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Interfaces in incompressible flows

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Abstract

The motion of both internal and surface waves in incompressible fluids under capillary and gravity forces is a major research topic. In particular, we review the derivation of some new models describing the dynamics of gravity-capillary nonlinear waves in incompressible flows. These models take the form of both bidirectional and unidirectional nonlinear and nonlocal wave equations. More precisely, with the goal of telling a more complete story, in this paper we present the results in the works (Cheng in Water Waves 1(1):71-130, 2019; Granero-Belinchón and Ortega in On the motion of gravity-capillary waves with odd viscosity. arXiv:2103.01062, 2021; Granero-Belinchón and Scrobogna in J Diff Equ 276:96–148 1921; SIAM J Appl Math 79(6):2530–2550, 2019; Proc Am Math Soc 148(12):5181–5191, 2020; Phys Fluids 33(10):102115, 2021; Granero-Belinchón and Shkoller in Multiscale Model Simul 15(1):274–308, 2017) together with some new results regarding the well-posedness of the resulting PDEs.

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Notes

  1. Note the convention for \(u^\perp \) that we are using in this paper is different than the one in [32].

References

  1. Abanov, A., Can, T., Ganeshan, S.: Odd surface waves in two-dimensional incompressible fluids. Sci. Post Phys. 5(010), 1 (2018)

    Google Scholar 

  2. Abanov, A.G., Can, T., Ganeshan, S., Monteiro, G.M.: Hydrodynamics of two-dimensional compressible fluid with broken parity: variational principle and free surface dynamics in the absence of dissipation. Phys. Rev. Fluids 5(10), 104802 (2020)

    Article  Google Scholar 

  3. Abanov, A.G., Monteiro, G.M.: Free-surface variational principle for an incompressible fluid with odd viscosity. Phys. Rev. Lett. 122(15), 154501 (2019)

    Article  Google Scholar 

  4. Akers, B., Milewski, P.A.: Dynamics of three-dimensional gravity-capillary solitary waves in deep water. SIAM J. Appl. Math. 70(7), 2390–2408 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Akers, B., Nicholls, D.P.: Traveling waves in deep water with gravity and surface tension. SIAM J. Appl. Math. 70(7), 2373–2389 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Avron, J.: Odd viscosity. J. Stat. Phys. 92(3), 543–557 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bae, H., Granero-Belinchón, R.: Global existence for some transport equations with nonlocal velocity. Adv. Math. 269, 197–219 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Banerjee, D., Souslov, A., Abanov, A.G., Vitelli, V.: Odd viscosity in chiral active fluids. Nat. Commun. 8(1), 1–12 (2017)

    Article  Google Scholar 

  9. Biler, P., Karch, G., Monneau, R.: Nonlinear diffusion of dislocation density and self-similar solutions. Commun. Math. Phys. 294(1), 145–168 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Boussinesq, J.: Lois de l’extinction de la houle en haute mer. CR Acad. Sci. Paris 121(15–20), 2 (1895)

    MATH  Google Scholar 

  11. Canfield, J., Denissen, N., Francois, M., Gore, R., Rauenzahn, R., Reisner, J., Shkoller, S.: A comparison of interface growth models applied to Rayleigh–Taylor and Richtmyer–Meshkov instabilities. J. Fluids Eng. 142(12), 121108 (2020)

    Article  Google Scholar 

  12. Castro, A., Córdoba, D.: Global existence, singularities and ill-posedness for a nonlocal flux. Adv. Math. 219(6), 1916–1936 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chae, D., Córdoba, A., Córdoba, D., Fontelos, M.A.: Finite time singularities in a 1d model of the quasi-geostrophic equation. Adv. Math. 194(1), 203–223 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cheng, C., Granero-Belinchón, R., Shkoller, S., Wilkening, J.: Rigorous asymptotic models of water waves. Water Waves 1(1), 71–130 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cordoba, A., Cordoba, D., Gancedo, F.: Interface evolution: water waves in 2-D. Adv. Math. 223(1), 120–173 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Coutand, D., Shkoller, S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Am. Math. Soc. 20(3), 829–930 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Craik, A.D.: The origins of water wave theory. Annu. Rev. Fluid Mech. 36 (2004)

