Constant-density incompressible Newtonian model

Let $I\subset \mathbb{R}$ be an open time interval. Use the Cartesian inertial-chart convention of the adopted continuum balances on Newtonian spacetime $I\times E$: $E$ is an oriented Euclidean affine three-space, and the event-coordinate map is $(t,X)\mapsto (t,x_t(X))$ with $x_t(X)=b(X)+tw$, where $b:E\to {\mathbb{R}}^3$ is an orientation-preserving affine isometry and $w\in {\mathbb{R}}^3$ is constant. Time is absolute and measured in seconds, spatial coordinates in metres, and $w$ in metres per second. All derivatives below use this fixed chart. Let $\mathrm{\Omega }\subset I\times {\mathbb{R}}^3$ be open. A smooth constant-density incompressible Newtonian branch has a constant ${\rho }_0>0$ in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, a constant dynamic shear viscosity $\mu \ge 0$ in $\mathrm{P}\mathrm{a}\mathrm{s}$, a velocity field $u\in C^2(\mathrm{\Omega },{\mathbb{R}}^3)$ measured in $\mathrm{m}{\mathrm{s}}^{-1}$, and a pressure multiplier $p\in C^1(\mathrm{\Omega },\mathbb{R})$ measured in $\mathrm{P}\mathrm{a}$. Define $L_{ij}={\mathrm{\partial }}_{x_j}{u}_i$, $D=(L+L^T)\mathrm{/}2$, and $D_{t}u={\mathrm{\partial }}_{t}u+(u\cdot \mathrm{\nabla })u$. The branch is specified by $\rho ={\rho }_0,\mathrm{div}u=0,\sigma =-p{I}_3+2\mu D,{\rho }_0{D}_{t}u=\mathrm{div}\sigma ,$ where $(\mathrm{div}\sigma {)}_i={\sum }_{j=1}^3{\mathrm{\partial }}_{x_j}{\sigma }_{ij}$. Equivalently, for constant $\mu$, its momentum equation is ${\rho }_0{D}_{t}u+\mathrm{\nabla }p=\mu \mathrm{\Delta }u,\mathrm{\Delta }{u}_i={\sum }_{j=1}^3{\mathrm{\partial }}_{x_j{x}_j}{u}_i\mathrm{.}$ Indeed, $\mathrm{div}(2\mu D)=\mu \mathrm{\Delta }u+\mu \mathrm{\nabla }(\mathrm{div}u)=\mu \mathrm{\Delta }u$. This is the incompressible specialization of the adopted Newtonian closure: $\theta =\mathrm{div}u=0$ implies $D_0=D$, so the bulk-viscosity and Fourier-flux terms do not enter this mechanical equation. The pressure is a constraint multiplier and is not identified with a compressible equation-of-state pressure. Define the kinematic viscosity $\nu =\mu \mathrm{/}{\rho }_0$, in ${\mathrm{m}}^2{\mathrm{s}}^{-1}$. Boundary and initial data are not part of this definition, and the item asserts no solution existence.

Adopted continuum balances and inherited Cartesian inertial-chart convention

David Tong, Fluid Mechanics, Part II lecture notes, §3.1.1, printed pp. 60–61

An Introduction to Continuum Mechanics, Second Edition, Chapter 6, §6.5.3, printed pp. 244–245