**2. Mathematical Formulation**

An incompressible laminar boundary layer two-dimensional flow of the micropolar fluid over the exponentially shrinking sheet has been considered. The corresponding velocities of *x* and *y-*axes are *u* and *v.* The shrinking velocity is assumed to be *uw*(*x*) = −*Uwe* 2*x* ` . The temperature of the sheet is taken to be *T <sup>w</sup>*(*x*) = *T*<sup>∞</sup> + *T*0*e x* <sup>2</sup>` , as shown in Figure 1. The *N* = *N*(*x,y*) is supposed as the angular velocity. The respective boundary layer movement equation, along with micro rotations and the heat transfer equations can be expressed as vectors in accordance with the abovementioned assumptions.

$$
\nabla \cdot \mathbf{V} = \mathbf{0} \tag{1}
$$

$$
\rho \frac{d\mathbf{V}}{dt} = -\nabla \mathbf{P} + (\mu + \kappa)\nabla^2 \mathbf{V} + \kappa(\nabla \times \mathbf{N})\tag{2}
$$

$$
\rho j \frac{d\mathbf{N}}{dt} = \gamma \nabla^2 \mathbf{N} - \kappa (2\mathbf{N} - \nabla \times \mathbf{V}) \tag{3}
$$

$$
\rho c\_p \frac{dT}{dt} = k \nabla^2 T \tag{4}
$$

in which velocity vector is *V* ≡ [*u*(*x*, *y*), *v*(*x*, *y*), 0], the micro-rotation vector is *N*, ρ stands for fluid density, µ for viscosity coefficient, κ for vertex viscosity, *j* is the density of micro-rotation, and γ stands for micropolar constant. We get following boundary layer equations according to the scale analysis.

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \tag{5}$$

$$u\,\frac{\partial u}{\partial \mathbf{x}} + v\,\frac{\partial u}{\partial y} = \left(\mathfrak{d} + \frac{\kappa}{\rho}\right)\frac{\partial^2 u}{\partial y^2} + \frac{\kappa}{\rho}\frac{\partial \mathcal{N}}{\partial y} \tag{6}$$

$$u\frac{\partial \mathcal{N}}{\partial \mathbf{x}} + v\frac{\partial \mathcal{N}}{\partial y} = \frac{\gamma}{\rho j} \frac{\partial^2 \mathcal{N}}{\partial y^2} - \frac{\kappa}{\rho j} \left(2\mathcal{N} + \frac{\partial u}{\partial y}\right) \tag{7}$$

$$
\mu \frac{\partial T}{\partial \mathbf{x}} + v \frac{\partial T}{\partial y} = \alpha \frac{\partial^2 T}{\partial y^2} \tag{8}
$$

Subject to these boundary conditions

$$\begin{aligned} v &= v\_w(\mathbf{x}); \quad u = u\_w(\mathbf{x}); \ N = -m \frac{\partial u}{\partial y}; \ T = T\_w(\mathbf{x}); \quad at \quad y = 0\\ u &\to 0; \ N \to 0; \ T \to T\_\infty \qquad \text{as} \quad y \to \infty \end{aligned} \tag{9}$$

Now, we look for similarity transformation variables in order to transform Equations (6)–(8) with boundary conditions (9)

$$\begin{split} u = \mathcal{U}\_{\rm w} e^{\frac{\tau}{2}} f'(\eta); v = -\sqrt{\frac{8\mathcal{U}\_{\rm w}}{2l}} e^{\frac{\tau}{2l}} (f(\eta) + \eta f'(\eta)); N = \mathcal{U}\_{\rm w} e^{\frac{3\tau}{2l}} \sqrt{\frac{\mathcal{U}\_{\rm w}}{2Sl}} h(\eta); \\ \theta(\eta) = \frac{(T - T\_{\rm co})}{(T\_{\rm w} - T\_{\rm co})}; \eta = \mathcal{Y} \sqrt{\frac{\mathcal{U}\_{\rm w}}{2Sl}} e^{\frac{\tau}{2l}} \end{split} \tag{10}$$

