降膜流中孤立脈波的互動理論

Abstract

人們已經觀察到薄膜方程中的解在某些參數條件下會呈現出空間, 時間, 或空間與時間的局部結構, 也就是所謂的相干結構. 其解表現為半平衡的孤立脈波並連續的互相吸引與排斥. 這些波最終成為束 縛態, 彼此以相同的速度前進並且不改變彼此間的距離. 本計劃目標為利用數值以及分析的方法來研究降膜流中孤立脈波的互動理論, 並更進一步建立一個理 論架構以期能系統性的研究一般色散-主動耗散系統中的相干結構. 為了分析這樣的現象, 我們需要更 準確並且有效率的數值方法以及一些分析的工具. 更明確的說, 本計劃第一年目標為: (1)研究發展薄 膜方程在不同幾何空間中的數值解法; (2)研究發展孤立脈波的弱互動理論. 第二年目標為: (1)發展數 值延拓法來尋找孤立脈波的束縛態; (2)驗證簡化系統中的預測是否與原Navier-Stokes 方程的解相符.

It has been observed that the solutions of the thin film equation, for certain parameter values, evolve into space, time or space-time localized structures, so-called coherent structures. The solution is described by quasi-stationary solitary pulses that attract and repel continuously with each other. The pulses eventually form bound states that separate from each other with fixed distances and travel together with the same speed. The goal of the proposed research is to analyze numerically and analytically the interaction of solitary pulses arising in the modeling of thin liquid film flows, and furthermore to develop a theoretical framework allowing a systematic investigation of coherent structures in general dispersion active-dissipative systems. It requires an accurate and efficient numerical method to capture the dynamics of the pulse interactions, as well as several analytical tools. More specifically, for the first year, the objectives are: (1) To develop numerical methods to solve the thin film equation on different geometries; (2) To develop a weak-interaction theory for the solitary pulse solutions and to investigate their interaction dynamics. For the second year, the objectives are: (1) To use numerical continuation techniques to obtain accurately the bound state of solitary pulses; (2) To validate the theoretical predictions of the simplified models with the numerical solutions of the full Navier-Stokes equations.