  We discuss the role of the extra force $$ \mathbf f_{\rm e}=-\frac 1
2(\nabla\cdot\mathbf v)\mathbf v $$ in the system of partial differential
equations \begin{equation*} \left\{ \begin{aligned} &\frac{\partial\mathbf
v}{\partial \rm t}+(\mathbf v\cdot\nabla)\mathbf v+\nabla \mathrm p-\frac 1
{\rm Re}\Delta\mathbf v=\mathbf f+\mathbf f_{\rm e},\\ &\frac 1 {\mathrm
K}\frac{\partial \mathrm p}{\partial \rm t}+\nabla\cdot\mathbf v=0,\qquad
\mathrm K>>1, \end{aligned}\right. \end{equation*} whose weak solution has been
proved in [Arch. Rat. Mech. Analysis, 32:135-153] to approximate, in the limit
$\mathrm K\to\infty$, the weak solution of the incompressible Navier-Stokes
system. Taking the cue from [Ann. Mat. Pura Appl. 172:103-124], we provide a
physical interpretation of the extra force $\mathbf f_{\rm e}$, showing that it
is a manifestation of inertia.
