二维电对流中的非线性时空不稳定性
Nonlinear spatiotemporal instabilities in two-dimensional electroconvective flows
摘要：
本文研究了通流对两平板间介电液体二维电流体力学（EHD）流动的影响，作为进一步理解电场存在下流体力学的模型问题。液体受到来自底板的强单极电荷注入和沿流向方向的压力梯度的影响。高精度的数值模拟和弱非线性稳定性分析的基础上的多尺度扩展和振幅扩展方法被用来解开非线性时空不稳定机制，在这个组合流。我们发现通流使EHD流动中的磁滞回线变窄。在脉冲响应的数值模拟中，非线性区域内的波包的前缘和后缘与线性波包的前缘和后缘一致，这一结果也与自然对流中的结果进行了验证。此外，由于EHD-Poiffille流的分叉具有亚临界性质，因此在亚临界区域存在非线性有限振幅解，并且我们的计算表明它们是对流不稳定的。Ginzburg-Landau方程（GLE）是由准静电极限下的纳维-斯托克斯方程和Maxwell方程的弱非线性展开式导出的，它可以作为一个物理降阶模型来研究这种流动的时空动力学。我们发现，使用振幅展开方法计算的GLE中的系数可以预测的绝对增长率，即使当参数远离线性临界条件，与本地色散关系相比，而来自多尺度展开方法的GLE的有效范围仅限于线性临界条件附近。
英文摘要：
This work studies the effects of a through-flow on two-dimensional electrohydrodynamic (EHD) flows of a dielectric liquid confined between two plane plates, as a model problem to further our understanding of the fluid mechanics in the presence of an electric field. The liquid is subjected to a strong unipolar charge injection from the bottom plate and a pressure gradient along the streamwise direction. Highly-accurate numerical simulations and weakly nonlinear stability analyses based on multiple-scale expansion and amplitude expansion methods are used to unravel the nonlinear spatiotemporal instability mechanisms in this combined flow. We found that the through-flow makes the hysteresis loop in the EHD flow narrower. In the numerical simulation of an impulse response, the leading and trailing edges of the wavepacket within the nonlinear regime are consistent with the linear ones, a result which we also verified against that in natural convection. In addition, as the bifurcation in EHD-Poiseuille flows is of a subcritical nature, nonlinear finite-amplitude solutions exist in the subcritical regime, and our calculation indicates that they are convectively unstable. The validity of the Ginzburg-Landau equation (GLE), derived from the weakly nonlinear expansion of Navier-Stokes equations and the Maxwell's equations in the quasi-electrostatic limit, serving as a physical reduced-order model for probing the spatiotemporal dynamics in this flow, has also been investigated. We found that the coefficients in the GLE calculated using amplitude expansion method can predict the absolute growth rates even when the parameters are away from the linear critical conditions, compared favourably with the local dispersion relation, whereas the validity range of the GLE derived from the multiple-scale expansion method is confined to the vicinity of the linear critical conditions.
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