Spontaneous Penetration of Liquids into Capillaries and Porous Membranes Revisited.

A critical review of the problem of spontaneous penetration of a wetting liquid into pore channels shows that no theory exists to quantitatively predict the initial stage of imbibition. Since C. H. Bosanquet (1923, Phil. Mag. 45, 525), the theory operates with an universal velocity U(Bosanquet)=(2gammacosstraight thetarhor(1/2), with gamma being the surface tension, straight theta the contact angle, r the capillary/pore radius, and rho the fluid density. It is assumed that the initial impulse of the liquid entering the pore is insignificant for the penetration dynamics. Though the importance of the outside flow pattern has been noted in many papers, a thorough mathematical analysis of this effect is lacking in the literature. We derived a generalized equation of the fluid front motion by averaging the Euler equations of flow inside and outside the pore space. This analysis shows the significance of the flow patterns at the pore entrance. The initial stage of liquid imbibition is studied in the inviscid approximation using the methods of dynamic systems. The phase portrait of the dynamic system reveals a multiplicity of penetration regimes. Remarkably, the Bosanquet solution represents a particular regime, with the apparent mass being set zero. The Bosanquet trajectory refers to a separatrix of the phase portrait. It is shown that the initial conditions affect the rate of uptake significantly. The initial conditions stem from the prehistory of the fluid motion outside the pores prior to the liquid-solid contact. The phase portrait method allows us to distinguish two groups of solutions for the capillary rise dynamics of an inviscid fluid. The first group of trajectories corresponds to the liquid front rebound; the second group includes cyclic trajectories which correspond to the periodic regimes with liquid front oscillations at the equilibrium position. The upper estimate of the oscillation amplitude is found. Copyright 2001 Academic Press.