Abstract

The problem of the nonlinear stability of the radial collapse of a cylindrical shell, which is filled with a viscous incompressible fluid of uniform density, is studied. A number of assumptions are made: (1) vacuum is contained inside the shell; (2) it is surrounded by a layer of compressed polytropic gas, which serves as a product of instant detonation and exerts constant pressure on the outer surface of the shell; (3) vacuum is also behind the gas layer. The absolute instability of the radial collapse of the considered viscous cylindrical shell with respect to finite perturbations of the same symmetry type is established by the direct Lyapunov method. A Lyapunov function that satisfies all of the conditions of the first Lyapunov instability theorem, regardless of the specific mode of radial convergence, is constructed. This result fully confirms Trishin’s corresponding hypothesis and is a rigorous mathematical proof that the cumulation of kinetic energy of a viscous incompressible fluid of uniform density in the process of radial collapse of the studied cylindrical shell to its axis occurs exclusively at its impulse stage.

中文翻译：

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由粘性不可压缩液体组成的圆柱壳径向收敛的稳定性

摘要

研究了填充有均匀密度的粘性不可压缩流体的圆柱壳径向坍塌的非线性稳定性问题。进行了许多假设：（1）外壳内部包含真空；（2）被一层压缩的多向性气体所包围，该气体用作瞬间爆炸的产物，并在壳体的外表面施加恒定的压力；（3）真空还在气体层后面。通过直接Lyapunov方法确定了考虑的粘性圆柱壳径向坍塌相对于相同对称类型的有限摄动的绝对不稳定性。构造一个满足第一个Lyapunov不稳定性定理的所有条件的Lyapunov函数，而不管径向收敛的具体模式如何。