Rib formation in inertial liquid entrainment

We present findings on pattern formation under high-speed liquid entrainment conditions on the outer surface of a rotating cylinder. Our observations for two liquids (water, and a 100 times more viscous mixture of water and UCON oil), reveal that this inertial coating flow produces a series of vertical parallel thin liquid sheets, as fluid is entrained and ballistically ejected by the rotating wall. The observed wavelength, of the order of a few centimeters, is surprisingly almost unchanged when velocity and the viscosity of the fluid are modified, even when the liquid sheets are strongly fragmented and droplet ejection occurs. We demonstrate that these axial patterns arise from a primary instability driven by an adverse pressure gradient in the meniscus region of the rotary Landau-Levich-Derjaguin flow. According to our model, the rib spacing is proportional to the capillary length l_c, even though the observed wavelength is itself much larger than l_c. Additionally, we conducted Direct Numerical Simulations using Basilisk, which evidence the central role of pressure in the entrained film in driving this instability. The numerical simulation replicates well the experimental data, and corroborate the predictions of our model when the density and surface tension are numerically varied.