# Navier Boundary Condtion for Stokes equation

**URL:** <https://community.freefem.org/t/navier-boundary-condtion-for-stokes-equation/2309>\
**Category:** General Discussion\
**Created:** [March 3, 2023, 11:08am UTC](https://community.freefem.org/t/navier-boundary-condtion-for-stokes-equation/2309 "2023-03-03T11:08:03Z")\
**Posts on this page:** 6\
**Page:** 1

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**Author:** ![zhugp123](https://avatars.discourse-cdn.com/v4/letter/z/71e660/32.png) [@zhugp123](https://community.freefem.org/u/zhugp123)\
**Post date:** [March 3, 2023, 11:08am UTC](https://community.freefem.org/t/navier-boundary-condtion-for-stokes-equation/2309/1 "2023-03-03T11:08:03Z")

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I am learning the document written by Prof. Frederic Hecht on boundary conditions for the Stokes equation.

 ![image](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/9/96a7bbaff9bc7c6f5963c182294c9f85af182d2a.png)  
 ![image](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/5/52bb4c38e54b22849a073055d09ad25aaf77cf43.png)

I have a question about the Navier boundary condition. How the following equation is derived, especially the negative symbol?

 ![image](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/8/8c84ca91ab3d8552d3c038dd20e28c5ca32a2dd7.png)

Thank you.

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<div class="post-metadata">

**Author:** ![zhugp123](https://avatars.discourse-cdn.com/v4/letter/z/71e660/32.png) [@zhugp123](https://community.freefem.org/u/zhugp123)\
**Post date:** [March 9, 2023, 3:53pm UTC](https://community.freefem.org/t/navier-boundary-condtion-for-stokes-equation/2309/2 "2023-03-09T15:53:05Z")

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@frederichecht professor, I try to derive the above identity (see the following figure). I have two questions: (1) in your side term \mathbf{\tau} \cdot \mathbf{v} is dropped for \mathbf{g}\cdot \mathbf{\tau}, it’s a typo or I missing something?  
(2) could you kindly tell me why the term marked in blue (the normal component, see the following figure) is dropped?

 ![image](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/7/7b601d4b119a334f4d12e89646d22afc5eb7aa26.png)

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**Author:** ![frederichecht](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/frederichecht/32/15_2.png) [@frederichecht](https://community.freefem.org/u/frederichecht)\
**Post date:** [March 10, 2023, 4:21pm UTC](https://community.freefem.org/t/navier-boundary-condtion-for-stokes-equation/2309/3 "2023-03-10T16:21:14Z")

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It is a Typo all term g.\tau must be multiplie by v so get g.\tau. v == (g.\tau).( v.\tau)

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<div class="post-metadata">

**Author:** ![zhugp123](https://avatars.discourse-cdn.com/v4/letter/z/71e660/32.png) [@zhugp123](https://community.freefem.org/u/zhugp123)\
**Post date:** [March 11, 2023, 7:15am UTC](https://community.freefem.org/t/navier-boundary-condtion-for-stokes-equation/2309/4 "2023-03-11T07:15:39Z")

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Dear Prof. Frédéric Hecht‬,  
Thanks for your kind reply. Could you kindly tell me why the term \int \left( \mathbf{f}\_\Gamma\cdot\mathbf{n} \right) \left( \mathbf{n} \cdot \mathbf{v}\right) \text{d}\Gamma is dropped? Thanks in advance.

Best,

Guangpu

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**Author:** ![frederichecht](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/frederichecht/32/15_2.png) [@frederichecht](https://community.freefem.org/u/frederichecht)\
**Post date:** [March 19, 2023, 3:02pm UTC](https://community.freefem.org/t/navier-boundary-condtion-for-stokes-equation/2309/5 "2023-03-19T15:02:20Z")

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because I miss this term, sorry.

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<div class="post-metadata">

**Author:** ![zhugp123](https://avatars.discourse-cdn.com/v4/letter/z/71e660/32.png) [@zhugp123](https://community.freefem.org/u/zhugp123)\
**Post date:** [March 20, 2023, 2:11am UTC](https://community.freefem.org/t/navier-boundary-condtion-for-stokes-equation/2309/6 "2023-03-20T02:11:03Z")

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Professor, thanks for your kind reply