Momentum flux tensor in an incompressible fluid

With density normalized to one, the advective momentum flux tensor is $u\otimes u$ , whose entry $u_i u_j$ transports momentum component $i$ in direction $j$ . Including pressure and constant Newtonian viscosity gives the total flux

J=u\otimes u+pI-2\nu D(u).

The sign of the viscous term corresponds to writing the balance law as $\partial_tu+\nabla\cdot J=f$ .

Flux across a surface

For a unit normal $n$ , the vector $Jn$ gives the total momentum flux across the oriented surface. In components it is $(Jn)_i=\sum_jJ_{ij}n_j$ , so the tensor convention agrees with row divergence.

Advective versus total flux

“Momentum flux” sometimes denotes just $u\otimes u$ . Pressure and viscous contributions must be included when converting the full momentum equation into conservative form.

References

Lenya Ryzhik, Lecture Notes for Math 256B (2024), Sections 1–3.