On oblique liquid curtains

In a recent paper ( J. Fluid Mech. , vol. 861, 2019, pp. 328–348), Benilov derived equations governing a laminar liquid sheet (a curtain) that emanates from a slot whose centreline is inclined to the vertical. The equations are valid for slender sheets whose characteristic length scale in the direction of flow is much larger than its cross-sectional thickness. For a liquid that leaves a slot with average speed, $u_{0}$ , volumetric flow rate per unit width, $q$ , surface tension, $\unicode[STIX]{x1D70E}$ , and density, $\unicode[STIX]{x1D70C}$ , Benilov obtains parametric equations that predict steady-state curtain shapes that bend upwards against gravity provided $\unicode[STIX]{x1D70C}qu_{0}/2\unicode[STIX]{x1D70E}<1$ . Benilov’s parametric equations are shown to be identical to those derived by Finnicum, Weinstein, and Ruschak ( J. Fluid Mech. , vol. 255, 1993, pp. 647–665). In the latter form, it is straightforward to deduce an alternative solution of Benilov’s equations where a curtain falls vertically regardless of the slot’s orientation. This solution is consistent with prior experimental and theoretical results that show that a liquid curtain can emerge from a slot at an angle different from that of the slot centreline.