  We investigate the transport of phonons between $N$ harmonic oscillators in
contact with independent thermal baths and coupled to a common oscillator, and
derive an expression for the steady state heat flow between the oscillators in
the weak coupling limit. We apply these results to an optomechanical array
consisting of a pair of mechanical resonators coupled to a single quantised
electromagnetic field mode by radiation pressure as well as to thermal baths
with different temperatures. In the weak coupling limit this system is shown to
be equivalent to two mutually-coupled harmonic oscillators in contact with an
effective common thermal bath in addition to their independent baths. The
steady state occupation numbers and heat flows are derived and discussed in
various regimes of interest.