  18. Craik, A.D.: George gabriel stokes on water wave theory. Annu. Rev. Fluid Mech. 37, 23–42 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dawson, L., McGahagan, H., Ponce, G.: On the decay properties of solutions to a class of schrödinger equations. Proc. Am. Math. Soc. 136(6), 2081–2090 (2008)

    Article  MATH  Google Scholar 

  20. Dias, F., Dyachenko, A.I., Zakharov, V.E.: Theory of weakly damped free-surface flows: a new formulation based on potential flow solutions. Phys. Lett. A 372(8), 1297–1302 (2008)

    Article  MATH  Google Scholar 

  21. Ganeshan, S., Abanov, A.G.: Odd viscosity in two-dimensional incompressible fluids. Phys. Rev. Fluids 2(9), 094101 (2017)

    Article  Google Scholar 

  22. Ganeshan, S., Monteiro, G.: Non-linear shallow water dynamics with odd viscosity. Bull. Am. Phys. Soc.

  23. Granero-Belinchón, R.: On a nonlocal differential equation describing roots of polynomials under differentiation. Commun. Math. Sci. 18(6), 1643–1660 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Granero-Belinchón, R.: Ondas no lineales en fluidos incompresibles. La Gaceta de la RSME 24(3), 507–531 (2021)

    MATH  Google Scholar 

  25. Granero-Belinchón, R., Ortega, A.: On the motion of gravity-capillary waves with odd viscosity. arXiv preprintarXiv:2103.01062, (2021)

  26. Granero-Belinchón, R., Scrobogna, S.: Well-posedness of the water-wave with viscosity problem. J. Diff. Equ. 276, 96–148 (1921)

    Article  MathSciNet  MATH  Google Scholar 

  27. Granero-Belinchón, R., Scrobogna, S.: Asymptotic models for free boundary flow in porous media. Phys. D 392, 1–16 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Granero-Belinchón, R., Scrobogna, S.: Models for damped water waves. SIAM J. Appl. Math. 79(6), 2530–2550 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Granero-Belinchón, R., Scrobogna, S.: On an asymptotic model for free boundary darcy flow in porous media. SIAM J. Math. Anal. 52(5), 4937–4970 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. Granero-Belinchón, R., Scrobogna, S.: Well-posedness of water wave model with viscous effects. Proc. Am. Math. Soc. 148(12), 5181–5191 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Granero-Belinchón, R., Scrobogna, S.: Global well-posedness and decay for viscous water wave models. Phys. Fluids 33(10), 102115 (2021)

    Article  MATH  Google Scholar 

  32. Granero-Belinchón, R., Shkoller, S.: A model for Rayleigh-Taylor mixing and interface turn-over. Multiscale Model. Simul. 15(1), 274–308 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Helmholtz, H.v.: Uber discontinuirliche flüssigkeits-bewegung, (1868)

  34. Jiang, L., Ting, C.-L., Perlin, M., Schultz, W.W.: Moderate and steep faraday waves: instabilities, modulation and temporal asymmetries. J. Fluid Mech. 329, 275–307 (1996)

    Article  MATH  Google Scholar 

  35. Kuznetsov, E., Spector, M., Zakharov, V.: Surface singularities of ideal fluid. Phys. Lett. A 182(4–6), 387–393 (1993)

    Article  MathSciNet  Google Scholar 

  36. Lamb, H.: Hydrodynamics. Cambridge Univ Press, (1932)

  37. Lannes, D.: Well-posedness of the water-waves equations. J. Am. Math. Soc. 18(3), 605–654 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lapa, M.F., Hughes, T.L.: Swimming at low reynolds number in fluids with odd, or hall, viscosity. Phys. Rev. E 89(4), 043019 (2014)

