Stationary Flows and Periodic Dynamics of Binary Mixtures in Tall Laterally Heated Slots

Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 50)


The steady and oscillatory dynamics of binary fluids contained in slots heated by the side is studied by using continuation methods, and stability analysis. The bifurcation points on the branches of solutions are determined with precision by calculating their spectra for a large range of Rayleigh numbers. It will be seen that continuation and stability methods are a powerful tool to analyze the origin of the hydrodynamic instabilities leading to steady and time periodic flows, and their dynamics. The role played by the shear stresses of the steady field, and the solutal and thermal buoyancies, at the onset of the oscillations is studied by means of the energy equation of the perturbations. With the parameters used, it is found that the shear is always the main responsible for the instabilities, and that the work done by the two buoyancies can even help to stabilize the fluid. The results also show that binary mixtures of Prandtl number order one, like pure gases, present multiple stable periodic flows coexisting in the same range of parameters, since several unstable leading multipliers remain attached to the unit circle and go back into it. However, at lower Prandtl numbers only the first branch of periodic orbits bifurcating directly from the steady state is found to be stable, because some of the unstable multipliers of the other branches quickly increase their modulus and never re-enter the unit circle.


Continuation methods Stability analysis Periodic orbits Binary mixtures Lateral heating 



This work has been supported by Spanish MCYT/FEDER and Catalan GENCAT grants FIS2016-76525-P and 2017-SGR-1374, respectively.


  1. 1.
    Allgower, E.L., Georg, K.: Numerical Continuation Methods: An Introduction. Computational Mathematics, vol. 13. Springer, Berlin (1990)zbMATHGoogle Scholar
  2. 2.
    Antonijoan, J., Marqués, F., Sánchez, J.: Nonlinear spirals in the Taylor–Couette problem. Phys. Fluids 10, 829–838 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aruliah, D.A., Veen, L.V., Dubitski, A.: Algorithm 956: PAMPAC, a parallel adaptive method for pseudo-arclength continuation. ACM Trans. Math. Softw. 42(1), 8:1–8:18 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barkley, D., Henderson, R.D.: Floquet stability analysis of the periodic wake of a circular cylinder. J. Fluid Mech. 322, 215–241 (1996)CrossRefGoogle Scholar
  5. 5.
    Beaume, C., Bergeon, A., Knobloch, E.: Convectons and secondary snaking in three-dimensional natural doubly diffusive convection. Phys. Fluids 25, 024105-1–024105-15 (2013)CrossRefGoogle Scholar
  6. 6.
    Bergeon, A., Knobloch, E.: Periodic and localized states in natural doubly diffusive convection. Phys. D 237, 1139–1150 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bergeon, A., Knobloch, E.: Spatially localized states in natural doubly diffusive convection. Phys. Fluids 20, 034102-1–034102-8 (2008)CrossRefGoogle Scholar
  8. 8.
    Blackburn, H.M., Barkley, D., Sherwin, S.J.: Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271–304 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Böhmer, K., Mei, Z., Schwarzer, A., Sebastian, R.: Path-following of large bifurcation problems with iterative methods. In: Doedel, E., Tuckerman, L.S. (eds.) Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems. The IMA Volumes in Mathematics and its Applications, vol. 119, pp. 35–65. Springer, Berlin (2000)Google Scholar
  10. 10.
    Borońska, K., Tuckerman, L.S.: Extreme multiplicity in cylindrical Rayleigh-Benard convection. II. Bifurcation diagram and symmetry classification. Phys. Rev. E 81, 036321 (2010)Google Scholar
  11. 11.
    Brown, P.N., Saad, Y.: Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comput. 11(3), 450–481 (1990)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer, Berlin (2007)zbMATHGoogle Scholar
  13. 13.
    Christon, M., Gresho, P., Sutton, S.: Computational predictibility of natural convection flows in enclosures. Int. J. Numer. Methods Fluids 40, 953–980 (2002)CrossRefGoogle Scholar
  14. 14.
    Cliffe, K.A.: Numerical calculations of two-cell and single-cell Taylor flows. J. Fluid Mech. 135, 219–233 (1983)CrossRefGoogle Scholar
  15. 15.
