The long cross-over dynamics of capillary imbibition

Spontaneous capillary imbibition is a classical problem in interfacial fluid dynamics with a broad range of applications, from microfluidics to agriculture. Here we study the duration of the cross-over between an initial linear growth of the imbibition front to the diffusive-like growth limit of Washburn's law. We show that local-resistance sources, such as the inertial resistance and the friction caused by the advancing meniscus, always limit the motion of an imbibing front. Both effects give rise to a cross-over of the growth exponent between the linear and the diffusive-like regimes. We show how this cross-over is much longer than previously thought - even longer than the time it takes the liquid to fill the porous medium. Such slowly slowing-down dynamics is likely to cause similar long cross-over phenomena in processes governed by wetting.