Abstract: Cumulative phenomena, which are accompanied by concentration at a point, along a straight line or on a plane of force, energy, or another physical quantity, are still of interest to specialists. In this report, we consider the nonlinear stability of the radial collapse of a cylindrical shell, which is filled with a viscous incompressible fluid of uniform density, with respect to perturbations of the same symmetry. A number of assumptions are made: (1) a 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) a vacuum is also behind the gas layer. On one hand, this problem is associated with the cumulative processes of energy concentration along a straight line. On the other hand, it is directly related to the acceleration of bodies via the detonation products of explosives, which, in turn, is a key issue in high energy density physics, the mechanics of impulse processes, and explosion physics. Of the wide range of explosive throwing methods, this work studies only the dynamic loading of a cylindrical cumulative lining by an explosion. This loading is carried out with powerful condensed explosives. Detonation products, impacting the shell, accelerate it. It is important that inertial forces are decisive in the deformation of a thin shell, while the effects of the strength make themselves felt for thicker shells. Hence, various problem statements are possible. In this report, the problem is considered in a 1D formulation. Also, the collapse of a cylindrical shell surrounded by a layer of compressed polytropic gas of finite thickness, which acts on the shell and expands into a vacuum, is studied in a pulse statement. To this end, a decrease in pressure in the gas on the outer surface of the cylindrical shell in the rarefaction wave that occurs when the shell converges is disregarded, and it is assumed that constant pressure is applied to the cylindrical shell during the time until the arrival of the second rarefaction wave from the free boundary of the polytropic gas and is instantly removed by this wave. Thus, two phases of shell collapse are distinguished: a pulsed stage of constant pressure on the cylindrical shell and an inertial stage with zero pressure and conservation of the kinetic energy of the shell material. At last, a model of a viscous incompressible fluid is applied to the cladding material. This model explains several characteristic physical effects that were first obtained experimentally: the stopping of a shell when its inner surface reaches a certain nonzero radius at the inertial phase of collapse, explosive evaporation of the shell as a result of the rapid transfer of all of its kinetic energy to heat due to the action of viscous forces, and dynamic loss of stability by shell shape. Eventually, 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 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.

Cite: Fursova D.A. , Gubarev Y.G.
On instability of radial collapse of cylindrical shell consisting of viscous incompressible fluid
The 5th International Conference on New Energy and Future Energy Systems 03-06 Nov 2020