On the dynamics of a meniscus inside capillaries during imbibition and drainage processes: A generalized model, effect of inertia, and a numerical algorithm

Abstract

In imbibition or drainage processes, a fluid displaces another immiscible one. If the displacing fluid is wetting, this is an imbibition process and is drainage if otherwise. While imbibition can proceed without the action of external force (e.g., pressure), drainage cannot unless sufficient external force is applied. One of the most important phenomena in this regard is related to the estimation of the location of the meniscus inside the tube and its velocity with time. This has been the topic of extensive research works for which analytical expressions exist for some special cases including the case in which the displaced fluid is air. Recently, a generalization to this approach has been developed, which accounts for the more general scenario in which the displaced fluid assumes considerable viscosity and density contrasts compared with the displacing one. However, in this recently developed model, and even in most of the previously studied special cases, an inherent assumption was made to ignore inertial effects. While this assumption is reasonable given the relatively slow advancement of the meniscus in capillaries, it results in the velocity to jump at the start of the imbibition process to a relatively higher value before declining as the meniscus advances. In fact, in actual imbibition experiments, velocity develops from zero to a maximum value in a short period of time before it declines as the meniscus continues to advance. In this work, a generalized model is developed, which accounts for the inertia of the fluids inside the tube. A nonlinear ordinary differential equation is developed, which accounts for the acceleration of the fluid and the contrasts of viscosity and density of the two fluids in capillaries. A numerical algorithm is also developed where the differential equation is linearized to facilitate the numerical solution. Verifications of the numerical algorithm are conducted to build confidence in the computational approach.