    Cliffe, K.A.: Numerical calculations of the primary-flow exchange process in the Taylor problem. J. Fluid Mech. 197, 57–79 (1988)CrossRefGoogle Scholar
  16. 16.
    Cliffe, K.A., Spence, A., Taverner, S.: The numerical analysis of bifurcation problems with applications to fluid mechanics. Acta Numer. 39–131 (2000)MathSciNetCrossRefGoogle Scholar
  17. 17.
    COMSOL Inc., Sweden: COMSOL Multiphysics Reference Manual, version 5.3 (2008)Google Scholar
  18. 18.
    Dankowicz, H., Schilder, F.: Recipes for Continuation: Computational Science and Engineering. SIAM, Philadelphia (2013)Google Scholar
  19. 19.
    Davidenko, D.F.: On a new method of numerical solution of systems of nonlinear equations. Dokl. Akad. Kauk SSSR. 88, 601–602 (1953)MathSciNetGoogle Scholar
  20. 20.
    Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19(2), 400–408 (1982)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Dijkstra, H.A., Wubs, F.W., Cliffe, A.K., Doedel, E., Dragomirescu, I.F., Eckhardt, B., Gelfgat, A., Hazel, A., Lucarini, V., Salinger, A., Sánchez, J., Schuttelaars, H., Tuckerman, L., Thiele, U.: Numerical bifurcation methods and their application to fluid dynamics: analysis beyond simulation. Commun. Comput. Phys. 15(1), 1–45 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Dinar, N., Keller, H.B.: Computation of Taylor vortex flows using multigrid continuation methods. In: Chao, C.C., Orszag, S.A., Shyy, W. (eds.) Recent Advances in Computational Fluid Dynamics. Lecture Notes in Engineering, vol. 43, pp. 191–262. Springer, Berlin (1989)CrossRefGoogle Scholar
  23. 23.
    Doedel, E.: AUTO: software for continuation and bifurcation problems in ordinary differential equations. Technical report, Applied Mathematics, California Institute of Technology, Pasadena, CA (1986)Google Scholar
  24. 24.
    Doedel, E.: Lecture notes on numerical analysis of nonlinear equations. Technical report, Concordia University, Canada (2007)Google Scholar
  25. 25.
    Doedel, E., Tuckerman, L.S. (eds.): Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems. IMA Volumes in Mathematics and its Applications, vol. 119. Springer, Berlin (2000)Google Scholar
  26. 26.
    Doedel, E., Govaerts, W., Kuznetsov, Y.A.: Computation of periodic solution bifurcations in ODEs using bordered systems. SIAM J. Numer. Anal. 41(2), 401–435 (2003)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Dorr, F.W.: The direct solution of the discrete poisson equation on a rectangle. SIAM Rev. 12(2), 248–263 (1970)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Duguet, Y., Pringle, C.C.T., Kerswell, R.R.: Relative periodic orbits in transitional pipe flow. Phys. Fluids 20(11), 114102 (2008)CrossRefGoogle Scholar
  29. 29.
    Edwards, W.S., Tuckerman, L.S., Friesner, R.A., Sorensen, D.C.: Krylov methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 110, 82–102 (1994)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Feigelson, R. (ed.): 50 years Progress in Crystal Growth. A Reprint Collection. Elsevier, Amsterdam (2004)Google Scholar
  31. 31.
    Feudel, F., Tuckerman, L.S., Gellert, M., Seehafer, N.: Bifurcations of rotating waves in rotating spherical shell convection. Phys. Rev. E 92, 053015 (2015)Google Scholar
  32. 32.
    Formica, G., Arena, A., Lacarbonara, W., Dankowicz, H.: Coupling FEM with parameter continuation for analysis of bifurcations of periodic responses in nonlinear structures. J. Comput. Nonlinear Dyn. 8(2), 021013-8 (2012)CrossRefGoogle Scholar
  33. 33.
    Gao, Z., Podvin, B., Sergent, A., Xin, S.: Chaotic dynamics of a convection roll in a highly confined, vertical, differentially heated fluid layer. Phys. Rev. E 91, 013006 (2015)Google Scholar
  34. 34.
    Gao, Z., Sergent, A., Podvin, B., Xin, S., Le Quéré, P., Tuckerman, L.S.: Transition to chaos of natural convection between two infinite differentially heated vertical plates. Phys. Rev. E 88, 023010 (2013)Google Scholar
  35. 35.
