Time-periodic convective patterns in a horizontal porous layer with through-flow
Authors:
Françoise Dufour and Marie-Christine Néel
Journal:
Quart. Appl. Math. 58 (2000), 265-281
MSC:
Primary 76S05; Secondary 35P15, 35Q35, 76E06, 76R05
DOI:
https://doi.org/10.1090/qam/1753399
MathSciNet review:
MR1753399
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We investigate the time-periodic convective patterns which set on in an infinite porous layer saturated by a fluid, under the influence of a vertical temperature gradient, superimposed to a horizontal seeping through-flow. When the seepage velocity keeps moderate, it obeys the Darcy law, while the heat equation rules the evolution of the temperature. The transition towards convection is governed by the filtration Rayleigh number, and the above-mentioned system of partial differential equations has Galilean invariance: travelling waves, stationary with respect to a moving frame, solve this problem near the threshold. This follows directly from the study of natural convection. With the help of the center manifold theory, we show that other kinds of time-periodic two-dimensional structures exist: upstream to a fixed zone, their amplitude vanishes. Downstream, they resemble the travelling waves. Numerical simulation of the governing equations reproduces elements of both sets of convective structures, indexed by definite values of the time-frequency.
- Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
G. S. Beavers and E. M. Sparrow, Non-Darcy flow through fibrous media, Journal of Applied Mechanics 36, 711-714 (1969)
J. L. Beck, Convection in a box of porous material saturated with fluid, Physics of Fluids 15, 1377-1383 (1972)
H. R. Brand, R. J. Deissler, and G. Ahlers, Simple model for the Bénard instability with horizontal flow near threshold, Physical Review A 43, 4262-4268 (1991)
- Pierre Collet and Jean-Pierre Eckmann, Instabilities and fronts in extended systems, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1990. MR 1109707
- P. Collet and J.-P. Eckmann, The existence of dendritic fronts, Comm. Math. Phys. 107 (1986), no. 1, 39–92. MR 861884
- Pierre Collet, Jean-Pierre Eckmann, Henri Epstein, and Joachim Stubbe, A global attracting set for the Kuramoto-Sivashinsky equation, Comm. Math. Phys. 152 (1993), no. 1, 203–214. MR 1207676
M. Combarnous, Convection Naturelle et Convection Mixte en Milieu Poreux, Thèse d’Etat, Paris, 1970
- J.-P. Eckmann, H. Epstein, and C. E. Wayne, Normal forms for parabolic partial differential equations, Ann. Inst. H. Poincaré Phys. Théor. 58 (1993), no. 3, 287–308 (English, with English and French summaries). MR 1222944
- Jean-Pierre Eckmann and C. Eugene Wayne, The nonlinear stability of front solutions for parabolic partial differential equations, Comm. Math. Phys. 161 (1994), no. 2, 323–334. MR 1266487
J. Ettefagh, K. Vafai and S. Kim, Non Darcian effects in open ended cavities filled with a porous medium, J. of Heat Transfer 113, 747-756 (1991)
- G. Iooss and J. Los, Bifurcation of spatially quasi-periodic solutions in hydrodynamic stability problems, Nonlinearity 3 (1990), no. 3, 851–871. MR 1067084
- G. Iooss and A. Mielke, Bifurcating time-periodic solutions of Navier-Stokes equations in infinite cylinders, J. Nonlinear Sci. 1 (1991), no. 1, 107–146. MR 1102833, DOI https://doi.org/10.1007/BF01209150
- G. Iooss, A. Mielke, and Y. Demay, Theory of steady Ginzburg-Landau equation, in hydrodynamic stability problems, European J. Mech. B Fluids 8 (1989), no. 3, 229–268. MR 1008662
- G. Iooss and M.-C. Pérouème, Perturbed homoclinic solutions in reversible $1:1$ resonance vector fields, J. Differential Equations 102 (1993), no. 1, 62–88. MR 1209977, DOI https://doi.org/10.1006/jdeq.1993.1022
R. M. Islam and K. Nandakumar, Multiple solutions for buoyancy induced flow in saturated porous media for large Peclet numbers, ASME Journal of Heat Transfer 108, 866-871 (1986)
M. Le Cotillec, Instabilités dans une Couche Plane Poreuse Horizontale en Convection Mixte, Université de Paris VI, 1983
- J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
- Alexander Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci. 10 (1988), no. 1, 51–66. MR 929221, DOI https://doi.org/10.1002/mma.1670100105
H. W. Müller, M. Lücke, and M. Kamps, Convective Patterns in Horizontal Flow, Europhysics Letters, vol. 10, no. 5, 1989, pp. 451-456
