# Understanding MP-PIC implementation in OpenFOAM

The multiphase particle-in-cell (MP-PIC) method is characterized by mapping particle properties from the Lagrangian coordinates to the Eulerian grid. However, the implementation of this method in OpenFOAM solves an interpolated equation on the Eulerian grid only in one "submodel", the implicit packing model, for which there are alternatives (the none or explicit packing models). How is this possible and why is it still considered MP-PIC?

Furthermore, OpenFOAM seems to solve this equation in order to generate the particle velocity correction: $$\frac{\partial\alpha}{\partial t} - \Delta\left(\tau^\prime(\alpha) \alpha\right) = 0,$$ where $$\alpha$$ is the particle phase volume fraction (averaged on the Eulerian grid) and $$\tau$$ is the interparticle stress. However, I can't find this equation in neither of the papers listed below. How does one derive the equation, and thus the velocity correction used in OpenFOAM?

References:

• Andrews, M.J. and O'Rourke, P.J. (1996). The Multiphase Particle-in-Cell (MP-PIC) Method for Dense Particle Flows. International Journal of Multiphase Flow, 22(2):379–402.
• Snider, D.M. (2001). An Incompressible Three-Dimensional Multiphase Particle-in-Cell Model for Dense Particle Flows. Journal of Computational Physics, 170:523–549.