    Garcia, F., Net, M., García-Archilla, B., Sánchez, J.: A comparison of high-order time integrators for the Boussinesq Navier–Stokes equations in rotating spherical shells. J. Comput. Phys. 229, 7997–8010 (2010)CrossRefGoogle Scholar
  36. 36.
    Garcia, F., Net, M., Sánchez, J.: Continuation and stability of convective modulated rotating waves in spherical shells. Phys. Rev. E 93, 013119 (2016)Google Scholar
  37. 37.
    García-Archilla, B., Sánchez, J., Simó, C.: Krylov methods and test functions for detecting bifurcations in one parameter-dependent partial differential equations. BIT 46(4), 731–757 (2006)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Gelfgat, A.Y.: Stability of convective flows in cavities: solution of benchmark problems by a low-order finite volume method. Int. J. Numer. Methods Fluids 53(3), 485–506 (2007)CrossRefGoogle Scholar
  39. 39.
    Gelfgat, A.Y., Bar-Yoseph, P.Z., Yarin, A.L.: Stability of multiple steady states of convection in laterally heated cavities. J. Fluid Mech. 388, 315–334 (1999)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Gelfgat, A.Y., Molokov, S.: Quasi-two-dimensional convection in a three-dimensional laterally heated box in a strong magnetic field normal to main circulation. Phys. Fluids 23, 034101-1–034101-13 (2011)CrossRefGoogle Scholar
  41. 41.
    Ghorayeb, K., Mojtabi, A.: Double diffusive convection in a vertical rectangular cavity. Phys. Fluids 9(8), 2339–2348 (1997)CrossRefGoogle Scholar
  42. 42.
    Gibson, J.F., Halcrow, J., Cvitanovic, P.: Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130 (2008)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Goto, K., van de Geijn, R.A.: Anatomy of high-performance matrix multiplication. ACM Trans. Math. Softw. 34(3), 1–25 (2008)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Govaerts, W.J.F.: Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM, Philadelphia (2000)CrossRefGoogle Scholar
  45. 45.
    Green, K.R., Van Veen, L.: Open-source tools for dynamical analysis of Liley’s mean-field cortex model. J. Comput. Sci. 5(3), 507–516 (2014)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Griewank, A., Reddien, G.: The calculation of Hopf points by a direct method. IMA J. Numer. Anal. 3, 295–303 (1983)MathSciNetCrossRefGoogle Scholar
  47. 47.
    de Groot, S.R., Mazur, P.: Non-Equilibrium Thermodynamics. Dover Publications, Amsterdam (1962)zbMATHGoogle Scholar
  48. 48.
    Heil, M., Hazel, A.L.: oomph-lib – an object-oriented multi-physics finite-element library. In: Schafer, M., Bungartz, H.J. (eds.) Fluid-Structure Interaction, pp. 19–49. Springer, Berlin (2006)CrossRefGoogle Scholar
  49. 49.
    Henry, D., Ben Hadid, H.: Multiple flow transitions in a box heated from the side in low-Prandtl-number fluids. Phys. Rev. E 76, 016314 (2007)Google Scholar
  50. 50.
    Henry, D., Bergeon, A. (eds.): Continuation Methods in Fluid Mechanics, Contributions to the ERCOFTAC/EUROMECH Colloquium. Notes on Numerical Fluid Mechanics, vol. 383. Vieweg (2000)Google Scholar
  51. 51.
    Kawahara, G., Uhlmann, M., van Veen, L.: The significance of simple invariant solutions in turbulent flows. Ann. Rev. Fluid Mech. 44(1), 203–225 (2012)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Ke, H., He, Y., Liu, Y., Cui, F.: Mixture working gases in thermoacoustic engines for different applications. Int. J. Thermophys. 33, 1143–1163 (2012)CrossRefGoogle Scholar
  53. 53.
    Keller, H.B.: Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Rabinowitz, P.H. (ed.) Applications of Bifurcation Theory, pp. 359–384. Academic Press, New York (1977)Google Scholar
  54. 54.
    Keller, H.B.: Lectures on Numerical Methods in Bifurcation Theory. Lectures on Mathematics and Physics. Tata Institute of Fundamental Research, Springer, New York (1987)Google Scholar
  55. 55.
