  We study low-speed flows of a highly compressible, single-phase fluid in the
presence of gravity, for example in a regime appropriate for modeling recent
space-shuttle experiments on fluids near the liquid-vapor critical point. In
the equations of motion, we include forces due to capillary stresses that arise
from a contribution made by strong density gradients to the free energy. We
derive formally simplified sets of equations in a low-speed limit analogous to
the zero Mach number limit in combustion theory.
  When viscosity is neglected and gravity is weak, the simplified system
includes: a hyperbolic equation for velocity, a parabolic equation for
temperature, an elliptic equation related to volume expansion, an
integro-differential equation for mean pressure, and an algebraic equation (the
equation of state). Solutions are determined by initial values for the mean
pressure, the temperature field, and the divergence-free part of the velocity
field. To model multidimensional flows with strong gravity, we offer an
alternative to the anelastic approximation, one which admits stratified fluids
in thermodynamic equilibrium, as well as gravity waves but not acoustic waves.
