Threshold speed for air entrainment in forced wetting

For a solid plunging at speed $U$ into a liquid of viscosity $\eta$ and surface tension $\gamma$, surrounded by gas of viscosity $\eta_g$ and pressure $p$, predict the critical capillary number $\mathrm{Ca}* = \eta U*/\gamma$ for dynamic wetting failure, including its dependence on the viscosity ratio $\eta_g/\eta$, on gas rarefaction when the mean free path approaches the air-film thickness, and on the large increase of $\mathrm{Ca}*$ when the flow pushes the liquid against the solid (curtain coating). An answer predicts $\mathrm{Ca}*$ within experimental error for $\eta_g/\eta$ from $10^{-6}$ to $10^{-2}$ and across reduced gas pressures.

A comparison of measured $\mathrm{Ca}*$ for plunging plates and curtain coating over wide viscosity ratios and gas pressures with a single model of the gas film and the contact line.