  In this paper, we perform analysis of the fluid velocity vector field
divergence $\nabla \cdot \vec{u}$ derived from the continuity equation, and we
explore its application in the Navier-Stokes equations for compressible fluids
$\rho (\vec{x},t)\ne const$, occupying all of $\vec{x}\in R^{3} $ space at any
$t\geq 0$. The resulting velocity vector field divergence $\nabla \cdot
\vec{u}=-\frac{1}{\rho } (\frac{\partial \rho }{\partial t} +\vec{u}\cdot
\nabla \rho )$ is a direct consequence of the fluid density rate of change over
time $\frac{\partial \rho }{\partial t}$ and over space $\nabla \rho $, in
addition to the fluid velocity vector field $\vec{u}(\vec{x},t)$ and the fluid
density $\rho (\vec{x},t)$ itself. We derive the conditions for the
divergence-free fluid velocity vector field $\nabla \cdot \vec{u}=0$ in
scenarios when the fluid density is not constant $\rho(\vec{x},t)\neq const$
over space nor time, and we analyze scenarios of the non-zero divergence
$\nabla \cdot \vec{u}\ne 0$. We apply the statement for divergence in the
Navier-Stokes equation for compressible fluids, and we deduct the condition for
vanishing (zero) viscosity term of the compressible Navier-Stokes equation:
$\nabla (\frac{1}{\rho } (\frac{\partial \rho }{\partial t} +\vec{u}\cdot
\nabla \rho ))=-3\Delta \vec{u}$. In addition to that, we derive even more
elementary condition for vanishing viscosity, stating that vanishing viscosity
is triggered once scalar function $d(\vec{x},t)=-\frac{1}{\rho }
(\frac{\partial \rho }{\partial t} +\vec{u}\cdot \nabla \rho )$ is harmonic
function. Once that condition is satisfied, the viscosity related term of the
Navier-Stokes equations for compressible fluids equals to zero, which is known
as related to turbulent fluid flows.
