Bead-break instability

Flow is considered in a “fluid bead” located in the nip between two contra-rotating rolls, and bounded by two curved menisci. Such a flow arises in meniscus roll coating where fluid is transferred from the lower applicator roll to a substrate, in contact and moving with the upper metering roll, by means of a transfer jet (snake). Equilibrium of the bead is maintained through a balance of hydrodynamic and capillary stresses, the stability of which is considered experimentally by increasing the speed of the metering roll while that of the applicator roll remains constant. At a critical speed ratio, the upstream meniscus becomes unstable; the bead contracts as the meniscus accelerates forward and merges with its downstream counterpart—giving rise to “bead-break.” A mathematical model, based on lubrication theory, exhibits multiple solutions and a limit point for the existence of steady solutions. A linear stability analysis identifies the stable solution and shows that the flow becomes unstable at the limit point—which is taken to be the onset of bead-break. The criterion for bead-break is ${\mathrm{(d/dX)[P+\sigma /R}}_u\text{]<0}$ at the upstream meniscus, where P, σ, and $R_u$ represent fluid pressure, surface tension, and radius of curvature of the upstream meniscus, respectively. The effect of the pressure gradient is shown to stabilize the fluid bead whereas that of the surface tension term is to destabilize once the upstream meniscus—and hence the whole bead—is located downstream of the nip. For a given geometry and flow rate, the critical speed ratio decreases with increasing characteristic capillary number. Theoretical predictions compare well with experimental data.