Stability and Bifurcation of a Soap Film Spanning a Flexible Loop
Stability and Bifurcation of a Soap Film Spanning a Flexible Loop
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跨越柔性环的皂膜的稳定性和分叉
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
E. Fried
中科院分区：
文献类型：
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作者：
Yi;E. Fried
欧拉高原问题，由Giomi和Mahadevan在Proc. R. Soc提出。Lond。爵士。一、数学。理论物理。Eng。科学进展，468,1851-1864(2012)，涉及一种跨越柔性环的肥皂膜。薄膜和线圈的形状是由这两个组成部分之间的相互作用决定的。在目前的工作中，欧拉高原问题被重新表述为产生一个向量场的边值问题，该问题参数化了跨越表面和边界环。利用相关自由能泛函的第一次和第二次变分，进行了详细的分岔和稳定性分析。对于能量密度为σ的跨面和长度为2πR、弯曲刚度为a的边界环，在R3σ/a=3时出现了跨面保持平坦但边界环变为非圆的第一次分岔，证实了之前通过能量比较得到的结果。所有从平面圆形解分支发散出来的分支，包括那些非平面解分支，都是不稳定的。
The Euler–Plateau problem, proposed by Giomi and Mahadevan in Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 468, 1851–1864 (2012), concerns a soap film spanning a flexible loop. The shapes of the film and the loop are determined by the interactions between the two components. In the present work, the Euler–Plateau problem is reformulated to yield a boundary-value problem for a vector field that parameterizes both the spanning surface and the bounding loop. Using the first and second variations of the relevant free-energy functional, detailed bifurcation and stability analyses are performed. For a spanning surface with energy density σ and a bounding loop with length 2πR and flexural rigidity a, the first bifurcation, during which the spanning surface remains flat but the bounding loop becomes noncircular, occurs at R3σ/a=3, confirming a result obtained previously via an energy comparison. All other bifurcation solution branches emanating from the flat circular solution branch, including those to nonplanar solution branches, are found to be unstable.