# How to implement 'stress-free' boundary conditions (stokes flow)?

I have to implement 'stress-free' boundary conditions for a stokes flow of a 2D-domain (for the left, bottom and right boundaries). The boundary condition are also described mathematically as $2 \eta \dot{\epsilon} = 0$. $\eta$ and $\dot{\epsilon}$ are the viscosity and strain rate.

How do I implement such boundary conditions? Would stress-free mean that for the left/right boundary the velocities are: $\frac{\delta v_x}{\delta x} = \frac{\delta v_y}{\delta x} = 0$? And for the bottom: $\frac{\delta v_x}{\delta y} = \frac{\delta v_y}{\delta y} = 0$?

• That depends on your discretization scheme. In general, indeed, the gradients should vanish at the boundary. – nluigi Nov 6 '15 at 15:21
• I am using finite elements...so in that case I don't have to implement any boundary condition, since 'stress-free' conditions are based only on Neumann conditions. Is that correct? – mar Nov 6 '15 at 15:36
• Can you describe the physical problem briefly? For the left and right boundaries, $\frac{\delta v_x}{\delta x}= 0$ will make the boundary an "outflow". Do you want fluid flowing out at the boundary? $\frac{\delta v_y}{\delta x} = 0$ is sufficient to set the tangential shear stress at the boundary to zero. – Abhilash Reddy M Nov 6 '15 at 15:48
• At least two of the comments above are wrong. It's not the normal component of the gradient that is vanishing, but $n \cdot \dot\varepsilon$ that is zero. Depending on your normal vector $n$, this will couple different derivatives of $u$. – Wolfgang Bangerth Nov 8 '15 at 18:43