# Electromagnetic and Thermal Phenomena Modeling of Electrical Discharges in Liquids
## 5 Modeling of the Discharge Development
The gas bubble model with spherical geometry of 10 \(\upmu\)m diameter was created around the selected node (Figure 8b). According to the Equation (5), the intensity of the electric field in the bubble is higher than in its direct vicinity [39]. The phenomenon was considered during calculations by introducing the value resulting from the formula (5) as an additional source of electric field in the numerical model (Figure 8c) [17]. \[E_{B}=\frac{3\varepsilon_{r}}{2\varepsilon_{r}+1}E_{ia}, \tag{5}\]  
where: \(E_{ia}\)--the intensity of the electric field in the vicinity of the gas bubble; \(\varepsilon_{r}\)--electrical dielectric permittivity  
The plasma channel can be developed only if the local electric field intensity on the streamer's head exceeds the critical value specified for breakdown of the dielectric material (juice model in the analyzed case) (E **) [17]. This condition was used to determine the streamer's propagation path, by determination of electric field intensity gradients (Figure 9a). However, in many cases, the main discharge streamer can be branched. The condition for creation of new streamers may be presented basing on the field fluctuation criterion, leading to formula (6) [17]. \[E_{i}>E_{*}-\delta_{i}, \tag{6}\]  
where: \(E_{i}\)--the gradient of the electric field intensity in the direction in which a new streamer can propagate; \(\delta_{i}\)--random variable, taking into account the fluctuations associated with the local heterogeneity of the critical values of the breakdown voltage (randomness of the discharge). Computations of streamers propagation scheme were performed by the analysis of electric field intensity gradients distributions between individual nodes located on the successive \"virtual\" surfaces of the discharge area (Figure 7). In cases, where gradients were characterized by similar values (differences lower than 10%) (Figure 9b), the condition (6) was analyzed and a new streamer was introduced (Figure 9c,d). The proposed procedure uses static calculations. Considerable disadvantage of this solution is the inability to use it to transient analysis of the pre-breakdown stage. In order to enable such calculations, the Equation (7) [17; 44] was used to determine the average speed of streamers propagation, based on the analysis of local values of the electric field intensity. \[v(E)=\frac{h}{\tau}e^{-(\frac{E_{*}}{\delta})}e^{(\frac{E}{\delta})} \tag{7}\]  
Figure 9: Vectors of the electric field intensity (**a**), (**b**) and streamer models (**c**), (**d**) in subsequent layers of the discharge space. where: \(\tau\)--time step; \(h\)--distance (between subsequent surfaces of the differential division of the discharge space); \(g\)--width of the streamer; \(E\)--electric strength, \(E\)--breakdown field strength  
Equation (7) was used to determine the time, after which, the streamer head reaches subsequent \"virtual\" surfaces dividing the area of the discharge propagation (Figure 7b). Such computations can be performed for all calculation steps to determine the temporal characteristics of the discharge propagation path. Procedure shown in Figure 9 was used to determine the locations of all nodes, where streamers heads occur. Around these nodes, models of the gas bubbles were created, similarly to the above described modeling of the first node located on the surface of the electrode (Figure 8b). The occurrence of the plasma channel between the gas bubble models is taken into account. The channel was simulated in the form of a cylinder with a diameter of 10 \(\upmu\)m (Figure 9c,d). Inside the channel, the model of the plasma of conductivity determined by Equation (8) was assumed [45; 46; 47]. This formula is useful for poorly ionized plasma. During calculations movements of plasma particles was neglected [46].