Thermodynamic Theory of Incompressible Hydrodynamics

The grand potential for open systems describes thermodynamics of fluid flows at low Mach numbers. A new system of reduced equations for the grand potential and the fluid momentum is derived from the compressible Navier-Stokes equations. The incompressible Navier-Stokes equations are the quasistationary solution to the new system. It is argued that the grand canonical ensemble is the unifying concept for the derivation of models and numerical methods for incompressible fluids, illustrated here with a simulation of a minimal Boltzmann model in a microflow setup.

Figure 1

Velocity profile of the 2D body-force driven Poiseuille flow at $\mathrm{K}\mathrm{n}=0.035$ and $\mathrm{M}\mathrm{a}=0.01$.