ロウソクの同期振動現象に対する数理モデルとその数理解析

金沢大学数物科学系複数のロウソクを束にするとロウソク火炎が振動する現象はよく知られている(以降、この振動するロウソクの束をロウソク振動子と呼ぶ).2006年に2組のロウソク振動子を並べたとき,その距離に依存して同位相振動と逆位相振動を起こすことが実験によって報告された.ロウソク振動子の振動現象は酸素の供給不足が原因となって起こることは知られているが,ロウソク振動子の同期現象に対してその機構は明らかになっておらず,結合振動子系に対する数理的理解という視点から同期現象の本質的機構を明らかにすることは重要である.そこでこの同期現象(同位相,逆位相)が引き起こされる機構に対して数理モデルを用いて明らかにすることを目的として研究を推進してきた.昨年度までに,逆相同期現象が起こる機構には幅射の相互作用が重要であることを示し,位相縮約法を用いて示すことができた.我々の提出した数理モデルでは,流体の効果は拡散相互作用を打ち消すことを示唆しただけであり,流体効果について詳しく検討していなかった.今年度は,流体がロウソク振動子の同期現象にどのような影響を与えるのか明らかにするために研究を推進した.燃焼反応拡散系モデルと流体方程式の結合した数理モデルを提案した.この数理モデルに対して,酸素供給不足による振動現象の再現を試みたが振動する現象を再現することができなかった.これは「空間3次元問題においては方程式に含まれるパラメータを同定することが非常に困難であることと「流体と反応拡散系の相互作用が流れだけでよいなのかわからないこと」に起因している.また,境界条件も自明ではなく今後の課題となってしまった.この経験からどのような視点に立って数理モデルを構成するかは非常に重要であり,現象論的な手法ではあるが本質だけを取り出すことに主眼をおく数理モデルの重要性を再認識することになった.研究課題/領域番号:19654017, 研究期間(年度):2007 – 2009出典：研究課題「ロウソクの同期振動現象に対する数理モデルとその数理解析」課題番号19654017（KAKEN：科学研究費助成事業データベース（国立情報学研究所）） （https://kaken.nii.ac.jp/ja/grant/KAKENHI-PROJECT-19654017/）を加工して作

Two floating camphor particles interacting through lateral capillary force

We consider a mathematical model for a two-particle system driven by the spatial gradient of a concentration field of chemicals with conservative attractive interactions in one dimension. This setup corresponds to an experimental system with floating camphor particles at a water surface. Repulsive interaction is introduced, as well as self-propelling force, through the concentration field of camphor molecules at the water surface. Here we newly adopt the attractive lateral capillary force due to the deformation of the water surface. The particles experience competing dissipative repulsion and conservative attraction. We numerically investigated the mathematical model, and found six different modes of motion. The theoretical approach revealed that some of such mode transitions can be understood in terms of bifurcation.Comment: 9 pages, 6 figure

Chemo-Sensitive Running Droplet

Chemical control of the spontaneous motion of a reactive oil droplet moving on a glass substrate under an aqueous phase is reported. Experimental results show that the self-motion of an oil droplet is confined on an acid-treated glass surface. The transient behavior of oil-droplet motion is also observed with a high-speed video camera. A mathematical model that incorporates the effect of the glass surface charge is built based on the experimental observation of oil-droplet motion. A numerical simulation of this mathematical model reproduced the essential features concerning confinement within a certain chemical territory of oil-droplet motion, and also its transient behavior. Our results may shed light on physical aspects of reactive spreading and a chemotaxis in living things.Comment: 17 pages, 10 figure

Bifurcation of self-motion depending on the reaction order

As a simple example of an autonomous motor, the characteristic features of self-motion coupled with the acid-base reaction were numerically and experimentally investigated at the air/aqueous interface. Oscillatory and uniform motion were categorized as a function of the reaction order by numerical computations using a mathematical model that incorporates both the distribution of the surface active layer developed from a material particle as the driving force and the kinetics of the acid-base reaction. The nature of the self-motion was experimentally observed for a boat adhered to a camphor derivative with a mono- or di-carboxylic acid on a phosphate aqueous phase as the base. © the Owner Societies

A simplified numerical model for evaluating sediment control by open-type sabo dams in the Joganji River basin

The present study proposes a method to estimate sediment runoff by introducing a dam function of the relationship between inflow sediment and sediment runoff through a slit dam. The model can process rainfall runoff, sediment yield and runoff of a mountainous basin, and the model is applied to the upper reaches of the Joganji River basin, which is known for its huge amount of sediment runoff and intense bed variation because of the sediment yield caused by the earthquake in 1858. The performance of the calculations of sediment control of the slit dam is evaluated by the model. The result indicates that sediment deposition is significantly changed by sediment runoff. The proposed method can be expected to evaluate sediment transport with sabo dams on a basin scale