Pattern formation in systems with slowly varying geometry

SIAM Journal on Applied Mathematics

Volumn 57, Issue 1, 1997, Pages 112-152

  Eckhaus, Wiktor ^a   Kuske, Rachel ^b  

^a UTRECHT UNIVERSITY   (Netherlands)

^b TUFTS UNIVERSITY   (United States)

Author keywords

Nonlinear modulation equations; Slowly varying geometry; Spatially localized patterns; Uniform asymptotic approximation

Indexed keywords

ASYMPTOTIC STABILITY; BIFURCATION (MATHEMATICS); DYNAMICS; GEOMETRY; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; NONLINEAR EQUATIONS; NUMERICAL ANALYSIS; PATTERN RECOGNITION; PROBLEM SOLVING; RIVERS; SEDIMENTATION;

NONLINEAR MODULATION EQUATIONS; PATTERN FORMATION; SLOWLY VARYING GEOMETRY; SPATIALLY LOCALIZED PATTERNS; UNIFORM ASYMPTOTIC APPROXIMATION;

EQUATIONS OF MOTION;

EID: 0031078822     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/s0036139994277531     Document Type: Article

References (31)

1. P. M. EAGLES, A Bénard problem with a perturbed lower wall, Proc. Roy. Soc. London Ser. A, 371 (1980), pp. 359-379.
2. I. C. WALTON, Effects of slow spatial variations on Bénard convection, Quart. J. Mech. Appl. Math., 35 (1982), pp. 33-48.
3. I. C. WALTON, On the onset of Rayleigh-Bénard convection in a fluid layer of slowly increasing depth, Stud. Appl. Math., 67 (1982), pp. 199-216.
4. P. M. EAGLES AND K. EAMES, Taylor vortices between almost cylindrical boundaries, J. Engrg. Math., 17 (1983), pp. 263-280.
5. P. G. DANIELS, Effects of geometric imperfection at the onset of convection in a shallow two dimensional cavity, International Journal of Heat and Mass Transfer, 25 (1982), pp. 337-343.
6. P. G. DANIELS, The onset of Bénard convection in a shallow sloping container, Quart. J. Mech. Appl. Math., 35 (1982), pp. 49-67.
7. D. A. S. REES, The effect of long-wavelength thermal modulations on the onset of convection in an infinite porous layer heated from below, Quart. J. Mech. Appl. Math., 43 (1990), pp. 189-214.
8. L. KRAMER, E. BEN-JACOB, H. BRAND, AND M. CROSS, Wavelength selection in systems far from equilibrium, Phys. Rev. Lett., 49 (1982), pp. 1891-1894.
9. L. KRAMER AND H. RIECKE, Wavelength selection in Rayleigh-Bénard convection, Z. Phys. B, 59 (1985), pp. 245-251.
10. H. RIECKE, Pattern selection by weakly pinning ramps, Europhys. Lett., 2 (1986), pp. 1-8.
11. H. RIECKE AND H.-G. PAAP, Perfect wave-number selection and drifting patterns in ramped Taylor vortex flow, Phys. Rev. Lett., 59 (1987), pp. 2570-2573.
12. H. RIECKE, Imperfect wave-number selection by space-dependent control parameters in a model for Taylor vortex flow, Phys. Rev. A, 37 (1988), pp. 636-638.
13. H.-G. PAAP AND H. RIECKE, Drifting vortices in ramped Taylor vortex flow: Quantitative results from phase equation, Phys. Fluids A, 3 (1991). pp. 1519-1532.
14. H. RIECKE AND H.-G. PAAP, Spatio-temporal chaos through ramp-induced Eckhaus instability, Europhys. Lett., 14 (1991), pp. 433-438.
15. I. REHBERG AND H. RIECKE, Rayleigh-number ramps cause moving convection patterns, in The Physics of Structure Formation, W. Guttinger and G. Dangelmayr, eds., Springer, Berlin, 1987.
16. H. RIECKE AND H.-G. PAAP, Chaotic phase diffusion through the interaction of phase slip processes, in Ordered and Turbulent Patterns in Taylor-Couette Flow, C. D. Andereck and F. Hayot, eds., Plenum Press, New York, 1992.
17. R. SCHIELEN, A. DOELMAN, AND H. E. DE SWART, On the nonlinear dynamics of free bars in straight channels, J. Fluid Mech., 252 (1993), pp. 325-356.
18. A. VAN HARTEN, On the validity of Ginzburg-Landau's equation, J. Nonlinear Sci., 1 (1991), pp. 397-422.
19. W. ECKHAUS, The Ginzburg-Landau manifold is an attractor, J. Nonlinear Sci., 3 (1993), pp. 329-348.
20. G. SCHNEIDER, The Validity of Ginzburg-Landau Approximation, Thesis, University of Stuttgart, 1992.
21. P. BOLLERMAN, Validity results for Ginzburg-Landau's equation, Tech. report 800, Mathematical Institute, University of Utrecht, 1993.
22. R. E. KELLY AND D. PAL, Thermal convection with spatially periodic non-uniform heating: Non-resonant wavelength excitation, J. Fluid Mech., 86 (1978), pp. 433-456.
23. D. A. S. REES AND D. S. RILEY, Free convection in an undulating saturated porous layer: Resonant wavelength excitation, J. Fluid Mech., 166 (1986), pp. 503.
24. D. A. S. REES AND D. S. RILEY, The effects of boundary imperfections on convection in a saturated porous layer: Near resonant wavelength excitation, J. Fluid Mech., 199 (1989), pp. 133-154.
25. D. N. RIAHI, Preferred pattern of convection in a porous layer with a spatially nonuniform boundary temperature, J. Fluid Mech., 246 (1993), pp. 529-543.
26. R. SCHIELEN AND A. DOELMAN, Modulation equations for spatially periodic systems: Derivation and solutions, SIAM J. Appl. Math., submitted.
27. P. MANNEVILLE, Dissipative Structures and Weak Turbulence, Academic Press, San Diego, 1990.
28. A. DOELMAN, W. ECKHAUS, R. KUSKE, AND R. SCHIELEN, Pattern formation in systems on spatially periodic domains, in Nonlinear Dynamics and Pattern Formation in the Natural Environment, A. Doelman and A. van Harten, eds., Longman, London, England, 1995.
29. G. DEWEL AND P. BORCKMANS, Effects of slow spatial variations on dissipative structures, Phys. Lett. A, 138 (1989), pp. 189-192.
30. P. BORCKMANS, A. DEWIT, AND G. DEWEL, Competition in ramped Turing structures, Phys. A, 188 (1992), pp. 137-157.
31. A. DEWIT, P. BORCKMANS, AND G. DEWEL, Turing structures in the presence of gradients, Instabilities and Nonequilibrium Structures, 4 (1993), pp. 247-258.