  The computation of damping rates of an oscillating fluid with a free surface
in which viscosity is small and surface tension high is numerically
challenging. A typical application requiring such computation is drop-on-demand
(DoD) microfluidic devices that eject liquid metal droplets, where accurate
knowledge of damping rates for the least-damped oscillation modes following
droplet ejection is paramount for assessing jetting stability at higher jetting
frequencies. Computational fluid dynamics (CFD) simulations often struggle to
accurately predict meniscus damping unless a very fine discretization is
adopted, so calculations are computationally expensive. The faster alternative
we adopt is to compute damping rates directly from the eigenvalues of the
linearized problem. The surface tension term in Stokes or sloshing problems
requires approximation of meniscus displacements, which introduces additional
complexity in their numerical solution. We consider the combined effects of
viscosity and surface tension, approximate the meniscus displacements, and
construct a finite element method to compute the fluid's oscillation modes. We
prove that the method is free of spurious modes with zero or positive damping
rates, and we implement it with Taylor-Hood elements for velocity and pressure,
and with continuous piecewise quadratic elements for meniscus displacement. We
verify the numerical convergence of the method by reproducing the solution to
an analytical benchmark problem and two more complex examples with axisymmetric
geometry. We obtain the spatial shape and temporal evolution (angular frequency
and damping rate) of the set of least-damped oscillation modes in minutes,
compared to days for a CFD simulation. The method's ability to quickly generate
accurate estimates of fluid oscillation damping rates makes it suitable for
integration into design loops for prototyping microfluidic nozzles.
