# Study of Deformation and Breakup of Submillimeter Droplets' Spray in a Supersonic Nozzle Flow
## 1 Introduction  
Experimental data on droplets behavior in high-speed flows is often required for such applications as combustion of liquid fuels, especially in ramjets, liquid atomization in airblast nozzles and many others. In general, the atomization of liquid droplets due to aerodynamic forces is a complex process that involves various physical mechanisms. Through the last several decades, a large number of experimental works aimed at revealing of influencing parameters, better understanding of breakup evolution and morphology, systematization of acquired data were performed. A widely-adopted breakup morphology classification was proposed in the paper by [PERSON] and [PERSON] [1]. Since this publication, a lot of attention was paid to the physical phenomena, governing the droplets morphology and transition between regimes. Some of the results were summarized in the works by [PERSON] [2] and [PERSON] et al. [3]. The proposed classification was expanded, and the role of different phenomena was revised through a series of works summarized in the paper of [PERSON] [4]. Generally, it is considered that Weber number (_We_), Ohnesorge number (_Olt_) and Eotvos or Bond number (_Bo_) govern the behavior of liquid drops in the flow, while an effect of Reynolds number (_Re_) is less apparent. The dimensionless parameters mentioned are expressed as:  
\[We=\frac{\rho_{g}\Delta v^{2}d}{\sigma} \tag{1}\]  
\[Bo=\frac{\rho_{d}ad^{2}}{\sigma} \tag{2}\]\[Oh=\frac{\mu_{d}}{\sqrt{\rho_{d}\omega d}} \tag{3}\]  
\[Re=\frac{\rho_{g}\Delta vd}{\mu_{d}} \tag{4}\]  
In these equations, \(\rho_{g}\) is the gas, \(\Delta v\) is the droplet-to-gas relative velocity, \(d\) is the initial droplet size (diameter), \(\rho_{d}\), \(\sigma\) and \(\mu_{d}\) are the density, surface tension and the dynamic viscosity of the liquid, respectively. The Weber number is generally considered the most important parameter. In connection with the classification of breakup regimes introduced in [1], Weber number ranges are often referred to as \"low\"--\(We<100\), \"medium\" (\(100<We<350\)) and \"high\" (\(We>350\)). Ohnesorge number expresses the measure of viscosity effect in breakup processes. Usually, transitional Weber numbers, demarking the breakup regimes, are considered stable for \(Oh<0.1\), while for a higher \(Oh\) increase in transitional \(We\) are typically observed (see [5]). At low Ohnesorge numbers, several breakup regimes are usually demarked according to \(We\) range: bag breakup for \(We>12\), multimode (multi-bag) or bag and stamen breakup for \(30<We<80\), sheet (or shear) stripping regime in the range of \(80<We<350\) and catastrophic regime at \(We>350\). A droplet flattening into the ellipsoid form precedes any breakup regime. [PERSON] and [PERSON] [6] provided an approximation for the maximum deformation of a droplet. \[d_{max,c}/d=\left(1+0.19\sqrt{We}\right) \tag{5}\]  
where \(d_{y,c}\) is the major axis of the ellipse (flattened droplet). [PERSON] and [PERSON] also found, that for low \(Oh\), total breakup time almost does not vary over the wide range of \(We\) and may be approximated as  
\[t_{b}=5t^{*} \tag{6}\]  
where \(t^{*}\) is a characteristic time of breakup initiation \(t^{*}\), proposed by [PERSON] and [PERSON] [7]. \[t^{*}=\frac{d}{\Delta v}\left(\frac{\rho_{d}}{\rho_{g}}\right)^{1/2} \tag{7}\]  
Other approximations taking into account \(t_{b}\) dependence on \(We\) for different breakup regimes were also provided in [1]. According to [3; 8], and many other papers, an effect of viscous shear stress, and therefore, Reynolds number, plays no significant role in the transition between breakup regimes. Some numerical studies suggest that \(Re\), although not governing the breakup mode, has an effect on the shape of the droplet during the breakup. For example, studies performed by [PERSON], and [PERSON] [9; 10], showed that the liquid film surface area is correlated with \(Re\). A similar conclusion can be drawn from the numerical investigation by [PERSON] et al. [11].