diff --git a/docs/axisymmetry.md b/docs/axisymmetry.md index 231e316..2c1a2f7 100644 --- a/docs/axisymmetry.md +++ b/docs/axisymmetry.md @@ -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 @@ -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