    Kim, K.M., Witt, A.F., Gatos, H.C.: Crystal growth from the melt under destabilizing thermal gradients. J. Electrochem. Soc. 119(9), 1218–1226 (1972)CrossRefGoogle Scholar
  56. 56.
    Kranenborg, J.: Double-diffusive convection due to lateral thermal forcing. Ph.D. thesis, Utrecht University (1996)Google Scholar
  57. 57.
    Krauskopf, B., Osinga, H.: Computing invariant manifolds via the continuation of orbit segments. In: Krauskopf, B., Osinga, H., Galán-Vioque, J. (eds.) Numerical Continuation Methods for Dynamical Systems: Path following and Boundary Value Problems, Understanding Complex Systems, pp. 117–154. Springer, New York (2007)CrossRefGoogle Scholar
  58. 58.
    Krauskopf, B., Osinga, H.M., Doedel, E.J., Henderson, M.E., Guckenheimer, J., Dellnitz, M., Junge, O.: A survey of methods for computing (un)stable manifolds of vector fields. Int. J. Bifurc. Chaos Appl. Sci. Eng. 15, 763–791 (2005)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Kubíček, M., Marek, M.: Computational Methods in Bifurcation Theory and Dissipative Structures. Springer, Berlin (1983)CrossRefGoogle Scholar
  60. 60.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, Berlin (1998)zbMATHGoogle Scholar
  61. 61.
    Lappa, M.: Thermal Convection: Patterns Evolution and Stability. Wiley, Singapore (2009)Google Scholar
  62. 62.
    Le Quéré, P.: Transition to unsteady natural convection in a tall water-filled cavity. Phys. Fluids A 2(4), 503–515 (1990)CrossRefGoogle Scholar
  63. 63.
    Le Quéré, P., Behnia, M.: From onset of unsteadiness to chaos in a differentially heated square cavity. J. Fluid Mech. 359, 81–107 (1998)MathSciNetCrossRefGoogle Scholar
  64. 64.
    Lee, J., Hyun, M., Kang, Y.: Confined natural convection due to lateral heating in a stably stratified solution. Int. J. Heat Mass Transf. 33(5), 869–875 (1990)CrossRefGoogle Scholar
  65. 65.
    Lehoucq, R.B., Sorensen, D.C.: Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM J. Matrix Anal. Appl. 17, 789–821 (1996)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK User’s Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. Software, Environments Tools. SIAM, Philadelphia (1998)CrossRefGoogle Scholar
  67. 67.
    Liu, J., Ahlers, G.: Rayleigh–bénard convection in binary-gas mixtures: thermophysical properties and the onset of convection. Phys. Rev. E 55, 6950–6968 (1997)CrossRefGoogle Scholar
  68. 68.
    Lo Jacono, D., Bergeon, A., Knobloch, E.: Localized traveling pulses in natural doubly diffusive convection. Phys. Rev. Fluids 2, 093501-1–093501-19 (2017)Google Scholar
  69. 69.
    Lopez, J.M., Marqués, F., Sánchez, J.: Oscillatory modes in an enclosed swirling flow. J. Fluid Mech. 439, 109–129 (2001)CrossRefGoogle Scholar
  70. 70.
    Lust, K., Roose, D., Spence, A., Champneys, A.: An adaptive Newton–Picard algorithm with subspace iteration for computing periodic solutions. SIAM J. Sci. Comput. 19(4), 1188–1209 (1998)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Mamun, C.K., Tuckerman, L.S.: Asymmetry and Hopf bifurcation in spherical Couette flow. Phys. Fluids 7, 80–91 (1995)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Meerbergen, K., Roose, D.: Matrix transformations for computing rightmost eigenvalues of large sparse non-symmetric eigenvalue problems. IMA J. Numer. Anal. 16(3), 297–346 (1996)MathSciNetCrossRefGoogle Scholar
  73. 73.
    Meyer-Spasche, R., Keller, H.B.: Computation of the axisymmetric flow between rotating cylinders. J. Comput. Phys. 35, 100–109 (1980)CrossRefGoogle Scholar
  74. 74.
