The vortex blob method as a second-grade non-Newtonian fluid
We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity, and initial data in the space of Radon measures ${\mathcal M}({\mathbb R}^2)$. The...
| Główni autorzy: | , |
|---|---|
| Format: | Journal article |
| Język: | English |
| Wydane: |
1999
|
| _version_ | 1826305928861843456 |
|---|---|
| author | Oliver, M Shkoller, S |
| author_facet | Oliver, M Shkoller, S |
| author_sort | Oliver, M |
| collection | OXFORD |
| description | We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity, and initial data in the space of Radon measures ${\mathcal M}({\mathbb R}^2)$. The solutions of this regularized PDE, also known as the averaged Euler or Euler-$\alpha$ equations, are geodesics on the volume preserving diffeomorphism group with respect to a new weak right invariant metric. We prove global existence of unique weak solutions (geodesics) for initial vorticity in ${\mathcal M}({\mathbb R}^2)$ such as point-vortex data, and show that the associated coadjoint orbit is preserved by the flow. Moreover, solutions of this particular vortex blob method converge to solutions of the Euler equations with bounded initial vorticity, provided that the initial data is approximated weakly in measure, and the total variation of the approximation also converges. In particular, this includes grid-based approximation schemes of the type that are usually used for vortex methods. |
| first_indexed | 2024-03-07T06:40:15Z |
| format | Journal article |
| id | oxford-uuid:f90ad55e-1472-443f-a083-25e58f28a64d |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:40:15Z |
| publishDate | 1999 |
| record_format | dspace |
| spelling | oxford-uuid:f90ad55e-1472-443f-a083-25e58f28a64d2022-03-27T12:55:01ZThe vortex blob method as a second-grade non-Newtonian fluidJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:f90ad55e-1472-443f-a083-25e58f28a64dEnglishSymplectic Elements at Oxford1999Oliver, MShkoller, SWe show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity, and initial data in the space of Radon measures ${\mathcal M}({\mathbb R}^2)$. The solutions of this regularized PDE, also known as the averaged Euler or Euler-$\alpha$ equations, are geodesics on the volume preserving diffeomorphism group with respect to a new weak right invariant metric. We prove global existence of unique weak solutions (geodesics) for initial vorticity in ${\mathcal M}({\mathbb R}^2)$ such as point-vortex data, and show that the associated coadjoint orbit is preserved by the flow. Moreover, solutions of this particular vortex blob method converge to solutions of the Euler equations with bounded initial vorticity, provided that the initial data is approximated weakly in measure, and the total variation of the approximation also converges. In particular, this includes grid-based approximation schemes of the type that are usually used for vortex methods. |
| spellingShingle | Oliver, M Shkoller, S The vortex blob method as a second-grade non-Newtonian fluid |
| title | The vortex blob method as a second-grade non-Newtonian fluid |
| title_full | The vortex blob method as a second-grade non-Newtonian fluid |
| title_fullStr | The vortex blob method as a second-grade non-Newtonian fluid |
| title_full_unstemmed | The vortex blob method as a second-grade non-Newtonian fluid |
| title_short | The vortex blob method as a second-grade non-Newtonian fluid |
| title_sort | vortex blob method as a second grade non newtonian fluid |
| work_keys_str_mv | AT oliverm thevortexblobmethodasasecondgradenonnewtonianfluid AT shkollers thevortexblobmethodasasecondgradenonnewtonianfluid AT oliverm vortexblobmethodasasecondgradenonnewtonianfluid AT shkollers vortexblobmethodasasecondgradenonnewtonianfluid |