  We construct stationary non-equilibrium black funnels locally asymptotic to
global AdS4 in vacuum Einstein-Hilbert gravity with negative cosmological
constant. These are non-compactly-generated black holes in which a single
connected bulk horizon extends to meet the conformal boundary. Thus the induced
(conformal) boundary metric has smooth horizons as well. In our examples, the
boundary spacetime contains a pair of black holes connected through the bulk by
a tubular bulk horizon. Taking one boundary black hole to be hotter than the
other ($\Delta T \neq 0$) prohibits equilibrium. The result is a so-called
flowing funnel, a stationary bulk black hole with a non-Killing horizon that
may be said to transport heat toward the cooler boundary black hole. While
generators of the bulk future horizon evolve toward zero expansion in the far
future, they begin at finite affine parameter with infinite expansion on a
singular past horizon characterized by power-law divergences with universal
exponents. We explore both the horizon generators and the boundary stress
tensor in detail. While most of our results are numerical, a semi-analytic
fluid/gravity description can be obtained by passing to a one-parameter
generalization of the above boundary conditions. The new parameter detunes the
temperatures $T_{bulk BH}$ and $T_{bndy BH}$ of the bulk and boundary black
holes, and we may then take \alpha = $T_\mathrm{bndy BH}/T_\mathrm{bulk BH}$
and \Delta T small to control the accuracy of the fluid-gravity approximation.
In the small \alpha, \Delta T regime we find excellent agreement with our
numerical solutions. For our cases the agreement also remains quite good even
for $\alpha \sim 0.8$. In terms of a dual CFT, our \alpha = 1 solutions
describe heat transport via a large N version of Hawking radiation through a
deconfined plasma that couples efficiently to both boundary black holes.
