Variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells. Part 1. Linear stability analysis

Journal of Fluid Mechanics

Volumn 721, Issue , 2013, Pages 268-294

  Talon, L ^a,b   Goyal, N ^b   Meiburg, E ^b  

^a Centre National de la Recherche Scientifique   (France)

^b University of California   (United States)

Author keywords

buoyancy driven instability; fingering instability; Hele Shaw flows

Indexed keywords

BUOYANCY; GRAVITATION; LINEAR STABILITY ANALYSIS; NONLINEAR EQUATIONS; VISCOSITY;

BUOYANCY DRIVEN INSTABILITY; COMPUTATIONAL INVESTIGATION; DENSITY STRATIFICATION; FINGERING INSTABILITIES; GRAVITATIONAL INSTABILITY; HELE-SHAW FLOW; MISCIBLE DISPLACEMENT; PARAMETRIC STABILITY;

INTERFACE STATES;

BUOYANCY; COMPUTER SIMULATION; DENSITY; DISPLACEMENT; STABILITY ANALYSIS; STOKES FORMULA; VISCOSITY;

EID: 84875030142     PISSN: 00221120     EISSN: 14697645     Source Type: Journal    
DOI: 10.1017/jfm.2013.63     Document Type: Article

References (44)

1. AUBERTIN, A., GAUTHIER, G., MARTIN, J., SALIN, D. & TALON, L. 2009 Miscible viscousfingering in microgravity. Phys. Fluids 21, 054107.
2. BACRI, J.-C, RAKOTOMALALA, N. & SALIN, D. 1991 Three-dimensional miscible viscous fingeringin porous media. Phys. Rev. Lett. 67, 2005.
3. CHEN, C.-Y. & MEIBURG, E. 1996 Miscible Displacement in capillary tubes. Part 2. Numericalsimulations. J. Fluid Mech. 326, 57-67.
4. D'OLCE, M., MARTIN, J., RAKOTOMALALA, N., SALIN, D. & TALON, L. 2008 Pearl andmushroom instability patterns in two miscible fluids core annular flow. Phys. Fluids 20,24104.
5. D'OLCE, M., MARTIN, J., RAKOTOMALALA, N., SALIN, D. & TALON, L. 2009Convective/absolute instability in miscible core-annular flow. Part 1: Experiments.J. Fluid Mech. 618, 305-311.
6. FERNANDEZ, J., KUROWSKI, P., PETITJEANS, P. & MEIBURG, E. 2002 Density-driven unstableflows of miscible fluids in a Hele-Shaw cell. J. Fluid Mech. 451, 239-260. (Pubitemid 34781216)
7. GOVINDARAJAN, R. 2004 Effect of miscibility on the linear instability of two-fluid channel flow.Intl J. Multiphase Flow 30, 1177.
8. GOYAL, N. & MEIBURG, E. 2004 Unstable density stratification of miscible fluids in a verticalHele-Shaw cell: influence of variable viscosity on the linear stability. J. Fluid Mech. 516,211-238. (Pubitemid 39384260)
9. GOYAL, N. & MEIBURG, E. 2006 Miscible displacements in Hele-Shaw cells: two-dimensional basestates and their linear stability. J. Fluid Mech. 558, 329-355. (Pubitemid 43997688)
10. GOYAL, N., PICHLER, H. & MEIBURG, E. 2007 Variable-density miscible displacements in avertical Hele-Shaw cell: linear stability. J. Fluid Mech. 584, 357-372. (Pubitemid 47215397)
11. GRAF, F., MEIBURG, E. & ḦARTEL, C. 2002 Density-driven instabilities of miscible fluids in aHele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations. J. FluidMech. 451, 261-282. (Pubitemid 34781217)
12. ḦARTEL, C., MEIBURG, E. & NECKER, F. 2000 Analysis and direct numerical simulation of theflow at a gravity current head. Part 1: flow topology and front speed for slip and boundaries.J. Fluid Mech. 418, 189-212.
13. HINCH, E. J. 1984 A note on the mechanism of the instability at the interface between two shearingfluids. J. Fluid Mech. 144, 463-465.
14. HOMSY, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19.
15. JOHN, M. O., OLIVEIRA, R. M., HEUSSLER, F. H. C. & MEIBURG, E. 2013 Variable density andviscosity, miscible displacements in horizontal Hele-Shaw cells. Part 2. Nonlinear simulations.J. Fluid Mech. 721, 295-323.
16. LAJEUNESSE, E., MARTIN, J., RAKOTOMALALA, N. & SALIN, D. 1997 3D instability of miscibledisplacements in a Hele-Shaw cell. Phys. Rev. Lett. 79, 5254-5257.
17. LAJEUNESSE, E., MARTIN, J., RAKOTOMALALA, N., SALIN, D. & YORTSOS, Y. C. 1999 Miscibledisplacement in a Hell-Shaw cell at high rates. J. Fluid Mech. 398, 299-319. (Pubitemid 129657157)
18. MANICKAM, F. J. & HOMSY, G. M. 1993 Stability of miscible displacements in porous media withnon-monotonic viscosity profiles. Phys. Fluids A.5, 1356-1367.
