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.

中文翻译：

---

液体流动 Navier-Stokes 方程的新公式

对于等温液体流动，不可压缩性条件为描述其机械特性提供了有用的简化。然而，它忽略了声学效应，并且无论采用何种类型的剪应力本构方程，它都存在速度扰动无限传播速度的令人不快的缺点。在本文中，我们推导了一种克服上述问题的液体流动纳维-斯托克斯方程的新公式。压力失去了其单纯的规范变量（或等效的不可压缩条件的拉格朗日乘数）的辅助地位，加强了速度场的螺线管性质，并获得了以其自身的时空演化为特征的流体动力场变量的适当物理意义。根据与非零体积粘度的出现相关的声音衰减的实验证据，在不引入任何可调参数的情况下导出了非平衡条件下压力的演变方程。建立了内应力传播过程中可压缩性和记忆效应之间的联系。还讨论了正常模式分析和一些初步模拟。