## 3 Results and Discussion
### Determining the Gas Velocity  
The gas velocity is an important parameter for the analysis of spray systems, since the relative velocity of droplets to the gas flow can be derived. This relative velocity, which is also called slip velocity, has a strong influence on droplet kinematics, secondary atomisation, and evaporation [PERSON] _et al._ (2019). Hence, the simultaneous measurement of gas and droplet velocity is a great advantage for spray diagnostics. Standard optical techniques for measuring the gas velocity are Laser Doppler Anemometry (LDA) and Particle Image Velocimetry (PIV). Both techniques require seeding the flow with small particles in the order of 1 mm in diameter. These seeding particles will follow the flow almost perfectly due to their low inertia. Microscopic imaging has already been demonstrated to be capable to detect such seeding particles if the droplet size is not of interest using Particle Shadow Velocimetry ([PERSON] _et al._, 2019). In addition to an often simpler optical setup, this technique can be advantageous in comparison to established diagnostics like PIV or LDA for small-scale applications or multiphase flows ([PERSON] _et al._, 2019; [PERSON] _et al._, 2019). However, the goal of this study is to detect not only the velocity, but also the diameter of individual droplets. Figure 11: Snapshot of the spray in the near nozzle region using an exposure time of approximately 0.14 ms. Therefore, one way to realise a simultaneous measurement of gas and droplet velocity is the utilisation of the smallest detectable droplets as seeding particles. Hence, no additional seeding is necessary and the non-intrusive and simultaneous measurement of the gaseous velocity field is possible as long as enough small droplets are present in the flow. This methodology was successfully applied with the PDA for detecting evaporating [31] and burning droplets [32]. In these studies, droplets of a size smaller than 3-5 \(\mathrm{\SIUnitSymbolMicro m}\) were utilised to derive the gas velocity. The present microscopic imaging setup also offers the possibility to detect droplets of such small size, which may be utilised as seeding. However, it is mandatory that the droplets of a selected size do follow the flow correctly in order to derive a sensible gas velocity. In this context, it is important to note that the droplet size cannot be specified in advance since it depends on the actual flow field of the gas. Subsequently, the timescales due to the inertia of the droplets will be estimated in order to decide up to which size the droplets are suitable as tracers of the gas flow. This gives a measure of the accuracy of the present diagnostics for determining the gas velocity. The following estimation will focus on the most critical part of the flow field of the experimental setup in this study: the strong deceleration of the air jet emerging from the atomiser nozzle. First of all, a Stokes flow is assumed for the gas flow around the droplet [43]. The relevant timescale is the response or relaxation time of a droplet  
\[\tau_{\mathrm{d}}=\frac{\rho_{\mathrm{d}}\ D_{\mathrm{d}}^{2}}{18\ \mu_{\mathrm{g}}}, \tag{3}\]  
which is dependent on the density of the droplet \(\rho_{\mathrm{d}}\), the droplet diameter \(D_{\mathrm{d}}\) and the dynamic viscosity of the gas \(\mu_{\mathrm{g}}\). Considering the relative velocity between droplet and the gas phase \((u_{\mathrm{g}}(t)-u_{\mathrm{d}}(t))\) and neglecting gravitation, the equation of motion can be written as  
\[\frac{\mathrm{d}u_{\mathrm{d}}}{\mathrm{d}t}=\frac{1}{\tau_{\mathrm{d}}}(u_{ \mathrm{g}}(t)-u_{\mathrm{d}}(t)). \tag{4}\]  
Taking the homogeneous solution of this differential equation, it is obvious that the droplet velocity recovers after a sudden jump according to  
\[u_{\mathrm{d}}(t)\sim e^{-\frac{t}{\tau_{\mathrm{d}}}}. \tag{5}\]  
The determination of an appropriate timescale of the gaseous phase is not straightforward, since it depends on the actual flow field.