**3. Stability Analysis**

According to Nasir et al., [28] and Rana et al., [25], we need to introduce the unsteady form of Equations (6)–(8) in order to perform stability test,

$$u\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = \left(\mathfrak{d} + \frac{\kappa}{\rho}\right)\frac{\partial^2 u}{\partial y^2} + \frac{\kappa}{\rho}\frac{\partial \mathcal{N}}{\partial y} \tag{17}$$

$$u\frac{\partial \mathbf{N}}{\partial t} + u\frac{\partial \mathbf{N}}{\partial \mathbf{x}} + v\frac{\partial \mathbf{N}}{\partial y} = \frac{\gamma}{\rho j}\frac{\partial^2 \mathbf{N}}{\partial y^2} - \frac{\kappa}{\rho j} \left(2\mathbf{N} + \frac{\partial u}{\partial y}\right) \tag{18}$$

$$\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial \mathbf{x}} + v \frac{\partial T}{\partial y} = \frac{k^\*}{\rho c\_p} \frac{\partial^2 T}{\partial y^2} \tag{19}$$

where time is denoted by *t*. The similarity transformations are written according to Ro¸sca and Pop [30], as,

$$\begin{split} \psi = \sqrt{2\mathcal{S}La}e^{\frac{\pi}{2L}}f(\eta,\tau); \ N = ae^{\frac{2\pi}{2L}}\sqrt{\frac{a}{2\mathcal{S}L}}h(\eta,\tau); \Theta(\eta,\tau) = \frac{(T-T\_{\infty})}{(T\_w-T\_{\infty})};\\ \eta = \mathcal{Y}\sqrt{\frac{a}{2\mathcal{S}L}}e^{\frac{\pi}{2L}}; \ \tau = \frac{a}{2L}e^{\frac{\pi}{2}}.\end{split} \tag{20}$$

By applying these similarity transformations, we reduced our Equations (17)–(19) into the following form

$$(1+K)\frac{\partial^3 f}{\partial \eta^3} + f \frac{\partial^2 f}{\partial \eta^2} - 2\left(\frac{\partial f}{\partial \eta}\right)^2 + K \frac{\partial \hbar}{\partial \eta} - \frac{\partial^2 f}{\partial \tau \partial \eta} = 2\pi \left(\frac{\partial f}{\partial \eta} \frac{\partial^2 f}{\partial \eta \partial \mathbf{x}} - \frac{\partial^2 f}{\partial \eta^2} \frac{\partial f}{\partial \mathbf{x}}\right) \tag{21}$$

$$\left(1+\frac{\mathrm{K}}{2}\right)\frac{\partial^2 \mathrm{h}}{\partial \eta^2} + f\frac{\partial \mathrm{h}}{\partial \eta} - 3\mathrm{h}\frac{\partial f}{\partial \eta} - 2\mathrm{K}\mathrm{h} - \mathrm{K}\frac{\partial^2 f}{\partial \eta^2} - \frac{\partial \mathrm{h}}{\partial \tau} = 2\pi \left(\frac{\partial f}{\partial \eta}\frac{\partial \mathrm{h}}{\partial \mathbf{x}} - \frac{\partial \mathrm{h}}{\partial \eta}\frac{\partial f}{\partial \mathbf{x}}\right) \tag{22}$$

$$\left(\frac{1}{Pr}\frac{\partial^2 \theta}{\partial \eta^2} + f\frac{\partial \theta}{\partial \eta} - \theta\frac{\partial f}{\partial \eta} - \frac{\partial \theta}{\partial \tau} = 2\pi \left(\frac{\partial f}{\partial \eta}\frac{\partial \theta}{\partial x} - \frac{\partial \theta}{\partial \eta}\frac{\partial f}{\partial x}\right) \tag{23}$$

with the following boundary conditions

$$f(0,\tau) = \text{S; } \frac{\partial f(0,\tau)}{\partial \eta} = -1; \ h(0,\tau) = -m \frac{\partial^2 f(0,\tau)}{\partial \eta^2}; \ \theta(0,\tau) = 1$$

$$\frac{\partial f(\eta,\tau)}{\partial \eta} \to 0; \ h(\eta,\tau) \to 0; \ \theta(\eta,\tau) \to 0 \qquad \text{as } \eta \to \infty$$

and then we applied perturbed with the disturbance (see Ro¸sca and Pop [30]) in Equations (21)–(24) with the following functions

$$\begin{cases} f(\eta,\tau) = f\_0(\eta) + e^{-\varepsilon\tau} F(\eta,\tau) \\ h(\eta,\tau) = h\_0(\eta) + e^{-\varepsilon\tau} G(\eta,\tau) \\ \theta(\eta,\tau) = \theta\_0(\eta) + e^{-\varepsilon\tau} H(\eta,\tau) \end{cases} \tag{25}$$

where *F*(η, τ), *G*(η, τ), and *H*(η, τ) are small relative to *f*0(η), *h*0(η), and θ0(η), respectively. Further, ε is the unknown eigenvalue. By substituting Equation (25) in Equations (21)–(23) by keeping τ = 0, we have following the linearized eigenvalue problems

$$(1+K)F\_{0}^{\prime\prime} + f\_{0}F\_{0}^{\prime\prime} + F\_{0}f\_{0}^{\prime\prime} - 4f\_{0}^{\prime}F\_{0}^{\prime} + KG\_{0}^{\prime} + \varepsilon F\_{0}^{\prime} = 0\tag{26}$$

$$\left(1+\frac{K}{2}\right)G\_0^{\prime\prime} + f\_0G\_0^{\prime} + F\_0h\_0^{\prime} - 3h\_0F\_0^{\prime} - 3h\_0F\_0^{\prime} - 2KG\_0 - KF\_0^{\prime\prime} + \varepsilon G\_0 = 0\tag{27}$$

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

$$\frac{1}{Pr}H\_0'' + f\_0 H\_0' + F\_0 \theta\_0' - f\_0' H\_0 - F\_0' \theta\_0 + \varepsilon H\_0 = 0\tag{28}$$

With the boundary conditions

$$\begin{array}{cccc} F\_0(0) = 0, & F\_0'(0) = 0, & G\_0(0) = -nF\_0''(0), & H\_0(0) = 0\\ F\_0'(\eta) \to 0, & G\_0(\eta) \to 0, & H\_0(\eta) \to 0, & \text{as} \ \eta \to \infty \end{array} \tag{29}$$

We have to solve above linearized Equations (10)–(13) with new relax boundary conditions in order to find the values of smallest eigenvalue. In this particular problem, we have relaxed *H*0(η) → 0 as η → ∞ into *H*<sup>0</sup> 0 (0) = 1, see [31–33].
