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Fix a sign error in step-22's introduction. #13277

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Jan 21, 2022
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7 changes: 4 additions & 3 deletions examples/step-22/doc/intro.dox
Original file line number Diff line number Diff line change
Expand Up @@ -43,7 +43,7 @@ we will focus on the simpler Stokes system.
Note that when deriving the more general compressible Navier-Stokes equations,
the diffusion is modeled as the divergence of the stress tensor
@f{eqnarray*}
\tau = - \mu (2\varepsilon(\textbf{u}) - \frac{2}{3}\nabla \cdot \textbf{u} I),
\tau = - \mu \left(2\varepsilon(\textbf{u}) - \frac{2}{3}\nabla \cdot \textbf{u} I\right),
@f}
where $\mu$ is the viscosity of the fluid. With the assumption of $\mu=1$
(assume constant viscosity and non-dimensionalize the equation by dividing out
Expand All @@ -60,7 +60,7 @@ are continuous), the formulations are equivalent:
@f{eqnarray*}
\textrm{div}\; \tau
= -2\textrm{div}\;\varepsilon(\textbf{u})
= -\triangle \textbf{u} + \nabla \cdot (\nabla\textbf{u})^T
= -\triangle \textbf{u} - \nabla \cdot (\nabla\textbf{u})^T
= -\triangle \textbf{u}.
@f}
This is because the $i$th entry of $\nabla \cdot (\nabla\textbf{u})^T$ is given by:
Expand All @@ -70,7 +70,8 @@ This is because the $i$th entry of $\nabla \cdot (\nabla\textbf{u})^T$ is given
= \sum_j \frac{\partial}{\partial x_j} [(\nabla\textbf{u})]_{j,i}
= \sum_j \frac{\partial}{\partial x_j} \frac{\partial}{\partial x_i} \textbf{u}_j
= \sum_j \frac{\partial}{\partial x_i} \frac{\partial}{\partial x_j} \textbf{u}_j
= \frac{\partial}{\partial x_i} \textrm{div}\; \textbf{u}
= \frac{\partial}{\partial x_i}
\underbrace{\textrm{div}\; \textbf{u}}_{=0}
= 0.
@f}
If you can not assume the above mentioned regularity, or if your viscosity is
Expand Down