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Variational Derivation of the Weak Form for Viscous Flows Involving Free Surfaces with Implications for Geodynamic Simulations — arXiv:2610.12152

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.12152
License (as archived): CC0-1.0
Archived: 2026-10-09T06:28:45.881Z
SHA-256 of the archived text: c67ed2c793cfe8d6e44dbb90ea640c198d6e72b06e85d3dde332d716b67bd255
Stable URL of this page: https://metraipolis.com/documento/dc_22184f128317

How to cite

Variational Derivation of the Weak Form for Viscous Flows Involving Free Surfaces with Implications for Geodynamic Simulations — arXiv:2610.12152 — arXiv (en), https://arxiv.org/abs/2610.12152 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_22184f128317 (Metraipolis archive dc_22184f128317, sha256 c67ed2c793cfe8d6e44dbb90ea640c198d6e72b06e85d3dde332d716b67bd255)

BibTeX

@misc{dc22184f128317,
  title = {Variational Derivation of the Weak Form for Viscous Flows Involving Free Surfaces with Implications for Geodynamic Simulations — arXiv:2610.12152},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.12152},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_22184f128317; sha256 c67ed2c793cfe8d6e44dbb90ea640c198d6e72b06e85d3dde332d716b67bd255},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.12152. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Geodynamic simulations involve solving the Stokes equations, where boundary conditions play a crucial role in governing internal dynamics and surface topography evolution. A common approach is to model the interface in contact with air as a free surface. This interface is stress-free and deforms in response to internal force balances, which is essential for understanding processes ranging from rifted margins to subduction. However, a free surface can introduce instabilities and oscillations in the numerical solution, commonly known as the "drunken sailor" instability, forcing the use of small time steps. Geodynamic models therefore often employ stabilization terms based on the gravitational effect of free-surface motion. To date, these terms have been derived primarily on a heuristic basis. Here, we apply variational calculus to the energy-minimization formulation of incompressible Stokes flow. Using the Gateaux derivative, we derive the complete weak form for a deforming free-surface domain and show that the stabilization terms arise naturally from the free-surface assumption. We consider the linear-viscosity case and identify two boundary-integral contributions: one associated with internal viscous deformation and the other with gravitational body force. Two benchmark cases demonstrate the consistency of the derived formulation with previously proposed stabilization approaches. Our results provide a mathematical foundation for the commonly used gravity stabilization and identify an additional viscous boundary contribution.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.NA, nuovi lavori del giorno).
Categorie arXiv: Numerical Analysis (math.NA) ; Fluid Dynamics (physics.flu-dyn); Geophysics (physics.geo-ph).
Pagina del lavoro: https://arxiv.org/abs/2610.12152 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-09T05:53:16.589Z dalla pagina https://arxiv.org/list/math.NA/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).