交互式放射狀注入對於混合效率的改善

Abstract

本研究利用數值模擬研究Hele-Shaw cell流場並進行分析，觀察交互式注入在放射狀Hele-Shaw flow流場中對於完全可互溶之兩流體的混合效率之改善。主要控制參數為(1)交互注入的時間間隔time interval(Δt)，(2)兩注入流體間之黏度比值Atwood number(A)，(3)對流效果以及擴散效果的比值Péclet number (Pe)。

在Jha學者等人的研究[2-4]中已證實黏性指狀物與交互式注入在矩形Hele-Shaw cell流場中有改善混合效率的作用，但在非線性的放射狀流場中由於擴張速度差異很大，交互式注入造成的影響尚未明瞭，因此以數值模擬進行研究。

本研究的模擬為在固定注入量Q的條件下，分析不同控制參數所造成的影響。模擬結果顯示交互注入的時間間隔time interval (Δt)的縮短會使得兩流體之間的界面增加，而隨著界面的增加，擴散作用因此得到強化，明顯的增加了兩種可完全互溶流體的混合效率，並因為多次的交互注入造成的多層構造，在兩流體間的黏度比例Atwood number (A)大於零(內部流體黏滯係數小於外部流體)時更加強化黏性指狀物之間的交互作用(指狀物的穿透(interpenetration)及混合(merging))，增加混合的效率。

Atwood number越高代表兩流體之間的黏度差異越大，當Atwood number的數值越大，黏性指狀物的產生也會更為劇烈。

而Péclet number的影響則會隨著Atwood number的變化而有差異，在低Péclet number的環境中擴散作用的效果較為顯著，因此在低Péclet number的案例中混合的進行是由擴散作用主導，在此環境下Atwood number的影響相對較不明顯。而高Péclet number的案例中由於對流效果較擴散作用強，強化了黏性指狀物之間的穿透與混合，因此在高Péclet number的環境下混合主要是由對流作用主導，高Péclet number的環境下Atwood number的變化對混合效率所造成的影響會變大，由於兩流體間黏滯係數差距大會造成兩流體界面會產生較多的黏性指狀物，並在且高Péclet number的較強對流作用的影響下黏性指狀物的相互作用(指狀物的穿透、結合等現象)會變得更劇烈，可以得到較好的混合效率，而在低Atwood number的情況下由於生成的黏性指狀物較少，難以觀察到黏性指狀物之間的交互作用造成的混合效果，對於混合效率的提升效果也有限。

藉由本研究，我們可以分析在低Reynolds number的環境下藉由交互式注入與黏性指狀物的生成強化混合效率的手段，可以應用於低Reynolds number但是卻難以外加擾動強化混合的環境，例如:微少流體元件或是微流道，在其中藉由黏性指狀物和交互式注入造成的多層結構達到提高混合的效率的功用。

The viscous fingering problem, or the so-called Saffman-Taylor instability considers the evolution of interfacial instability when a less viscosity fluid displaces another fluid of higher viscosity in porous medium or between the narrowly-spaced plates of a Hele-Shaw cell. Which can be treated in lieu of a two-dimensional homogeneous porous media. We consider a Hele-Shaw cell of constant gap thickness b containing two incompressible viscous fluids, which are miscible to each other. The cell is initially fully occupied by the more viscous fluid 2, whose viscosity is denoted η2. The viscosities of the fluids are denoted as η1(injected fluid),and η2(displaced fluid), respectively, and assume that η2>η1. Equal amount of a less viscous fluid 1(viscosity η1), and the more viscous fluid 2 are injected alternatively in sequence. The process continues up to a time t=tf , when the area of the total injected fluid expands to πDf /4 in a stable injection condition without fingering instability. Initially, the fluid-fluid interface is a small circular core of diameter D0, and a Cartesian coordinate system (x,y) is defined in such a way that its origin is located at the center of this core region. Consequently, the injection strength can be obtained as Q=π(〖D_f〗^2-〖D_0〗^2)/4t_f,in which the total injection duration for both the less viscous fluid 1 and the more viscous fluid 2 is tf /2.The interface becomes unstable during the injection of the less viscous fluid 1,while expands stably followed by the injections of the more viscous fluid 2. The injection are carried out by evenly injecting the less viscous fluid and the more viscous fluid in sequence till the completion at t=1. Several alternative injection scheme associated with different injection interval are applied, which yield the same total amount of injection. The duration of each injection interval is constant, denoted as Δt, and ΣΔt=0.5 for each fluid. We can observe that the concentration variations are apparently reduced by smaller injection intervals. And better mixing efficiencies are found in conditions of vigorous fingering associated with active interactions between the injected fluids, e.g. larger Atwood number associated with smaller injection interval. It is interesting to found out that, better mixing occurs at higher Pe if the Atwood number is sufficiently large to trigger fingering. e.g. A≥0.848. This better mixing efficiency is due to stronger interactions between the injection fluids. On the mechanism. So that slightly better mixing occurs for large Pe.