具導電性差異之雙層不互溶流體之流動穩定性數學模式推導

Abstract

本研究針對位在雙層電極板之間，具備不同導電性質且不互溶的 雙層平板流體，在受到外加垂直電場下之流體介面穩定度三維建模。 由於在微流體混合器中，以壓力梯度驅動流體是相當沒有效率的方 式，再者，在如此微小的尺度，微流道由於強度因素，亦無法承受過 大的內部壓力，因此以外加電場的方式驅動流體，較為妥當。而由於 微流道中的Reynolds 數通常相當低，流體多以層流方式流動，導致流 體內部的混合僅能以分子擴散達成，故外加電壓場亦可提高流體的紊 亂度，以增強相異流體的混合度。 相較於單層流體，在雙層流體之中，存在著流體介面。此一介面 對於流體的混合有著關鍵性的角色，若能提高此介面的不穩定度，混 合流體的目的便能達成。本研究將以弱介電質模型(Leaky Dielectric Model)，將雙層流體與外加的電場在流體界面上做偶合，由最基本的 動量方程式、連續方程式以及電壓勢能方程式出發，並對方程式作微 擾處理與進ㄧ步的簡化，以得到往後能用以分析此雙層流系統穩定性 的擾動方程式。

This study focuses on the stability system formulation of plane double-layered flows with different conductivities between two electrodes under external applied perpendicular electric field. In microfluidic mixers, it is quite inefficient to drive fluids by the mean of pressure gradient, and the micro channel could not sustain large internal pressure due to structure weakness under such small scale. Thus, driving the fluids with external electric field is more preferable. Also, the Reynolds number in the micro channels is usually very small, making the flows laminar. Under such condition, the mixing within the flows could only be done by molecule diffusion. In order to enhance the mixing, external electric field could help making the flows become turbulence. Compared to single-fluid flows, there exists a fluid interface in double-layered flows. This interface plays a crucial role in mixing problems. The mixing could be enhanced if the instability of this interface is raised. The external applied electric field and the double-layered flows are coupled with Leaky Dielectric Model on the interface in this study. Starting with the most fundamental equations, including Navier-Stoke equations, continuity and voltage potential equations, small amplitude perturbation and further simplifications will be done, in order to obtain a set of perturbation equations for further stability analysis of the double-layered flows.