Restricted incompressible creeping-flow Stokes model

Use the Newtonian Cartesian inertial chart convention of the earlier classical-frame definition, with spatial coordinates $x$ in metres. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be open, ${\rho }_0>0$ constant density in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $\mu >0$ constant dynamic viscosity in $\mathrm{P}\mathrm{a}\mathrm{s}$, $u\in C^2(\mathrm{\Omega };{\mathbb{R}}^3)$ velocity in $\mathrm{m}{\mathrm{s}}^{-1}$, $p\in C^1(\mathrm{\Omega })$ pressure in pascals, and $b\in C^0(\mathrm{\Omega };{\mathbb{R}}^3)$ specified body acceleration in $\mathrm{m}{\mathrm{s}}^{-2}$. Set $D=(\mathrm{\nabla }u+\mathrm{\nabla }{u}^{\mathsf{T}})\mathrm{/}2$ in ${\mathrm{s}}^{-1}$, $(\mathrm{div}\sigma {)}_i={\sum }_j{\mathrm{\partial }}_j{\sigma }_{ij}$, and $(\mathrm{\Delta }u{)}_i={\sum }_j{\mathrm{\partial }}_{jj}{u}_i$; $\mathrm{\nabla }$ has units ${\mathrm{m}}^{-1}$ and $\mathrm{\Delta }$ has units ${\mathrm{m}}^{-2}$. The stress $\sigma =-pI+2\mu D$ has units Pa. The selected steady creeping Stokes branch is $\mathrm{div}u=0,\sigma =-pI+2\mu D,\mathrm{div}\sigma +{\rho }_{0}b=0.$ Equivalently, $-\mathrm{\nabla }p+\mu \mathrm{\Delta }u+{\rho }_{0}b=0$. The definition uses the zero-inertia alternative force balance and does not also impose the full material-acceleration equation on the same field. Boundary, far-field, geometry and decay conditions are separate. The force-free sphere application selects $b=0$. No error estimate comparing this steady branch with the full constant-density momentum equation retaining material acceleration is asserted.

Classical frames, primitive quantities and SI units

Stewart, MIT 8.09, Fluid Mechanics, §6.8, printed pp. 135–139