The Stability of a Radial Convergence of a Cylindrical Shell Consisting of Viscous Incompressible Liquid

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.