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In this paper, we perform analysis of the fluid velocity vector fielddivergence $\nabla \cdot \vec{u}$ derived from the continuity equation, and weexplore 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 overtime $\frac{\partial \rho }{\partial t}$ and over space $\nabla \rho $, inaddition to the fluid velocity vector field $\vec{u}(\vec{x},t)$ and the fluiddensity $\rho (\vec{x},t)$ itself. We derive the conditions for thedivergence-free fluid velocity vector field $\nabla \cdot \vec{u}=0$ inscenarios 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 theNavier-Stokes equation for compressible fluids, and we deduct the condition forvanishing (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 moreelementary condition for vanishing viscosity, stating that vanishing viscosityis triggered once scalar function $d(\vec{x},t)=-\frac{1}{\rho }(\frac{\partial \rho }{\partial t} +\vec{u}\cdot \nabla \rho )$ is harmonicfunction. Once that condition is satisfied, the viscosity related term of theNavier-Stokes equations for compressible fluids equals to zero, which is knownas related to turbulent fluid flows.

Analysis of fluid velocity vector field divergence $\nabla\cdot\vec{u}$  in function of variable fluid density $\rho(\vec{x},t)\neq const$ and  conditions for vanishing viscosity of compressible Navier-Stokes equations