Mean-field dynamics of free surface flows through obstacle arrays in a narrow passage: amendments of the Washburn model

This paper is devoted to a moving-boundary problem of the creeping flow of a Newtonian liquid in a narrow passage, driven by capillary action at a free liquid–gas interface. We are specifically interested in the effects of an array of cylindric obstacles on that flow, which is of direct relevance to electronics manufacturing applications. We solve the related hydrodynamic equations by the efficient Monte Carlo technique of random walk simulation, capturing the time evolution of a global shape of the liquid–gas interface and the corresponding mean-field (averaged) dynamics of the flow. This dynamics represents the known Washburn dependence amended by a certain factor that reflects geometric parameters of the obstacle array. As a main contribution of this investigation, we calculate Washburn's factor for a wide range of rectangular and rectangular centered arrays. The results of our calculations demonstrate good agreement with available experimental data.