Conservative incompressible momentum equation

Core idea

For smooth divergence-free velocity and constant $\nu$ , the Navier–Stokes momentum equation is equivalent to

\partial_tu+\nabla\cdot \bigl(u\otimes u+pI-\nu(\nabla u+(\nabla u)^{\mathsf T})\bigr)=f.

The divergence of a matrix is taken row by row: $(\nabla\cdot J)_i=\sum_j\partial_jJ_{ij}$ .

Verification

The product rule yields

\nabla\cdot(u\otimes u)=(u\cdot\nabla)u+u(\nabla\cdot u),

while equality of mixed derivatives gives

\nabla\cdot(\nabla u+(\nabla u)^{\mathsf T}) =\Delta u+\nabla(\nabla\cdot u).

Both extra terms vanish under incompressibility. Also $\nabla\cdot(pI)=\nabla p$ .

Simpler equivalent flux

For divergence-free velocity, $u\otimes u+pI-\nu\nabla u$ has the same divergence as the symmetric-stress flux. The flux tensors differ, but their local momentum equations coincide under this constraint.