## 3 Methods
### Computational Fluid Dynamics  
The system was modeled within the OpenFOAM(r) environment [45; 46; 47]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Journal Diameter D [mm]** & **Bearing Length** & **Clearance** & **Speed** \\  & **L [mm]** & **e [\(\upmu\)m]** & **\(\upomega\) [rad/s]** \\ \hline
4 & 13 & 10 & 328–3000 \\ \hline
**Oil Density (15 \({}^{\circ}\)C) [kg/m\({}^{3}\)]** & **Oil Viscosity (40 \({}^{\circ}\)C) [m\({}^{2}\)/s]** & **Oil Viscosity (100 \({}^{\circ}\)C) [m\({}^{2}\)/s]** \\ \hline
1060 & \(220\times 10^{-6}\) & \(40\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Bearing parameters. Figure 2: The studied bearing. The method relied on finite volumes (FV) [48]. The computational domain, namely the clearance between ring and journal, was discretized into small cells by means of the mesh generator _blockMesh_ with about 500k hexahedrons (15 cells in the radial direction). Equilibrium equations are applied to each cell of the grid in order to ensure the mass and the momentum conservations [49]. \[\frac{\partial\rho}{\partial t}+\
abla(\rho v)=0 \tag{1}\]  
\[\frac{\partial\rho}{\partial t}+\
abla(\rho v)=-\
abla p+\
abla\left[\mu\big{(} \
abla v+\
abla v^{T}\big{)}\right]+\rho g+F \tag{2}\]  
where \(\rho\) is the lubricant density, \(v\) is the velocity vector, \(p\) is the pressure, \(g\) is the gravitational acceleration and \(F\) represents the external forces. The mean temperature was imposed as a variation of the fluid properties according, for the density, to the following formula [50]. \[\rho_{\theta}=\rho_{15^{\circ}\mathrm{C}}-(\theta-15)\cdot 0.0007 \tag{3}\]  
and by using a logarithmic relation between the known viscosity values at 40 \({}^{\circ}\mathrm{C}\) and 100 \({}^{\circ}\mathrm{C}\)[51]. \[v_{\theta}=v_{40^{\circ}\mathrm{C}}+\left(\frac{\theta-313}{60}\right)\left( v_{100^{\circ}\mathrm{C}}-v_{40^{\circ}\mathrm{C}}\right) \tag{4}\]  
The fluid is regarded as a continuum and its behavior is described by means of its macroscopic properties. The Boundary Conditions (BC) are reported in Table 2.  
The partial differential equations are solved using a PIMPLE (merged PISO-SIMPLE) scheme. The PIMPLE scheme combines the structure of the SIMPLE algorithm allowing the under-relaxation and a fast convergence of the solution with the velocity correction loops that are typically of the PISO mode. The solver uses a phase-fraction Eulerian fixed-grid technique to describe the interface between the two phases. The tracking of the interface is accomplished by the solution of a continuity equation for the volume fraction of one of the phases (in this case the vapor-sub-index v). \[\frac{\partial\alpha}{\partial t}+\
abla\cdot(\alpha_{v}v)+\
abla\cdot(v_{ \mathrm{C}}\alpha_{v}(1-\alpha_{v}))=\dot{m} \tag{5}\]  
Several different approaches to define the compressive velocity field are reported in literature, e.g., CICSAM [14] and the currently used multi-dimensional universal limiter with explicit solution [15], which limits the flux of the variables to guarantee a bounded solution. \(\dot{m}\) is a source term that will mimic the phase-change rate and is defined as the difference between vaporization \(\dot{m^{+}}\) and condensation \(\dot{m^{-}}\) defined according to [PERSON] [22]. \[\dot{m^{+}}=C^{+}\rho_{v}(1-\alpha)\frac{\min[0,\overline{p}-p_{v}]}{\frac{1}{ 2}\rho_{l}U_{\infty}^{2}t_{\infty}} \tag{6}\]  
\[\dot{m^{-}}=\frac{C^{-}\rho_{v}\alpha_{v}\alpha_{l}^{2}}{t_{\infty}} \tag{7}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Volume Fraction** & **Pressure** & **Velocity** \\ \hline
**Journal** & \(\
abla\alpha=0\) & \(\
abla\mathrm{p}=0\) & \(\omega=0\) \\
**Bearing** & \(\
abla\alpha=0\) & \(\
abla\mathrm{p}=0\) & \(\omega=\mathrm{U}\) \\
**Sides** & \(\alpha_{v}=0\) & \(\mathrm{p}=10^{5}\) kg/ms\({}^{2}\) & \(\
abla\mathrm{u}=0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: BC. The constants C\({}^{+}\) and C\({}^{-}\) are set to 100.