Stability of oblique liquid curtains with surface tension

Abstract

Oblique (non-vertical) liquid curtains are examined under the assumption that the Froude number is large. As shown previously [E. S. Benilov, “Oblique liquid curtains with a large Froude number,” J. Fluid Mech. 861, 328 (2019)], their structure depends on the Weber number: if We < 1 (strong surface tension), the Navier–Stokes equations admit asymptotic solutions describing curtains bending upward, i.e., against gravity. In the present paper, it is shown that such curtains are unstable with respect to small perturbations of the flow parameters at the outlet: they give rise to a disturbance traveling downstream and becoming singular near the curtain's terminal point (where the liquid runs out of the initial supply of kinetic energy). It is argued that, since the instability is spatially localized, the curtain can be stabilized by a properly positioned collection nozzle. All curtains with We > 1 bend downward and are shown to be stable.