  We present a study of the elliptic flow and $R_{AA}$ of D and anti-D mesons
in Au+Au collisions at FAIR energies. We propagate the charm quarks and the D
mesons following a previously applied Langevin dynamics. The evolution of the
background medium is modeled in two different ways: (I) we use the UrQMD
hydrodynamics + Boltzmann transport hybrid approach including a phase
transition to QGP and (II) with the coarse-graining approach employing also an
equation of state with QGP. The latter approach has previously been used to
describe di-lepton data at various energies very successfully. This comparison
allows us to explore the effects of partial thermalization and viscous effects
on the charm propagation. We explore the centrality dependencies of the
collisions, the variation of the decoupling temperature and various
hadronization parameters. We find that the initial partonic phase is
responsible for the creation of most of the D mesons elliptic flow and that the
subsequent hadronic interactions seem to play only a minor role. This indicates
that D mesons elliptic flow is a smoking gun for a partonic phase at FAIR
energies. However, the results suggest that the magnitude and the details of
the elliptic flow strongly depend on the dynamics of the medium and on the
hadronization procedure, which is related to the medium properties as well.
Therefore, even at FAIR energies the charm quark might constitute a very useful
tool to probe the Quark-Gluon Plasma and investigate its physics.
