This study concerns a two-dimensional liquid drop surrounded by gas and attached to a sinusoidally vibrating wall. Gravity is neglected and the moving contact lines are modelled using a Navier-type boundary condition at the wall and a prescribed contact angle, $\overline{\theta }$, which can take any value in the range $0<\overline{\theta }<\pi$. The vibration amplitude, and hence the departure from equilibrium of the drop, is assumed sufficiently small that the problem can be linearized. Wall vibration can have components both normal and tangential to the wall. The solution of the linear problem can be expressed as the sum of two decoupled components corresponding to the response to purely normal and purely tangential vibration, which are respectively symmetric and antisymmetric with respect to reflection in the symmetry plane of the equilibrium drop. Asymptotic analysis of the drop oscillations for small Ohnesorge number, $\text{Oh}$, brings out two distinct damping mechanisms, both of which are accounted for. One, arising from viscous dissipation in the regions near the contact lines, is characterized by a parameter $\beta$. The other comes from the boundary layer at the wall and is of order ${\text{Oh}}^{1/2}$. The small-$\text{Oh}$ problem has been implemented numerically. As expected, lightly damped normal modes are found to have resonant response close to their inviscid oscillation frequencies. Damping coefficients for each of the two damping mechanisms and lightly damped modes are determined as a function of contact angle. The relative importance of boundary-layer and contact-line damping is quantified and found to depend both on the contact angle and on $\beta /{\text{Oh}}^{1/2}$. Cases can be found in which the two damping mechanisms have comparable effects, as well as others for which one or other of the mechanisms is dominant. Comparison with DNS, which allows for nonlinear effects and has the same contact-line model, shows agreement for a particular case having small $\text{Oh}$ and small wall displacement amplitude.

这项研究涉及被气体包围并附着在正弦振动壁上的二维液滴。忽略重力，使用壁上的 Navier 型边界条件和规定的接触角对移动接触线进行建模，$\overline{\theta }$, 可以取范围内的任何值 $0<\overline{\theta }<\pi$. 振动幅度，以及因此对液滴平衡的偏离，被假定为足够小，问题可以被线性化。壁振动可以具有与壁垂直和相切的分量。线性问题的解可以表示为对应于对纯法向振动和纯切向振动的响应的两个解耦分量之和，这两个分量分别相对于平衡液滴对称平面中的反射对称和反对称。小 Ohnesorge 数下降振荡的渐近分析，$\text{哦}$，带来了两种不同的阻尼机制，这两种机制都被考虑在内。一种是由接触线附近区域的粘性耗散引起的，其特征在于参数$\beta$. 另一个来自壁面的边界层，是有序的${\text{哦}}^{1/2}$. 小——$\text{哦}$问题已经在数值上实现了。正如预期的那样，发现轻微阻尼的正常模式具有接近其无粘性振荡频率的共振响应。两种阻尼机制和轻阻尼模式中每一种的阻尼系数被确定为接触角的函数。边界层和接触线阻尼的相对重要性被量化并发现取决于接触角和$\beta /{\text{哦}}^{1/2}$. 可以找到两种阻尼机制具有类似效果的情况，以及其中一种或其他机制占主导地位的其他情况。与允许非线性效应并具有相同接触线模型的 DNS 的比较表明，对于具有较小$\text{哦}$ 壁位移幅度小。