Reynolds defect system

A Reynolds defect system, with the right-hand-side sign convention, is

\partial_t v+\operatorname{div}(v\otimes v)+\nabla p =\nu\Delta v+f+\operatorname{div}R,\qquad \operatorname{div}v=0.

Here $R(x,t)$ is a symmetric matrix field, its divergence is $(\operatorname{div}R)_i=\sum_j\partial_jR_{ij}$ , and $v\otimes v$ is an outer product. For $\nu>0$ this is called Navier–Stokes–Reynolds; for $\nu=0$ , Euler–Reynolds. The Navier–Stokes equation is recovered when the defect divergence vanishes.

The pressure term is its gradient and $\Delta$ is the componentwise Laplacian.

Pressure and trace

Writing $R=R^\circ+rI$ with $r=\operatorname{tr}R/n$ , replace $p$ by $p-r$ to use the trace-free tensor $R^\circ$ . Thus trace parts can be put into pressure. The sign of a tensor called Reynolds stress must be checked against its displayed equation.