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Diffusive instabilities and spatial patterning from the coupling of reaction–diffusion processes with Stokes flow in complex domains
- Robert A. Van Gorder, Hyunyeon Kim, Andrew L. Krause
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- Journal:
- Journal of Fluid Mechanics / Volume 877 / 25 October 2019
- Published online by Cambridge University Press:
- 27 August 2019, pp. 759-823
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- Article
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We study spatial and spatio-temporal pattern formation emergent from reaction–diffusion–advection systems formed by considering reaction–diffusion systems coupled to prescribed fluid flows. While there have been a number of studies on the planar dynamics of such systems and the resulting instabilities and spatio-temporal patterning in the plane, less has been done on complicated flows in complex domains. We consider a general approach for the study of bounded domains in order to model two- and three-dimensional geometries which are more likely to be of relevance for modelling dynamics within fluid vessels used in experiments. Considering a variety of problem geometries with finite cross-sections, such as two-dimensional channels, three-dimensional ducts and three-dimensional pipes, we demonstrate the role cross-section geometry plays in pattern formation under such systems. We find that the generic instability is that of an oscillatory or wave Turing instability, resulting in patterns which change in time, often being advected with the fluid flow. As in previous works, we observe a change in patterns formed when progressing from zero to weak to strong advection for uniform advection across the domain, with particularly strong advection destroying patterns. One novel finding is that heterogeneous fluid flow can induce qualitatively different patterns across the domain. For instance, Poiseuille flow with maximal advection in the centre of a vessel and zero advection at the boundary of a vessel is shown to exhibit patterns in the centre of the vessel which are different from patterns near the boundary, with differences attributed to the differential local advection within each region of the vessel. Additionally, we observe sheared patterns, which appear due to gradients in the fluid velocity, and cannot be obtained via any kind of uniform flow. Finally we also explore flow in more complex domains, including wavy-walled channels, continuous stirred-tank reactors, U-shaped pipes and a toroidal domain, in order to demonstrate behaviours when the flow is both heterogeneous and bidirectional, as well as to demonstrate that our results still apply for complex finite domains. Our analysis suggests that such non-trivial advection results in moving patterns which are more complex than observed in simpler reaction–diffusion–advection, and may be more characteristic of realistic flow regimes in biological media.
Contributors
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- By Leonard A. Adler, Henrik Anckarsäter, L. Eugene Arnold, Philip J. Asherson, Russell Barkley, Joseph Biederman, Andrew D. Blackwell, Jessica Bramham, Thomas E. Brown, Richard Bruggeman, Jan K. Buitelaar, C. Keith Conners, Jonathan H. Dowson, Steve V. Faraone, Christopher Gibbins, Christopher Gillberg, I. Carina Gillberg, Ylva Ginsberg, Laurence L. Greenhill, Julia D. Hunter, Cornelis C. Kan, Ronald C. Kessler, Scott H. Kollins, J. J. Sandra Kooij, Johanna Krause, Jonna Kuntsi, Florence Levy, Stephen P. McDermott, Gráinne McLoughlin, Mitul A. Mehta, Asko Niemela, Eleni Paliokosta, Yannis Paloyelis, Vangelis Pappas, Patricia Quinn, Maria Råstam, Doris Ryffel, David Shaw, Seija Sirviö, Thomas Spencer, Lacramioara Spetie, Siegfried Tuinier, Fiona E. van Dijk, Anne M. D. N. van Lammeren, Wim J. C. Verbeeck, Margaret Weiss, Timothy E. Wilens, Kiriakos Xenitidis
- Edited by Jan K. Buitelaar, Cornelis C. Kan, Philip Asherson, Institute of Psychiatry, London
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- Book:
- ADHD in Adults
- Published online:
- 04 April 2011
- Print publication:
- 03 March 2011, pp vii-ix
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