The multiphase particle-in-cell method assumes an incompressible fluid phase with the corresponding continuity equation, .
\frac{\partial \theta_f \bold{u}_f}{\partial t} + \nabla \cdot ( \theta_f \bold{u}_f \bold{u}_f ) = - \frac{\nabla p}{\rho_f} - \frac{\bold{F}}{\rho_f}+\theta_f \bold{g} .