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

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.

Bespoke Turing Systems

“…3.3, are strictly a two-dimensional phenomenon. Although the resulting dynamics can be much richer in more complex cases (for example, through: higher domain dimensions; advective transport; and increased numbers of species (Klika et al 2012;Marcon et al 2016;Arcuri and Murray 1986;Aragón et al 2012;Woolley 2014Woolley , 2017Van Gorder et al 2019)), the proposed Fig. 1 Flowchart for constructing a Turing system that will work in any specified parameter region and provide a desired pattern.…”

“… 2012 ; Woolley 2014 , 2017 ; Van Gorder et al. 2019 )), the proposed framework can be generalised to these cases. Indeed, with cases of higher complexity there are often even more degrees of freedom to design kinetics that match observed patterns.…”

“…Turing-type patterning is also studied in a variety of other settings, such as in nonlinear optics (Oppo 2009;Ardizzone et al 2013;Chembo et al 2017), geochemistry (Baurmann et al 2004;McBride and Picard 2004), astrophysics (Smolin 1996), reaction-advection-diffusion systems (Klika et al 2018;Krause et al 2018a;Van Gorder et al 2019), network-organised media (Nakao and Mikhailov 2010;Asllani et al 2014), spatial ecology (Sherratt 2012;Hata et al 2014;Taylor et al 2020), and social dynamics (Wakano et al 2009). Whilst it has been known for decades that chemical systems can be engineered to produce specific Turing patterns (Vanag and Epstein 2001), recently Tan et al (2018) employed such designed Turing systems to manufacture a porous filter for use in water purification.…”

“… 2018a ; Van Gorder et al. 2019 ), network-organised media (Nakao and Mikhailov 2010 ; Asllani et al. 2014 ), spatial ecology (Sherratt 2012 ; Hata et al.…”

Influence of temperature on Turing pattern formation

“…Analysis of the Turing instability and resulting pattern formation in activator-inhibitor systems advected with a Stokes flow such as the Poiseuille flow is also possible, and was considered in detail in [40]. A generalization of [40] to include thermodynamics would be interesting but is beyond the scope of this paper, and instead we consider such cases through simulations. In figure 6, we plot dispersion sets corresponding to unstable wavenumbers satisfying (5.5).…”