AI 中文总结
本文研究两粗糙平板间液体的毛细行为，推导能量Γ-展开得到降维模型，并用于相场近似数值模拟，为相关毛细模型提供理论依据。
AI 中文摘要
我们研究了限制在两个粗糙固体平板之间的给定体积液体的行为。当平板间距相对于液体体积较小时，预计会形成毛细桥，这些毛细桥局部极小化高斯毛细能量。我们推导了当平板间距趋近于零时能量的$\Gamma$-展开，从而得到一个关于平板上润湿区域的降维问题。在首阶，能量由润湿区域的面积决定，而下一阶项则由其周长给出，并乘以相对粘附系数的适当函数加权。这为随后的相场近似提供了框架，该近似被用于数值模拟，以研究在平板正常移动下液滴的演化。该理论也为假设液体填充粗糙地形至开尔文半径的毛细模型提供了合理性依据。
英文摘要
We study the behaviour of a given volume of liquid confined between two rough solid plates. When the separation between the plates is small relative to the liquid volume, capillary bridges are expected to form, which minimise Gauss's capillary energy locally. We derive a $Γ$-expansion for the energy as the plate separation approaches zero, yielding a dimensionally reduced problem in terms of the wetted regions on the plates. At leading order, the energy is determined by the area of the wetted regions, while the next order term is given by their perimeter, weighted by appropriate functions of the relative adhesion coefficients. This provides a framework for a subsequent phase-field approximation, which is employed in numerical simulations to study the evolution of the droplets under the normal movement of the plates. The theory also gives justification to models for capillarity that assume that liquids fill up the rough topography to the Kelvin radius.
Comments29 pages, 7 figures
