## 3 Numerical Results and Discussion
### Laplace Law  
The Laplace law, which considers a light bubble (density \(\rho_{L}\)) immersed in a heavier fluid (density \(\rho_{H}\)), was used to test whether the proposed method would have a good influence on mass conservation and interface correction. The law of the pressure and surface tension is written as:  
\[p_{in}-p_{out}=\frac{\sigma}{R}, \tag{30}\]  
Here, \(p_{in}\) is the average pressure inside the bubble, while \(p_{out}\) is the average pressure outside, \(\sigma\) is the surface tension coefficient, and \(R\) is the radius of the bubble. The order parameter \(C\) along the normal direction of the phase interface can be obtained with Equation (6). In this study, the mesh size of \(301\times 301\) was adopted for simulation, The natural boundary condition was applied on all walls. The natural boundary condition is that the gradient of all flow variables and the order parameter in the phase field is zero along the outer normal direction of the boundary. In addition, the following parameters were applied in the numerical simulation: \(\rho_{H}=10\), \(\rho_{L}=1\), and \(\
u_{H}=\
u_{L}=0.002\), while \(R\) was varied from 20 to 60, \(\sigma\) was defined as 0.001 and 0.005, and \(\xi=3\) for the solutions obtained in Figures 4-6. The coefficient \(k\) was defined as \(k=k_{1}\Gamma\left(\frac{\Delta x}{L}\right)^{2}\) in Equation (11), where \(k_{1}\) was set as \(k_{1}=1\). The characteristic length \(L=2R\). For the solution obtained in Figure 7, we set the interface thickness \(\xi=3,4,5,6\) in the initial condition to examine the effects of the interface correction terms in the improved C-H equation. It should be noted that Equation (6) was applied to set the order parameter \(C(d)\) in the initial phase field and the theoretical value of the interface thickness of \(\xi=3\) was applied to calculate the surface tension. Figure 4 shows the pressure difference in and out of the bubble at \(\sigma=0.001,\ 0.005\). It is obvious that the obtained results correspond with the prediction of the law. Figure 5 shows the variation in the mass for the whole bubble with the present method and the original DI approach for the case of \(\sigma=0.005\), and Figure 6 shows the contrast of the final interface for the static bubble with \(R=20\) at \(\sigma=0.005\). It is clear that the present method conserved the fluid mass well. However, the bubble mass obtained with the original DI method decreased, and the speed at which the mass decreased obtained by the original method increased as the radius grew. Without the correction, up to 20% of the mass of the bubble could be lost. This characteristic agrees well with a previous study [37]. Figure 7 compares the profile of the order parameter \(C\) along the y-centerline of the flow domain and shows the results of interface shapes from the setting of the initial order parameter with different interface thicknesses \(\xi=3,4,5,6\), obtained with the present interface correction  
method and the original DI method at \(\sigma=0.001\). It should be noted that the surface force was calculated by setting the interface thickness \(\xi=3\), which means that the final solution of the interface shape should be that described by Equation (6) with \(\xi=3\), regardless of the initial conditions. As can be seen from Figure 7, the order parameter with different thicknesses from the original method cannot give an accurate theoretical value, while the present method was able to obtain correct interface shapes that agree well with theoretical solutions. This feature shows that, when the interface shape is compressed or diffused in the initial condition, the present method is able to adjust and, finally, maintain the interface shape acquired with the theory, which shows its effectiveness and reliability. Figure 4: Pressure difference inside and outside the bubble at \(\sigma=0.001\) and \(0.005\). Figure 5: Variation in the mass for the bubble at \(\sigma=0.005\). Figure 6: The final interface shapes for the static bubble of R = 20 at \(\sigma=0.005\).