Models of low-speed flow for near-critical fluids with gravitational and capillary effects

We study low-speed flows of a highly compressible, single-phase fluid in th e presence of gravity, for example, in a regime appropriate for modeling re cent space-shuttle experiments on fluids near the liquid-vapor critical poi nt. In the equations of motion, we include forces due to capillary stresses that arise from a contribution made by strong density gradients to the fre e energy. We derive formally simplified sets of equations in a low-speed li mit analogous to the zero Mach number limit in combustion theory. When visc osity is neglected and gravity is weak, the simplified system includes: a h yperbolic equation for velocity, a parabolic equation for temperature, an e lliptic equation related to volume expansion, an integro-differential equat ion for mean pressure, and an algebraic equation (the equation of state). S olutions are determined by initial values for the mean pressure, the temper ature field, and the divergence-free part of the velocity field. To model m ulti-dimensional flows with strong gravity, we offer an alternative to the anelastic approximation, one which admits stratified fluids in thermodynami c equilibrium, as well as gravity waves but not acoustic waves.