We conduct a rigorous mathematical analysis of an oscillatory behaviour during the capillary flow. Our investigation concerns a nonlinear ODE modelling rise in a narrow vertical tube. This equation has been proposed by many authors as a generalization and improvement over the classical Lucas-Washburn model. The extension includes components related to a non-zero immersion length and to a discontinuous variable describing the exit/entrance effect at the bottom of the tube. We propose an arbitrarily accurate regularization of the non-smooth component in order to prove several results concerning dynamical behaviour of the exact solution: existence and uniqueness, estimates, and switch of the behaviour from monotone to oscillatory. We have found that the latter is independent of the chosen regularization which makes the model robust. Furthermore, we applied the Lyapunov stability analysis to deliver a comprehensive description of the asymptotic behaviour of trajectories

我们对毛细管流动过程中的振动行为进行了严格的数学分析。我们的研究涉及在狭窄的垂直管中进行非线性ODE建模的过程。许多作者提出了该方程，作为对经典Lucas-Washburn模型的推广和改进。扩展包括与非零浸入长度和描述管底部出射/进入效应的不连续变量有关的组件。我们提出了一个非光滑分量的任意精确的正则化方法，以证明有关精确解的动力学行为的若干结果：存在性和唯一性，估计以及行为从单调到振荡的转换。我们发现后者独立于所选择的正则化，这使得模型更加健壮。此外，