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Diffusive instabilities and spatial patterning from the coupling of reaction–diffusion processes with Stokes flow in complex domains

Published online by Cambridge University Press:  27 August 2019

Robert A. Van Gorder Open the ORCID record for Robert A. Van Gorder [Opens in a new window] ,
Hyunyeon Kim  and
Andrew L. Krause
Robert A. Van Gorder*
Affiliation:
Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin 9054, New Zealand
Hyunyeon Kim
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
Andrew L. Krause
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
*
†Email address for correspondence: rvangorder@maths.otago.ac.nz
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Abstract

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.

JFM classification

Nonlinear Dynamical Systems: Pattern formation Reacting Flows: Laminar reacting flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 877 , 25 October 2019 , pp. 759 - 823
Copyright
© 2019 Cambridge University Press 

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