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: $\frac{\delta v_x}{\delta y} = \frac{\delta v_y}{\delta y} = 0$?

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    $\begingroup$ That depends on your discretization scheme. In general, indeed, the gradients should vanish at the boundary. $\endgroup$
    – nluigi
    Nov 6, 2015 at 15:21
  • $\begingroup$ 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? $\endgroup$
    – mar
    Nov 6, 2015 at 15:36
  • $\begingroup$ 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. $\endgroup$ Nov 6, 2015 at 15:48
  • $\begingroup$ 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 $u$. $\endgroup$ Nov 8, 2015 at 18:43

1 Answer 1


My take on boundary conditions for the Stokes equations are given here. It also links to some video lectures that describe the procedure by which you arrive at it. The thing to note is that "stress-free" also involves the pressure.


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