Alternative restricted creeping-flow force balance

For a smooth constant-density incompressible Newtonian continuum in an inertial Cartesian chart, select the steady zero-inertia alternative force balance on an open region $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$. Let ${\rho }_0>0$ be constant mass density, $\mu >0$ constant dynamic viscosity, $u\in C^2(\mathrm{\Omega };{\mathbb{R}}^3)$ velocity, $p\in C^1(\mathrm{\Omega })$ pressure and $b\in C^0(\mathrm{\Omega };{\mathbb{R}}^3)$ specified body acceleration. In SI units, ${\rho }_0$ is $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $\mu$ is $\mathrm{P}\mathrm{a}\mathrm{s}$, $u$ is $\mathrm{m}{\mathrm{s}}^{-1}$, $b$ is $\mathrm{m}{\mathrm{s}}^{-2}$, and $p$ is in pascals. Adopt $\mathrm{div}u=0,\sigma =-pI+2\mu D,\mathrm{div}\sigma +{\rho }_{0}b=0,D=\frac{1}{2}(\mathrm{\nabla }u+\mathrm{\nabla }{u}^{\mathsf{T}})\mathrm{.}$ Here $(\mathrm{\nabla }u{)}_{ij}={\mathrm{\partial }}_j{u}_i$ and $(\mathrm{div}\sigma {)}_i={\sum }_j{\mathrm{\partial }}_j{\sigma }_{ij}$. This is an alternative zero-inertia branch: the full material-acceleration momentum equation is not also imposed on the same field. For constant $\mu$, the adopted balance is equivalent to $-\mathrm{\nabla }p+\mu \mathrm{\Delta }u+{\rho }_{0}b=0$. The Postulate does not assert that small Reynolds number alone yields this branch, provide an approximation error or solution-existence result, or establish empirical validity. Boundary and far-field data remain separate.

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

University of Washington, An Introduction to Continuum Mechanics, Chapter 6, §§6.5.1–6.5.3, printed pp. 242–245

The steady zero-inertia balance is an adopted alternative branch, not a deduction from the full material-acceleration equation or from experimental data. For constant $\mu$, componentwise differentiation gives $\mathrm{div}(2\mu D)=\mu \mathrm{\Delta }u+\mu \mathrm{\nabla }\mathrm{div}u$. Incompressibility therefore makes the adopted force law equivalent to $-\mathrm{\nabla }p+\mu \mathrm{\Delta }u+{\rho }_{0}b=0$. Boundary and far-field data remain separate inputs; this algebraic equivalence supplies no approximation-error or solvability claim.