# Has this boundary condition been used in fluid flow?

I would like to know whether anyone has seen a boundary condition used in a fluid flow problem, of the following type. Suppose viscous incompressible fluid is to the left of a plane $x_1=a$, so the outer normal to the fluid region is $n=(1,0,0)$. The condition specifies that the $x_1$ momentum flux is a nonnegative multiple of the velocity: $$m\begin{bmatrix} u_1\cr u_2\cr u_3\end{bmatrix} = \begin{bmatrix}\rho u_1^2+p-2\mu u_{1,1} \cr \rho u_2u_1-\mu(u_{1,2}+u_{2,1}) \cr \rho u_3u_1-\mu(u_{1,3}+u_{3,1}) \end{bmatrix}$$ where $m$ could be a number $\ge0$ or even a nonnegative 3 by 3 matrix. (This is connected with my question at http://www.mathoverflow.net/q/163772/) Thanks for any references.

• Your notation confuses me. Are you actually mixing subscript for components AND derivatives? Please don't do that. – Bernhard Apr 21 '14 at 16:12
• Can you clarify this with an edit to your question? – Bernhard Apr 26 '14 at 10:01