    Article  Google Scholar 

  39. Longuet-Higgins, M.S.: Theory of weakly damped stokes waves: a new formulation and its physical interpretation. J. Fluid Mech. 235, 319–324 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  40. Matsuno, Y.: Nonlinear evolutions of surface gravity waves on fluid of finite depth. Phys. Rev. Lett. 69(4), 609 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  41. Matsuno, Y.: Nonlinear evolution of surface gravity waves over an uneven bottom. J. Fluid Mech. 249, 121–133 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  42. Matsuno, Y.: Two-dimensional evolution of surface gravity waves on a fluid of arbitrary depth. Phys. Rev. E 47(6), 4593 (1993)

    Article  Google Scholar 

  43. Ngom, M., Nicholls, D.P.: Well-posedness and analyticity of solutions to a water wave problem with viscosity. J. Diff. Equ. 265(10), 5031–5065 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  44. Ramani, R., Shkoller, S.: A multiscale model for Rayleigh–Taylor and Richtmyer–Meshkov instabilities. J. Comput. Phys. 405, 109177 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  45. Rayleigh, L.: On the instability of jets. Proc. Lond. Math. Soc. 1(1), 4–13 (1878)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ruvinsky, K., Feldstein, F., Freidman, G.: Numerical simulations of the quasi-stationary stage of ripple excitation by steep gravity-capillary waves. J. Fluid Mech. 230, 339–353 (1991)

    Article  MATH  Google Scholar 

  47. Soni, V., Bililign, E.S., Magkiriadou, S., Sacanna, S., Bartolo, D., Shelley, M.J., Irvine, W.T.: The odd free surface flows of a colloidal chiral fluid. Nat. Phys. 15(11), 1188–1194 (2019)

    Article  Google Scholar 

  48. Souslov, A., Dasbiswas, K., Fruchart, M., Vaikuntanathan, S., Vitelli, V.: Topological waves in fluids with odd viscosity. Phys. Rev. Lett. 122(12), 128001 (2019)

    Article  MathSciNet  Google Scholar 

  49. Stokes, G.G.: Report on recent researches in hydrodynamics. Brit. Assoc. Rep 1, 1–20 (1846)

    Google Scholar 

  50. Stokes, G.G.: On the theory of oscillatory waves. Transactions of the Cambridge philosophical society, (1880)

  51. Taylor, G.: The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 201, 192–196 (1950)

  52. Thomson, W.: Hydrokinetic solutions and observations. Lond. Edinburgh Dublin Philos. Magaz. J. Sci. 42(281), 362–377 (1871)

    Article  Google Scholar 

  53. Wu, G., Liu, Y., Yue, D.K.: A note on stabilizing the Benjamin–Feir instability. J. Fluid Mech. 556, 45–54 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  54. Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9(2), 190–194 (1968)

    Article  Google Scholar 

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Acknowledgements

I thank Professor Jon Wilkening for providing his data for the simulations of the full water wave system. The author was supported by the project “Mathematical Analysis of Fluids and Applications” Grant PID2019-109348GA-I00 funded by MCIN/AEI/ 10.13039/501100011033 and acronym “MAFyA”. This publication is part of the project PID2019-109348GA-I00 funded by MCIN/ AEI /10.13039/501100011033. This publication is also supported by a 2021 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation. The BBVA Foundation accepts no responsibility for the opinions, statements, and contents included in the project and/or the results thereof, which are entirely the responsibility of the authors.

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  1. Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Santander, Spain

    Rafael Granero-Belinchón

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Correspondence to Rafael Granero-Belinchón.

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Granero-Belinchón, R. Interfaces in incompressible flows. SeMA 80, 1–25 (2023). https://doi.org/10.1007/s40324-021-00283-w

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