具有记忆的惯性粒子的渐近动力学
Asymptotic dynamics of inertial particles with memory
摘要：
最近的实验和数值观测表明，Basset-Boussinesq记忆项的意义上的小球形刚性粒子（或惯性粒子）悬浮在周围的流体流动的动力学。这些观察表明，代数衰减到一个渐近状态，而不是指数收敛的记忆项的情况下。在这里，我们证明了所观察到的代数衰减是Maxey-Riley方程的一个普遍性质。具体地，颗粒速度在时间上代数地衰减到接近流体速度的极限$\mathcal O(ε)$，其中$0<ε\ll 1$与颗粒半径与流体特征长度尺度之比的平方成比例。这些结果如下从一个尖锐的分析上限，我们推导出的粒子速度。为了完整性，我们还提出了第一个证明的存在性和唯一性的整体解的Maxey-Riley方程，一个非线性系统的分数阶微分方程。
英文摘要：
Recent experimental and numerical observations have shown the significance of the Basset--Boussinesq memory term on the dynamics of small spherical rigid particles (or inertial particles) suspended in an ambient fluid flow. These observations suggest an algebraic decay to an asymptotic state, as opposed to the exponential convergence in the absence of the memory term. Here, we prove that the observed algebraic decay is a universal property of the Maxey--Riley equation. Specifically, the particle velocity decays algebraically in time to a limit that is $\mathcal O(ε)$-close to the fluid velocity, where $0<ε\ll 1$ is proportional to the square of the ratio of the particle radius to the fluid characteristic length-scale. These results follows from a sharp analytic upper bound that we derive for the particle velocity. For completeness, we also present a first proof of existence and uniqueness of global solutions to the Maxey--Riley equation, a nonlinear system of fractional-order differential equations.
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