    Molemaker, M.J., Dijkstra, H.A.: Multiple equilibria and stability of the North-Atlantic wind-driven ocean circulation. In: Doedel, E., Tuckerman, L.S. (eds.) Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems. The IMA Volumes in Mathematics and its Applications, vol. 119, pp. 35–65. Springer, Berlin (2000)CrossRefGoogle Scholar
  75. 75.
    Moore, G., Spence, A.: The calculation of turning points of nonlinear equations. SIAM J. Numer. Anal. 17(4), 567–576 (1980)MathSciNetCrossRefGoogle Scholar
  76. 76.
    Net, M., Sánchez, J.: Continuation of bifurcations of periodic orbits for large-scale systems. SIAM J. Appl. Dyn. Syst. 14(2), 674–698 (2015)MathSciNetCrossRefGoogle Scholar
  77. 77.
    Net, M., Sánchez Umbría, J.: Periodic orbits in tall laterally heated rectangular cavities. Phys. Rev. E 95, 023102 (2017)Google Scholar
  78. 78.
    Pawlowski, R.P., Shadid, J.N., Simonis, J.P., Walker, H.F.: Globalization techniques for Newton–Krylov methods and applications to the fully coupled solution of the Navier–Stokes equations. SIAM Rev. 48, 700–721 (2006)MathSciNetCrossRefGoogle Scholar
  79. 79.
    Pozzo, M., Davies, C., Gubbins, D., Alfè, D.: Transport properties for liquid silicon-oxygen-iron mixtures at earth’s core conditions. Phys. Rev. B 87, 014110-1–014110-10 (2013)Google Scholar
  80. 80.
    Puigjaner, D., Herrero, J., Simó, C., Giralt, F.: From steady solutions to chaotic flows in a Rayleigh–Bénard problem at moderate Rayleigh numbers. Phys. D 240, 920–934 (2011)MathSciNetCrossRefGoogle Scholar
  81. 81.
    Rheinboldt, W.C.: Numerical Analysis of Parametrized Nonlinear Equations. Wiley, New York (1986)zbMATHGoogle Scholar
  82. 82.
    Riks, E.: The application of Newton’s method to the problem of elastic stability. ASME J. Appl. Mech. 39(4), 1060–1065 (1971)CrossRefGoogle Scholar
  83. 83.
    Roache, P.J.: Computational Fluid Dynamics. Hermosa Publishers, Albuquerque (1972)zbMATHGoogle Scholar
  84. 84.
    Roose, D., Hlavaček, V.: A direct method for the computation of Hopf bifurcation points. SIAM J. Appl. Math. 45(6), 879–894 (1985)MathSciNetCrossRefGoogle Scholar
  85. 85.
    Saad, Y.: Numerical Methods for Large Eigenvalue Problems. Manchester University Press, Manchester (1992)zbMATHGoogle Scholar
  86. 86.
    Saad, Y.: Preconditioned Krylov subspace methods for CFD applications. Technical report, UMSI-94-171, Minnesota Supercomputer Institute, Minneapolis, MN 55415 (1994)Google Scholar
  87. 87.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS Pub. Co., New York (1996)zbMATHGoogle Scholar
  88. 88.
    Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)MathSciNetCrossRefGoogle Scholar
  89. 89.
    Salinger, A.G., Bou-Rabee, N.M., Pawlowsky, R.P., Wilkes, E.D., Burroughs, E.A., Lehoucq, R.B., Romero, L.A.: LOCA 1.1. Library of Continuation Algorithms: Theory and Implementation Manual. Sandia National Laboratories, Albuquerque, NM (2002)Google Scholar
  90. 90.
    Salinger, A.G., Lehoucq, R.B., Pawlowski, R.P., Shadid, J.N.: Computational bifurcation and stability studies of the 8:1 thermal cavity problem. Int. J. Numer. Methods Fluids 40(8), 1059–1073 (2002)CrossRefGoogle Scholar
  91. 91.
    Sánchez, J., Net, M.: On the multiple shooting continuation of periodic orbits by Newton–Krylov methods. Int. J. Bifurc. Chaos Appl. Sci. Eng. 20(1), 1–19 (2010)MathSciNetCrossRefGoogle Scholar
  92. 92.
    Sánchez, J., Net, M.: A parallel algorithm for the computation of invariant tori in large-scale dissipative systems. Phys. D 252(1), 22–33 (2013)MathSciNetCrossRefGoogle Scholar
  93. 93.
