Erbaş A, Podgornik R, Netz R R
Physics Department, Technical University Munich, Garching, Germany.
Eur Phys J E Soft Matter. 2010 Jun;32(2):147-64. doi: 10.1140/epje/i2010-10610-7. Epub 2010 Jun 25.
We consider the linearized time-dependent Navier-Stokes equation including finite compressibility and viscosity. We first constitute the Green's function, from which we derive the flow profiles and response functions for a plane, a sphere and a cylinder for arbitrary surface slip length. For high driving frequency the flow pattern is dominated by the diffusion of vorticity and compression, for low frequency compression propagates in the form of sound waves which are exponentially damped at a screening length larger than the sound wave length. The crossover between the diffusive and propagative compression regimes occurs at the fluid's intrinsic frequency omega approximately c2rho0/eta, with c the speed of sound, rho0 the fluid density and eta the viscosity. In the propagative regime the hydrodynamic response function of spheres and cylinders exhibits a high-frequency resonance when the particle size is of the order of the sound wave length. A distinct low-frequency resonance occurs at the boundary between the propagative and diffusive regimes. Those resonant features should be detectable experimentally by tracking the diffusion of particles, as well as by measuring the fluctuation spectrum or the response spectrum of trapped particles. Since the response function depends sensitively on the slip length, in principle the slip length can be deduced from an experimentally measured response function.
我们考虑包含有限可压缩性和粘性的线性时变纳维-斯托克斯方程。我们首先构造格林函数，由此推导出任意表面滑移长度下平面、球体和圆柱体的流动剖面及响应函数。对于高频驱动，流动模式由涡度扩散和压缩主导；对于低频，压缩以声波形式传播，且在大于声波波长的屏蔽长度处呈指数衰减。扩散压缩 regime 与传播压缩 regime 之间的转变发生在流体的固有频率ω约为c²ρ₀/η处，其中c为声速，ρ₀为流体密度，η为粘性。在传播 regime 中，当颗粒尺寸与声波波长量级相当时，球体和圆柱体的流体动力学响应函数呈现高频共振。在传播 regime 与扩散 regime 的边界处出现明显的低频共振。通过追踪颗粒的扩散，以及测量捕获颗粒的涨落谱或响应谱，这些共振特征在实验上应是可检测的。由于响应函数对滑移长度敏感，原则上可从实验测量的响应函数推导出滑移长度。
