  The sign change of the slope of the directed flow of baryons has been
predicted as a signal for a first order phase transition within fluid dynamical
calculations. Recently, the directed flow of identified particles has been
measured by the STAR collaboration in the beam energy scan (BES) program. In
this article, we examine the collision energy dependence of directed flow $v_1$
in fluid dynamical model descriptions of heavy ion collisions for
$\sqrt{s_{NN}}=3-20$ GeV. The first step is to reproduce the existing
predictions within pure fluid dynamical calculations. As a second step we
investigate the influence of the order of the phase transition on the
anisotropic flow within a state-of-the-art hybrid approach that describes other
global observables reasonably well. We find that, in the hybrid approach, there
seems to be no sensitivity of the directed flow on the equation of state and in
particular on the existence of a first order phase transition. In addition, we
explore more subtle sensitivities like e.g. the Cooper-Frye transition
criterion and discuss how momentum conservation and the definition of the event
plane affects the results. At this point, none of our calculations matches
qualitatively the behavior of the STAR data, the values of the slopes are
always larger than in the data.
