Abstract
The stability of thin viscous sheets has been studied so far in the special case where the base flow possesses a direction of invariance: the linear stability is then governed by an ordinary differential equation. We propose a mathematical formulation and a numerical method of solution that are applicable to the linear stability analysis of viscous sheets possessing no particular symmetry. The linear stability problem is formulated as a non-Hermitian eigenvalue problem in a 2D domain and is solved numerically using the finite-element method. Specifically, we consider the case of a viscous sheet in an open flow, which falls in a bath of fluid; the sheet is mildly stretched by gravity and the flow can become unstable by ‘curtain’ modes. The growth rates of these modes are calculated as a function of the fluid parameters and of the geometry, and a phase diagram is obtained. A transition is reported between a buckling mode (static bifurcation) and an oscillatory mode (Hopf bifurcation). The effect of surface tension is discussed.
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