  We construct an approximate static gravitational solution of the Einstein
equations, with negative cosmological constant, describing a test black string
stretching from the boundary of the Schwarzschild-AdS_5 black brane toward the
horizon. The construction builds on a derivative expansion method, assuming
that the black brane metric changes slowly along the black string direction. We
provide a solution up to second order in derivatives and it implies, in
particular, that the black string must shrink to zero size at the horizon of
the black brane. In the near horizon region of the black brane, where the two
horizons intersect, we provide an exact solution of a cone that describes two
intersecting horizons at different temperatures. Moreover, we show that this
solution equally describes a thin and long black droplet.
