Compressible and immiscible fluids with arbitrary density ratio

For the water-air system, the bulk density ratio is as high as about 1000; no model can fully tackle such a high density ratio system. In the Navier-Stokes and Euler equations, the density $\rho$ within the water-air interface is assumed to be a constant based on the Boussinesq approximation namely $\rho (\mathrm{d} \mathbf u/\mathrm{d} t)$, which does not account for the true momentum evolution $\mathrm{d} (\rho \mathbf u)/\mathrm{d} t$ ($\mathbf u$-fluid velocity). Here, we present an alternative theory for the density evolution equations of immiscible fluids in computational fluid dynamics, differing from the concept of Navier-Stokes and Euler equations. Our derivation is built upon the physical principle of energy minimization from the aspect of thermodynamics. The present results provide a generalization of Bernoulli's principle for energy conservation and a general formulation for the sound speed. The present model can be applied for immiscible fluids with arbitrarily high density ratios, thereby, opening a new window for computational fluid dynamics both for compressible and incompressible fluids.