Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-22T08:54:18.244Z Has data issue: false hasContentIssue false

Pattern formation on time-dependent domains

Published online by Cambridge University Press:  07 October 2019

M. Ghadiri
Affiliation:
Departments of Mathematics and Mechanical Engineering, University of Alberta, Edmonton, AB T6G 2R3, Canada
R. Krechetnikov*
Affiliation:
Departments of Mathematics and Mechanical Engineering, University of Alberta, Edmonton, AB T6G 2R3, Canada
*
Email address for correspondence: [email protected]

Abstract

In the quest to understand the dynamics of distributed systems on time-dependent spatial domains, we study experimentally the response to domain deformations by Faraday wave patterns – standing waves formed on the free surface of a liquid layer due to its vertical vibration – chosen as a paradigm owing to their historical use in testing new theories and ideas. In our experimental set-up of a vibrating water container with controlled positions of lateral walls and liquid layer depth, the characteristics of the patterns are measured using the Fourier transform profilometry technique, which allows us to reconstruct an accurate time history of the pattern three-dimensional landscape and reveal how it reacts to the domain dynamics on various length and time scales. Analysis of Faraday waves on growing, shrinking and oscillating domains leads to a number of intriguing results. First, the observation of a transverse instability – namely, when a two-dimensional pattern experiences an instability in the direction orthogonal to the direction of the domain deformation – provides a new facet to the stability picture compared to the one-dimensional systems in which the longitudinal (Eckhaus) instability accounts for pattern transformation on time-varying domains. Second, the domain evolution rate is found to be a key factor dictating the patterns observed on the path between the initial and final domain aspect ratios. Its effects range from allowing the formation of complex sequences of patterns to impeding the appearance of any new pattern on the path. Third, the shrinkage–growth process turns out to be generally irreversible on a horizontally evolving domain, but becomes reversible in the case of a time-dependent liquid layer depth, i.e. when the dilution and convective effects of the domain flow are absent. These experimentally observed enigmatic effects of the domain size variations in time are complemented here with appropriate theoretical insights elucidating the dynamics of two-dimensional pattern evolution, which proves to be more intricate compared to one-dimensional systems.

