Nonequilibrium reaction-diffusion structures in rigid and viscoelastic media: Knots and unstable noninertial flows.

*(English)*Zbl 0673.76100Summary: A simple reaction-diffusion model is used to demonstrate the existence of asymptotic (i.e. long time) knotted solutions of reaction-transport problems. The knots are attained with respect to the surface of constant concentration. These solutions cannot be mapped continuously onto the plane and as such have no two-dimensional analogue - they are strictly three-dimensional structures.

The existence of knotted solutions is first argued for intuitively using the properties of a simple reaction-diffusion system. A variational theorem for this system is then derived. Extrema of the associated “energy” functional with knotted topology are obtained numerically. The existence of a rich class of knotted and other strictly three dimensional solutions is also discussed. When the reaction-diffusion medium is subject to mechanical stresses, flows may result. These flows may interact with emerging dissipative structure when the time scales for flow and reaction are comparable. Imposed shears may orient compositional patterns. If the rheologic properties of the medium depend on composition, vortices may emerge under conditions far below the critical Taylor shear rate.

The existence of knotted solutions is first argued for intuitively using the properties of a simple reaction-diffusion system. A variational theorem for this system is then derived. Extrema of the associated “energy” functional with knotted topology are obtained numerically. The existence of a rich class of knotted and other strictly three dimensional solutions is also discussed. When the reaction-diffusion medium is subject to mechanical stresses, flows may result. These flows may interact with emerging dissipative structure when the time scales for flow and reaction are comparable. Imposed shears may orient compositional patterns. If the rheologic properties of the medium depend on composition, vortices may emerge under conditions far below the critical Taylor shear rate.

##### MSC:

76R50 | Diffusion |

76A10 | Viscoelastic fluids |

35Q99 | Partial differential equations of mathematical physics and other areas of application |

##### Keywords:

viscoelastic media; unstable noninertial flows; simple reaction-diffusion model; reaction-transport problems; constant concentration; existence of knotted solutions; variational theorem; strictly three dimensional solutions; critical Taylor shear rate##### References:

[1] | P. ORTOLEVA, Knots and tangles in Reaction Diffusion Systems (to appear in JIMA). |

[2] | R. SULTAN and P. ORTOLEVA, J. Chem. Phys. 84, 6781 (1986). |

[3] | R. SULTAN and P. ORTOLEVA, J. Chem. Phys. 85, 5068 (1986). |

[4] | C. H. CHENG and P. ORTOLEVA, < Knots in Reaction-Diffusion Systems with Folded Slow Manifolds > (in preparation); |

[5] | P. Ortoleva, The Variety and Structure of Chemical Waves (Manchester Umversity Press, 1989). · Zbl 0673.76100 |

[6] | T. DEWERS and P. ORTOLEVA, Mechano-Chemical Coupling via Texture Dependent Solubility in Stressed Rocks (Geochimica Cosmochimica Acta) (submitted for publication). |

[7] | T. DEWERS and P. ORTOLEVA, Geochemical Self-Organization III : A Mean Field, Pressure Solution Model of Spaced Cleavage and Metamorphic Seg-regational Layering (to appear in the Am. Jour, of Sci.). |

[8] | C. WEI and P. ORTOLEVA, A Linear Stability Analyses of a Visco-Elastic Model of Metamorphic Differentiation (in préparation). |

[9] | P. ORTOLEVA (1988), Geochemical Self-Organization (Oxford University Press, N. Y.). |

[10] | C. WEI and P. ORTOLEVA, Numerical Simulation of Metamorphic Differentiation in Two Spatial Dimensions (in préparation). |

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