PHASE TRANSITIONS IN CONVECTION

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 SwiftHohenberg 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.

Below, we discuss the possibility of phase transitions in a thin layer of liquid or gas, between walls with poorly conducting heat. The layer is limited only from below and above, in other directions there are no borders.
We first consider the Proctor-Sivashinsky equation with allowance for the temperature dependence of viscosity [1,2] in the case of the proximity of the Rayleigh number to the critical value of the occurrence of convection c Ra , that is, ) 1 is a two-dimensional operator, and, i  , j  are single unit orthogonal to each other unit vectors in the ( ϑ ζ , ) plane of the media separation, and we will assume 1 0 ≈ k , since we restrict ourselves to the case of a weak excess over the threshold of convective instability. Indeed, for any deviation of the wavenumber from unity, the perturbation amplitude rapidly decreases. Here φ is the relative temperature at the upper boundary of the layer. An increase in this value indicates an increase in the thermal conductivity of the layer as a whole. Table 1.
Transition from used variables to real physical quantities Equation (1) contains vector quadratic nonlinearity (that is, dependent on the orientation of perturbations and derived quantities), and the cubic nonlinearity, which takes into account the influence of the temperature field at the upper boundary of the layer, responsible for the change in the topology of spatial structures of convection.
To describe such convection under the same conditions, the simplified Swift-Hohenberg equation is often used [3] 35 Phase Transitions in Convection EEJP. 4 (2019) where the vector character of the nonlinear terms is replaced by the scalar one. Here , and the dimensionless heat transfer coefficient between the fluid and the boundary, here equals the number of Biot and is equal to . The growth of disturbance amplitudes in case of instability } exp{Im T ⋅ ∝ ω φ occurs with an increment . For 0 > γ gas flow (this corresponds to gas convection) it goes back to the center of the cell, with 0 < γ (which corresponds to the movement of the liquid) -vice versa. The Swift-Hohenberg equation, as noted in the review [4], describes, after the formation of an amorphous state of disorderly convection, a system of distinct hexahedral cells as a result of a soft (for liquid) and hard (for gas) instability regime, observed in particular in [5]. Moreover, the nature of the instability demonstrates all the features of the firstorder phase transition -the formation of a clear spatial structure of convection from an amorphous state.
The purpose of the work is the analysis of the soft and hard regimes of structural-phase transitions in the Proctor-Sivashinsky model in the conditions of temperature dependence of viscosity.

DESCRIPTION OF PHASE TRANSITIONS IN THE MODEL OF THE PROCTOR-SIVASHINSKY IN THE ABSENCE OF TEMPERATURE DEPENDENCE OF VISCOSITY
In contrast to the traditionally used Swift-Hohenberg equations, we use the 3D Proctor-Sivashinsky equation that meets the real conditions. This task is obviously three-dimensional in space and non-stationary, which at first glance creates significant problems. However, the Proctor-Sivashinsky model makes it possible to reduce the dimension of the description and focus on topological aspects, that is, appearance, size and development time of spatial structures.
In the case of a more realistic model of convection, described by the Proctor-Sivashinsky equation (1), both the first-order phase transition and the second-order phase transition can be observed, and the state function, which is responsible for the topology of the resulting convective structure, was detected [6].
The equation that determines the dynamics of the temperature field of this process in the horizontal plane (x, y) is: , fis a random function describing external noise, θthe temperature deviation from the equilibrium one (varying according to a linear law), and the magnitudeε determining the excess of the convection development threshold, we suppose, as before, to be quite small ( .We present the solution in the form of For replacements t T = ⋅ 2 ε , for slow amplitudes in the absence of noise, we obtain a mathematical expression for the Proctor-Sivashinsky model for describing convection. where the interaction coefficients are defined by the relations The analysis of the Proctor-Sivashinsky model in the absence of the dependence of viscosity on temperature ( 0) γ = was studied in detail by authors of this work earlier [7]. It was shown that after the first-order phase transition, a quasistable system of convective rolls (the form of which is shown in Fig. 1a) is formed from amorphous random convection of a state. Later, as a result of rolls modulation within the framework of a second-order phase transition, form a stable field of square convective cells (the form of which is shown in Fig. 1b).
The first-order structural-phase transition, noted earlier in [4], corresponds to the transition from an amorphous state of convection to a state that has the form of a pronounced spatial structure. It should be emphasized that such a spatial clarity of the structure is observed only in conditions of proximity to the instability threshold. If, as a result of instability, the topology of the structure changes, we can speak of a second-order structural-phase transition.
In this case, the phase transition times are inversely proportional to the difference in the values of the state function It was the fulfillment of the last relation that allowed us to consider the value as a state function, since each spatial structure of convection uniquely corresponded to its value of this function, besides, the phase transition time was also related to the changes of the magnitudes of this function (these changes can be seen in Fig. 2). The fragments of the spatial structure of the temperature field distribution on the surface of layer are presented in Fig. 3. a b

PHASE TRANSITIONS IN THE PROJECT-SIVASHINSKY MODEL UNDER THE CONDITIONS OF VISCOSITY DEPENDENCE ON TEMPERATURE Soft mode of excitation of a six-sided convective structure.
Taking into account the temperature dependence of viscosity demonstrates the ability to implement soft (when 0 < γ ) and hard (that is, setting the initial perturbation already in the form of the desired structure 20% higher than the average values of the amorphous state upon 0 γ > ) excitation of six-sided convective cells, the state function of which is almost equal to the state function of the rolls system. The time of the first-order structural-phase transition from the amorphous state is almost the same. When negative 0 < γ is observed, as in the Swift-Hohenberg model, the mode of soft excitation of six-sided convective cells (Fig. 4). The time interval of the first-order phase transition is equal 2 τ , but despite the fact that the values of the state function are equal to unity, which in the former case of the absence of the temperature viscosity dependence ( 0), γ = corresponded to a system of convective rolls. But in that case the system of six-sided cells is formed, as can be seen in Fig. 5, which depicts the dynamics of the instability spectrum. To implement a hard mode against the background of an amorphous state, a structure was formed whose amplitude was 20% higher than the average value of the amplitudes of the modes (Fig. 6).   is presented in Fig. 7 and Fig. 8 respectively.

CONCLUSIONS
The peculiarity of the models describing convective instability of thin layers of liquid and gas, bounded by poorly conducting walls of Swift-Hohenberg and Proctor-Sivashinsky, is that they 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 of convective instability 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. It was in the case of a more realistic model of convection, described by the Proctor-Sivashinsky equation (1), that we were able to observe both the first-order phase transition and the second-order phase transition and detect the form of the state function that is responsible for the topology of the resulting convective structures.
It should be noted that such a description of phase transitions did not use phenomenological approaches and various speculative considerations, which makes it possible to more closely examine the nature of the transient processes, which arouses the greatest interest of researchers. It is necessary to pay attention that presented on Fig. 3c. a metastable state in the vicinity of a second-order phase transition needs more careful analysis and will help clarify not only the particular nature of this transition, but it is possible to see some common features of such transitions.