论文标题
关于时间周期性边界数据在粘性液体中对刚体的自我塑造
On the Self-Propulsion of a Rigid Body in a Viscous Liquid by Time-Periodic Boundary Data
论文作者
论文摘要
考虑一个刚体的身体，即$ \ Mathscr b $，它被限制在无界粘性液体中的翻译运动中被限制。驾驶机制是时间周期性速度场的给定分布，$ \ bfv _*$，在接口体液体上，幅度为$δ$（在适当的功能类中）。主要目的是在$ \ bfv _*$上找到条件，以确保$ \ mathscr b $执行非零净运动，即，$ \ mathscr b $可以在有限的时间内覆盖任何给定的距离。问题的方法取决于一段时间内的平均值$ \ bfv _*$是（案例（b））还是不是（案例（a））相同的零。在（a）的情况下，我们以相对直接的方式解决了该问题，通过表明，对于小$δ$，它减少了研究合适的AMD良好评估的时间依赖性的Stokes（线性）问题。但是，在（b）的情况下，问题要复杂得多，因为我们表明{\ em不能}将其带入线性问题。因此，在（b）的情况下，自我推测是我们通过矛盾论证方法直接在非线性系统上解决的真正非线性问题。通过这种方式，我们也能够在（b）中给出足够的自我推测条件（对于小$δ$）。最后，我们通过反例证明，这种情况通常是必要的。
Consider a rigid body, $\mathscr B$, constrained to move by translational motion in an unbounded viscous liquid. The driving mechanism is a given distribution of time-periodic velocity field, $\bfv_*$, at the interface body-liquid, of magnitude $δ$ (in appropriate function class). The main objective is to find conditions on $\bfv_*$ ensuring that $\mathscr B$ performs a non-zero net motion, namely, $\mathscr B$ can cover any given distance in a finite time. The approach to the problem depends on whether the averaged value of $\bfv_*$ over a period of time is (case (b)) or is not (case (a)) identically zero. In case (a) we solve the problem in a relatively straightforward way, by showing that, for small $δ$, it reduces to the study of a suitable amd well-investigated time-dependent Stokes (linear) problem. In case (b), however, the question is much more complicated, because we show that it {\em cannot} be brought to the study of a linear problem. Therefore, in case (b), self-propulsion is a genuinely nonlinear issue that we solve directly on the nonlinear system by a contradiction argument approach. In this way, we are able to give, also in case (b), sufficient conditions for self-propulsion (for small $δ$). Finally, we demonstrate, by means of counterexamples, that such conditions are, in general, also necessary.
