Computations of spontaneous rise of a rivulet in a corner of a vertical square capillary

In this study, the spontaneous rise of a Newtonian liquid in a square capillary with completely or partially wetted walls is investigated using numerical simulations. The flow is modelled using volume-of-fluid method with adaptive mesh refinement to resolve the interface for high accuracy. The computations show that for contact angles smaller than 45 degrees, rivulets appear in the corners of the capillary. At large times the length of the capillary growth approaches the one-third power function of time. The same asymptotic behavior has been identified in the existing experimental observations for corners of different geometries. The computations predict the dependence of the rate of the rivulet growth on the liquid viscosity, gravity, width of the capillary and the contact angle. The flow in the rivulet is described using a long-wave approximation which considers three regions of the rivulet flow: the flow near the rivulet tip which is described by a similarity solution, an intermediate region approaching a static rivulet shape, and the bulk meniscus. Finally, a scaling analysis is proposed which predicts rivulet growth rates for the given parameters.