Capillary rise of a liquid into a deformable porous material

Abstract

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.

Dynamics of elastocapillary rise

“…sap flow, spore release), and it is of fundamental and practical importance to estimate the time to reach equilibrium in such systems, as well as to understand their transient dynamics. Recent examples of such elastocapillary dynamics include studies of capillary-driven flow in deformable porous media (Siddique et al 2009) and in flexible channels where gravitational effects could be neglected (van Honschoten et al 2007;Aristoff et al 2010). Here, we consider a model system to study elastocapillary dynamics with gravity, namely capillary rise between flexible sheets that are clamped at their upper ends.…”

A Review of Mixture Theory for Deformable Porous Media and Applications

“…In the presence of gravity effects, in the beginning both curves follow similarity solution and then deviate from this trend and reach to equilibrium solution. The limit of this presentation of these graphical results is up to this and readers are referred to Siddiqui et al [20] for more details where steady state solution along with experimental comparison can be found. Anderson [29] also studied the imbibition of a liquid droplet on a deformable porous substrate using mixture theory approach.…”

“…In their model the gravity effects were not present and it is reported that imbibition of liquid into deformable porous material results into swelling, swelling relaxation, and shrinking. Later on, Siddique et al [19,20,30] studied the capillary rise phenomena into a deformable sponge like material by including the gravity, power law and MHD fluid effects. On the similar lines, Sommer and Mortensen [31] studied unidirectional forced infiltration in an initially dry sponge like material.…”

“…Equations (19) and (21) together with boundary conditions Equation (20) are now solved numerically for the fluid pressure p(r, t) and solid displacement u(r, t). The details of the solution procedure are not presented here.…”

“…Number of authors [11,[15][16][17][18][19][20][21] have used different forms of permeability functions while analyzing the specifics of their research.…”