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Home » Issues » Archive of Issues » 2014 - 4 issue » pp. 95-108

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Russia Vladivostok
Year
2014
Issue
4
Pages
95-108
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Section Mathematics
Title An asymptotic study of multi-ring pattern formation in axisymmetric two-layer creeping flow with variable layer thicknesses, and some geophysical applications
Author(-s) Pak V.V.a
Affiliations Pacific Oceanological Institute, Far Eastern Branch of the Russian Academy of Sciencesa
Abstract The axisymmetric model based on the Stokes equations is proposed to investigate the multi-ring pattern formation in two-layer creeping flow with variable thickness of layers. Each layer has uniform density and viscosity. The upper layer is lighter than the lower layer. The flow is generated by both surface and interface geometry. The effect of surface tension is supposed to be negligible. We apply the method of multiple scales to obtain the governing equations describing instability in the form of wave in the flow. Using the Fourier-Laplace method, we analyze the small-amplitude leading behavior of the instability. The asymptotic study reveals that this kind of instability manifests itself as axisymmetric wave which length is comparable with layer thickness; moreover, layer thicknesses play a major role in spatial distribution of wave extrema. The other model parameters alter mostly the wave amplitude. The equation relating extrema distribution to layer thicknesses is derived. We apply the obtained results to study a ring spacing rule observed for most multi-ring basins on the Moon. Using parameters of some lunar multi-ring basins we calculate the consecutive crest radii of the unstable wave and compare the results of simulation with the measured ring radii.
Keywords multi-layer creeping flow, Stokes equations, method of multiple scales, inertialess instability, ring basins
UDC 532.5.032
MSC 76D50
DOI 10.20537/vm140408
Received 10 September 2014
Language Russian
Citation Pak V.V. An asymptotic study of multi-ring pattern formation in axisymmetric two-layer creeping flow with variable layer thicknesses, and some geophysical applications, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 4, pp. 95-108.
References
  1. Arkani-Hamed J. Viscosity of the Moon, The Moon, 1973, vol. 6, pp. 112-124.
  2. Bender C.M., Orszag S.A. Advanced mathematical methods for scientists and engineers, McGraw-Hill Inc., USA, 593 p.
  3. Chadderton L.T., Krajenbrink F.G., Katz R., Poveda A. Standing waves on the Moon, Nature, 1969, vol. 223, pp. 259-263.
  4. Chen D.-H., Yue Z.-Y., Bin Z. A tentative mathematical description of the oscillating peak model of multi-ring basin formation, Earth, Moon, and Planets, 1990, vol. 49, pp. 241-252.
  5. Chen K.P., Wave formation in gravity-driven low Reynolds number flow of two liquid films down an incline plane, Phys. Fluids A, 1993, vol. 5, pp. 3038-3048.
  6. Craster R.V., Matar O.K. Dynamics and stability of thin liquid films, Reviews of Modern Physics, 2009, vol. 81, no. 3, pp. 1131-1198.
  7. Hikida H., Wieczorek M.A. Crustal thickness of the Moon: new constraints from gravity inversions using polyhedral shape models, Icarus, 2007, vol. 192, pp. 150-166.
  8. Hu J., Millet S., Botton V., Ben Hadid H., Henrya D. Inertialess temporal and spatio-temporal stability analysis of the two-layer film flow with density stratification, Phys. Fluids, 2006, vol. 18, 104101.
  9. Jiang W.Y., Helenbrook B., Lin S.P. Inertialess instability of a two-layer liquid film flow, Phys. Fluids, 2004, vol. 16, no. 3, 054105.
  10. Kaab A., Weber M. Development of transverse ridges on rock glaciers: field measurements and laboratory experiments, Permafrost and Periglacial Processes, 2004, vol. 15, no. 4, pp. 379-391.
  11. Loewenherz D.S., Lawrence C.J. The effect of viscosity stratification on the stability of a free surface flow at low Reynolds number, Physics of Fluids A: Fluid Dynamics, 1989, vol. 1, issue 10, pp. 1686-1693.
  12. Merkt D., Pototsky A., Bestehorn M. Long-wave theory of bounded two-layer films with a free liquid–liquid interface: short- and long-time evolution, Phys. Fluids, 2005, vol. 17, 064104.
  13. Mohit P.S., Phillips R.J. Viscoelastic evolution of lunar multiring basins, J. Geophys. Res., 2006, vol. 111, E12001.
  14. Murray J.B. Oscillating peak model of basin and crater formation, Moon Planets, 1980, vol. 22, pp. 472-476.
  15. Nakamura Y. Seismic velocity structure of the lunar mantle, J. Geophys. Res., 1983, vol. 88, pp. 677-686.
  16. Nayfeh A.H. Perturbation methods, New York: John Wiley, Inc., 1973, 425 p.
  17. Pike R.J., Spudis P.D. Basin-ring spacing on the Moon, Mercury, and Mars, Earth, Moon, and Planets, 1987, vol. 39, pp. 129-194.
  18. Pozrikidis C. Instability of two-layer creeping flow in a channel with parallel-sided walls, J. Fluid Mech., 1997, vol. 351, pp. 139-165.
  19. Severtson Y.C., Aidun C.K. Stability of two-layer stratified flow in inclined channels: applications to air entrainment in coating systems, J. Fluid Mech., 1996, vol. 312, pp. 173-200.
  20. Shi J.-C., Ma Y.-H., Chen D.-H., Bao G. Analysis of origin of multi-ring basins by theory of deep water waves, Chin. Phys. Lett., 2008, vol. 25, no. 2, pp. 787-789.
  21. Tilley B.S., Davis S.H., Bankoff S.G. Linear stability theory of two-layer fluid flow in an inclined channel, Phys. Fluids., 1994, vol. 6, no. 12, pp. 3906-3922.
  22. Turcotte D.L., Schubert G. Geodynamics: applications of continuum physics to geological problems, New York: John Wiley, Inc., 1982, 450 p.
  23. Wieczorek M.A., Phillips R.J. Lunar multiring basins and the cratering process, Icarus, 1999, vol. 139, pp. 246–259.
  24. Wünnemann K., Ivanov B.A. Numerical modelling of the impact crater depth-diameter dependence in an acoustically fluidized target, Planetary and Space Science, 2003, vol. 51, pp. 831-845.
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