Nonlinear Dynamics and Wall Touch-Up in Unstably Stratified Multilayer Flows in Horizontal Channels under the Action of Electric Fields
Nonlinear Dynamics and Wall Touch-Up in Unstably Stratified Multilayer Flows in Horizontal Channels under the Action of Electric Fields
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电场作用下水平通道不稳定分层多层流的非线性动力学和壁修补
DOI：
10.1137/140968070
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发表时间：
2015
影响因子：
1.9
通讯作者：
Barannyk L
中科院分区：
文献类型：
--
作者：
Barannyk L
本研究考虑了当电场垂直于重力方向作用时分层不混溶流体的非线性动力学。详细研究了一个特定的设置，即无限范围的水平通道内的两种分层流体。流体被认为是完美的电介质，并且沿着通道施加恒定的水平场。分隔两种流体的尖锐界面可能支持也可能不支持表面张力，当较重的流体位于顶部时，通常会出现瑞利-泰勒不稳定性。对描述流体层中界面位置和主阶水平速度的新型偏微分方程组进行了分析和计算研究。该系统在一层与第二流体层相比渐近薄的渐近极限下是有效的，因此由于物理设置的多尺度性质而出现非局域静电项。数值求解了空间周期域上的初值问题，结果表明，足够强的电场可以线性稳定瑞利-泰勒不稳定性，从而在时间上产生非常接近驻波的非线性准周期振荡。在存在不稳定性的情况下，系统通常会演化为接触奇点，界面在有限时间内接触上壁，同时主阶水平速度爆炸。精确的数值解与渐近分析相结合表明，如果存在或不存在表面张力，但存在电场，则终端态遵循自相似结构，该结构会有所不同。在存在表面张力的情况下，发现修饰以有界界面梯度但无界曲率发生，静电效应降级为更高阶。然而，如果不存在表面张力，电场就会支持局部尖点结构的修饰，从而使界面梯度本身不受限制。自相似解属于第二类，并且使用大量模拟来提取缩放指数。描述和实现了不同且独立的方法，并且它们之间的一致性非常好。
This study considers the nonlinear dynamics of stratified immiscible fluids when an electric field acts perpendicular to the direction of gravity. A particular setup is investigated in detail, namely, two stratified fluids inside a horizontal channel of infinite extent. The fluids are taken to be perfect dielectrics, and a constant horizontal field is imposed along the channel. The sharp interface separating the two fluids may or may not support surface tension, and the Rayleigh--Taylor instability is typically present when the heavier fluid is on top. A novel system of partial differential equations that describe the interfacial position and the leading order horizontal velocity in the fluid layers is studied analytically and computationally. The system is valid in the asymptotic limit of one layer being asymptotically thin compared to the second fluid layer, and as a result nonlocal electrostatic terms arise due to the multiscale nature of the physical setup. The initial value problem on spatially periodic domains is solved numerically, and it is shown that a sufficiently strong electric field can linearly stabilize the Rayleigh--Taylor instability to produce nonlinear quasi-periodic oscillations in time that are quite close to standing waves. In situations when the instability is present, the system is shown to generically evolve to touch-up singularities with the interface touching the upper wall in finite time while the leading order horizontal velocity blows up. Accurate numerical solutions allied with asymptotic analysis show that the terminal states follow self-similar structures that are different if surface tension is present or absent, but with the electric field present. In the presence of surface tension, the touch-up is found to take place with bounded interfacial gradients but unbounded curvature, with electrostatic effects relegated to higher order. If surface tension is absent, however, the electric field supports touch-up with a local cusp structure so that the interfacial gradients themselves are unbounded. The self-similar solutions are of the second kind and extensive simulations are used to extract the scaling exponents. Distinct and independent methods are described and implemented, and agreement between them is excellent.
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