  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.
