Nitsche’s method for Navier–Stokes equations with slip boundary conditions

Abstract:

We formulate Nitsche’s method to implement slip boundary conditions for flow problems in domains with curved boundaries. The slip boundary condition, often referred to as the Navier friction condition, is critical for understanding and simulating a wide range of phenomena such as turbulence, droplet spread and flow through microdevices. In this work, we highlight the role of the approximation of the normal and tangent vector. In particular, we show that using the normal and tangent vectors with respect to the discretized domain $\Omega _h$, denoted $\mathbf {n}_h$ and $\boldsymbol {\tau }_h$, is suboptimal. Taking instead a projection of the normal and tangent vectors with respect to $\Omega$, denoted $\mathbf {n}_\pi$ and $\boldsymbol {\tau }_\pi$, gives the best convergence rate that can be expected for a polygonal approximation of a curved boundary.

Finally we also prove that, if you use instead the exact slip with $\mathbf {n}_h$ and $\boldsymbol {\tau }_h$, the approximation converges to the wrong solution. This is known as the Babuška-Sapondzhyan Paradox. Thus Nitsche’s method relaxes the slip condition and avoids the lack of convergence.