Oscillatory behaviour analysis of a liquid rise in cylindrical capillaries

Abstract 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