A classical problem in fluid mechanics is the motion of an axisymmetric vortex sheet evolving under the action of surface tension, surrounded by an inviscid fluid. Lagrangian descriptions of these dynamics are well-known, involving complex nonlocal expressions for the radial and longitudinal velocities in terms of elliptic integrals. Here we use these prior results to arrive at a remarkably compact and exact Eulerian evolution equation for the sheet radius $r(z,t)$ in an explicit flux form associated with the conservation of enclosed volume. The flux appears as an integral involving the pairwise mutual induction formula for vortex loop pairs first derived by Helmholtz and Maxwell. We show how the well-known linear stability results for cylindrical vortex sheets in the presence of surface tension and streaming flows [A.M. Sterling and C.A. Sleicher, $J.~Fluid~Mech.$ ${\bf 68}$, 477 (1975)] can be obtained directly from this formulation. Furthermore, the inviscid limit of the empirical model of Eggers and Dupont [$J.~Fluid~Mech.$ $\textbf{262}$ 205 (1994); $SIAM~J.~Appl.~Math.$ ${\bf 60}$, 1997 (2000)], which has served as the basis for understanding singularity formation in droplet pinchoff, is derived within the present formalism as the leading order term in an asymptotic analysis for long slender axisymmetric vortex sheets, and should provide the starting point for a rigorous analysis of singularity formation.

中文翻译：

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轴对称无粘涡片动力学的紧凑欧拉表示

流体力学中的一个经典问题是在表面张力作用下演化的轴对称涡旋片的运动，该涡旋片被无粘性流体包围。这些动力学的拉格朗日描述是众所周知的，涉及以椭圆积分表示的径向和纵向速度的复杂非局部表达式。在这里，我们使用这些先前的结果，以与封闭体积守恒相关的显式通量形式，为片材半径 $r(z,t)$ 得出非常紧凑和精确的欧拉演化方程。通量表现为一个积分，其中涉及由亥姆霍兹和麦克斯韦首先推导出的涡环对的成对互感应公式。我们展示了在存在表面张力和流动的情况下，圆柱形涡流片的众所周知的线性稳定性是如何产生的 [AM Sterling 和 CA Sleicher，$J.~Fluid~Mech.$ ${\bf 68}$, 477 (1975)] 可以直接从这个公式中获得。此外，Eggers 和 Dupont [$J.~Fluid~Mech.$ $\textbf{262}$ 205 (1994); $SIAM~J.~Appl.~Math.$ ${\bf 60}$, 1997 (2000)]，作为理解液滴夹断中奇点形成的基础，是在当前形式主义中作为主要顺序推导出来的细长轴对称涡旋片渐近分析中的术语，应该为奇异形成的严格分析提供起点。