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27 changes: 12 additions & 15 deletions docs/axisymmetry.md
Original file line number Diff line number Diff line change
Expand Up @@ -68,26 +68,23 @@ For CNS, let $\mathbf{u}=(u_z,u_r)$ and use the swirl-free cylindrical velocity
divergence

$$
\nabla\!\cdot\mathbf{u}
= \frac{\partial u_z}{\partial z}
+ \frac{\partial u_r}{\partial r}
+ \frac{u_r}{r}.
\nabla\cdot\mathbf{u} = \frac{\partial u_z}{\partial z} + \frac{\partial u_r}{\partial r} + \frac{u_r}{r}
$$

With the Stokes hypothesis used by Theseus, the stress components needed by the
meridional operator are

$$
\begin{aligned}
\tau_{zz} &= \mu\left(2\frac{\partial u_z}{\partial z}
- \frac{2}{3}\nabla\!\cdot\mathbf{u}\right), \\
\tau_{rr} &= \mu\left(2\frac{\partial u_r}{\partial r}
- \frac{2}{3}\nabla\!\cdot\mathbf{u}\right), \\
\tau_{\theta\theta} &= \mu\left(2\frac{u_r}{r}
- \frac{2}{3}\nabla\!\cdot\mathbf{u}\right), \\
\tau_{zr}=\tau_{rz} &= \mu\left(
\frac{\partial u_z}{\partial r}+\frac{\partial u_r}{\partial z}\right).
\end{aligned}
\tau_{zz} = \mu\left(2\frac{\partial u_z}{\partial z} - \frac{2}{3}\nabla\cdot\mathbf{u}\right),
$$
$$
\tau_{rr} = \mu\left(2\frac{\partial u_r}{\partial r} - \frac{2}{3}\nabla\cdot\mathbf{u}\right),
$$
$$
\tau_{\theta\theta} = \mu\left(2\frac{u_r}{r} - \frac{2}{3}\nabla\cdot\mathbf{u}\right),
$$
$$
\tau_{zr}=\tau_{rz} = \mu\left(\frac{\partial u_z}{\partial r}+\frac{\partial u_r}{\partial z}\right).
$$

The radial heat flux is $q_r=-\kappa\,\partial_r T$. The Cartesian-like
Expand All @@ -114,7 +111,7 @@ These source terms are volume terms and do not depend on a boundary normal. On
an off-axis curved or oblique boundary, the numerical flux uses the actual
meridional normal $\mathbf{n}=(n_z,n_r)$ through
$F_n=F_z n_z+F_r n_r$; no additional source correction involving
$\hat{\mathbf r}\!\cdot\mathbf{n}$ is needed.
$\hat{\mathbf r}\cdot\mathbf{n}$ is needed.

## Axis regularity

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