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

In this paper, we perform analysis of the fluid velocity vector field divergence $\mathrm{\nabla }\cdot \vec{u}$ derived from the continuity equation, and we explore its application in the Navier-Stokes equations for compressible fluids $\rho (\vec{x},t)\mathrm{\ne }const$, occupying all of $\vec{x}\in R^3$ space at any $t\ge 0$. 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 $\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}$ and over space $\mathrm{\nabla }\rho$, 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 $\mathrm{\nabla }\cdot \vec{u}=0$ in scenarios when the fluid density is not constant $\rho (\vec{x},t)\mathrm{\ne }const$ over space nor time, and we analyze scenarios of the non-zero divergence $\mathrm{\nabla }\cdot \vec{u}\mathrm{\ne }0$. 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.