# An Analysis of Bubble Migration in Horizontal Thermo-Capillarity Using the VOF Modeling
## 3 Problem Definition and Numerical Method  
In Figure 1, the schematic diagram illustrates the problem definition and boundary conditions in this work. The computational domain is a 0.12 m \(\times\) 0.04 m rectangle, and the time step size is chosen as \(\Delta t=2.5\times 10^{-4}\) s. One spherical vapor bubble is initially positioned near the bottom wall at the center region of the liquid. The distance of the bubble centroid from the bottom wall is \(R_{\text{bubble}}+0.005\) m. All solid walls are set to the no-slip boundary conditions. The surface tension is defined as a function of temperature \(\sigma=f(T)\). The temperature of the left wall is fixed at \(T_{L}=373\) K, and that of the right wall is varied from \(T_{R}=273\) K to \(373\) K. The computations are performed for two-dimensional, unsteady, and incompressible flow with no mass change. The governing equations are solved using the finite volume method performed on ANSYS Fluent 2020 R1. The PISO scheme is chosen for pressure-velocity coupling. The 2 nd order upwind scheme is used for the conservation of both momentum and energy. The PRESTOI and the Geo-Reconstruct schemes are adopted for the pressure and the volume fraction interpolation, respectively. The Courant number is set at 0.25. The solution is converged well below the residuals of \(10^{-5}\). The numerical simulation settings are listed in Table 1.  
To ensure numerical stability, the density ratio of liquid to vapor can't be large in the VOF method [14]. The working fluid properties used in the simulations are listed in Table 2. Further comparison between the properties of working fluid and water can be done based on non-dimensional parameters. The primary non-dimensional parameters are listed below:  
\[Re=\frac{\rho_{l}\sqrt{\mathbf{g}_{y}\lambda_{0}}\lambda_{0}}{\mu_{l}},\ Pr=\frac{C_{pl}}{\mu_{l}k_{l}},\ We=\frac{\mathbf{g}_{y}\lambda_{0}^{2}\rho_{l}}{\sigma},\ Pe=\frac{C_{pl}\lambda_{0}\sqrt{ \mathbf{g}_{y}\lambda_{0}}\rho_{l}}{k_{l}},\ Ja=\frac{C_{pl}\Delta T}{h_{l_{p}}}.\]  
where \(\lambda_{0}\) is the capillary length, \(\rho_{l}\) is the liquid density, \(\mathbf{g}_{y}\) is the gravity acceleration, \(\mu_{l}\) is the liquid viscosity, \(k_{l}\) is the liquid conductivity, \(C_{pl}\) is the liquid specific heat, \(\sigma\) is the surface tension, \(h_{l_{p}}\) is the latent heat, and the \(\Delta T\) is the temperature difference. The calculated non-dimensional settings are \(R_{e}=3.85\), \(P_{r}=1\), \(W_{e}=1.023\), \(P_{e}=3.849\), and \(J_{a}=8\times 10^{-2}\) to \(4\). The Jakob number (\(J_{a}\)) of the working fluid in the simulations is larger than that of real water, which represents a better ability to phase change. The Peclet number (\(P_{e}\)) of the working fluid is smaller than that of water, representing lower convection. The Reynolds numbers (\(R_{e}\)) in both fluids are chosen in the laminar regime. The Weber numbers (\(W_{e}\)), which depend on surface tension, are set close to unity in both fluids. The Prandtl numbers (\(P_{r}\)) in both fluids are within the same order of magnitude. The parameters chosen for the working fluid in our simulations are based on previous work [22; 23], as listed in Table 2. The only difference is the surface tension now is chosen according to the temperature and is in the same order as the previous one. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Liquid** & **Vapor** \\ \hline \(\rho\) Density (kg/m\({}^{3}\)) & \(2.0\times 10^{2}\) & \(5.0\times 10^{0}\) \\ \(u\) Viscosity (Pa\(\cdot\)s) & \(1.0\times 10^{-1}\) & \(5.0\times 10^{-3}\) \\ \(k\) Thermal conductivity (W/m\(\cdot\)K) & \(4.0\times 10^{1}\) & \(1.0\) \\ \(C_{p}\) Specific heat (J/kg\(\cdot\)K) & \(4.0\times 10^{2}\) & \(2.0\times 10^{2}\) \\ \(h_{l_{p}}\) Latent heat (J/kg) & \(1.0\times 10^{4}\) & \\ \(\sigma\) Surface tension (N/m) & (\(-0.001\)\(\times\) Temperature + \(0.473\)) \(\times\) 15 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of the working fluid in simulations.