Outline of the paper

The Navier-Stokes (N-S) system, whose most popular version is

constitutes a basic model in fluid dynamics. It governs the spatio-temporal evolution of the fields fields $\varrho(x, t)$, $\boldsymbol{v}(x, t)$, and $p(x,t)$, respectively, the mass density $\left(\mathrm{kg} / \mathrm{m}^{3}\right)$, the velocity $(\mathrm{m} / \mathrm{s})$ and the thermodynamic pressure $(\mathrm{Pa})$. Moreover, $\boldsymbol{f}(x, t)$ is the non-inertial body force field $\left(\mathrm{N} / \mathrm{m}^{3}\right) .$ The term $(\boldsymbol{v} \cdot \nabla) \boldsymbol{v}=(\nabla \boldsymbol{v}) \boldsymbol{v},$ that is, the application of the linear operator $\nabla \boldsymbol{v}$ to $\boldsymbol{v} .$ The terms $\Delta \boldsymbol{v}$ and $\nabla(\nabla \cdot \boldsymbol{v})$ are, respectively, the Laplacian of the velocity field and the gradient of its divergence. These terms carry dissipative effects proportional to the dynamic viscosity $\mu(\mathrm{Pa} / \mathrm{s})$ and the volumetric viscosity $\zeta$, assumed to be constant.

For fluids, the density can be considered equal to a constant $\varrho_{*}$, which may be taken, for example, to be the density measured at the atmospheric pressure $p_{*}$, so that density may be expressed in terms of pressure, giving an equation of the form:

This equation allows to remove the density from the list of unknowns, so that the system $\eqref{eq:1}$ becomes:

Notice that $\eqref{eq:2}$ is a consequence of the second of $\eqref{eq:3}$. On the basis of $\eqref{eq:2}$, one may expect that, if the bulk modulus $K$ tends to infinity, then the density tends to a constant:

and one may expect that the behavior of the solutions of $\eqref{eq:3}$ could be be captured by that of the solutions of ¹

This is the so-called quasi-incompressible Navier-Stokes system, which was introduced by Temam as a mathematical regularization of the notoriously stiff fully-incompressible system:

and has been recently been re-discovered as an independent model capable, without to many complications, to describe phenomena such as pressure waves. The problem with $\eqref{eq:5}$ is that, since the divergence of $\boldsymbol v$ does not vanish, balance of mass is not satisfies. In fact, by Reynolds transport theorem asserts:

When $\dot\varrho+\varrho\operatorname{\nabla\cdot}\boldsymbol v=0$, we find that the time derivative of the kinetic energy is

Thus, the power of the inertial force is minus the time derivative of the kinetic energy.

If, however, the mass density is constant, but the velocity field has non-null divergence, then this balance is not satisfied.

To overcome this inconvenience, an additional force is added on the right-hand side

Note that this extra term is exactly what is needed to restore the balance of energy. An interpretation of this extra term is given in the paper on arxiv, published on APPLES.