AI 中文总结
该研究通过实验与理论，将经典最速降线问题扩展到粘性流体中的球形粒子，发现时间与能量最小化路径偏离经典摆线，存在St_p参数，最快路径为近乎直道加局部摆线，能量受限的最快路径为S形，统一了耗散介质输运的时间与能量最优性。
AI 中文摘要
我们通过实验和理论研究经典最速降线问题的流体力学类比：在粘性流体中沿斜面滚动的球形粒子的「时间最小化」和「能量最小化」路径。研究表明，在粘性耗散存在时，最小值路径从经典摆线偏离为曲率相反的曲线，其特征为无量纲有效参数St_p，代表粒子粘性响应时间尺度与重力时间尺度的比值。利用广义变分框架，我们发现最快路径简化为「近乎笔直的斜坡」，但起始和终止于局部曲率摆线，曲率κ_c~St_p⁻²。值得注意的是，给定能量预算下的最快下降路径需要沿非单调的「S形」路径运动，且存在内部拐点。我们的发现揭示了耗散介质中输运的时间最优性与能量最优性的统一，并将著名的最速降线解扩展到流体力学领域。
英文摘要
We experimentally and theoretically study the hydrodynamic analog of the classical brachistochrone problem: the {\it time-} and {\it energy-minimizing} paths for a spherical particle rolling down an incline within a viscous fluid. We show that in the presence of viscous dissipation, the paths of minima diverge from the classical cycloid, into curves of opposing curvature for time and energy, and are characterized by an effective dimensionless parameter, $St_p$, representing the ratio of the particle's viscous response time scale to its gravitational time scale. Using a generalized variational framework, we show that the fastest path reduces to {nearly straight ramps}, however, beginning and terminating in localized cycloids of curvature, $κ_c \sim St_p^{-2}$. Remarkably, the path of fastest descent on a given energy budget requires navigating a non-monotonic path ({\it``S-shaped''}) with an interior point of inflection. Our findings reveal a unification of temporal and energetic optimality for transport through dissipative media, and expand the celebrated brachistochrone solutions to the hydrodynamic regime.
Comments5 pages, 4 figures
