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| Autores principales: | , , , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2310.07085 |
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| _version_ | 1866929303538630656 |
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| author | Sanders, John W. DeVoria, Adam C. Washuta, Nathan J. Elamin, Gafar A. Skenes, Kevin L. Berlinghieri, Joel C. |
| author_facet | Sanders, John W. DeVoria, Adam C. Washuta, Nathan J. Elamin, Gafar A. Skenes, Kevin L. Berlinghieri, Joel C. |
| contents | This paper presents a novel Hamiltonian formulation of the isotropic Navier-Stokes problem based on a minimum-action principle derived from the principle of least squares. This formulation uses the velocities $u_{i}(x_{j},t)$ and pressure $p(x_{j},t)$ as the field quantities to be varied, along with canonically conjugate momenta deduced from the analysis. From these, a conserved Hamiltonian functional $H^{*}$ satisfying Hamilton's canonical equations is constructed, and the associated Hamilton-Jacobi equation is formulated for both compressible and incompressible flows. This Hamilton-Jacobi equation reduces the problem of finding four separate field quantities ($u_{i}$,$p$) to that of finding a single scalar functional in those fields--Hamilton's principal functional $\text{S}^{*}[t,u_{i},p]$. Moreover, the transformation theory of Hamilton and Jacobi now provides a prescribed recipe for solving the Navier-Stokes problem: Find $\text{S}^{*}$. If an analytical expression for $\text{S}^{*}$ can be obtained, it will lead via canonical transformation to a new set of fields which are simply equal to their initial values, giving analytical expressions for the original velocity and pressure fields. Failing that, if one can only show that a complete solution to this Hamilton-Jacobi equation does or does not exist, that will also resolve the question of existence of solutions. The method employed here is not specific to the Navier-Stokes problem or even to classical mechanics, and can be applied to any traditionally non-Hamiltonian problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_07085 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A canonical Hamiltonian formulation of the Navier-Stokes problem Sanders, John W. DeVoria, Adam C. Washuta, Nathan J. Elamin, Gafar A. Skenes, Kevin L. Berlinghieri, Joel C. Fluid Dynamics This paper presents a novel Hamiltonian formulation of the isotropic Navier-Stokes problem based on a minimum-action principle derived from the principle of least squares. This formulation uses the velocities $u_{i}(x_{j},t)$ and pressure $p(x_{j},t)$ as the field quantities to be varied, along with canonically conjugate momenta deduced from the analysis. From these, a conserved Hamiltonian functional $H^{*}$ satisfying Hamilton's canonical equations is constructed, and the associated Hamilton-Jacobi equation is formulated for both compressible and incompressible flows. This Hamilton-Jacobi equation reduces the problem of finding four separate field quantities ($u_{i}$,$p$) to that of finding a single scalar functional in those fields--Hamilton's principal functional $\text{S}^{*}[t,u_{i},p]$. Moreover, the transformation theory of Hamilton and Jacobi now provides a prescribed recipe for solving the Navier-Stokes problem: Find $\text{S}^{*}$. If an analytical expression for $\text{S}^{*}$ can be obtained, it will lead via canonical transformation to a new set of fields which are simply equal to their initial values, giving analytical expressions for the original velocity and pressure fields. Failing that, if one can only show that a complete solution to this Hamilton-Jacobi equation does or does not exist, that will also resolve the question of existence of solutions. The method employed here is not specific to the Navier-Stokes problem or even to classical mechanics, and can be applied to any traditionally non-Hamiltonian problem. |
| title | A canonical Hamiltonian formulation of the Navier-Stokes problem |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2310.07085 |