    Sánchez, J., Net, M.: Numerical continuation methods for large-scale dissipative dynamical systems. Eur. Phys. J. Spec. Top. 225(13), 2465–2486 (2016)Google Scholar
  94. 94.
    Sánchez, J., Net, M.: Prandtl number dependence of convective fluids in tall laterally heated slots. Eur. J. Phys. Special Top. (under review) (2018)Google Scholar
  95. 95.
    Sánchez, J., Marqués, F., López, J.M.: A continuation and bifurcation technique for Navier–Stokes flows. J. Comput. Phys. 180, 78–98 (2002)MathSciNetCrossRefGoogle Scholar
  96. 96.
    Sánchez, J., Net, M., García-Archilla, B., Simó, C.: Newton–Krylov continuation of periodic orbits for Navier–Stokes flows. J. Comput. Phys. 201(1), 13–33 (2004)MathSciNetCrossRefGoogle Scholar
  97. 97.
    Sánchez, J., Net, M., García-Archilla, B., Simó, C.: Continuation of periodic orbits in large-scale dissipative systems. In: Dumortier, F., Broer, H., Mawhin, J., Vanderbauwhede, A., Lunel, S.V. (eds.) Proceedings of the Equadiff-2003 Conference, pp. 625–630. World Scientific, Singapore (2005)Google Scholar
  98. 98.
    Sánchez, J., Net, M., Vega, J.: Amplitude equations close to a triple-(\(+1\)) bifurcation point of \({D}_4\)-symmetric periodic orbits in \({O}(2)\)-equivariant systems. Discret. Contin. Dyn. Syst. B 6(6), 1357–1380 (2006)Google Scholar
  99. 99.
    Sánchez, J., Net, M., Simó, C.: Computation of invariant tori by Newton–Krylov methods in large-scale dissipative systems. Phys. D 239, 123–133 (2010)MathSciNetCrossRefGoogle Scholar
  100. 100.
    Sánchez, J., Garcia, F., Net, M.: Computation of azimuthal waves and their stability in thermal convection in rotating spherical shells with application to the study of a double-Hopf bifurcation. Phys. Rev. E 87, 033014 (2013)Google Scholar
  101. 101.
    Seydel, R.: Numerical computation of branch points in nonlinear equations. Numer. Math. 33(3), 339–352 (1979)MathSciNetCrossRefGoogle Scholar
  102. 102.
    Seydel, R.: Practical Bifurcation and Stability Analysis. From Equilibrium to Chaos. Springer, New York (1994)zbMATHGoogle Scholar
  103. 103.
    Shroff, G.M., Keller, H.B.: Stabilization of unstable procedures: the recursive projection method. SIAM J. Numer. Anal. 30(4), 1099–1120 (1993)MathSciNetCrossRefGoogle Scholar
  104. 104.
    Sleijpen, G.L.G., Fokkema, D.R.: BICGSTAB(L) for linear equations involving unsymmetric matrices with complex spectrum. ETNA 1, 11–32 (1993)MathSciNetzbMATHGoogle Scholar
  105. 105.
    Thurlow, M.S., Brooks, B.J., Lucas, P.G.J., Ardron, M.R., Bhattacharjee, J.K., Woodcraft, A.L.: Convective instability in rotating liquid 3He-4He mixtures. J. Fluid Mech. 313, 381–407 (1996)CrossRefGoogle Scholar
  106. 106.
    Tiesinga, G., Wubs, F., Veldman, A.: Bifurcation analysis of incompressible flow in a driven cavity by the Newton–Picard method. J. Comput. Appl. Math. 140(1–2), 751–772 (2002)MathSciNetCrossRefGoogle Scholar
  107. 107.
    Tsitverblit, N.: Bifurcation phenomena in confined thermosolutal convection with lateral heating: commencement of the double-diffusive region. Phys. Fluids 7(4), 718–736 (1995)CrossRefGoogle Scholar
  108. 108.
    Tuckerman, L.S.: Steady-state solving via Stokes preconditioning; recursion relations for elliptic operators. In: Dwoyer, D., Hussaini, M., Voigt, R. (eds.) 11th International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes in Physics, pp. 573–577. Springer, Berlin (1989)Google Scholar
  109. 109.
