The flow of a thin film coating the underside of an inclined substrate is studied. We measure experimentally spatial growth rates and compare them to the linear stability analysis of a flat film modelled by the lubrication equation. When forced by a stationary localized perturbation, a front develops that we predict with the group velocity of the unstable wave packet. We compare our experimental measurements with numerical solutions of the nonlinear lubrication equation with complete curvature. Streamwise structures dominate and saturate after some distance. We recover their profile with a one-dimensional lubrication equation suitably modified to ensure an invariant profile along the streamwise direction and compare them with the solution of a purely two-dimensional pendent drop, showing overall a very good agreement. Finally, those different profiles agree also with a two-dimensional simulation of the Stokes equations.

中文翻译：

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在倾斜基板下流动的粘性薄膜的不稳定性：稳定模式

研究了倾斜基板下侧的薄膜涂层流动。我们通过实验测量空间增长率，并将它们与由润滑方程建模的平膜的线性稳定性分析进行比较。当受到固定局部扰动的强迫时，会形成我们用不稳定波包的群速度预测的前沿。我们将我们的实验测量结果与具有完整曲率的非线性润滑方程的数值解进行比较。流向结构在一定距离后占主导地位并饱和。我们使用适当修改的一维润滑方程恢复它们的轮廓，以确保沿流向的轮廓不变，并将它们与纯二维悬滴的解进行比较，总体上显示出非常好的一致性。最后，