Frequency and damping of non-axisymmetric surface oscillations of a viscous cylindrical liquid bridge

Abstract

We present a semi-analytic solution for the non-axisymmetric oscillations of a viscous cylindrical free-standing liquid bridge formed between two coaxial discs of radius R. Even though a streamfunction does not exist, a Helmholtz decomposition is used to obtain an analytic representation of the velocity field. An eigenvalue problem is formulated by projecting the free-surface boundary conditions onto a suitable space of test functions. This is then solved iteratively along with the dispersion relation obtained from the satisfaction of endwall boundary conditions. Extensive comparison with previous theoretical and numerical results, for a range of Reynolds number and bridge slenderness ratio, shows very good agreement in most cases. The present solution generalises that of Tsamopoulos, Chen & Borkar (J. Fluid Mech., vol. 235, 1992, p. 579), which employed a streamfunction formulation and was for the axisymmetric case.