# Physical and Mathematical Modeling of the Interaction of Water Droplets and High-Speed Gas Flow
## 1 Introduction
The most relevant approach at the moment is described in [5; 12]. It is assumed that intense gas-dynamic breakup occurs when Weber numbers are greater than the critical one. At the same time, for \(12\leq\)\(We\leq 35\), classical (single-mode) bag breakup prevails, and for \(35<\)\(We\leq 80\), multimode bag breakup occurs. It is generally accepted that the breakup occurs in the shear stripping regime due to the emerging instabilities on the droplet surface at \(We\) from 80 to 350 [6]. For large Weber numbers (\(We>350\)), catastrophic breakup occurs. Some sources [15] also describe the possibility of vibrational breakup of droplets at Weber numbers below critical values (\(We<12\)). This type of breakup occurs only under certain conditions of the gas flow, and currently it is not completely understood. A description of the driving forces and causes of vibrational breakup is given in [16]. Mathematical modeling of the processes of gas-dynamic breakup of droplets is one of the most actual areas of research. Nowadays, several approaches of modeling allow to estimate the intensity of gas-dynamic fragmentation with various precision. The usage of a detailed three-dimensional modeling of gas-dynamic fragmentation taking account of the effects at the interface [2; 3; 16; 17] requires significant computational resources and complex validation of the results. However, at this stage the models used in the calculations (especially for finely dispersed liquid droplets) can give results that differ significantly (up to 50%) from the experimental ones [18; 19; 20; 21]. In addition, performing a large number of parametric calculations requires significant computing resources which complicates the operational solution of a number of practical problems [22; 23]. At the moment criteriad of gas-dynamic breakup of droplets based on experimental data are relevant. Such models can be used for the analysis of processes as well as for the numerical simulation of two-phase flows using the combined Lagrange-Euler approach [24; 25; 26]. In this case the problems of determining the spatial distribution of the concentration and size of droplets as well as the parameters of the movement of gaseous and dispersed media can be solved. However, the known parametric models [11; 19; 20; 27] have a limited ranges of application in terms of velocities, Weber and Reynolds numbers. In particular, the analysis of works [1; 23] shows that the widespread cone-shaped dependence [27] for describing gas-dynamic fragmentation can give more than a twofold difference with experiment [24] for regimes with Weber numbers less than 100. The latter it is explained by the fact that the cone-shaped dependence is more than suitable for multimode bag mode and does not take into account the unsteady flow regime (flow structure localized near the drop) and drop deformation (deviation from the spherical shape). Previous studies [18] made it possible to form the criterion equation for the derivative of the change in the droplet mass m with respect to time t because of its gas-dynamic fragmentation:  
\[\frac{dm}{dt}\ =\ -m_{1}\cdot\frac{K_{1}\cdot(We)^{\psi}\cdot(Re)^{\omega}}{t_{i }}\cdot\left[1-\exp\left(-K_{2}\cdot\frac{t}{t_{i}}\right)\right] \tag{4}\]  
where \(t_{i}\) is the characteristic time of separation induction; \(m_{1}\) is the mass of a single particle separated from the drop; \(K_{1}\); \(K_{2}\); \(\psi\); \(\omega\) empirically obtained coefficients _We_ [PERSON]'s number; \(Re\) is the Reynolds number. However, it should be emphasized that the empirical coefficients \(K_{1}\); \(K_{2}\); \(\psi\); \(\omega\) have a certain dependence on the realizable conditions of gas-dynamic fragmentation. Establishing these dependencies on the two-phase flow conditions requires detailed experimental research. Taking into consideration all mentioned above, it can be argued that now the processes of interaction of a liquid with a high-speed flow have not been completely studied.