New formulation of the Navier–Stokes equations for liquid flows

For isothermal liquid flows, the condition of incompressibility provides a useful simplification for describing their mechanical properties. Nevertheless, it overlooks acoustic effects, and it provides the unpleasant shortcoming of infinite propagation speed of velocity perturbations, no matter the type of constitutive equation for the shear stresses is adopted. In this paper, we provide a derivation of a new formulation of the Navier–Stokes equations for liquid flows that overcomes the above issues. The pressure looses its ancillary status of mere gauge variable (or equivalently Lagrange multiplier of the incompressibility condition) enforcing the solenoidal nature of the velocity field, and attains the proper physical meaning of hydrodynamic field variable characterized by its own spatiotemporal evolution. From the experimental evidence of sound attenuation, related to the occurrence of a non-vanishing bulk viscosity, the evolution equation for pressure in out-of-equilibrium conditions is derived without introducing any adjustable parameters. The connection between compressibility and memory effects in the propagation of internal stresses is established. Normal mode analysis and some preliminary simulations are also discussed.