## 2 Numerical Calculation of Oil Dispersion Flow in Flowing Air
### Mathematical Formulation  
Ansys CFX 19.1 software was used for the numerical calculation of the multiphase fluid flow represented by the oil fraction in the flowing air. Software solves the conservation equations for momentum, mass and energy by the finite volume formulation. The equations are expressed in terms of the Cartesian coordinate system. Continuity equation for multiphase non-homogeneous flow without mass sources and mass flow rate per unit volume from phase \(\beta\) to phase \(\alpha\) is as follows:  
\[\frac{\partial(r_{\alpha}\rho_{\alpha})}{\partial t}+\
abla\cdot(r_{\alpha} \rho_{\alpha}\mathbf{U}_{\alpha})=0 \tag{1}\]  
where \(r_{\alpha}\) is the volume fraction of phase \(\alpha\), \(\rho_{\alpha}\) is the density of the \(\alpha\)- phase as the phase of the fluid, \(t\) is time, and \(\mathbf{U}_{\alpha}\) is the velocity of the phase \(\alpha\) as the phase of the fluid. The momentum conservation equation is given by:  
\[ \begin{array}{l}\frac{\partial(r_{\alpha}\rho_{\alpha}\mathbf{U}_{\alpha})}{ \partial t}+\
abla\cdot(r_{\alpha}(\rho_{\alpha}\mathbf{U}_{\alpha}\otimes\mathbf{U}_ {\alpha}))=\\ -r_{\alpha}\
abla\rho_{\alpha}+\
abla\cdot\left(r_{\alpha}\mu_{\alpha}\left( \
abla\mathbf{U}_{\alpha}+(\
abla\mathbf{U}_{\alpha})^{T}\right)\right)+\mathbf{M}_{ \alpha}\end{array}  \tag{2}\]  
where \(p\) is the static pressure, \(\mu_{\alpha}\) is dynamic viscosity, \(M_{\alpha}\) describes the interphase forces acting on the \(\alpha\) phase due to the presence of other phases. An interphase drag with drag coefficient \(C_{D}=0.44\) was used in Interphase Momentum Transfer Models. Lift coefficient \(C_{L}=0.5\) was defined for the lift force. The wall lubrication force was activated according to the numerical solution of the two-phase flow which acts on the oil fraction and is described by:  
\[\mathbf{F}_{wl}=C_{wl}\rho_{q}\cdot\alpha_{p}\cdot\left|\left(\mathbf{U}_{q}-\mathbf{U}_{ p}\right)_{t}\right|^{2}\cdot\mathbf{u}_{w} \tag{3}\]  
where \(C_{wl}\) is the wall lubrication coefficient (1), \(\rho_{q}\) is the primary phase density, \(\alpha_{p}\) is the oil volume fraction, \(\left|\left(\mathbf{U}_{q}-\mathbf{U}_{p}\right)_{t}\right|\) is the phase relative velocity component tangential to the wall surface (m\(\cdot\)s\({}^{-1}\)), and \(\mathbf{n}_{w}\) is a normal unit pointing away from the wall. [PERSON] et al. Model [21] was used to calculate the wall lubrication coefficient:  
\[C_{wl}=\max\left(0,\frac{C_{wl}}{d_{b}}+\frac{C_{wl}}{y_{w}}\right) \tag{4}\]  
where \(C_{wl}=-0.01\) and \(C_{wl}=0.05\) are non-dimensional coefficients, \(d_{b}\) is the particle diameter, and \(y_{w}\) is the distance to the nearest wall. The homogeneous \(k\)-\(\omega\) SST (Share Stress Transport) turbulence model was used for the numerical calculation of the multiphase flow with the corresponding \(k\)- and \(\omega\)- equations containing the buoyancy effects:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial}{\partial x_{j}}\left(\rho U _{j}k\right)=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{ \alpha_{k}}\right)\frac{\partial k}{\partial x_{j}}\right]+P_{k}-\beta^{\prime }\rho k\omega+P_{kb} \tag{5}\]  
\[\begin{array}{l}\frac{\partial(  
where \(k\) is the turbulent kinetic energy, \(\omega\) is specific dissipation rate, \(\mu\)t is turbulent dynamic viscosity calculated by (7), \(x\) is the space coordinate, and \(P_{k}\) is the shear production of turbulence.  
\[\mu_{t}=\frac{a_{1}k\rho}{\max(a_{1}\omega,SF_{2})} \tag{7}\]  
The coefficients \(\sigma_{k3}\) and \(\sigma_{\omega 3}\) are calculated by a linear combination of the respective coefficients according to the following Equations (8) and (9). \[\sigma_{k3}=F_{1}\sigma_{k1}+(1-F_{1})\sigma_{k2} \tag{8}\]  
\[\sigma_{\omega 3}=F_{1}\sigma_{\omega 1}+(1-F_{1})\sigma_{\omega 2} \tag{9}\]  
The blending functions (10) are critical to the success of the method.