# Computing vorticity axisymmetric

**URL:** https://community.freefem.org/t/computing-vorticity-axisymmetric/3637
**Category:** General Discussion
**Created:** [December 16, 2024, 4:55pm UTC](https://community.freefem.org/t/computing-vorticity-axisymmetric/3637 "2024-12-16T16:55:37Z")
**Posts on this page:** 2
**Page:** 1

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### Author: ![zese](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/zese/32/917_2.png) [@zese](https://community.freefem.org/u/zese)
#### Post date: [December 16, 2024, 4:55pm UTC](https://community.freefem.org/t/computing-vorticity-axisymmetric/3637/1 "2024-12-16T16:55:37Z")

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Dear all

I want to compute vorticity in axisymmetric flow. The components are as follows:

 ![2024-12-16_11-55-12](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/6/6cb4142aa938504ef949a09f3ad80ab3b330ff4d.png)  
what to do to overcome singularity for axial vorticity at r=0 in FF++?

BR

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### Author: ![RdMl](https://avatars.discourse-cdn.com/v4/letter/r/a88e57/32.png) [@RdMl](https://community.freefem.org/u/RdMl)
#### Post date: [December 17, 2024, 8:46am UTC](https://community.freefem.org/t/computing-vorticity-axisymmetric/3637/2 "2024-12-17T08:46:55Z")

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Bonjour Mademoiselle,

In axisymmetric coordinates, the singularity at r=0 is not a true problem if you use the weak form (i.e variationnal if the operator is symmetric).

The tricks is the following :

- considere that the FF++ coordinates (x,y) are the axisymmetric coordinates (r,z)
- so, the differential element (Lebesgue’s measure) which is dV(r,q,z) = r dr dq fz (q = theta) becomes x dx dy dq
- now , integrate over q
- then, in these conditions, in the meridian plane, dV(r,z) = 2 pi dr dv, i.e., in FF++ coordinnates, dv(r,z) = x dx dy

Morality, you just must ponderate the integrands by x. Is the same to work in weighted Sobolev spaces.

For exemple, the weak form of the Laplace operator is :

```
int2d(Th) ( (dx(uh)*dx(vh) + dy(uh)*dy(vh))*x )

```

Please, note the presence of the weighting function (x,y) → x.

Very good and simple presentation of the axisymmetric formulation of FEM can be find in Krizek and Neittaanmaki, Finite Elements Approximation of Variationnal Problems and Applications, Pittman (1990). A more mathematically complex one in Bernardi, Dauge and Maday, Spectral Methods for Axisymmetric Domains, RMA 4, Gauthier-Villard (1999).

Je vous souhaite une bonne journée.