Notes on Symmetry Boundary Conditions

The symmetry boundary condition prescribes no penetration and vanishing shear stresses. The boundary condition is a combination of a Dirichlet condition and a Neumann condition:

\[\begin{aligned} \mathbf{u} \cdot \mathbf{n}=\mathbf{0}, &\quad \left(-p \mathbf{I}+\mu\left(\nabla \mathbf{u}+(\nabla \mathbf{u})^{\mathrm{T}}\right)\right) \mathbf{n}=\mathbf{0} \end{aligned}\]

for the incompressible formulation.

The Dirichlet condition takes precedence over the Neumann condition, and the above equations are equivalent to the following equation :

\[\begin{array}{c} \mathbf{u} \cdot \mathbf{n}=\mathbf{0}, \quad \mathbf{K}-(\mathbf{K} \cdot \mathbf{n}) \mathbf{n}=\mathbf{0} \\ \mathbf{K}=\mu\left(\nabla \mathbf{u}+(\nabla \mathbf{u})^{\mathrm{T}}\right) \mathbf{n} \end{array}\]

This means that the tangential components of the normal stress \(K\) (called the shear stress, often noted \(\tau\)) are 0 and the normal component of the velocity is 0.

These conditions can also be interpreted as a "slip'' wall. No-slip wall corresponds to full Diricihlet condition \(\mathbf{u}=0\).

These conditions are not implemented in Feel++ CFD toolbox, but we have already the framework for that.

For the Stokes problem, the variationnal formulation is :

Mathematician version
\[\int_\Omega \nabla u : \nabla v -\int_{\Omega} p \nabla \cdot v + \int_{\partial \Omega} \left(p n - \frac{\partial u}{\partial n}\right) \cdot v = f\]
\[\left(-p n + \frac{\partial u}{\partial n}\right) = 0 \mbox{ Neumann}\]
Mecanician version
\[\sigma (u,p)= -p I + 2 \mu D(u), \quad D(u)=\frac{1}{2}\left(\nabla u + \nabla u^T\right)\\ \int_\Omega \sigma(u,p) : \nabla v + \int_{\partial \Omega} (\sigma(u,p) n) \cdot v = f\]
\[\sigma(u,p) n = 0 \mbox{ Neumann}\]

X-Y surface:

\[n=e_z(0,0,1) \quad \tau_1 = (1,0,0) \quad \tau_2(0,1,0)\]
\[K \tau_1 = 0 \\ K \tau_2 = 0 \\ K = \begin{bmatrix} 2 \partial_x u_x& \partial_y u_x+\partial_x u_y & \partial_z u_x+\partial_x u_z\\ \partial_x u_y+\partial_y u_x & 2\partial_y u_y & \partial_z u_y+\partial_y u_z\\ \partial_x u_z+\partial_z u_x& \partial_y u_z+\partial_z u_y +\partial_x u_y & 2\partial_z u_z\\ \end{bmatrix} \cdot \begin{bmatrix}0\\0\\1\end{bmatrix} = \begin{bmatrix} \partial_z u_x+\partial_x u_z\\ \partial_z u_y+\partial_y u_z\\ 2\partial_z u_z\\ \end{bmatrix}\]
\[K \tau_1 = \partial_z u_x+\partial_x u_z\\ K \tau_2 = \partial_z u_y+\partial_y u_z\]

in the formulation we have only

\[\int_{XY} (-p + 2\partial_z u_x) v_z = \int_{XY} ( ( -p + n \cdot K ) n )\cdot v\]
We need to verify that the term above is indeed implemented in Feel++
\[u \cdot n = 0 = u_z\]
we can already set the Dirichlet unknowns component - wise