Stability of thin liquid curtains

We investigate the stability of thin liquid curtains with respect to two-dimensional perturbations. The dynamics of perturbations with wavelengths exceeding (or comparable to) the curtain's thickness are examined using the lubrication approximation (or a kind of geometric optics). It is shown that, contrary to the previous theoretical results, but in agreement with the experimental ones, all curtains are stable with respect to small perturbations. Large perturbations can still be unstable, however, but only if they propagate upstream and, thus, disrupt the curtain at its outlet. This circumstance enables us to obtain an effective stability criterion by deriving an existence condition for upstream propagating perturbations.

Figure 1

The setting: a free-falling liquid curtain.

Figure 2

The solution of the boundary-value problem (35, 36, 37) for $\overline{z}=-2$. The curtain's outlet is located at $z=0$. (a) $\overline{w}$ vs $z$; (b) $\overline{P}$ [defined by (41)] vs $z.z_{\mathrm{cr}}$ is the coordinate of the critical point such that $\overline{P}\left(z_{\mathrm{cr}}\right)=0$.