The paper presents the results of the study of the models of convective instability near its threshold of thin layers of liquid and gas bounded by poorly conducting walls. These models single out one spatial scale of interaction, leaving the possibility for the evolution of the system to choose the symmetry character. This is due to the fact that the conditions for the realization of the modes of convective instability near the threshold are chosen. All spatial perturbations of the same spatial scale, but of different orientations, interact with each other. It turned out that the presence of minima of the interaction potential of the Proctor-Sivashinsky equation modes, the absolute value of the wave number vectors of which is unchanged, determines the choice of symmetry and, accordingly, the characteristics of the spatial structure. In the case of a more realistic model of convection described by the Proctor-Sivashinsky equation, it was possible to observe both the first-order phase transition and the second-order phase transition and detect the form of the state function, which is responsible for the topology of the resulting convective structures: metastable rolls and stable square cells. In this paper, it is shown that the nature of the structural-phase transition in a liquid when taking into account the dependence of viscosity on temperature in the Proctor-Sivashinsky model is similar to the case of the absence of such a dependence. The transition time turns out to be the same, despite the fact that a different structure is formed -hexagonal convective cells. As in the Swift-Hohenberg model, a hard mode for the formation of hexagonal cells in a gas medium is possible only for a sufficiently noticeable dependence of its viscosity on temperature. The phase transition times are inversely proportional to the difference in the values of this function for two consecutive states. A similar description of phase transitions did not use phenomenological approaches and various speculative considerations, which allows for a closer look at the nature of transients.
The paper presents the description of a demonstration bench, which includes a mathematical model and analysis tools for understanding the features of phase transitions of the first and second kind. The advantage of this demonstration bench is the rejection of all phenomenology and the obvious limitation of the application of various approximations and hypotheses. The description is formed on the well-known equations of hydrodynamics, which are well-tested and are a reliable basis for the construction of realistic models. The Proctor-Sivashinsky model, which was used to describe the process of convection development in a thin layer of liquid with poorly conductive heat boundaries, is the basis for the demonstration bench. Exactly this model allows to observe phase transitions of the first and second kind. The feature of the model is that it allocates one spatial scale of interaction, leaving for the evolution of the system the possibility to choose the nature of symmetry. All spatial disturbances of the same size but of different orientation interact with each other. This allows us not to distract from the main task of this work, which is to demonstrate the process of structure formation as a result of a cascade of phase transitions. The mechanism of phase transitions associated with the presence of minimums of the interaction coefficients of modes of the spectrum of the instability. There are a large number of structural defects, which appear as attributes of phase transition. The instability spectrum modes interference is the reason of the high rate of correlations in the propagation of a new phase.
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