# A Near-Far-Field Model for Bubbles Influenced by External Electrical Fields
## 5 Coupling Near-Field and Far-Field
### Bubble Formation: Validation of the Near-Field  
In the following, we validate the near-field model, which is a model based on the formation of bubbles. We can obtain an analytical expression, see [29]. Based on the mechanical equilibrium approach, we can develop the analytical expressions and predict the bubble departure volume. This departure volume is based on the force balance analysis under different assumptions, see also [38]. The relationship is given as:  
\[V=N\frac{Q^{6/5}}{g^{3/5}}, \tag{58}\]  
\(g\) is the gravity, \(N\) is a constant with \(N=1.378\). The departure volume is given as \(V\) and the gas-flow rate is \(Q\). We apply different experiment to validate with the analytical expression:  
* Test example 1: We apply the following test-example with the circumference of the ellipse, which are given as \(L_{i}=2\pi a_{i}^{2}/4\), where \(L_{1}=1,\ L_{2}=2,L_{3}=3,L_{4}=3.5,L_{5}=4.0\), while a special case is the circle with \(L_{circ}=\pi/2\) and radius \(r=1\). Furthermore, we assume \(N_{i}=L_{i}/\Delta s\), where the arc-length step for the computations is \(\Delta s=2\pi\cdot 0.001\) and \(\frac{\Delta p}{\sigma}=0\). Here, we measure prediction of the volume of the ellipsoid, while we assume a constant gas-flow-rate: \[Q_{L_{i}}=\frac{dV_{L_{i}}}{dt},\] (59) means we apply: \[Q_{L_{i}}=\frac{V_{L_{i}}(t_{2})-V_{L_{i}}(t_{1})}{t_{2}-t_{1}},\] (60) where \(i=1,2,3,4\), which we could compute with our MATLAB simulations. We assume that \(Q\) is a constant and approximated to \(V\). Then, we can predict the departure volume with the experimental formula: \[V_{L_{i}}=N\frac{Q_{L_{i}}^{6/5}}{g^{3/5}},\] (61) where the constants are defined in Equation (58) and different volumes of the ellipsoids are given with the indices \(i=1,2,3,4\). In the following Figure 12, we present the validation of the numerical computed volumes \(V_{num,L_{i}}/V_{num,L_{i-1}}\) and the experimental computed volumes \(V_{L_{i}}/V_{L_{i-1}}\). The experimental computed volumes are given with the Equation (61), see the experiments in [29]. * Test example 2: We have \(N=L/\Delta s\), where the arc-length step for the computations is \(\Delta s=0.001\) and \(L=2\) and we apply the following pressure relations \(\frac{\Delta p_{1}}{\sigma}=0\), \(\frac{\Delta p_{2}}{\sigma}=0.5\), \(\frac{\Delta p_{3}}{\sigma}=1.0\), \(\frac{\Delta p_{4}}{\sigma}=1.5\) and \(\frac{\Delta p_{5}}{\sigma}=2.0\). Here, we can predict the different gas-flow-rates, based on the different pressure relations. \[Q_{\Delta p_{i}/\sigma}=\frac{V_{\Delta p_{i}/\sigma}(t_{2})-V_{\Delta p_{i}/ \sigma}(t_{1})}{t_{2}-t_{1}}.\] (62) where \(i=1,2,3,4,5\), which we could compute with our MATLAB simulations and we assume \(Q\) is a constant and approximated to \(V\). Then, we can predict the departure volume with the experimental formula: \[V_{\Delta p_{i}/\sigma}=N\frac{Q_{\Delta p_{i}/\sigma}^{6/5}}{g^{3/5}},\] (63) where the constants are defined in Equation (58) and different pressure relations are given with the indices \(i=1,2,3,4\). In the following Figure 13, we present the validation of the numerical computed volumes \(V_{num,\Delta p_{i}/\sigma}/V_{num,\Delta p_{i-1}/\sigma}\) and the experimental computed volumes \(V_{\Delta p_{i}/\sigma}/V_{\Delta p_{i-1}/\sigma}\). The experimental computed volumes are given with the Equation (63), see the experiments in [29]. We have applied the validation of the near-field model simulation with the experimentally results of the heuristic formulas (61) and (63). We compared the experimental and numerical results and  
Figure 12: Validation for the relative departure volume related to the bubble length \(L_{i}\), which are based on the numerical volume \(V_{num,L_{i}}\) and the experimental volume \(V_{L_{i}}\). Figure 13: Validation for the relative departure volume related to the bubble length \(L_{i}\), which are based on the numerical volume \(V_{num,L_{i}}\) and the experimental volume \(V_{L_{i}}\). obtain relatively the same results.