Correlation of the dynamic contact angle with the capillary number and its hysteresis

The moving contact-line problem is of both theoretical and practical interest. The dynamic contact angle changes with the capillary number defined by the contact-line speed, and the correlation also depends on the equilibrium contact angle measured at the static state. This correlation is usually required as an input to the traditional solvers based on the Navier-Stokes-like equations, but it is simulated as an output in the current study using the lattice Boltzmann method (LBM) in a displacement process of two-immiscible fluids. The macroscopic theory and the molecular dynamics (MD) simulation had shown a linear scaling law for the cosine of dynamic contact angle, which is also observed in the previous LBM study in a short range of small capillary numbers and for two neutral wetting conditions. However, our study shows that this linear scaling law holds in the whole range of capillary numbers and is universal for all wetting conditions. In a special case of complete wetting (spreading) with a zero equilibrium contact angle, a thin film of the wetting fluid occurs when the wettability is very strong, which leads to a hysteresis that substantial capillary number is required to initiate the deviation of the dynamic contact angle from its equilibrium state. This observation is consistent with the previous report on a new mechanism for the static contact angle hysteresis due to the presence of free liquid films. With an increasing capillary number, the fluid-fluid interface starts oscillating before fingering. Different fingering patterns are observed for cases with different equilibrium contact angles.