Capillary rise of a liquid into a deformable porous material

We examine the effects of capillarity and gravity in a model of one-dimensional imbibition of an incompressible liquid into a deformable porous material. We focus primarily on a capillary rise problem but also discuss a capillary/gravitational drainage configuration in which capillary and gravity forces act in the same direction. Models in both cases can be formulated as nonlinear free-boundary problems. In the capillary rise problem, we identify time-dependent solutions numerically and compare them in the long time limit to analytically obtain equilibrium or steady state solutions. A basic feature of the capillary rise model is that, after an early time regime governed by zero gravity dynamics, the liquid rises to a finite, equilibrium height and the porous material deforms into an equilibrium configuration. We explore the details of these solutions and their dependence on system parameters such as the capillary pressure and the solid to liquid density ratio. We quantify both net, or global, deformation of the material and local deformation that may occur even in the case of zero net deformation. In the model for the draining problem, we identify numerical solutions that quantify the effects of gravity, capillarity, and solid to liquid density ratio on the time required for a finite volume of fluid to drain into the deformable porous material. In the  Appendix, experiments on capillary rise of water into a deformable sponge are described and the measured capillary rise height and sponge deformation are compared with the theoretical predictions. For early times, the experimental data and theoretical predictions for these interface dynamics are in general agreement. On the other hand, the long time equilibrium predicted theoretically is not observed in our experimental data.