta f' - f\,\Theta'\right) = 0\tag{15}$$

$$
\Phi'' + \mathcal{S}c(Sr - \tau\phi)\Theta'' + \mathcal{S}r(f - \tau\theta')\phi' - 2\mathcal{S}c\phi \, f' = 0\tag{16}
$$

$$f(0) = \emptyset(0) = 0,\\ f'(0) = \emptyset(0) = \emptyset(0) = 1 \tag{17}$$

$$f''(\mathfrak{k}) = f(\mathfrak{k}) = \mathfrak{g}'(\mathfrak{k}) = \mathfrak{g}'(\mathfrak{k}) = \mathfrak{g}'(\mathfrak{k}) = 0 \tag{18}$$

where *f* is a dimensionless velocity function and *g* is a dimensionless microrotation angular velocity function, θ is the temperature function, *φ* is the concentration function, β is the non-dimensional thickness of the liquid film, Δ = *<sup>k</sup>*<sup>1</sup> *<sup>υ</sup>* is the vortex–viscosity parameter, *Mr* <sup>=</sup> *Ka* <sup>2</sup>*φυ* is the permeability parameter, *Nr* = <sup>2</sup>*φCrUw <sup>a</sup>* is the inertia coefficient parameter, *Gr* <sup>=</sup> *<sup>G</sup>*1*<sup>a</sup> <sup>υ</sup>* represents the microrotation parameter, *Pr* = <sup>ρ</sup>*υcp <sup>k</sup>* represents the Prandtl number, *<sup>R</sup>* <sup>=</sup> <sup>4</sup>σ∗*T*<sup>3</sup> 1 *<sup>k</sup>*∗*<sup>k</sup>* represents the radiation parameter, *Sc* = *<sup>υ</sup> Dm* represents the Schmidt number, *Sr* <sup>=</sup> *DmkT*(*Tw*−*T*0) *<sup>υ</sup>Tm*(*Cw*−*C*0) represents the Soret number, and <sup>τ</sup> <sup>=</sup> *kU*<sup>2</sup> *w* 2*υa* is the thermophoretic parameter and is same as in the works of [17–20].

The important physical quantities are skin friction coefficient *Cf* , local Nusselt number *Nu*, and Sherwood number, which are defined as follows:

$$\mathbb{C}\_{f} = \frac{\mu \left(\mu\_{\mathcal{Y}}\right)\_{y=0}}{\frac{1}{2} \mathfrak{gl} \mathcal{U}\_{\mathcal{w}}^{2}}, \mathcal{N}\mathfrak{u} = \frac{-k \left(T\_{\mathcal{Y}}\right)\_{y=0} \mathfrak{x}}{k \left(T\_{\mathcal{w}} - T\_{0}\right)}, \mathcal{S}\mathfrak{h} = \frac{-D\_{\mathfrak{m}} \left(\mathbb{C}\_{\mathcal{Y}}\right)\_{y=0} \mathfrak{x}}{D\_{\mathfrak{m}} \left(\mathbb{C}\_{\mathcal{w}} - \mathbb{C}\_{0}\right)}.$$

where μ *uy <sup>y</sup>*=0, −*k Ty <sup>y</sup>*=0, and −*Dm Cy <sup>y</sup>*=<sup>0</sup> are shear stress, heat, and mass fluxes at the surface, respectively. Using the variables in (11), the expressions for dimensionless skin friction, Nusselt number, and Sherwood number are obtained as follows:

$$\mathcal{C}\_f\left(\frac{\text{Re}}{2}\right)^{\frac{1}{2}} = -f''(0),\\Nu\left(\frac{\text{Re}}{2}\right)^{-\frac{3}{2}} = -\theta'(0),\\Sh\left(\frac{\text{Re}}{2}\right)^{-\frac{3}{2}} = -\phi'(0) \tag{19}$$

Here, Re = *Uwx <sup>υ</sup>* represents the Reynold number based on the stretching velocity. The calculated values for the skin friction coefficient and local Nusselt number are shown in Tables 1–3.

**Table 1.** Values for the skin friction coefficient, when *h* = −0.2, *Mr* = *Gr* = 0.8, *Nr* = *R* = Δ = *Sc* = *Sr* = τ = *Pr* = 0.3, β = 1.


**Table 2.** Values of rate of heat transfer or the local Nusselt number, when *h* = −0.2, *Mr* = *Gr* = 0.8, *Nr* = *R* = Δ = *Sc* = *Sr* = τ = *Pr* = 0.3, β = 1.


**Table 3.** Values of the Sherwood number, when *h* = −0.2, *Mr* = *Gr* = 0.8, *Nr* = *R* = Δ = *Sc* = *Sr* = τ = *Pr* = 0.3, β = 1.

