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 the presence of gravity, for example, in a regime appropriate for modeling recent spaceshuttle 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 multi-dimensional 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.