Type
JFM Papers
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bard, J. & Lauder, I. 1974 How well does Turing’s theory of morphogenesis work? J. Theor. Biol. 45, 501531.Google Scholar
Batson, W., Zoueshtiagh, F. & Narayanan, R. 2013 The Faraday threshold in small cylinders and the sidewall non-ideality. J. Fluid Mech. 729, 496523.Google Scholar
Bechhoefer, J., Ego, K., Manneville, S. & Johnson, B. 1995 An experimental study of the onset of parametrically pumped surface waves in viscous fluids. J. Fluid Mech. 288, 325350.Google Scholar
Benjamin, T. B. & Scott, J. C. 1979 Gravity-capillary waves with edge constraints. J. Fluid Mech. 92, 241267.Google Scholar
Benjamin, T. B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Phil. Trans. R. Soc. Lond. 225, 5054515.Google Scholar
Binks, D. & van de Water, W. 1997 Nonlinear pattern formation of Faraday waves. Phys. Rev. Lett. 78, 40434046.Google Scholar
Castets, V., Dulos, E., Boissonade, J. & Kepper, P. D. 1990 Experimental evidence of a sustained Turing-type equilibrium chemical pattern. Phys. Rev. Lett. 64, 29532956.Google Scholar
Cerda, E. A. & Tirapegui, E. L. 1998 Faradays instability in viscous fluid. J. Fluid Mech. 368, 195228.Google Scholar
Christiansen, B., Alstrom, P. & Levinsen, M. 1995 Dissipation and ordering in capillary waves at high aspect ratio. J. Fluid Mech. 291, 323341.Google Scholar
Ciliberto, S. & Gollub, J. P. 1984 Pattern competition leads to chaos. Phys. Rev. Lett. 52, 922925.Google Scholar
Ciliberto, S. & Gollub, J. P. 1985a Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381398.Google Scholar
Ciliberto, S. & Gollub, J. P. 1985b Phenomenological model of chaotic mode competition in surface waves. Il Nuovo Cimento D 6, 309316.Google Scholar
Cobelli, P., Maurel, A., Pagneux, V. & Petitjeans, P. 2009 Global measurement of water waves by Fourier transform profilometry. Exp. Fluids 46, 10371047.Google Scholar
Cobelli, P. J., Pagneux, V., Maurel, A. & Petitjeans, P. 2011 Experimental study on water-wave trapped modes. J. Fluid Mech. 666, 445476.Google Scholar
Craik, A. D. D. & Armitage, J. G. M. 1995 Faraday excitation, hysteresis and wave instability in a narrow rectangular wave tank. Fluid Dyn. Res. 15, 129143.Google Scholar
Crampin, E. J., Gaffney, E. A. & Maini, P. K. 1999 Reaction and diffusion on growing domains: Scenarios for robust pattern formation. Bull. Math. Biol. 61, 10931120.Google Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 8511112.Google Scholar
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383409.Google Scholar
Douady, S. & Fauve, S. 1988 Pattern selection in Faraday instability. Eur. Phys. Lett. 6, 221226.Google Scholar
Eckhaus, W. 1965 Studies in Non-Linear Stability Theory. Springer.Google Scholar
Edwards, W. S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123148.Google Scholar
Ezerskii, A. B., Rabinovich, M. I., Reutov, V. P. & Starobinets, I. M. 1986 Spatiotemporal chaos in the parametric excitation of a capillary ripple. Sov. Phys. JETP 64, 12281236.Google Scholar
Faraday, M. 1831 On a peculiar class of acoustical figures, and on the forms of fluids vibrating on elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 299340.Google Scholar
Fauve, S. 1998 Pattern forming instabilities. In Hydrodynamics and Nonlinear Instabilities (ed. Godréche, C. & Manneville, P.), pp. 387492. Cambridge University Press.Google Scholar
Feng, Z. C. & Sethna, P. R. 1989 Symmetry-breaking bifurcations in resonant surface waves. J. Fluid Mech. 199, 495518.Google Scholar
Francesco, P., Mathieu, P. & Sénéchal, D. 1997 Conformal Field Theory. Springer.Google Scholar
Gollub, J. P. 2006 Patterns and chaotic dynamics in Faraday surface waves. In Dynamics of Spatio-Temporal Cellular Structures. Henri Bénard Centenary Review (ed. Mutabazi, I., Wesfreid, J. E. & Guyon, E.), pp. 213224. Springer.Google Scholar
Grassberger, P. & Procaccia, I. 1983a Characterization of strange attractors. Phys. Rev. Lett. 50, 346349.Google Scholar
Grassberger, P. & Procaccia, I. 1983b Estimation of the Kolmogorov entropy from a chaotic signal. Phys. Rev. A 28, 25912593.Google Scholar
Grassberger, P. & Procaccia, I. 1983c Measuring the strangeness of strange attractors. Physica D 9, 189208.Google Scholar
Gu, X. M., Sethna, P. R. & Narain, A. 1988 On three-dimensional non-linear subharmonic resonant surface waves in a fluid. Part I: theory. Trans. ASME E: J. Appl. Mech. 55, 213219.Google Scholar
Hartong-Redden, R. & Krechetnikov, R. 2011 Pattern identification in systems with S(1) symmetry. Phys. Rev. E 84, 056212.Google Scholar
Henderson, D. M. & Miles, J. W. 1990 Single-mode Faraday waves in small cylinders. J. Fluid Mech. 213, 95109.Google Scholar
Henry, B., Lovell, N. & Camacho, F. 2012 Nonlinear dynamics time series analysis. Nonlinear Biomed. Signal Process. Dyn. Anal. Modeling 2, 139.Google Scholar
Hocking, L. M. 1987a The damping of capillary–gravity waves at a rigid boundary. J. Fluid Mech. 179, 253266.Google Scholar
Hocking, L. M. 1987b Waves produced by a vertically oscillating plate. J. Fluid Mech. 179, 267281.Google Scholar
Hoyle, R. B. 1993 Long wavelength instabilities of square patterns. Physica D 67, 198223.Google Scholar
Hoyle, R. B. 2006 Pattern Formation: An Introduction to Methods. Cambridge University Press.Google Scholar
Kidambi, R. 2009 Capillary damping of inviscid surface waves in a circular cylinder. J. Fluid Mech. 627, 323340.Google Scholar
Knobloch, E. & Krechetnikov, R. 2014 Stability on time-dependent domains. J. Nonlinear Sci. 24, 493523.Google Scholar
