Nonlinear dynamics of capillary bridges: theory

Abstract

Finite-amplitude, forced and free oscillations of capillary bridges are studied. They are characterized by a resonant frequency and a damping rate which, in turn, depend on fluid properties, dimensions of the bridge, gravitational force relative to surface tension and amplitude of the external disturbance. The Navier–Stokes equations are solved numerically using the Galerkin/finite-element methodology for discretization in space and implicit finite differences with adaptive time stepping for discretization in time. It is found that the resonant frequency decreases and the damping rate increases almost linearly with the oscillation amplitude. Their relative changes from their corresponding values at infinitesimal amplitude depend on fluid properties and dimensions of the bridge. Moreover, careful measurement of the resonant frequency and damping rate in a well-controlled experiment may provide quite accurate values for properties of the liquid over a wide range of modified Reynolds numbers.

Weakly nonlinear nonaxisymmetric oscillations of capillary bridges at small viscosity

“…Thus the coefficients of the direct forcing term are the same in Eqs. (12) and (13). Different forcing coefficients result from applying non reflection-symmetric forcing.…”

“…Nonlinear effects have been only considered in the strictly axisymmetric case in a few works, most of them based on direct numerical simulations, which become quite expensive as viscous effects decrease. 13 A weakly nonlinear analysis of resonant, axisymmetric oscillations of capillary bridges at small viscosity and small driving amplitude was performed by two of us. 1415 The outcome in Ref.…”

Chaotic oscillations in a nearly inviscid, axisymmetric capillary bridge at 2:1 parametric resonance

“…Their specific feature here is that they include a two-term approximation to the damping rate, a 4k \¡C + a 5k C. In so doing, we are anticipating that a 4k is typically about 10~2 times a 5k ; thus, if only the leading-order term were considered, then the resulting approximation would be useful only for extremely small valúes of C (roughly C=s 10~5) while the two-term approximation gives reasonably good results for CslO -2 . A straightforward orders-of-magnitude analysis readily shows that the first term, a 4k^C , accounts for viscous dissipation in the Stokes boundary layers and the second term, a 5k C, comes from viscous dissipation in the bulk and a first correction of viscous dissipation in the Stokes boundary layers; viscous dissipation in the interface boundary layer and in the córner tori provides higher-order terms [of orders eC 32 and e(C log Q 2 , respectively] in (18). Notice also that if HOT are neglected, then Eqs.…”

“…Notice also that if HOT are neglected, then Eqs. (18) are decoupled from the secondary streamingflow. This is so because the streaming velocity is roughly of the order of e 2 (see Ref 19) and thus its effect on (18) cannot be larger than e 3 (in fact, it is even smaller than that because, the oscillating flow being axisymmetric, the leading e 3 effect identically vanishes, as in the analysis in Ref.…”

“…7 Theoretical analyses of mechanical oscillations of liquid bridges have been mainly based on one-dimensional approximations 81: (which exhibit several limitations 12 ) and/or concerned with the linear approximation. 71317 Among the few exceptions to this rule, there are two recent papers by Chen and Tsamopoulos 18 and Nicolás and Vega, 19 who analyzed fully nonlinear oscillations by direct numerical simulation and weakly nonlinear oscillations by asymptotic methods, respectively. These two papers may be seen as complementary of each other.…”