Cooling flow bulk motion corrections to the Sunyaev Zel'dovich effect

We study the influence of converging cooling flow bulk motions on the Sunyaev-Zel'dovich (SZ) effect. To that purpose we derive a modified Kompaneets equation which takes into account the contribution of the accelerated electron media of the cooling flow inside the cluster frame. The additional term is different from the usual kinematic SZ-effect, which depends linearly on the velocity, whereas the contribution described here is quadratic in the macroscopic electron fluid velocity, as measured in the cluster frame. For clusters with a large cooling flow mass deposition rate and/or a small central electron density, it turns out that this effect becomes relevant.