Stability analyses of long liquid bridges in the presence of gravity and flow

Abstract

The stability of liquid bridges under terrestrial conditions can be enhanced when a recirculating flow is introduced inside the bridge. The Plateau–Rayleigh (PR) limit sets the stability limit of a liquid bridge in the absence of gravity and flow. The PR limit corresponds to a cylindrical bridge of length L equal to its circumference, L = 2π R . At lengths exceeding this first onset point, antisymmetric disturbances grow and result in disconnection. In recent experiments involving flow in the bridge, stable lengths greater than the PR limit were obtained by suppressing the antisymmetric disturbances. The observed lengths are close to the second bifurcation or onset point, where centro–symmetric disturbances become unstable. The second onset point occurs at L ≈ 8.987 R . In this paper, the problem is modelled using the lubrication approximation and a rectilinear flow within the bridge with zero flow rate across the cross–section is assumed. The bifurcation analysis indicates that while gravity or flow acting alone is destabilizing, a combination of the two can increase the region of stability. Local bifurcation analysis using the Liapunov–Schmidt approach is complemented by global computations of the nonlinear model equations using pseudoarclength continuation with the AUTO software. Symmetries affect the stability of the bridge solutions in a key way. Singularity theory provides a framework for the unfolding of the two bifurcations (antisymmetric and centro–symmetric branches). A detailed analysis of the governing equation obtains the coefficients in the unfolding. A surprising result is that there is a maximal influence that flow can have on the centro–symmetric disturbances. That is, as flow is increased, imperfection increases up to a certain value, beyond which the imperfection diminishes with increasing flow rate. Singularity theory provides a framework for understanding this and other reports of stabilization of the PR instability.