# An Analysis of Bubble Migration in Horizontal Thermo-Capillarity Using the VOF Modeling
## 2 Governing Equations  
Various methodologies have been proposed to model the vapor-liquid two-phase problems, such as the molecular dynamic (MD) simulation [13], lattice Boltzmann method [14; 15], immersed boundary methods [16], level set (LS) [17] methods, and volume-of-fluid method [18; 19]. The VOF method proposed by [PERSON] and [PERSON] in 1981 tracks the interface between two phases [20], and it has been used widely in studying various two-phase flow systems. The VOF method is adopted here since the fluid mass can be appropriately conserved, and it can be applied on a larger scale compared with the LBM and MD methods. Also, it's computationally and timely cheap. Therefore, the VOF model is chosen in this work to characterize the liquid/vapor interface to illustrate the bubble dynamics. In the VOF model, \(\alpha\) is the volume fraction of two phases with subscripts v and l denoting vapor and liquid. The volume fractions of the two phases in the VOF model can be defined as:  
\[\alpha_{v}+\alpha_{l}=1, \tag{1}\]  
The continuity equation describes the conservation of mass as:  
\[\left\{\begin{array}{l}\frac{\partial}{\partial t}(\alpha_{v}\rho_{v})+ \
abla\!\cdot\!(\mathbf{u}\alpha_{v}\rho_{v})=0,\\ \frac{\partial}{\partial t}(\alpha_{l}\rho_{l})+\
abla\!\cdot\!(\mathbf{u} \alpha_{l}\rho_{l})=0,\end{array}\right. \tag{2}\]  
where \(t\) is time and \(\mathbf{u}\) is the average fluid velocity. The momentum equation is expressed as:  
\[\frac{\partial}{\partial t}(\rho\mathbf{u})+\
abla\!\cdot\!(\rho\mathbf{u} \mathbf{u})=-\
abla p+\
abla\!\cdot\!(\mu\
abla\mathbf{u})+\rho\mathbf{g}+ \mathbf{F}, \tag{3}\]where \(p\) is the pressure, \(\mathbf{g}\) is the gravitational acceleration, and \(\mathbf{F}\) is the external force. The non-dimensional fluid properties of the mixture density \(\rho\), dynamic viscosity \(\mu\), specific heat \(C_{p}\), and thermal conductivity \(k\) can be expressed as:  
\[\left\{\begin{array}{rl}\rho=\rho_{l}\alpha_{l}+\rho_{v}\alpha_{v},\\ \mu=\mu_{l}\alpha_{l}+\mu_{v}\alpha_{v},\\ C_{p}=C_{p}\alpha_{l}+C_{pv}\alpha_{v}\\ k=k_{l}\alpha_{l}+k_{v}\alpha_{v}\end{array}\right. \tag{4}\]  
The conservation of energy is expressed as:  
\[\frac{\partial}{\partial t}\big{(}\rho C_{p}T\big{)}+\
abla\cdot\big{(}\rho \mathfrak{u}C_{p}T\big{)}=-\
abla\cdot(k\
abla T)+S_{h} \tag{5}\]  
where \(T\), \(C_{p}\), \(k\), and \(S_{h}\) are the temperature, the specific heat, the mixture thermal conductivity, and the heat source, respectively. [PERSON] [21] proposed the continuum surface force (CSF) model, which assumes that surface tension is constant at the same temperature, where the forces are typical to the two-phase interface. In the CSF model, the surface tension is considered the source term in the momentum equation in the VOF model. With the surface tension \(\sigma\) and two radii orthogonal to the interface \(R_{1}\) and \(R_{2}\), the equation for the pressure drop, \(p_{2}-p_{1}\), is expressed as:  
\[p_{2}-p_{1}=\sigma\bigg{(}\frac{1}{R_{1}}+\frac{1}{R_{2}}\bigg{)} \tag{6}\]  
In previous research, surface tension \(\sigma\) is considered a constant. However, the surface tension \(\sigma\) is a function of temperature in this work. Hence, in the momentum equation, \(\mathbf{F}\) is the surface tension force between the two phases that are expressed as a volume force density:  
\[\mathbf{F}=\sigma(T)\frac{\alpha_{l}\rho_{l}\kappa_{l}\
abla\alpha_{l}+\alpha _{v}\rho_{v}\kappa_{v}\
abla\alpha_{v}}{\frac{1}{2}(\rho_{l}+\rho_{v})}, \tag{7}\]  
where \(\kappa\) is the interface curvature denoted as:  
\[\kappa_{l}=-\kappa_{v}=-\
abla\cdot\bigg{(}\frac{\
abla\alpha_{l}}{|\
abla \alpha_{l}|}\bigg{)}. \tag{8}\]