On hydrodynamic instabilities qua nonequilibrium (Cahn–Pillard) phase transitions

E. V. Radkevich ; O. A. Vasil'eva ; E. A. Lukashev

Eurasian journal of mathematical and computer applications, Tome 7 (2019) no. 2, pp. 20-61.

Résumé

For the laminar–turbulent transition, we construct a model of reconstruction of the initial stage of instability qua a nonequilibrium transition with diffusion separation mechanism. It is shown that the free Gibbs energy of departure from the homogeneous state (withrespect to the instability under consideration) is an analogue of the Ginzburg–Landau potential. Numerical experiments for self-excitation of the homogeneous state with control of the boundary condition of velocity increase were carried out, which showed the appearance of the laminar–turbulent transition and its development from regular forms (the so-called dissipative structures) with subsequent transition to irregular flows via chaotization of the process. An external action (an increase in velocity) results in a transition to chaos in terms of period-doubling bifurcations similarly to the Feigenbaum cascade of period-doubling bifurcations. The chaotization of the process transforms regular forms (dissipative structures) into the two-velocity regime (the regime of two shock waves), which was called the Riemann–Hugoniot catastrophe by Prigogine and Nicolis. This transformation depends substantially on gravitation. The perturbation is shown to be nonlocal, which indications that the classical perturbation theory is inapplicable in this case.

Détail

Keywords: the laminar–turbulent transition, nonequilibrium phase transitions, the Kahn Hillard model, diffusion fibration, the Ginzburg-Landau potential, Gibbs free energy.