A gravity-driven, thin, incompressible liquid film flow on a non-uniformly heated, slippery inclined plane is considered within the framework of the long-wave approximation method. A mathematical model incorporating variation in surface tension with temperature has been formulated by coupling the Navier–Stokes equation, governing the flow, with the equation of energy. For the slippery substrate, the Navier slip boundary condition is used at the solid–liquid interface. An evolution equation is formed in terms of the free surface, which includes the effects of inertia, thermocapillary as well as slip length. Using the normal mode approach, linear stability analysis is carried out and a critical Reynolds number is obtained, which reflects its dependence on the Marangoni number $Mn$ as well as slip length $\delta$. This depicts that $\delta$ and $Mn$ both have the destabilization effect on the flow field. The linear study also reveals that the inertia force has a negligible effect compare to the thermocapillary or slip. In addition, the study highlights a critical Marangoni number at which the instability sets in when the thermocapillary stress attains a critical value. The method of multiple scales is used to investigate the weakly nonlinear stability analysis of the flow. The study interprets that the variation of $Mn$ and $\delta$ have substantial effects on different stable/unstable zones. It also shows that within a considered parametric domain, the unconditional stable zone completely vanishes for any value of $Mn$, when the slip length $\delta$ attains a critical value. The study also divulges that in the subcritical unstable (supercritical stable) zone the threshold amplitude $\left(\zeta a\right)$ decreases (increases) with the increment of $Mn$ and $\delta$. Further, we discussed the spatial uniform solution of the complex Ginzburg–Landau equation for sideband disturbances. Employing the Crank–Nicolson method, the nonlinear evolution equation of the free surface is solved numerically in a periodic domain, considering the sinusoidal initial perturbation of small amplitude. The nonlinear simulations are found to be in good agreement with the linear and weakly nonlinear stability analysis. The evolution of the maximum $\left(h_{\text{max}}\right)$ and minimum $\left(h_{\text{min}}\right)$ thickness amplifies, for small change of $Mn$ and $\delta$. Further, it shows that the influence of the thermocapillary force amplifies the destabilizing nature of $\delta$. The traveling wave solution confirms the existence of a fixed point for the considered parametric domain, chosen from the experimental data. Finally, the Hopf bifurcation of the fixed point exhibits that the nonlinear wave speed at the transcritical point increases as $\delta$ increases but decreases as $Mn$ increases. The noteworthy result which arises from the study is that a transcritical Hopf bifurcation exists if the slip length $\delta >max\left\{\left(\frac{1}{6}Mn-\frac{1}{3}\right),\frac{1}{2}\left(\frac{Mn-2}{3-Mn}\right)\right\}$.

在长波近似方法的框架内，可以考虑在非均匀加热的，光滑的倾斜平面上的重力驱动的，薄的，不可压缩的液膜流。通过将控制流量的Navier–Stokes方程与能量方程耦合，建立了一个包含表面张力随温度变化的数学模型。对于湿滑的基材，在固液界面使用了Navier滑移边界条件。根据自由表面形成一个演化方程，其中包括惯性，热毛细管以及滑移长度的影响。使用法线模式方法，进行了线性稳定性分析，并获得了一个关键的雷诺数，这反映了它对马兰哥尼数的依赖性$\mathrm{中号}ñ$ 以及滑移长度 $\delta$。这说明$\delta$ 和 $\mathrm{中号}ñ$两者都对流场产生不稳定作用。线性研究还表明，与热毛细管或滑移相比，惯性力的影响可忽略不计。此外，该研究还强调了一个临界Marangoni数，当热毛细应力达到临界值时，该数值将导致不稳定性。采用多尺度方法研究了流动的弱非线性稳定性分析。该研究解释说，$\mathrm{中号}ñ$ 和 $\delta$对不同的稳定/不稳定区域具有重大影响。它还表明，在一个已考虑的参数域内，对于以下任何值，无条件稳定区域都将完全消失。$\mathrm{中号}ñ$，当滑移长度 $\delta$达到临界值。研究还揭示了在亚临界不稳定（超临界稳定）区域中的阈值幅度$（\zeta \mathrm{一种}）$ 随（）的增加而减少（增加） $\mathrm{中号}ñ$ 和 $\delta$。此外，我们讨论了针对边带干扰的复杂Ginzburg-Landau方程的空间均匀解。考虑到小振幅的正弦初始扰动，采用Crank-Nicolson方法，在周期域内对自由表面的非线性演化方程进行了数值求解。发现非线性仿真与线性和弱非线性稳定性分析非常吻合。最大的演变$\left(H_{\text{最大限度}}\right)$ 和最小 $\left(H_{\text{分}}\right)$ 厚度放大，变化很小 $\mathrm{中号}ñ$ 和 $\delta$。此外，它表明，热毛细作用力的影响会放大热稳定性的不稳定因素。$\delta$。行波解确定了从实验数据中选择的参数域的固定点的存在。最后，定点的霍普夫分叉表明，跨临界点处的非线性波速随$\delta$ 增加但减少 $\mathrm{中号}ñ$增加。该研究得出的值得注意的结果是，如果滑移长度存在跨临界霍普夫分叉$\delta >最大限度\left\{\left(\frac{\mathrm{1个}}{6}\mathrm{中号}ñ-\frac{\mathrm{1个}}{3}\right)，\frac{\mathrm{1个}}{\mathrm{2个}}\left(\frac{\mathrm{中号}ñ-\mathrm{2个}}{3-\mathrm{中号}ñ}\right)\right\}$。