19. MARTIN, J., RAKOTOMALALA, N. & SALIN, D. 2002 Gravitational instability of miscible fluids ina Hele-Shaw cell. Phys. Fluids 14, 902-905. (Pubitemid 34147426)
20. MARTIN, J., RAKOTOMALALA, N., TALON, L. & SALIN, D. 2011 Viscous lock-exchange indifferent geometries. J. Fluid Mech. 673, 132-146.
21. OLIVEIRA, R. M. & MEIBURG, E. 2011 Miscible displacements in Hele-Shaw cells: threedimensionalNavier-Stokes simulations. J. Fluid Mech. 687, 431-460.
22. PETITJEANS, P. & MAXWORTHY, T. 1996 Miscible displacements in capillary tubes. Part 1.Experiments. J. Fluid Mech. 326, 37-56. (Pubitemid 126457537)
23. RAKOTOMALALA, N., SALIN, D. & WATZKY, P. 1997a Fingering in 2D parallel viscous flow.J. Phys. II France 7, 967-972. (Pubitemid 127806304)
24. RAKOTOMALALA, N., SALIN, D. & WATZKY, P. 1997b Miscible displacement between two parallelplates: BGK lattice gas simulations. J. Fluid Mech. 338, 277-297.
25. SAHU, K. C., DING, H., VALLURI, P. & MATAR, O. K. 2009a Linear stability analysis andnumerical simulation of miscible two-layer channel flow. Phys. Fluids 21, 042104.
26. SAHU, K. C., DING, H., VALLURI, P. & MATAR, O. K. 2009b Pressure-driven miscible two-fluidchannel flow with density gradients. Phys. Fluids 21 (4).
27. SELVAM, B., MERK, S., GOVINDARAJAN, R. & MEIBURG, E. 2007 Stability of misciblecore-annular flow with viscosity stratification. J. Fluid Mech. 592, 23-49. (Pubitemid 350154120)
28. SELVAM, B., TALON, L., LESSHAFT, L. & MEIBURG, E. 2009 Convective/absolute instability inmiscible core-annular flow. Part 2: numerical simulation and nonlinear global modes. J. FluidMech. 618, 323-348.
29. ŚEON, T., HULIN, J. P., SALIN, D., PERRIN, B. & HINCH, E. J. 2004 Buoyant mixing of misciblefluids in tilted tube. Phys. Fluids 16, 103.
30. ŚEON, T., HULIN, J. P., SALIN, D., PERRIN, B. & HINCH, E. J. 2005 Buoyancy driven misciblefront dynamics in tilted tubes. Phys. Fluids 17 (3).
31. ŚE ON, T., HULIN, J. P., SALIN, D., PERRIN, B. & HINCH, E. J. 2006 Laser-induced fluorescencemeasurements of buoyancy driven mixing in tilted tubes. Phys. Fluids 18 (4).
32. ŚEON, T., ZNAIEN, J., PERRIN, B., HINCH, E. J., SALIN, D. & HULIN, J. P. 2007a Frontdynamics and macroscopic diffusion in buoyant mixing in a tilted tube. Phys. Fluids 19 (12).
33. ŚEON, T., ZNAIEN, J., SALIN, D., HULIN, J. P., HINCH, E. J. & PERRIN, B. 2007b Transientbuoyancy-driven front dynamics in nearly horizontal tubes. Phys. Fluids 19, 123603.
34. TAGHAVI, S. M., ALBA, K., SEON, T., WIELAGE-BURCHARD, K., MARTINEZ, D. M. &FRIGAARD, I. A. 2012 Miscible displacement flows in near-horizontal ducts at low Atwoodnumber. J. Fluid Mech. 696, 175-214.
35. TAGHAVI, S. M., SEON, T., MARTINEZ, D. M. & FRIGAARD, I. A. 2009 Buoyancy-dominateddisplacement flows in near-horizontal channels: the viscous limit. J. Fluid Mech. 639, 1-35.
36. TAGHAVI, S. M., SEON, T., MARTINEZ, D. M. & FRIGAARD, I. A. 2010 Influence of an imposedflow on the stability of a gravity current in a near horizontal duct. Phys. Fluids 22 (3),031702.
37. TAGHAVI, S. M., SEON, T., WIELAGE-BURCHARD, K., MARTINEZ, D. M. & FRIGAARD, I. A.2011 Stationary residual layers in buoyant Newtonian displacement flows. Phys. Fluids 23 (4),044105.
38. TALON, L. & MEIBURG, E. 2011 Plane Poiseuille flow of miscible layers with different viscosities:instabilities in the Stokes flow regime. J. Fluid Mech. 686, 484-506.
39. TAYLOR, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R.Soc. Lond. A 219, 186-203.
40. VANAPARTHY, S. H. & MEIBURG, E. 2008 Variable density and viscosity, miscible displacements incapillary tubes. Eur. J. Mech. (B/Fluids 27 (3), 268-289. (Pubitemid 351474392)
41. VEDERNIKOV, A., SCHEID, B., ISTASSE, E. & LEGROS, J. C. 2001 Viscous fingering in miscibleliquids under microgravity conditions. Phys. Fluids 13 (9), S12.
42. WOODING, R. 1969 Growth of fingers at an unstable diffusing interface in a porous medium orHele-Shaw cell. J. Fluid Mech. 39, 477-495.
43. YIH, C. S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27, 337.
44. YORTSOS, Y. C. & ZEYBEK, M. 1988 Dispersion driven instability in miscible displacement inporous media. Phys. Fluids 31, 3511-3518.