  Oscillation of sessile drops is important to many applications. In the
present study, the natural oscillation of a sessile drop on flat surfaces with
free contact lines (FCL) is investigated through numerical and theoretical
analysis. The FCL condition represents a limit of contact line mobility, i.e.
the contact angle remains constant when the contact line moves. In the
numerical simulation, the interfaces are captured by the volume-of-fluid method
and the contact angle at the boundary is specified using the height-function
method. The oscillation frequencies for sessile drops with FCL are mainly
controlled by the contact angle and the Bond number and a parametric study is
carried out to characterize their effects on the frequencies for the first and
high-order modes. Particular attention is paid to the frequency of the first
mode, since it is usually the dominant mode. An inviscid theoretical model for
the first mode is developed. The model yields an explicit expression for the
first-mode frequency as a function of the contact angle and the Bond number,
with all parameters involved fully determined by the equilibrium drop theory
and the simulation. The predicted frequencies for a wide range of contact
angles agree very well with the simulation results for small Bond numbers. The
frequencies for both the first and high-order modes decrease with the contact
angle and increase with the Bond number. For the high-order modes, the
frequencies for different modes generally scale with the Rayleigh frequencies.
The scaling relation performs better for small Bond numbers and large contact
angles. A simple model is proposed to predict the frequencies of high-order
modes for large contact angles and a good agreement with the simulation results
is observed.
