Numerical simulations of spontaneous capillary rises with very low capillary numbers using a front-tracking method combined with generalized Navier boundary condition

We propose a front-tracking method that considers a moving contact line with a generalized Navier boundary condition (GNBC) and with a delta function distribution approximated on the grid scale. A method of evaluating the interfacial balance at the contact line in the form of a body force, which is straightforward with the front-tracking method, cannot give a natural flow field. In contrast, the proposed method using the GNBC, which includes the unbalanced Young’s force as stress on the wall, can give a very stable and reasonable flow field. The proposed front-tracking method was applied for the capillary rise of a liquid in a tube, in which the velocity-dependent contact angle dominates the dynamic characteristics. The validity of the proposed method was confirmed by comparing simulation results with experimental measurements and simple theoretical models. The results of the present simulations with adjusted non-dimensional slip parameters agreed very well with experimental measurements. Under the present simulation conditions, the linearity of the GNBC allows the correlation between the dynamic contact angle and the contact line’s velocity to follow a simple linear expression that involves the difference of the cosine with the capillary number. The non-dimensional slip parameter, which represents the dynamic nature of the moving contact line, can therefore be easily adjusted to reproduce experimental observations under small-capillary-number conditions.