### Inertial Particles

A one-way coupled non-reacting iso-thermal Lagrangian tracking model is currently implemented. Inertial droplets/particles are dispersed by the gaseous phase, while their interaction/reaction with the background flow is neglected. Buoyancy and drag, as the carrier phase forces acting on the particles, are currently implemented. The position, \(\boldsymbol{x}_p\), and velocity, \(\boldsymbol{u}_p\), of particles with diameter \(d_p\) and density of \(\rho_p\) are obtained as,

\[\begin{eqnarray} \frac{d\boldsymbol{x}_p}{dt} = \boldsymbol{u}_p \\ \frac{d\boldsymbol{u}_p}{dt} = \frac{f}{\tau_p}\left(\boldsymbol{u}_f - \boldsymbol{u}_p\right) + \boldsymbol{g}\left(1-\frac{\rho_f}{\rho_p}\right) \end{eqnarray}\]where, \(f=1+0.15Re_p^{0.687}\) is the Schiller-Naumann correction factor to Stokes drag expression (see the reference) to account for the effect of finite particle Reynolds number (denoted by \(Re_p=\rho_f|\boldsymbol{u}_f-\boldsymbol{u}_p|d_p/\mu\)). In addition, \(\tau_p=\rho_pd^2_p/18\mu\) is the particle relaxation time obtained based on Stokes drag expression. \(\mu\) and \(\rho_f\) are the carrier phase dynamic viscosity and density, respectively.

The upstream fluid velocity for each particle, \(\boldsymbol{u}_f\), is obtained by interpolating the surrounding fluid velocity to the particle location.

In the limit of small \(Re_p\) (thus \(f{=}1\)), analytical solution for the particle velocity is obtained as,

\[\begin{eqnarray} \boldsymbol{u}_p(t) = \boldsymbol{u}_{st}\left(1-\exp\left(\frac{-t}{\tau_p}\right)\right) \end{eqnarray}\]where,

\(\boldsymbol{u}_{st}=\tau_p\boldsymbol{g}(1-\frac{\rho_f}{\rho_p})\) is the particle terminal (settling) velocity.

## References

- Clift, R., Grace, J.R., & Weber, M.E. (2005). Bubbles, drops, and particles. Dover Publications.