标题： 薄层直线 Couette 流的屈曲不稳定性 显示英文标题

标题： Buckling instability of a thin-layer rectilinear Couette flow

摘要： 我们分析了当受到简单剪切时，薄粘性片材的屈曲稳定性，使用两种模型来提供主导的面外模式开始的条件：(i) 粘性板动力学的渐近理论，以及(ii) 全部的斯托克斯方程。 在任何情况下，板通过粘性阻力、表面张力和相对于下方密度较大的流体的浮力的组合而稳定。 在厚度趋于零的极限下，板在剪切速率$\gamma/(\mu d)$时发生屈曲，与浮力无关，其中 2d 是板的厚度，$\gamma$是上下表面之间的平均表面张力，$\mu$是流体粘度。 对于由上下表面相等的表面张力稳定的较厚板，在开始时及之后，最不稳定的模式具有中等波长，在中心线框架中是静止的，跨越板的宽度，波峰和波谷大约与墙壁成$45^\circ$的角度，并且非常类似于弹性剪切模式。 能够发生屈曲的最厚板的纵横比（厚度/宽度）约为$\EC$，并且仅由内部粘性阻力稳定。 我们表明，粘性板模型只能准确描述极薄板的屈曲开始，但对于开始后的最不稳定模式提供了极好的描述。 最后，我们表明，通过修改板模型以包含对流并使模型符合材料参考系不变性，可以扩展其预测能力，以描述相对较短的行进波。 显示英文摘要

摘要： We analyse the buckling stability of a thin, viscous sheet when subject to simple shear, providing conditions for the onset of the dominant out-of-plane modes using two models: (i) an asymptotic theory for the dynamics of a viscous plate and (ii) the full Stokes equations. In either case, the plate is stabilised by a combination of viscous resistance, surface tension and buoyancy relative to an underlying denser fluid. In the limit of vanishing thickness, plates buckle at a shear rate $\gamma/(\mu d)$ independent of buoyancy, where 2d is the plate thickness, $\gamma$ is the average surface tension between the upper and lower surfaces and $\mu$ is the fluid viscosity. For thicker plates stabilised by an equal surface tension at the upper and lower surfaces, at and above onset, the most unstable mode has moderate wavelength, is stationary in the frame of the centre-line, spans the width of the plate with crests and troughs aligned at approximately $45^\circ$ to the walls and closely resembles elastic shear modes. The thickest plates that can buckle have an aspect ratio (thickness/width) approximately $\EC$ and are stabilised only by internal viscous resistance. We show that the viscous plate model can only accurately describe the onset of buckling for vanishingly thin plates but provides an excellent description of the most unstable mode above onset. Finally, we show that by modifying the plate model to incorporate advection and make the model material frame-invariant, it is possible to extend its predictive power to describe relatively short, travelling waves.