Absolute and convective instability of a radially expanding liquid sheet

Abstract

A radially expanding liquid sheet of a finite radius can be formed by impacting a liquid jet on a circular disk [Savart, Ann. Chim. Phys. 54, 55 (1833); Ann. Chim. 54, 113 (1833)] or by impinging two jet heads on against each other [Savart, Ann. Chim. Phys. 55, 257 (1833)]. The breakup of the liquid sheet first observed by Savart at the outer rim of the sheet is explained from the point of view of absolute and convective instability. Whether the sheet is convectively or absolutely unstable depends on the local Weber number We=ρU2H/S, where ρ and S are, respectively, the liquid density and the surface tension, and H and U are, respectively, the local half-sheet thickness and the local average liquid velocity. It is shown that absolute instability occurs at We=1, and convective instability occurs in the region where We>1. In the radially expanding liquid sheet, H decreases inversely with the radial distance from the center of the sheet, but U remains constant. Thus, the local Weber number that is greater than one near the center must decrease radially, and eventually reaches one where the sheet encounters absolute instability. In absolute instability, the disturbance propagates upstream to terminate the liquid sheet. This is one of the two regimes of breakup. The onset of absolute instability in this regime always occurs at the broken free edge, where the local Weber number is 1, regardless of the sheet radius. Taylor [Proc. R. Soc. London, Ser. A 253, 296 (1959)] obtained the same criterion from a balance of the surface tension force with a time rate of change of momentum experienced by the fluid at the broken edge. In regime II, it is shown that the sheet may break up at local Weber numbers greater than 1, due to convective instability. The global implications of the local absolute instability are discussed in connection with the related phenomena of breakups of planar sheets, water bells and annular jets.