Derivation of phenomenological equations of hydromechanics of multi-phase flows

In the article, a multi-phase (non-homogeneous, heterogeneous) medium is considered as a macrosystem (continuum) composed of several (at least two) phases, such as a carrier phase (liquid, vapor or gas) and a carried phase (solid particles, bubbles or drops).The masses and mixtures of these phases undergo continuous changes over time due to the addition or separation of new masses to or from both phases. The model takes into account interphase transitions, discontinuities inside the mixture, and the possibility of phases being either continuous or discrete, depending on their location. A method for preliminary smoothing of discontinuities has been developed, leveraging the fact that the location in space, as well as the shape and size of the discrete phase are random. A function, denoted as φi (x, y, z, t), has been introduced, which indicates the probability of the presence of the i-th phase in the vicinity of a given point in space at time t, or that the given point of space x, y, z at time t belongs to the set of points of the i-th phase. On the other hand, this probability can be interpreted as the volumetric concentration of the i-th phase at a given point in space (i.e., the ratio of the measure of the set of points belonging to the i-th phase in the vicinity of the point under consideration at time t to the measure of the entire set of points in the surrounding area). This hypothetical medium, being equivalent to the original one, serves as a model for a multi-phase (inhomogeneous, heterogeneous, two-phase) medium. The uniqueness of the model arises from its construction. In addition, this paper considers several main areas of theoretical and experimental research concerning the hydrodynamics of a multi-phase (two-phase suspension-carrying) flow of a continuous medium. It also discusses the most important results achieved in existing works. A critical analysis of known theories for mathematically describing the motion of multi-phase (two-phase) systems and methods for averaging the hydrodynamic characteristics of a turbulent flow are given. The procedure for closing the equations systems of hydromechanics of multi-phase flows proposed in existing works is carried out within the framework of semi-empirical theories of turbulence. In nature, the vast majority of multi-phase (two-phase, inhomogeneous) mixtures exhibit turbulent behavior, making its study a crucial practical task. The mathematical description of the motion of a turbulent multi-phase flow relies on stylized laws of mechanics. The methods of operational analysis proposed at various times by different researchers for the mathematical description of the motion of a multi-phase (two-phase) flow have varying degrees of approximation and certain limited areas of application. One of the main challenges in formulating differential equations for the motion of a turbulent multi-phase (two-phase, suspension-carrying) flow is the fact that in a turbulent flow of a mixture, where the characteristics of the flow change chaotically and randomly over time and at each point in space, both in magnitude and in direction, there are surfaces with weak and strong discontinuities. Consequently, the actual values of velocity and pressure of a multi-phase flow, strictly speaking, cannot be considered continuous functions of the coordinates of space and time throughout the entire region occupied by the mixture.