# Fix incorrect treatment of body force density in FSLBM module

This merge request fixes the incorrect treatment of external body forces, that is, body force densities in the FSLBM module.
In the FSLBM, we relax the particle distribution functions to their compressible state. However, external body force densities were incorrectly assumed as if they were relaxed to an incompressible equilibrium. More specifically, the force density was `\boldsymbol{F} = \boldsymbol{a} \rho_0`

, where `\boldsymbol{a}`

is a user-chosen acceleration and `\rho_0=1`

is the LBM reference density.

Now, the force density is correctly taken into account by `\boldsymbol{F}(\boldsymbol{x}, t) = \boldsymbol{a} \rho(\boldsymbol{x}, t) \varphi(\boldsymbol{x}, t)`

, where `\rho(\boldsymbol{x},t)`

is the density and `\varphi(\boldsymbol{x}, t)`

is the fill level of a cell located at `\boldsymbol{x}`

.

This merge request also updates all showcases accordingly.