  The Hartle-Hawking state of $\mathcal{N}=4$ SYM at strong coupling and large
$N$ on a fixed black hole background has two proposed gravitational duals: a
black funnel or a black droplet. We construct the black funnel solutions that
are dual to the Hartle-Hawking state on a Schwarzschild black hole and on a
class of three-dimensional asymptotically flat black hole backgrounds. We
compute their holographic stress tensor and argue for the stability of these
solutions.