By applying Equation (10) in Equations (6)–(9), we have the following system of similarity transformed ordinary differential equations *Crystals* **2020**, *10*, x FOR PEER REVIEW 4 of 14

$$(\mathbf{1} + \mathbf{K})f'''' + ff'' - 2f^{'\mathbf{2}} + \mathbf{K}h' = \mathbf{0} \tag{11}$$

$$\left(1 + \frac{K}{2}\right)h'' + fh' - \mathfrak{H}'h - \mathbb{K}(2h + f'') = 0\tag{12}$$

$$1$$

$$\frac{1}{Pr}\theta'' + f\theta' - f'\theta = 0\tag{13}$$

(13) 0 = ߠ<sup>ᇱ</sup>

Subject to boundary conditions

2

$$\begin{array}{ll} f(0) = \text{S}; \ f'(0) = -1; \ h(0) = -nf''(0); \ \theta(0) = 1\\ f'(\eta) \to 0; \ h(\eta) \to 0; \ \theta(\eta) \to 0; \quad \text{as } \eta \to \infty \end{array} \tag{14}$$

where prime denotes the differentiation with respect to η, the micropolar material parameter is *K* = <sup>κ</sup> µ , Prandtl number is *Pr* = <sup>ϑ</sup> α , and suction is *S* = − <sup>√</sup> *vw* ϑ*Uw*/2*l* . ݂ ᇱ ∞ → ߟ ݏܽ ;0 → (ߟ)ߠ ;0 → (ߟ)ℎ; 0) → ߟ) (14) where prime denotes the differentiation with respect to ߟ, the micropolar material parameter is ܭ= ణ

The physical quantities of interest include skin friction, the stress of local couples, and the local number of Nusselt, which are described as ఓ , Prandtl number is ܲݎ= ఈ , and suction is ܵ = − ௩ೢ ඥణೢ⁄ଶ . The physical quantities of interest include skin friction, the stress of local couples, and the local

$$\mathbf{C}\_{f} = \frac{\left[ (\mu + \kappa) \frac{\partial \mu}{\partial y} + \kappa N \right]\_{y=0}}{\rho u\_{w}^{2}};\\M\_{x} = \frac{-\gamma \left[ \frac{\partial N}{\partial y} \right]\_{y=0}}{\rho x u\_{w}^{2}};\\N\_{u} = \frac{-\mathbf{x} \left[ \frac{\partial T}{\partial y} \right]\_{y=0}}{(T\_{w} - T\_{\infty})} \tag{15}$$

By applying similarity transformation variables (10) in Equation (15), we have By applying similarity transformation variables (10) in Equation (15), we have

ߠݎܲ

ߠ݂ + ᇱᇱ

<sup>ᇱ</sup> − ݂

$$\mathcal{C}\_f(\text{Re}\_\mathbf{x})^{\frac{1}{2}}\sqrt{2l/\mathbf{x}} = (1 + (1 - m)\mathcal{K})f''(0),\\M\_\mathbf{x}\text{Re}\_\mathbf{x} = \left(1 + \frac{\mathcal{K}}{2}\right)h'(0),$$

$$N\_\mathbf{u}(\text{Re}\_\mathbf{x})^{-\frac{1}{2}}\sqrt{2l/\mathbf{x}} = -\theta'(0)$$

where *Re<sup>x</sup>* = *xuw*/ϑ is the local Reynolds number. where ܴ݁<sup>௫</sup> = ݑݔ௪⁄ߴ is the local Reynolds number.

**3. Stability Analysis**

**Figure 1.** Flow of problem and coordinate system.

**Figure 1.** Flow of problem and coordinate system.

According to Nasir et al., [28] and Rana et al., [25], we need to introduce the unsteady form of

*Crystals* **2020**, *10*, 283