- M.-C. Néel, Convection in a horizontal porous layer of infinite extent, European J. Mech. B Fluids 9 (1990), no. 2, 155–176. MR 1096157
X. Nicolas, A. Mojtabi, and J.-K. Platten, Two-dimensional analysis of the Poiseuille-Bénard flow in a rectangular channel heated from below, Physics of Fluids 9, 337-348 (1997)
D. Roth, P. Büchel, M. Lücke, H. W. Müller, M. Kamps, and R. Schmitz, Influence of boundaries on pattern selection in throughflow, Physica D 97, 253-263 (1996)
- C. Eugene Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains, Arch. Rational Mech. Anal. 138 (1997), no. 3, 279–306. MR 1465095, DOI https://doi.org/10.1007/s002050050042
C. H. Yu, M. Y. Chang, and T. F. Lin, Structures of moving transverse and mixed rolls in mixed convection of air in a horizontal plane channel, Internat. J. of Heat and Mass Transfer 40, 333-346 (1997)
R. A. Adams, Sobolev Spaces, Academic Press, 1975
G. S. Beavers and E. M. Sparrow, Non-Darcy flow through fibrous media, Journal of Applied Mechanics 36, 711-714 (1969)
J. L. Beck, Convection in a box of porous material saturated with fluid, Physics of Fluids 15, 1377-1383 (1972)
H. R. Brand, R. J. Deissler, and G. Ahlers, Simple model for the Bénard instability with horizontal flow near threshold, Physical Review A 43, 4262-4268 (1991)
P. Collet and J.-P. Eckmann, Instabilities and fronts in extended systems, Princeton Series in Physics, 1987
P. Collet and J.-P. Eckmann, The existence of dendritic fronts, Comm. Math. Physics 107, 39-92 (1986)
P. Collet, J.-P. Eckmann, H. Epstein, and J. Stubbe, A global attracting set for the Kuramoto-Shivasinski equation, Comm. Math. Physics 153, 203–215 (1993)
M. Combarnous, Convection Naturelle et Convection Mixte en Milieu Poreux, Thèse d’Etat, Paris, 1970
J.-P. Eckmann and C. E. Wayne, Normal forms for parabolic partial differential equations, Ann. Inst. H. Poincaré Phys. Théor. 58, 287-308 (1993)
J.-P. Eckmann and C. E. Wayne, The nonlinear stability of fronts for parabolic partial differential equations, Comm. Math. Physics 161, 323-334 (1994)
J. Ettefagh, K. Vafai and S. Kim, Non Darcian effects in open ended cavities filled with a porous medium, J. of Heat Transfer 113, 747-756 (1991)
G. Iooss and J. Los, Bifurcation of Spatially Quasi-Periodic Solutions in Hydrodynamic Stability Problems, Nonlinearity, vol. 3, 1990, pp. 851-871
G. Iooss and A. Mielke, Bifurcating time-periodic solutions of Navier-Stokes equations in infinite cylinders, Journal of Nonlinear Science 1, 107-146 (1991)
G. Iooss, A. Mielke, and Y. Demay, Theory of steady Ginzburg-Landau equations in hydrodynamic stability problems, European Journal of Mechanics B/Fluids 8, 229-268 (1989)
G. Iooss and M.-C. Pérouème, Perturbed homoclinic solutions in reversible 1:1 resonance vector fields, Journal of Differential Equations 102, 62-88 (1993)
R. M. Islam and K. Nandakumar, Multiple solutions for buoyancy induced flow in saturated porous media for large Peclet numbers, ASME Journal of Heat Transfer 108, 866-871 (1986)
M. Le Cotillec, Instabilités dans une Couche Plane Poreuse Horizontale en Convection Mixte, Université de Paris VI, 1983
J. L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, vol. 1, edited by Dunod, 1968
A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Mathematical Methods in the Applied Sciences 10, 51-66 (1988)
H. W. Müller, M. Lücke, and M. Kamps, Convective Patterns in Horizontal Flow, Europhysics Letters, vol. 10, no. 5, 1989, pp. 451-456
M.-C. Néel, Convection in a horizontal porous layer of infinite extent, European Journal of Mechanics B/Fluids 9, 155-176 (1990)
X. Nicolas, A. Mojtabi, and J.-K. Platten, Two-dimensional analysis of the Poiseuille-Bénard flow in a rectangular channel heated from below, Physics of Fluids 9, 337-348 (1997)
D. Roth, P. Büchel, M. Lücke, H. W. Müller, M. Kamps, and R. Schmitz, Influence of boundaries on pattern selection in throughflow, Physica D 97, 253-263 (1996)
C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains, Arch. Rat. Mech. Anal. 158, 279-306 (1997)
C. H. Yu, M. Y. Chang, and T. F. Lin, Structures of moving transverse and mixed rolls in mixed convection of air in a horizontal plane channel, Internat. J. of Heat and Mass Transfer 40, 333-346 (1997)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
76S05,
35P15,
35Q35,
76E06,
76R05
Retrieve articles in all journals
with MSC:
76S05,
35P15,
35Q35,
76E06,
76R05
Additional Information
Article copyright:
© Copyright 2000
American Mathematical Society