AI 中文总结
研究可压缩毛细液滴临界半径，通过推导粘性可压缩方程和自由表面应力条件得出径向色散关系，用热力学论证其物理意义，还与Cahn-Hilliard自由能模型比较，揭示临界半径与扩散界面长度尺度关系及对不稳定性的影响。
AI 中文摘要
可压缩毛细液滴与不可压缩液滴不同，其半径是一个动态自由度。当液滴足够小时，球形模式可能变得不稳定，由表面张力和可压缩性之间的竞争确定临界半径。本文研究该临界半径的稳健性和物理意义。从粘性可压缩方程和自由表面应力条件出发，推导径向色散关系，表明剪切和体粘度改变特征值但不改变起始半径。热力学论证表明相同半径是均匀压缩液滴能量从局部稳定变为不稳定的点。该能量解释也适用于狭义相对论流体。然后将临界半径与两个Cahn-Hilliard自由能模型中的扩散界面长度尺度比较，确定在尖锐界面描述的有效性范围内是否会发生不稳定性。在对称四次模型中，临界半径远小于界面厚度，在整个范围内不存在不稳定性。在浅阱模型中，临界半径可能参数性地大于界面厚度，存在一系列不稳定的尖锐界面液滴。因此是否存在这样一个范围取决于扩散界面自由能模型。
英文摘要
A compressible capillary drop differs from an incompressible one in that its radius is a dynamical degree of freedom. When the drop is sufficiently small, this spherical mode can become unstable, defining a critical radius set by the competition between surface tension and compressibility. This paper examines the robustness and physical meaning of that critical radius. For a non-relativistic viscous fluid, starting from the viscous compressible equations and the free-surface stress condition, we derive the radial dispersion relation and show that shear and bulk viscosities change the eigenvalues but not the onset radius. A thermodynamic argument identifies the same radius as the point where the energy of a uniformly compressed drop changes from locally stable to unstable, explaining why the threshold is not set by viscous dissipation. This energetic interpretation can also be applied to a special-relativistic fluid, where the non-relativistic mass-density factor is replaced by the corresponding enthalpy-density factor. We then compare the critical radius with diffuse-interface length scales in two Cahn--Hilliard free-energy models to determine whether the instability can occur within the range of validity of the sharp-interface description. In a symmetric quartic model the critical radius is much smaller than the interface thickness, so the instability is absent throughout that range. In a shallow-well model, however, the critical radius can become parametrically larger than the interface thickness, leaving a range of sharp-interface drops that are unstable. Whether such a range exists therefore depends on the diffuse-interface free-energy model.
Comments24 pages, 7 figures, no table
