The equations of nearly incompressible fluids. I - Hydrodynamics, turbulence, and waves

An attempt is made to develop a more general theory of nearly incompressible fluids that can then be applied to many different fields. A perturbation expansion is developed for the fully compressible fluid equations which, in the limit of low Mach number (sound or Alfvenic Mach number), reduce to the appropriate incompressible fluid equations. The method developed derives modified systems of fluid equations in which the compressibility effects are admitted only weakly in terms of the incompressible hydrodynamic solutions ('nearly incompressible hydrodynamics'). Molecular viscosity is included self-consistently, and the role of thermal conduction in an ideal fluid is also considered. With heat conduction included, two distinct routes to incompressibility are found to be possible, distinguished according to the relative magnitudes of the temperature, density, and pressure fluctuations.