主動粒子集體運動的數值模擬

Abstract

本研究可分為模擬與理論兩部分，在模擬上，我們使用固液二相流Eulerian-Lagrangian模型模擬懸浮主動粒子，在這套數值模式中，流體運動以Navier-Stokes方程式在尤拉網格上進行解析，顆粒運動則為牛頓第二運動定律，且追蹤每個顆粒的動向，在模擬設置上，對於初始狀態、邊界條件、流場面積、流體與顆粒性質都與本研究參考的實驗設置相同進行模擬，我們將模擬結果與實驗結果做比較，確認因顆粒運動會有巨觀流場的產生，且利用自相關性分析(Autocorrelation analysis)，分析顆粒的同步性隨時間的變化。在理論上，以我們模擬的模式下引入動力理論(Kinetic theory)描述此研究的主動粒子，根據此理論可推導出顆粒相整體(Bulk)的數密度與速度方程式，我們將所得出之顆粒相方程式與流體動量方程式做耦合，進行線性穩定性分析，探討在何種機制下，流場會不穩定，即巨觀流場的產生。

This study is divided into two parts. In the first part, we use the Eulerian-Lagrangian model to simulate the active suspension of self-propelled small particles. The numerical model solves the momentum equations for the carrier liquid on the Eulerian meshes. The motion of the particle is governed by the Newton’s second law. We set up the simulation that is the same as the experimental set-up. We further compare the simulation results with the experimental data. The simulation results confirm the occurrence of the macroscopic flow field. We use the autocorrelation analysis to analyze the collective motion of particles. In the second part, we use the kinetic theory to obtain the governing equations of the number density and bulk velocity for groups of particles. The equations are then coupled with the Navier-Stokes equations, and the liner stability analysis is conducted for the present system.