通道中Yih黏度跳跃界面的镇定

中文摘要 AI 辅助

两种不混溶的黏性流体在压力驱动的通道流中，当它们的黏度不同时，即使雷诺数任意小，其界面也可能变得不稳定——这是Yih在1967年发现的不稳定性。不稳定状态正是界面本身：它是域内的一条曲线，将两种流体分隔开，其位移由自身的偏微分方程控制，并带动流体域随之移动。执行器仅位于一个壁面上，因此被控状态需要穿过流体才能达到。我们通过该壁面两个速度分量的静态反馈，结合作用于两层流体的Fredholm反步变换，以低于流向平均流所设限值的任意速率镇定该界面。目标系统是平移后的两流体Stokes问题，保留流体间的应力传递，并移除界面与流体的耦合。关于两流体谱的两个发现支撑了该设计：在界面处连接的两层特征值序列，当层间解耦时会在稠密的层比集合上碰撞，但它们在均匀地相互回避，因此无需指定退化；留给黏度的最慢模态不是体模态，而是界面的毛细弛豫，其速率在波数上是线性的而非二次的，这设定了可执行波数的频带。该控制器在非线性两流体通道上进行了测试，并在饱和界面波出现时启动。

英文摘要

Two immiscible viscous fluids in pressure-driven channel flow can be unstable through their interface when their viscosities differ, at arbitrarily small Reynolds number --- the instability Yih found in 1967. The unstable state is the interface itself: a curve inside the domain, separating the two fluids, whose displacement is governed by a partial differential equation of its own and which moves the fluid domain with it. Actuation is at one wall only, so the state to be controlled is reached across a fluid. We stabilize this interface at any rate below the limit set by the streamwise mean flow, by static feedback of the two velocity components of that wall, through a Fredholm backstepping transformation acting jointly on the two layers. The target is the two-fluid Stokes problem, shifted, with the transmission of stress between the fluids kept and the coupling of the interface into them removed. Two findings about the two-fluid spectrum carry the design: joined at the interface, the eigenvalue sequences of the two layers, which collide for a dense set of layer ratios when the layers are uncoupled, avoid each other uniformly, so no degeneracy has to be assigned; and the slowest mode left to viscosity is not a bulk mode but the capillary relaxation of the interface, whose rate is linear rather than quadratic in the wavenumber, which sets the band of actuated wavenumbers. The controller is tested on the nonlinear two-fluid channel, switched on against a saturated interfacial wave.

发表机构

• University of California, San Diego（加利福尼亚大学圣地亚哥分校）

机构由 AI 辅助整理，请以论文原文为准。