Time-dependent invasion laws for a liquid-liquid displacement system

Capillary-driven flow of fluids occurs frequently in nature and has a wide range of technological applications in the fields of industry, agriculture, medicine, biotechnology, and microfluidics. By using the Onsager variational principle, we propose a model to systematically study the capillary imbibition in titled tubes, and find different laws of time-dependent capillary invasion length for liquid-liquid displacement system other than Lucas-Washburn type under different circumstances. The good agreement between our model and experimental results shows that the imbibition dynamics in a capillary tube with a prefilled liquid slug can be well captured by the dynamic equation derived in this paper. Our results bear important implications for macroscopic descriptions of multiphase flows in microfluidic systems and porous media.