Capillary flow in corners slowed down by gravity and evaporation

Abstract Capillary flow in corner geometries in the presence of gravity and evaporation is relevant for numerous natural phenomena and industrial applications. In the absence of gravity, the length of the rivulet in the corner follows the t 1/2 asymptotic law (Lucas-Washburn kinetics), where t is the time. If the liquid flows against gravity, the propagation of the rivulet tip decelerates to follow the t 1/3 asymptotic law. In this paper, we present a model for simulation of the rivulet shape evolution in a corner with an arbitrary cross-section shape. Gravity and evaporation are taken into account. Several exact and asymptotic solutions are presented. In particular, a simple expression for the proportionality coefficient in the t 1/3 asymptotic law is derived, as well as an expression for the cross-over time moment corresponding to change from the t 1/2 to t 1/3 asymptotic behavior. In the presence of evaporation, the rivulet length reaches a maximal value, at which the rate of evaporation is balanced by the rate of the capillary flow. We derive expressions for the maximal rivulet length in the limiting cases of “strong” and “weak” evaporation. In the case of “strong” evaporation, the maximal rivulet length behaves as E −1/2 , where E denotes the dimensionless evaporation rate. In the case of “weak” evaporation, uniform evaporation rate and triangular groove geometry, the maximal rivulet length is proportional to E −1/5 Bo −3/5 , where Bo denotes the Bond number.