XIX. On the instability of cylindrical fluid surfaces

Weakly nonlinear nonaxisymmetric oscillations of capillary bridges at small viscosity

“…4 for the first two modes. The functions <p 3 , <p 4 , and <p 5 diverge at the rim of the disks, but this singularity is logarithmic and does not present major difficulty. Note that <p 5 is larger than <p u ..., and <p 4 , especially for n = 1 and for higher-order modes not shown here.…”

“…The functions <p 3 , <p 4 , and <p 5 diverge at the rim of the disks, but this singularity is logarithmic and does not present major difficulty. Note that <p 5 is larger than <p u ..., and <p 4 , especially for n = 1 and for higher-order modes not shown here. This is because <p¡,..., and <p 4 account for the forcing effect of the Stokes boundary layers attached to the disks, which are much weaker than the interface boundary layer responsible for <p 5 .…”

“…where we have taken into account the boundary condition (25) and the expressions (27) and (28) 4 , and a 5 are plotted in Fig. 2.…”

“…a 4 , a 5 , and R s , and on the real functions g, <p¡,..., and <p 5 ; all these are fixed once the liquid and the geometry are selected. The detuning parameter, S, and the re-scaled/orcing amplitude, a, instead can be varied independently through the forcing frequency and amplitude.…”

“…II we shall perform a weakly nonlinear, multiscale analysis in the limit (1), (4). The velocity and the stagnation pressure in the bulk (outside the boundary layers) and the free surface deflection will be given by [cf (22) and (23) -n (18) are the damping rate and detuning (with respect to the natural frequency ft) of the capillary waves and satisfy Q<d<\ and \S\<\.…”