  It has recently been proposed that the strong coupling behaviour of quantum
field theories on a non-dynamical black hole background can be described, in
the context of the AdS/CFT correspondence, by a competition between two gravity
duals: a black funnel and a black droplet. We present here thermal equilibrium
solutions which represent such spacetimes, providing the first example where
the thermal competition between the gravity duals can be studied. The solutions
correspond to a special family of charged AdS C-metrics. We compute the
corresponding Euclidean actions and find that the black funnel always dominates
the canonical ensemble in our example, meaning that the field theory does not
undergo a phase transition.
