We propose a simple model for the evolution of an inviscid vortex sheet in a potential flow in a channel with parallel walls. This model is obtained by augmenting the Birkhoff–Rott equation with a potential field representing the effect of the solid boundaries. Analysis of the stability of equilibria corresponding to flat sheets demonstrates that in this new model the growth rates of the unstable modes remain unchanged as compared to the case with no confinement. Thus, in the presence of solid boundaries the equilibrium solution of the Birkhoff–Rott equation retains its extreme form of instability with the growth rates of the unstable modes increasing in proportion to their wavenumbers. This linear stability analysis is complemented with numerical computations performed for the nonlinear problem which show that confinement tends to accelerate the growth of instabilities in the nonlinear regime.

中文翻译：

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受限涡流片的稳定性

我们提出了一个简单的模型，用于在具有平行壁的通道中的势流中演化无粘性涡旋片。该模型是通过用代表固体边界效应的势场来增加 Birkhoff-Rott 方程获得的。与平板对应的平衡稳定性分析表明，在这个新模型中，与没有限制的情况相比，不稳定模式的增长率保持不变。因此，在存在固体边界的情况下，Birkhoff-Rott 方程的平衡解保持其极端形式的不稳定性，不稳定模式的增长率与其波数成比例地增加。