微尺度电渗流体动力学隐身与屏蔽的数学理论
A mathematical theory of microscale hydrodynamic cloaking and shielding by electro-osmosis
摘要：
在本文中，我们开发了一个通用的数学框架，完美的和近似的流体动力学隐身和屏蔽的电渗流，这是由耦合PDE系统通过场效应电渗。我们首先利用层位技术建立了耦合系统解的表示式。基于傅里叶级数，导出了控制区截面形状为圆环或共焦椭圆时理想的流体动力学隐身和屏蔽条件。然后，我们进一步提出了一个优化方案，在一般几何形状的近似斗篷和盾牌的设计。证明了该最优化问题的适定性。特别地，还建立了一般几何条件下近似隐身和屏蔽的条件。我们的理论研究结果得到验证和补充的各种数值结果。本文的结果也为更复杂的流体动力隐身和屏蔽提供了数学基础。
英文摘要：
In this paper, we develop a general mathematical framework for perfect and approximate hydrodynamic cloaking and shielding of electro-osmotic flow, which is governed by a coupled PDE system via the field-effect electro-osmosis. We first establish the representation formula of the solution of the coupled system using the layer potential techniques. Based on Fourier series, the perfect hydrodynamic cloaking and shielding conditions are derived for the control region with the cross-sectional shape being annulus or confocal ellipses. Then we further propose an optimization scheme for the design of approximate cloaks and shields within general geometries. The well-posedness of the optimization problem is proved. In particular, the condition that can ensure the occurrence of approximate cloaks and shields for general geometries are also established. Our theoretical findings are validated and supplemented by a variety of numerical results. The results in this paper also provide a mathematical foundation for more complex hydrodynamic cloaking and shielding.
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