We study the dynamics of the interface between two immiscible fluids in contact with a chemically homogeneous moving solid plate. We consider the generic case of two fluids with any viscosity ratio and of a plate moving in either directions (pulled or pushed in the bath). The problem is studied by a combination of two models, namely, an extension to finite viscosity ratio of the lubrication theory and a Lattice Boltzmann method. Both methods allow to resolve, in different ways, the viscous singularity at the triple contact between the two fluids and the wall. We find a good agreement between the two models particularly for small capillary numbers. When the solid plate moves fast enough, the entrainment of one fluid into the other one can occur. The extension of the lubrication model to the case of a non-zero air viscosity, as developed here, allows us to study the dependence of the critical capillary number for air entrainment on the other parameters in the problem (contact angle and viscosity ratio).
Chan, T. S., Srivastava, S. K., Marchand, A., Andreotti, B., Biferale, L., Toschi, F., & Snoeijer, J. H. (2013). Hydrodynamics of air entrainment by moving contact lines. Physics of fluids, 25(7), 074105-1-074105-19. . https://doi.org/10.1063/1.4814466