# An Analysis of Bubble Migration in Horizontal Thermo-Capillarity Using the VOF Modeling
## 4 Results and Discussion
### Force Analysis  
In fluid mechanics, it is important to correlate parameters to meaningful dimensionless numbers. It is well known that the drag force is one of the four primary forces acting on an object moving through a fluid. In this way, the physics can be observed and discussed on a scaled-down model, e.g., an airfoil model in the wind tunnel, as long as the \(Re\) number is kept the same. In this work, we are goanna plot \(C_{Th}\) against the Weber number (\(We\)) because this information might be helpful for future development of the correlation of \(C_{Th}\) as a function of the \(We\) number. The drag coefficient of \(C_{D}\) is defined as  
\[C_{D}=\frac{F_{D}}{\frac{1}{2}\rho_{l}V^{2}A} \tag{12}\]  
where \(F_{D}\) is the drag force, \(\rho_{l}\) is the density of the liquid, \(V\) is the velocity, and \(A\) is the reference area. \(C_{D}\) can be represented as a function of dimensionless parameters such as Reynolds number, Froude number, Mach number, and surface roughness. That is  
\[C_{D}=f(\text{shape},\ \text{Re},\ \text{Fr},\ \text{Ma},\ldots). \tag{13}\]  
In this work, we are interested in finding the relationship between the coefficient of lateral force (\(C_{Th}\)) and the Weber number since the thermo-capillary force is dominant in this system. Weber number is defined as:  
\[W_{e}=\frac{2\rho_{l}V^{2}R}{\sigma}, \tag{14}\]  
Figure 9: The transient state of the bubble with radius \(R=0.006\) m and \(\Delta T=40\) K (**a**) Velocity field (**b**) Temperature field, and (**c**) pressure field. where \(R\) is the bubble radius, and \(\sigma\) is the surface tension. Hence, to determine the lateral force acting on the vapor bubble, the following assumptions have been made: (1) there is no phase change between the liquid and vapor, and (2) the dominant forces applied to the bubble in the \(x\)-direction are the drag force and thermo-capillary force (lubrication force due to the wall is neglected). \[\Sigma F_{x}=F_{D}+F_{La}=\frac{1}{2}C_{D}\rho V^{2}A+C_{th}\
abla\sigma A=ma_{ x}. \tag{15}\]  
This is the force balance equation in the \(x\)-direction, where the \(C_{D}\) represents the drag coefficient, \(V\) is the bubble velocity, \(A\) is the projection area, and \(C_{Th}\) is the lateral force coefficient due to the thermo-capillary force, and \(\
abla\sigma\) is the surface tension gradient. For a two-dimensional problem, the bubble mass is \(\rho\pi R^{2}\), and the projection area is 2 \(R\). By substituting the above quantities into the force balance Equation (15), the coefficient of lateral force \(C_{Th}\) is then derived as:  
\[C_{Th}=\frac{F_{La}}{\
abla\sigma A}=\frac{ma_{x}-F_{D}}{\
abla\sigma A}=\frac {ma_{x}-\frac{1}{2}C_{D}\rho V^{2}A}{\
abla\sigma A}. \tag{16}\]  
Figure 10 shows the relationship between the thermo-capillary coefficient (\(C_{Th}\)) and the Weber number (_We_). The bubble sizes range from \(R=0.0045\) m \(-0.03\) m, which are smaller than the Capillary length. The Weber numbers then fall between \(10^{-4}\) - \(10^{0}\). It is found that as the bubble radius increases, the Weber number increases. Since a larger bubble size induces a higher drag force, the surface tension effect is not significant. In addition, the thermo-capillary coefficient is large due to a large thermo-capillary force. For bubble sizes in the range of 0.0045 m \(<R\leq\) 0.007 m, Figure 11 shows the thermo-capillary coefficient (\(C_{Th}\)) vs. the Weber number (_We_) for \(R=0.0045\) m. It is noted that both the Weber number and thermo-capillary coefficient increase when \(\Delta T\) increases. Figure 10: Relation between thermo-capillary coefficient (\(C_{Th}\)) and the Weber number (_We_). For bubble sizes in the range of \(0.008\,\mathrm{m}\leq R<\)0.01 m, Figure 12 shows the thermocapillary coefficient (\(C_{Th}\)) vs. the Weber number (\(We\)) for \(R=0.008\,\mathrm{m}\). It is found that the temperature difference \(\Delta T\) does not affect the Weber number much. However, the thermo-capillary coefficient decreases when \(\Delta T\) increases except for \(\Delta T=20\,\mathrm{K}\).