Qualitative analysis of the self-similar solutions of hydrodynamical
equations, describing media with memory
V.A. Danylenko and V.A. Vladimirov
Division of Geodynamics od Explosion,
Subbotin Institute of Geophysics
of the NAS of Ukraine, Kiev
In recent decades synergetical approach has been successfully applied to the description of selforganization phenomena in nonlinear media, simulated by parabolic-type equations, which describe well enough processes of moderate intensivity but loss their applicability in the cases when gradients are changing during correlation time and inside correlation length. In this case essential dispersive effects take place and one needs to take the nonlocality and memory effects into account [1,2].
In this work we investigate a modelling system aimed at describing explosion mechanics problems for multicomponent relaxing media (soils, rocks, etc.). We derive the governing (constituti) equation [3] on the basis of generalized Onsager-Luikov relations between thermodynamical forces and fluxes [4] and employ this equation to close the balance of mass and momentum system, taken in single-velocity hydrodynamical approximation.
The modelling system is investigated by means of qualitative theory methods. The group theory technique is employed to make a passage from PDE to subsequent system of ODE, describing travelling wave solutions.
The main results obtained are as follows. In contrast to classical (gas-) hydro-dynamical system, the modelling system proposed posesses periodic, quasiperiodic and stochastic self-similar solutions. The existence of these regimes is possible due to the complex interaction of nonlinear terms with the terms, describing relaxing properties of the medium, that are expressed in terms of physically measurable quantities. The results of qualitative investigations, based on the canonical forms technique are backed by numerical simulations.