Asymptotic models for free boundary flow in porous media

Physica D: Nonlinear Phenomena

Volumn 392, Issue , 2019, Pages 1-16

  Granero Belinchón, Rafael ^a   Scrobogna, Stefano ^b  

^a UNIVERSITY OF CANTABRIA   (Spain)

^b BCAM BASQUE CENTER FOR APPLIED MATHEMATICS   (Spain)

Author keywords

Darcy law; Forchheimer flow; Free boundary problems; Moving interfaces; Muskat problem

Indexed keywords

POROUS MATERIALS;

DARCY LAW; FORCHHEIMER; FREE-BOUNDARY PROBLEMS; MOVING INTERFACE; MUSKAT PROBLEM;

FLOW OF FLUIDS;

EID: 85063264613     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2019.02.013     Document Type: Article

References (37)

1. Whitaker, Stephen, Flow in porous media i: A theoretical derivation of darcy's law. Transp. Porous Media 1:1 (1986), 3–25.
2. Muskat, M., Botset, H.G., Flow of gas through porous materials. Physics 1:1 (1931), 27–47.
3. Wyckoff, R.D., Botset, H.G., Muskat, M., Flow of liquids through porous media under the action of gravity. Physics 3:2 (1932), 90–113.
4. Muskat, Morris, The flow of compressible fluids through porous media and some problems in heat conduction. Physics 5:3 (1934), 71–94.
5. Cerminara, M., Fasano, A., Modelling the dynamics of a geothermal reservoir fed by gravity driven flow through overstanding saturated rocks. J. Volcanol. Geotherm. Res. 233 (2012), 37–54.
6. Muskat, Morris, The flow of fluids through porous media. J. Appl. Phys. 8:4 (1937), 274–282.
7. Hele-Shaw, Henry Selby, The flow of water. Nature, 1898.
8. Saffman, P.G., Taylor, Geoffrey, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. Ser. A 245 (1958), 312–329.
9. Constantin, P., Pugh, M., Global solutions for small data to the Hele-Shaw problem. Nonlinearity 6:3 (1993), 393–415.
10. Cheng, C.H., Coutand, D., Shkoller, S., Global existence and decay for solutions of the Hele-Shaw flow with injection. Interfaces Free Bound. 16 (2014), 297–338.
11. Córdoba, D., Gancedo, F., Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. Comm. Math. Phys. 273:2 (2007), 445–471.
12. Constantin, Peter, Córdoba, Diego, Gancedo, Francisco, Strain, Robert M., On the global existence for the Muskat problem. J. Eur. Math. Soc. 15:1 (2013), 201–227.
13. Gómez-Serrano, J., Granero-Belinchón, R., On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof. Nonlinearity 27:6 (2014), 1471–1498.
14. 2 initial data. Adv. Math. 286 (2016), 32–104.
15. 3∕2. https://arxiv.org/abs/1803.07528.
16. Forchheimer, Philipp, Hydraulik. 1914, Leipzig : B.G. Teubner.
17. Stokes, George Gabriel, On the Theory of Oscillatory Waves Cambridge Library Collection - Mathematics, vol. 1, 1847, Cambridge University Press.
18. Akers, Benjamin, Nicholls, David P., Traveling waves in deep water with gravity and surface tension. SIAM J. Appl. Math. 70:7 (2010), 2373–2389.
19. Akers, Benjamin, Nicholls, David P., Spectral stability of deep two-dimensional gravity water waves: repeated eigenvalues. SIAM J. Appl. Math. 72:2 (2012), 689–711.
20. Akers, Benjamin, Nicholls, David P., The spectrum of finite depth water waves. Eur. J. Mech. B Fluids 46 (2014), 181–189.
21. Nicholls, David P., Reitich, Fernando, On analyticity of travelling water waves. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461:2057 (2005), 1283–1309.
22. Nicholls, David P., Reitich, Fernando, Stable, high-order computation of traveling water waves in three dimensions. Eur. J. Mech. B Fluids 25:4 (2006), 406–424.
23. Matsuno, Y., Nonlinear evolutions of surface gravity waves on fluid of finite depth. Phys. Rev. Lett., 69(4), 1992, 609.
24. Matsuno, Yoshimasa, Two-dimensional evolution of surface gravity waves on a fluid of arbitrary depth. Phys. Rev. E, 47(6), 1993, 4593.
25. Matsuno, Yoshimasa, Nonlinear evolution of surface gravity waves over an uneven bottom. J. Fluid Mech. 249 (1993), 121–133.
26. Granero-Belinchón, Rafael, Shkoller, Steve, A model for Rayleigh-Taylor mixing and interface turn-over. Multiscale Model. Simul. 15:1 (2017), 274–308.
27. Arthur Cheng, Rafael Granero-Belinchón, Steve Shkoller, Jon Wilkening, Rigorous Asymptotic Models of Water Waves. To appear in Water Waves, https://arxiv.org/pdf/1807.00176.pdf.
28. Coutand, Daniel, Shkoller, Steve, On the finite-time splash and splat singularities for the 3-d free-surface euler equations. Comm. Math. Phys. 325:1 (2014), 143–183.
29. Coutand, Daniel, Shkoller, Steve, Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Amer. Math. Soc. 20:3 (2007), 829–930.
30. Lannes, David, The Water Waves Problem. Mathematical Surveys and Monographs, 2013, American Mathematical Society, Providence, RI Mathematical analysis and asymptotics.
31. Castro, Angel, Lannes, David, Fully nonlinear long-wave models in the presence of vorticity. J. Fluid Mech. 759 (2014), 642–675.
32. Castro, Angel, Lannes, David, Well-posedness and shallow-water stability for a new Hamiltonian formulation of the water waves equations with vorticity. Indiana Univ. Math. J. 64:4 (2015), 1169–1270.
33. Rafael Granero-Belinchón, Stefano Scrobogna, On an asymptotic model for free boundary Darcy flow in porous media. Submitted, https://arxiv.org/abs/1810.11798.
34. Platzer, Bernd, Adrian bejan: Convection heat transfer. john wiley & sons, hoboken, new jersey, 4th ed. ZAMM J. Appl. Math. Mech. Z. Angew. Math. Mech., 94(9), 2013 729–729.
35. Spivak, Michael, A Comprehensive Introduction to Differential Geometry. Vol. III. second ed., 1979, Publish or Perish, Inc., Wilmington, Del., xii+466.
36. Craig, W., Sulem, C., Sulem, P.-L., Nonlinear modulation of gravity waves: a rigorous approach. Nonlinearity 5:2 (1992), 497–522.
37. Craig, Walter, Schanz, Ulrich, Sulem, Catherine, The modulational regime of three-dimensional water waves and the Davey-Stewartson system. Ann. Inst. H. Poincaré Anal. Non Linéaire 14:5 (1997), 615–667.