Knobloch, E. & Krechetnikov, R. 2015 Problems on time-varying domains: formulation, dynamics, and challenges. Acta Appl. Math. 137, 123157.Google Scholar
Kondo, S. & Asai, R. 1995 A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus. Nature 376, 765768.Google Scholar
Krechetnikov, R. & Knobloch, E. 2017 Stability on time-dependent domains: convective and dilution effects. Physica D 342, 1623.Google Scholar
Krechetnikov, R. & Marsden, J. E. 2009 On the origin and nature of finite-amplitude instabilities in physical systems. J. Phys. A 42, 412004.Google Scholar
Kudrolli, A. & Gollub, J. P. 1996 Patterns and spatiotemporal chaos in parametrically forced surface waves: a systematic survey at large aspect ratio. Physica D 97, 133154.Google Scholar
Kumar, K. 1996 Linear theory of Faraday instabilty in viscous liquids. Proc. R. Soc. Lond. A 452, 11131126.Google Scholar
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 4968.Google Scholar
Kumar, S. & Matar, O. K. 2004 On the Faraday instability in a surfactant-covered liquid. Phys. Fluids 16, 3946.Google Scholar
Lamb, H. 1994 Hydrodynamics. Cambridge University Press.Google Scholar
Langer, J. S. & Ambegaokar, V. 1967 Intrinsic resistive transition in narrow superconducting channels. Phys. Rev. 164, 498510.Google Scholar
LeBlond, P. H. & Mainardi, F. 1987 The viscous damping of capillary–gravity waves. Acta Mechanica 68, 203222.Google Scholar
Madzvamuse, A., Gaffney, E. A. & Maini, P. K. 2010 Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains. J. Math. Biol. 61, 133164.Google Scholar
Madzvamuse, A., Maini, P. K. & Wathen, A. J. 2003 A moving grid finite element method applied to a model biological pattern generator. J. Comput. Phys. 190, 478500.Google Scholar
Matthiessen, L. 1868 Akustische Versuche, die kleinsten Transversal wellen der Flüssigskeiten betressend. Ann. Phys. 134, 107117.Google Scholar
Maurel, A., Cobelli, P., Pagneux, V. & Petitjeans, P. 2009 Experimental and theoretical inspection of the phase-to-height relation in Fourier transform profilometry. Appl. Opt. 48, 380392.Google Scholar
Miles, J. W. 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. 297, 459475.Google Scholar
Miles, J. W. & Henderson, D. M. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22, 143165.Google Scholar
Milner, S. T. 1991 Square patterns and secondary instabilities in driven capillary waves. J. Fluid Mech. 225, 81100.Google Scholar
Moisy, F., Rabaud, M. & Salsac, K. 2009 A synthetic Schlieren method for the measurement of the topography of a liquid interface. Exp. Fluids 46, 10211036.Google Scholar
Ouyang, Q. & Swinney, H. L. 1991 Transition from a uniform state to hexagonal and striped Turing patterns. Nature 352, 610612.Google Scholar
Painter, K. J., Maini, P. K. & Othmer, H. G. 1999 Stripe formation in juvenile pomacanthus explained by a generalized Turing mechanism with chemotaxis. Proc. Natl Acad. Sci. USA 96, 55495554.Google Scholar
Périnet, N., Falcón, C., Chergui, J. & Juric, D. 2016 Hysteretic Faraday waves. Phys. Rev. E 93, 063114.Google Scholar
Périnet, N., Gutiérrez, P., Urra, H., Mujica, N. & Gordillo, L. 2017 Streaming patterns in Faraday waves. J. Fluid Mech. 819, 285310.Google Scholar
Przadka, A., Cabane, B., Pagneux, V., Maurel, A. & Petitjeans, P. 2011 Fourier transform profilometry for water waves: how to achieve clean water attenuation with diffusive reflection at the water surface? Exp. Fluids 52, 519527.Google Scholar
Rayleigh, L. 1883 On maintained vibrations. Phil. Mag. 15, 229235.Google Scholar
Sauer, T. & Yorke, J. A. 1991 Rigorous verification of trajectories for computer simulation of dynamical systems. Nonlinearity 4, 961979.Google Scholar
Simonelli, F. & Gollub, J. 1988 Stability boundaries and phase space measurements for spatially extended dynamical systems. Rev. Sci. Instrum. 59, 280284.Google Scholar
Simonelli, F. & Gollub, J. 1989 Surface wave mode interactions: effects of symmetry and degeneracy. J. Fluid Mech. 199, 471494.Google Scholar
Simonelli, F. & Gollub, J. P. 1987 The masking of symmetry by degeneracy in the dynamics of interacting modes. Nucl. Phys. B 2, 8795.Google Scholar
Swinney, H. L. & Gollub, J. P. 1986 Characterization of hydrodynamic strange attractors. Physica D 18, 448454.Google Scholar
Takens, F. 1981 Detecting strange attractors in turbulence. In Dynamical Systems and Turbulence (ed. Rand, D. A. & Young, L. S.), Lecture Notes in Mathematics, vol. 898, pp. 366381. Springer.Google Scholar
Tuckerman, L. S. & Barkley, D. 1990 Bifurcation analysis of the Eckhaus instability. Physica D 46, 5786.Google Scholar
Turing, A. M. 1952 The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 3772.Google Scholar
Ueda, K.-I. & Nishiura, Y. 2012 A mathematical mechanism for instabilities in stripe formation on growing domains. Physica D 241, 3759.Google Scholar
Ursell, F., Dean, R. G. & Yu, Y. S. 1960 Forced small-amplitude water waves: a comparison of theory and experiment. J. Fluid Mech. 7, 3352.Google Scholar
Vasil, G. M. & Proctor, M. R. E. 2011 Dynamic bifurcations and pattern formation in melting-boundary convection. J. Fluid Mech. 686, 77108.Google Scholar
Wright, W. B., Budakian, R. & Putterman, S. J. 1996 Diffusing light photography of fully developed isotropic ripple turbulence. Phys. Rev. Lett. 76, 45284531.Google Scholar
Xia, H., Maimbourg, T., Punzmann, H. & Shats, M. 2012 Oscillon dynamics and rogue wave generation in Faraday surface ripples. Phys. Rev. Lett. 109, 114502.Google Scholar