Update navierstokes.md (#247)

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* Dokken/update fspace and wmtgs (#127)

* fix runs-on

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* Revert fundamentals

* Membrane code

* More updates due to autoformatting

* Lagrange -> CG back conversion

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* Update petsc arches

* Fix bounding boxes and more (#135)

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* rerun all files

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* PETSc python API update (#137)

* Resolve #136

* Fixes related to: FEniCS/dolfinx#2703

* Update changelog

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* Api changes related to: FEniCS/dolfinx#2763 (#142)

* Bump version numbers

* Bump version numbers (#149)

* Fix petsc arch

* Prepare v0.7.1 (#154)

* Merge branch 'main' into release and bump versions

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* Tabulation

* Updates compatible with nightly branch of DOLFINx (#156)

* Replace VectorElement and FiniteElement with `basix.ufl.element`.
Replace `dolfinx.fem.FunctionSpace` with `dolfinx.fem.functionspace`

* More updates

* And more

* Last vector element updates

* Bump pyvista and dolfinx to v0.7.2 (#159)

* Bump pyvista and dolfinx to v0.7.2

* remove -r

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* Bump versions

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* Fix typo and issue Issue on page /fem.html #122 (#161)

* Compute entity-cell connectivity before calling locate_dofs_topological.
Remove soon to be deprecated pyvista syntax for updating time dependent fields.
Fix range->np.arange conversion

* Update backend in workflow

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* Run book build prior to parallel run

* Add back write frame

* Fix wrong link

* Fix typo in changelog

* Merge main into release (#189)

* Merge main into release

* Apply suggestions from code review

* Various fixes when reading through the diff

* Further fixes

* add missing checkout (#191)

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* Add test docker on release PR and release (only push on tag

* Remove unused import

* Update text to resolve #194 (#195)

* Dokken/update nonlin options (#203)

* Update options for non-linear solver to sensible choice, ref #200

* Update docker path

* Update python file as well

* Update config ptr (#207)

* Pressure correction equation fix (#196)

* Update fundamentals.md (#199)

on line 110, updated a typesetting error in equation (3)

* Change from vector to petsc_vec (#206)

* dolfinx.fem.Form changed to dolfinx.fem.form (#213)

* Update compiler_parameters.ipynb

In the text it was mentioned that `dolfinx.fem.Form` was used to compile the form, instead of `dolfinx.fem.form`. Not matching the code cell.

* jupytext sync

* Form -> form in both py and ipynb and LinearProblem -> fem.petsc.LinearProblem

* Update book_stable.yml

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* Fix CI (#227)

* Bump docker/build-push-action from 5 to 6 (#221)

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* Bump actions/configure-pages from 4 to 5 (#222)

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* Update book_stable.yml (#225)

* Update book_stable.yml

* Update test_stable.yml

* resolve #224 (#226)

* Update test_stable.yml

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* Update test_stable.yml (#235)

* Add libgl flag to ci (#238)

* Update navierstokes.md

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Signed-off-by: dependabot[bot] <[email protected]>
Co-authored-by: Jørgen Schartum Dokken <[email protected]>
Co-authored-by: Jørgen S. Dokken <[email protected]>
Co-authored-by: Julius Herb <[email protected]>
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Author: 6 people

chapter2/navierstokes.md

@@ -59,7 +59,7 @@ $\langle -\nabla \cdot \sigma, v\rangle$. Just as for the [linear elasticity pro
 where $T=\sigma \cdot n$ is the boundary traction. If we solve a problem with a free boundary, we can take $T=0$ on the boundary. However, if we compute the flow through a channel or a pipe and want to model flow that continues into an "imaginary channel" at the outflow, we need to treat this term with some care. 
 The assumption we then can make is that the derivative of the velocity in the direction of the channel is zero at the outflow, corresponding to that the flow is "fully developed" or doesn't change significantly downstream at the outflow.
 Doing so, the remaining boundary term at the outflow becomes 
-$pn - \mu \nabla u \cdot n$, which is the term appearing in the variational problem [](ipcs-one). Note that this argument and the implementation depends exact on the definition of $\nabla u$, as either the  matrix with components $\frac{\partial u_i}{\partial x_j}$ or $\frac{\partial u_j}{\partial x_i}$.
+$pn - \mu \nabla u \cdot n$, which is the term appearing in the variational problem [](ipcs-one). Note that this argument and the implementation depend exactly on the definition of $\nabla u$, as either the  matrix with components $\frac{\partial u_i}{\partial x_j}$ or $\frac{\partial u_j}{\partial x_i}$.
 We here choose the  latter, $\frac{\partial u_j}{\partial x_i}$,
 which means that we must use the UFL-operator `nabla_grad`. If we use the operator `grad` and the definition $\frac{\partial u_i}{\partial x_j}$, we must instead keep the terms $pn-\mu(\nabla u)^T \cdot n$.