## 3 Results and Discussion  
Three PDFs that are widely used for theoretical interpretations of experimentally measured droplet radii distributions are the Rosin-Rammler distribution [28], the Nukiyama-Tanasawa distribution [29], and the well-known gamma distribution. The distribution parameters fit most of the experimental data. The proposed model was applied to arbitrary general experimental data of a polydispersed fuel spray produced by a plain-jet air-blast atomizer [18] and its approximations. All calculations were performed numerically using MATLAB software. The experimental distribution and its approximations based on known theoretical distributions that are used and normalised are presented in Figure 1.  
In the region close to \(r_{m}=1\), we observed an almost linear behaviour of the maximal radius \(r_{m}\) as a function of time. This corresponds to initial fast motion (evaporation). Close to this almost \"linear\" region, our model is valid; thus, we can study the various types of dynamics that describe the PSD. The integrand of \(F\) represents the PSD. This conclusion immediately follows from the definition of F (10). On the other side, the integrand of \(\tilde{F}\) is proportional to the PSD (it can be seen by Equation (13)). The dynamics of the PSD are based on the initial PSD (initial measurement) and its different approximations by various commonly used PDFs. As shown in Figure 2, the behaviour of different integrands of \(\tilde{F}\) is quantitatively very similar. The experimental PSD is discontinuous, because it is discrete and divided into several parcels, whereas all approximations are smooth. The integrands of F consist of \(P_{0}(G(R))\Pi_{R_{m}}(R)\), which can be described with the help of PDFs (more accurately, it is proportional to the PDFs). Recall that the function \(F\) is an integral that depends on \(r_{m}\). In Figure 3, we can see graphs of the integrand of \(F\), and the area under the graph is a visualisation of \(F\). The area can also be interpreted as a function of the droplet mean radius. As the maximum radius decreases, it is reasonable to suppose that the mean radius will decrease as well. With the help of the current more accurate model (compared with its previous version [24]), we can observe the dynamical behaviour of the probability functions. The results are shown in Figure 4.  
Our computations demonstrate a different type of behaviour for the mean radius in the initial time period. We observe an increase in the mean radius for a short period of time, followed by the expected behaviour (i.e., the decrease in the mean radius). A similar behaviour was also observed for a different probability function (Figure 5). This phenomenon is stable numerically. We have a heuristic explanation only, and hopefully, one will be derived analytically in the future. Figure 1: Normalized experimental probability distributions and various approximations by probability density functions (Nukiyama–Tanasawa, Rosin—Rammler, and gamma distributions) of the droplets’ radii. Figure 3: Integrand of F given by Equation (10). Figure 2: The integrand of \(\bar{F}\) given by Equation (13) in the region near \(r_{m}=1\) for experimental data and its approximations (Nukiyama–Tanasawa, Rosin–Rammler, and gamma distributions), at various values of \(r_{m}\). (**a**) The integrand of \(\bar{F}\) at \(r_{m}=1\). (**b**) The integrand of \(\bar{F}\) at \(r_{m}=0.975\). (**c**) The integrand of \(\bar{F}\) at \(r_{m}=0.95\). Figure 4: Nukiyama–Tanasawa, Rosin–Rammler, and gamma probability for various \(r_{m}\) values. (**a**) Probability at \(r_{m}=1\). (**b**) Probability at \(r_{m}=0.975\). (**c**) Probability at \(r_{m}=0.95\). (**d**) Probability at \(r_{m}=0.925\). Figure 5: The area of the probability integrand as a function of the \(r_{m}\) for the various distributions (Nukiyama–Tanasawa, Rosin—Rammler, and gamma). We can observe this numerical phenomenon of mean radius increase for any studied PDF.