  Small solids embedded in gaseous protoplanetary disks are subject to strong
dust-gas friction. Consequently, tightly-coupled dust particles almost follow
the gas flow. This near conservation of dust-to-gas ratio along streamlines is
analogous to the near conservation of entropy along flows of (dust-free) gas
with weak heating and cooling. We develop this thermodynamic analogy into a
framework to study dusty gas dynamics in protoplanetary disks. We show that an
isothermal dusty gas behaves like an adiabatic pure gas; and that finite
dust-gas coupling may be regarded as an effective heating/cooling. We exploit
this correspondence to deduce that 1) perfectly coupled, thin dust layers
cannot cause axisymmetric instabilities; 2) radial dust edges are unstable if
the dust is vertically well-mixed; 3) the streaming instability necessarily
involves a gas pressure response that lags behind dust density; 4) dust-loading
introduces buoyancy forces that generally stabilizes the vertical shear
instability associated with global radial temperature gradients. We also
discuss dusty analogs of other hydrodynamic processes (e.g. Rossby wave
instability, convective overstability, and zombie vortices), and how to
simulate dusty protoplanetary disks with minor tweaks to existing codes for
pure gas dynamics.
