Annular liquid jets: exact constraint absorption and finite-time loss of transversality

Abstract

An annular liquid jet encloses a volume of gas and, when surface tension dominates inertia, closes on the symmetry axis at a finite distance. In the one-dimensional model of this configuration that distance is not a boundary condition but an algebraic constraint on the state at the free end, and the established schemes differentiate it in time and march the resulting equation for the domain length. We show that this index reduction leaves the constraint unenforced and, in the steady limit, degenerate, and we avoid it by absorbing the constraint identically in the choice of dependent variable. The reduced system carries no constraint and is discretised by Chebyshev collocation. Convergence is geometric, published steady data are recovered, and a residual evaluated off the collocation nodes verifies the unsteady solution a posteriori. Three results follow. For jets of finite thickness the convergence length decreases strictly with the thickness-to-radius ratio at the nozzle whenever the free end is transversal, which contradicts a published table and localises the discrepancy in the treatment of the free end. The forced response is resonant in the pressure coefficient but not in the convergence length. And under forcing the free end ceases to be transversal in finite time, so that the reduction of the free boundary to a single scalar fails, at parameter values published as periodic responses. In one of them the computed failure time coincides with a near-vertical segment of the published record, and coarse discretisations do not detect the failure at all.