Immiscible-fluid displacement: Contact-line dynamics and the velocity-dependent capillary pressure

The dynamics of immiscible-fluid displacement is studied in the simple geometry of a capillary tube. Here the interesting physics lies in the breakdown of the no-slip boundary condition near the contact line, defined as the intersection of the fluid-fluid interface with the solid wall. Through numerical hydrodynamic calculations, we link macroscopic-flow behavior to the microscopic parameters governing the contact-line region. It is shown that the moving contact line generates two types of frictional forces. One, the viscous stress, is responsible for the observed deformation of the fluid-fluid interface as the flow velocity U increases. Our calculation is in excellent agreement with prior analytic works on this aspect. In particular, our results reproduce Hoffman's scaling relation as well as the logarithmic dependence of the viscous friction on slipping length. Identical macroscopic-flow behaviors are also found to result from three different slipping models provided that their slipping lengths are each renormalized by a model-dependent constant. Besides the viscous stress, however, comparison with experiments revealed a second frictional force that varies as U^x, with 0x0. 5, which is dominant at capillary numbers 10^-3. We propose that the source of this new friction is the excitation of damped capillary waves at the fluid-fluid interface due to contact-line motion over wall roughness. Consideration of this mechanism yields not only the correct range of x values, but also good agreement with the measured magnitude of the second frictional force. The paper concludes with an analysis of the frequency-dependent pressure response to an imposed ac velocity perturbation. An expression is derived for the critical frequency that separates the low-frequency behavior from that of the high-frequency regime.