Forced constant-density incompressible Newtonian model

Let $I\subset \mathbb{R}$ be a bounded open time interval measured in seconds and let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a fixed bounded open spatial domain measured in metres whose boundary is locally a Lipschitz graph after rigid coordinate changes. Take ${\rho }_0>0$ kilograms per cubic metre and ${\mu }_0>0$ pascal seconds as primitive branch-local physical parameters: this forced branch adopts them as constant density and dynamic shear viscosity inputs and derives neither from another framework nor from observations. Let $u:I\times \mathrm{\Omega }\to {\mathbb{R}}^3$ be a twice continuously differentiable velocity in metres per second, $p:I\times \mathrm{\Omega }\to \mathbb{R}$ a once continuously differentiable pressure multiplier in pascals, and $b:I\times \mathrm{\Omega }\to {\mathbb{R}}^3$ a continuous body-acceleration field in metres per second squared. Define $D=(\mathrm{\nabla }u+\mathrm{\nabla }{u}^{\mathsf{T}})\mathrm{/}2$ and the stress tensor $\sigma =-p{I}_3+2{\mu }_{0}D$ in pascals, where $I_3$ is the $3\times 3$ identity and $(\mathrm{div}\sigma {)}_i={\sum }_{j=1}^3{\mathrm{\partial }}_{x_j}{\sigma }_{ij}$.

The forced incompressible Newtonian branch adopts

$\mathrm{div}u=0,{\rho }_0({\mathrm{\partial }}_{t}u+(u\cdot \mathrm{\nabla })u)={\rho }_{0}b+\mathrm{div}\sigma \mathrm{.}$

Equivalently, since ${\mu }_0$ is constant and $\mathrm{div}u=0$, its momentum equation is

${\rho }_0({\mathrm{\partial }}_{t}u+(u\cdot \mathrm{\nabla })u)+\mathrm{\nabla }p={\mu }_0\mathrm{\Delta }u+{\rho }_{0}b\mathrm{.}$

The derived kinematic viscosity is ${\nu }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\mu }_0\mathrm{/}{\rho }_0$ in square metres per second. Pressure is a constraint multiplier, not an equation-of-state pressure. When the stationary no-slip branch is selected, additionally require that $u$ admit a continuous extension $\bar{u}:I\times \bar{\mathrm{\Omega }}\to {\mathbb{R}}^3$ of its interior values, and define its classical wall trace by $u_{\mathrm{\Gamma }}(t,x):=\bar{u}(t,x)$ for $(t,x)\in I\times \mathrm{\partial }\mathrm{\Omega }$. This extension, when it exists, is unique because $\mathrm{\Omega }$ is dense in $\bar{\mathrm{\Omega }}$. The separately selected no-slip boundary datum is $u_{\mathrm{\Gamma }}=0$; the extension requirement is additional boundary regularity, and neither it, this equality, nor the body force follows from the interior balance. This is a forced branch of the Newtonian model: it adds the prescribed acceleration field to the unforced branch and makes no claim of solution existence or empirical validity.

Unforced constant-density Newtonian model and stress convention

Adopted material momentum balance with body acceleration

Selected stationary no-slip boundary data

Tong, Fluid Mechanics, §§3.1.1 and 3.2.1

Powers, Lecture Notes on Intermediate Fluid Mechanics, §5.4.1