The response of a thin film flowing under an inclined plane, modelled using the lubrication equation, is studied. The flow at the inlet is perturbed by the superimposition of a spanwise-periodic steady modulation and a decoupled temporally periodic but spatially homogeneous perturbation. As the consequence of the spanwise inlet forcing, the so-called rivulets grow downstream and eventually reach a streamwise-invariant state, modulated along the direction perpendicular to the flow. The linearized dynamics in the presence of a time-harmonic inlet forcing shows the emergence of a time-periodic flow characterized by drop-like structures (so-called lenses) that travel on the rivulet. The spatial evolution is rationalized by a weakly non-parallel stability analysis. The occurrence of the lenses, their spacing and thickness profile, is controlled by the inclination angle, flow rate, and the frequency and amplitude of the time-harmonic inlet forcing. The faithfulness of the linear analyses is verified by nonlinear simulations. The results of the linear simulations with inlet forcing are combined with the computations of nonlinear travelling lenses solutions in a double-periodic domain to obtain an estimate of the dripping length, for a large range of conditions.

研究了使用润滑方程建模的在斜面下流动的薄膜的响应。入口处的流动受到展向周期性稳定调制和解耦的时间周期但空间均匀扰动的叠加扰动。作为展向入口强迫的结果，所谓的小溪在下游生长并最终达到流向不变状态，沿垂直于流动的方向调制。在存在时谐入口强迫的情况下，线性化动力学显示了时间周期流动的出现，其特征是在小溪上传播的滴状结构（所谓的透镜）。空间演化通过弱非平行稳定性分析合理化。镜片的出现，它们的间距和厚度分布，由倾角、流速以及时谐入口强迫的频率和幅度控制。线性分析的真实性通过非线性模拟得到验证。入口强迫线性模拟的结果与双周期域中非线性行进透镜解的计算相结合，以获得大范围条件下滴水长度的估计值。