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Mathematical Analysis of helical waves arising in some-reaction diffusion systems

部分反应扩散系统中产生的螺旋波的数学分析

基本信息

批准号：
12440032
负责人：
IKEDA Tsutomu
金额：
$ 7.23万
依托单位：
Ryukoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440032/
关键词：
self-propagating high-temperature syntheses planar traveling wave planar pulsating wave regular helical wave irregular helical wave wave patterns apparent activation energy bifurcation theory

项目摘要

A helical wave is observed in self-propagating high-temperature syntheses (SHS), for instance. One can create a high-quality uniform product by the SHS when a combustion wave keeps its profile and propagates at a constant velocity. When experimental conditions are changed, however, the planar traveling wave may lose its stability and give place some non-uniform ones. Actually, a planar pulsating wave appears through the Hopf bifurcation of planar traveling wave. Moreover, we observe a wave that propagates in the form of spiral encircling the cylindrical sample with several reaction spots. This wave is called a helical wave since it has been shown by our 3D numerical simulation that the isothermal surface of the wave has some wings and it helically rotates down as time passes on. Similar helical waves are observed also in propagation fronts of polymerizations in laboratory and they are obtained also by numerical simulation of some autocatalytic reactions as well as the SHS.We have been studied the existing condition of helical wave and the transition process of wave patterns from traveling mode to pulsating mode and/or helical mode, and we have obtained the following results:1. A stable helical wave can bifurcate directly from a planar traveling wave.2. Even if a traveling wave is stable in R, the corresponding planar traveling wave can be unstable in the band domain as well as in the cylindrical domain, and a helical wave takes the place of planar traveling wave.3. There are no stable helical wave when the band width L is small or the radius R of cylindrical domain is small.4. Helical waves with different numbers of reaction spots can coexist stably.

例如，在自传播高温合成（SHS）中观察到螺旋波。当燃烧波保持其轮廓并以恒定速度传播时，可以通过SHS产生高质量的均匀产品。然而，当实验条件改变时，平面行波可能会失去稳定性，并产生一些非均匀波。实际上，通过平面行波的霍普夫分叉，出现了一个平面脉动波。此外，我们观察到一个波的传播形式的螺旋环绕的圆柱形样品与几个反应点。我们称这种波为螺旋波，因为我们的三维数值模拟表明，这种波的等温表面有一些翅膀，随着时间的推移，它螺旋地向下旋转。在实验室的聚合反应传播前沿也观察到了类似的螺旋波，并通过对一些自催化反应和SHS的数值模拟也得到了它们。我们研究了螺旋波的存在条件波以及波型从行波到脉动波和螺旋波的转变过程，得到了以下结果：1.稳定的螺旋波可以从平面行波直接分叉.即使一个行波在R中是稳定的，对应的平面行波在带域和圆柱域中也可能是不稳定的，螺旋波取代了平面行波.当频带宽度L较小时或圆柱畴半径R较小时，不存在稳定的螺旋波.不同反应斑数的螺旋波可以稳定共存。