AI 中文总结
该研究推导了雷诺数远小于1时，具$C_{2v}$和$S_4$对称性的刚性颗粒在粘性流体中沉降的精确解，明确了其准周期运动特性、轨道特征及相关参数的闭式表达式。
AI 中文摘要
我们针对均匀密度、具有$C_{2v}$和$S_4$对称性的刚性颗粒，在雷诺数远小于1的粘性流体中受重力沉降的动力学问题，提供了精确且完整的解。$S_4$对称性使该问题可完全积分，所有轨道由单一守恒量$0\le C\le 1$标记。我们证明，当$0<C<1$时，存在两个不同的时间尺度，导致准周期演化，伴随显著的随时间变化的水平位移。我们将欧拉角的倾斜角$\theta$和自旋角$\psi$表示为时间的周期性雅可比椭圆函数，而方位角$\phi$则通过具有复特征的第三类不完全椭圆积分表示，该积分将$\phi$分解为均匀漂移和严格周期调制。取向周期和漂移率（对应绕重力方向的恒定角速度）以闭式形式推导得出。质心的垂直位移通过时间的周期性不完全椭圆积分表示，周期轨道平均沉降速度简化为完全椭圆积分的单一比值。实验室参考系中运动的水平分量通过三个欧拉角以代数方式精确求得，因此为准周期运动。质心位置的水平分量通常描绘出玫瑰花结状的开放曲线，被两个同心“包络”圆所限制。我们确定了两个包络半径的非常简单的精确表达式，并证明对于旋转-平移耦合减小至零的一类形状，包络半径趋于无穷大。我们还解释了玫瑰花结状轨迹中尖点的起源，并提供了选择严格周期玫瑰花结的可公度性条件。
英文摘要
We provide an exact and complete solution for the dynamics of a rigid particle of uniform density with $C_{2v}$ and $S_4$ symmetry, settling under gravity in a viscous fluid at a Reynolds number much smaller than unity. The $S_4$ symmetry renders the problem exactly integrable, with all orbits labelled by a single conserved quantity $0\le C\le 1$. We show that, for $0<C<1$, there are two different time scales, which lead to quasi-periodic evolution, with a significant time-dependent horizontal displacement. We obtain the tilt $θ$ and spin $ψ$ Euler angles as periodic Jacobi elliptic functions of time, and the azimuthal angle $ϕ$ through an incomplete elliptic integral of the third kind with a complex characteristic, which splits $ϕ$ into a uniform drift and a strictly periodic modulation. The orientation period and the drift rate (corresponding to a constant angular velocity around the gravity direction) follow in a closed form. The vertical centre-of-mass displacement is obtained in terms of periodic in time incomplete elliptic integrals, and the periodic orbit-averaged settling velocity reduces to a single ratio of complete elliptic integrals. The horizontal component of the motion in the laboratory frame of reference is obtained exactly and algebraically in terms of all three Euler angles, so it is quasi-periodic. The horizontal component of the centre-of-mass position traces rosette-like, in general open curves confined by two concentric `envelope' circles. We determine very simple exact expressions for radii of both envelopes and demonstrate that they tend to infinity for a family of shapes with the rotation-translation coupling decreasing to zero. We also explain the origin of cusps at the rosette-like trajectory and provide a commensurability condition selecting strictly periodic rosettes.
Comments31 pages, 9 figures