    Tuckerman, L.S., Barkley, D.: Bifurcation analysis for timesteppers. In: Doedel, E., Tuckerman, L.S. (eds.) Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems. IMA Volumes in Mathematics and its Applications, vol. 119, pp. 453–466. Springer, Berlin (2000)CrossRefGoogle Scholar
  110. 110.
    Uecker, H., Wetzel, D., Rademacher, J.: pde2path - a matlab package for continuation and bifurcation in 2D elliptic systems. Numer. Math. Theory, Methods Appl. 7, 58–106 (2014)Google Scholar
  111. 111.
    van Noorden, T.L., Verduyn Lunel, S.M., Bliek, A.: The efficient computation of periodic states of cyclically operated chemical processes. IMA J. Appl. Math. 68, 149–166 (2003)MathSciNetCrossRefGoogle Scholar
  112. 112.
    van Noorden, T.L., Verduyn Lunel, S.M., Bliek, A.: A Broyden rank p update continuation method with subspace iteration. SIAM J. Sci. Comput. (2004)Google Scholar
  113. 113.
    van Veen, L., Kawahara, G., Atsushi, M.: On matrix-free computation of 2D unstable manifolds. SIAM J. Sci. Comput. 33(1), 25–44 (2011)MathSciNetCrossRefGoogle Scholar
  114. 114.
    Viswanath, D.: Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358 (2007)MathSciNetCrossRefGoogle Scholar
  115. 115.
    Wakitani, S.: Flow patterns of natural convection in an air-filled vertical cavity. Phys. Fluids 10(8), 1924–1928 (1998)CrossRefGoogle Scholar
  116. 116.
    Wales, C., Gaitonde, A.L., Jones, D.P., Avitabile, D., Champneys, A.R.: Numerical continuation of high reynolds number external flows. Int. J. Numer. Methods Fluids 68(2), 135–159 (2012)MathSciNetCrossRefGoogle Scholar
  117. 117.
    Waugh, I., Illingworth, S., Juniper, M.: Matrix-free continuation of limit cycles for bifurcation analysis of large thermoacoustic systems. J. Comput. Phys. 240, 225–247 (2013)MathSciNetCrossRefGoogle Scholar
  118. 118.
    Waugh, I.C., Kashinath, K., Juniper, M.P.: Matrix-free continuation of limit cycles and their bifurcations for a ducted premixed flame. J. Fluid Mech. 759, 1–27 (2014)MathSciNetCrossRefGoogle Scholar
  119. 119.
    Werner, B., Spence, A.: The computation of symmetry-breaking bifurcation points. SIAM J. Numer. Anal. 21, 388–399 (1984)MathSciNetCrossRefGoogle Scholar
  120. 120.
    Winters, K.H.: Oscillatory convection in liquid metals in a horizontal temperature gradient. Int. J. Numer. Methods Eng. 25, 401–414 (1988)CrossRefGoogle Scholar
  121. 121.
    Wriggers, P., Wagner, W., Miehe, C.: A quadratically convergent procedure for the calculation of stability points in finite element analysis. Comput. Methods Appl. Mech. Eng. 70(3), 329–347 (1988)CrossRefGoogle Scholar
  122. 122.
    Xin, S., Le Quéré, P.: Natural-convection flows in air-filled differentially heated cavities with adiabatic horizontal walls. Numer. Heat Transf. Part A 50, 437–466 (2006)CrossRefGoogle Scholar
  123. 123.
    Xin, S., Le Quéré, P.: Stability of two-dimensional (2D) natural convection flows in air-filled differentially heated cavities: 2D/3D disturbances. Fluid Dyn. Res. 44(3), 031419 (2012)MathSciNetCrossRefGoogle Scholar
  124. 124.
    Xin, S., Le Quéré, P., Tuckerman, L.: Bifurcation analysis of doubly-diffusive convection with opposing horizontal thermal and solutal gradients. Phys. Fluids 10(4), 850–858 (1998)CrossRefGoogle Scholar
  125. 125.
    Yahata, H.: Stability analysis of natural convection in vertical cavities with lateral heating. J. Phys. Soc. Jpn. 66(11), 3434–3443 (1998)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departament de FísicaUniversitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations