The mathematical models for the capillary-driven flow of fluids in tubes typically assume a static contact angle at the fluid–air contact line on the tube walls. However, the dynamic evolution of the fluid–air interface is an important feature during capillary rise. Furthermore, inertial effects are relevant at early times and may lead to oscillations. To incorporate and quantify the different effects, a fundamental description of the physical processes within the tube is used to derive an upscaled model of capillary-driven flow in circular cylindrical tubes. The upscaled model extends the classical Lucas–Washburn model by incorporating a dynamic contact angle and slip. It is then further extended to account for inertial effects. Finally, the solutions of the different models are compared to experimental data. In contrast to the Lucas–Washburn model, the models with dynamic contact angle match well the experimental data, both the rise height and the contact angle, even at early times. The models have a free parameter through the dynamic contact angle description, which is fitted using the experimental data. The findings here suggest that this parameter depends only on the properties of the fluid but is independent of geometrical features, such as the tube radius. Therefore, the presented models can predict the capillary-driven flow in tubular systems upon knowledge of the underlying dynamic contact-angle relation.

We consider a model for the flow of two immiscible fluids in a two-dimensional thin strip of varying width. This represents an idealization of a pore in a porous medium. The interface separating the fluids forms a freely moving interface in contact with the wall and is driven by the fluid flow and surface tension. The contact-line model incorporates Navier-slip boundary conditions and a dynamic and possibly hysteretic contact angle law. We assume a scale separation between the typical width and the length of the thin strip. Based on asymptotic expansions, we derive effective models for the two-phase flow. These models form a system of differential algebraic equations for the interface position and the total flux. The result is Darcy-type equations for the flow, combined with a capillary pressure-saturation relationship involving dynamic effects. Finally, we provide some numerical examples to show the effect of a varying wall width, of the viscosity ratio, of the slip boundary condition as well as of having a dynamic contact angle law.

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