I did a lot of googling but could not find a satisfying answer to my question, hence a post here. Question: How to solve (or close) the isothermal incompressible Navier-Stokes equations for an isothermal compressible fluid? Situation: We have a compressible fluid, for example a gas. The flow of this compressible fluid is in the incompressible regime (i.e. it has low Mach number) Context: In three-dimensional space the isothermal incompressible Navier-Stokes equations consists of four equations: 1 mass equation 3 momentum equations (one for each for spatial dimension) There are five variables: Mass density rho Pressure P Velocity (u_x, u_y, u_z) So, to close the system of equations we need additional information. The simplest approach is to assume we are dealing with an incompressible fluid: Mass density rho then becomes a user-specifiable constant, i.e. constant over space and time. This assumption is acceptable if we are dealing with a liquid, since liquids are generally considered incompressible fluids. The remaining variables are then the pressure and velocity. The pressure is usually computed by solving the Poisson equation for the pressure. However, if we are dealing with an isothermal compressible fluid (e.g. a gas), then we generally cannot assume the mass density is constant over space and time. We can assume the flow of this compressible fluid is incompressible if the Mach number is low. To close the system of equations here we probably need to introduce an equation of state relating the existing variables with each other somehow. But which equation of state is generic? That is, is there an equation of state valid for many gases? Would it also be valid for liquids? And do we still solve the pressure from the Poisson equation for pressure? Questions I hope one of you guys can answer, even a partial answer is welcome.