非球形氣膠粒子之熱泳及緩慢運動

Abstract

本論文對非球形粒子在一氣體中的熱泳及緩慢運動作理論之探討。無論是驅動粒子熱泳運動之均勻溫度梯度，或是定溫下使粒子作緩慢運動之外加作用力，其方向都可以是任意的。由於紐森數〈Knudsen number〉較小，因此氣體可視為連續體，而在粒子表面考慮有溫度躍差、熱滑移及摩擦滑移的現象。在皮克列數及雷諾數很小的假設下，求解系統之能量及動量主導方程式。文中非球形粒子的形狀討論了數種情形，包括稍微偏離球形之粒子、軸對稱粒子及橢球粒子。 第二章探討稍微偏離球形之粒子在任意方向之熱泳及定溫下之移動和轉動。系統之能量及動量方程式以微擾展開的方法來漸進求解。對於一般形狀的粒子運動，其能量及動量主導方程式以及邊界條件，皆對於微小形狀偏離之參數作二階微擾展開。最後求出特定形狀的粒子，如長橢球及扁橢球，其熱泳速度及在定溫下緩慢運動所受流體作用之拖曳力和扭力之漸進展開解。將本研究得到的橢球之漸進展開解與文獻中的解析解及數值解比對，可以說是十分吻合，即使是橢球偏離球形不小的情況也是如此。 第三章探討橢球粒子沿著對稱軸方向之熱泳及定溫下的緩慢移動。在橢球座標下，溫度通解為變數分離形式之無窮級數，而流場的通解則是半變數分離形式之無窮級數。吾人將溫度躍差、熱滑移、及速度滑移的邊界條件代入通解，以決定通解中前幾項之未知係數，對此有兩種求解方法，分別是邊界取點法以求得數值解，以及直接求解析解。最後計算橢球粒子之熱泳速度及定溫運動中受到流體施加的拖曳力，在各種流體、粒子、及表面特性的參數下都能得到很好的收斂值，即使是粒子的長寬比十分遠離1的情況亦然。另外，本研究得到的結果與文獻及第二章中所得到的值比對十分吻合。 第四章探討軸對稱粒子沿著對稱軸方向之熱泳運動。在此使用奇點分佈法求得能量及動量主導方程式之通解，對於長形的粒子，奇點分佈在粒子內部對稱軸上，而對於扁形的粒子，奇點則分佈於粒子內部垂直對稱軸之基本面上。吾人將溫度躍差、熱滑移、及速度滑移的邊界條件代入能量及動量主導方程式之通解，並使用邊界取點法，以推導通解之未知係數。最後求得球形及橢球形粒子之熱泳速度，其數值在一定的橢球長寬比內都收斂得很好，而且與文獻及前兩章中得到的數值能夠吻合。 第五章探討軸對稱且前後對稱粒子垂直於對稱軸方向之熱泳運動及定溫下之緩慢移動。在此使用與第四章中相同的奇點分佈及邊界取點法，求解流體之溫度及速度分佈，最後求得球形及橢球形粒子之熱泳速度及定溫下移動所受到的拖曳力之數值結果，其都能收斂到一定的程度，而且能與文獻中的相關解析解以及第二章得到的近似解相符。 本論文中的數值結果皆以正規化後的值來呈現，其中，正規化的熱泳速度是橢球熱泳速度除以相關球形粒子的熱泳速度，此球形粒子的半徑與橢球赤道面半徑相同，而正規化的拖曳力及扭力則是橢球粒子的拖曳力及扭力分別除以不考慮摩擦滑移之相關球形粒子的拖曳力及扭力，此球形粒子的半徑亦與橢球赤道面半徑相同。由研究結果可以發現，在大部分的情況下，正規化的熱泳速度是橢球長寬比的單調變化函數，然而有例外存在。對於長寬比固定的粒子，熱泳可動度大致上都不是溫度躍差係數、速度滑移係數、以及熱傳導係數比的單調函數。另一方面，正規化的拖曳力及扭力是否為橢球長寬比之單調函數必須由摩擦滑移係數的值決定。而對於長寬比固定的橢球粒子，其正規化的拖曳力及扭力隨著摩擦滑移程度的提升而單調遞減。

In this thesis, the steady thermophoresis and creeping motion of non-spherical particles in a gaseous medium are theoretically studied. Either the applied uniform temperature gradient driving the thermophoresis of the particle or the external force driving the creeping motion of the particle in the absence of the temperature gradient can be in an arbitrary direction. The Knudsen number is assumed to be small so that the fluid flow is described by a continuum model with a temperature jump, a thermal slip, and a frictional slip at the surface of the particle. In the limit of small Peclet and Reynolds numbers, the appropriate energy and momentum equations governing the systems are solved for some general cases of non-spherical particles, including the slightly deformed spheres, the axisymmetric particles, and the spheroidal particles. In Chapter 2, the thermophoresis, translation, and rotation of a slightly deformed aerosol sphere in an arbitrary direction are analyzed. The energy and momentum equations governing the system are solved asymptotically using a method of perturbed expansions. To the second order in the small parameter characterizing the deformation of the aerosol particle from the spherical shape, the thermal and hydrodynamic problems are formulated for the general case, and explicit expressions for the thermophoretic velocity of the particle and the drag and torque exerted on the particle by the fluid due to its isothermal creeping motion are obtained for the special cases of prolate and oblate spheroids. The agreement between our asymptotic results for a spheroid and the relevant analytical and numerical solutions in the literature is quite good, even if the particle deformation from the spherical shape is not very small. In Chapter 3, the thermophoresis and translation of a spheroidal particle along the axis of revolution of the particle are studied. The general solutions in prolate and oblate spheroidal coordinates can be expressed in infinite-series forms of separation of variables for the temperature distribution and of semi-separation of variables for the stream function. The jump/slip boundary conditions on the particle surface are applied to these general solutions to determine the unknown coefficients of the leading orders, which can be numerical results obtained from a boundary collocation method or explicit formulas derived analytically. Numerical results for the thermophoretic velocity of the particle and the drag force exerted on the particle translating in an isothermal fluid are obtained in a broad range of its aspect ratio with good convergence behavior for various cases. The agreement between our results and the available numerical results in the literature and those in the previous chapter is very good. In Chapter 4, the thermophoresis of an axisymmetric particle along its axis of revolution is analyzed. A method of distribution of a set of spherical singularities along the axis of revolution within a prolate particle or on the fundamental plane within an oblate particle is used to find the general solutions for the temperature distribution and fluid velocity field. The jump/slip conditions on the particle surface are satisfied by applying a boundary-collocation technique to these general solutions. Numerical results for the thermophoretic velocity are obtained with good convergence behavior for the spherical and spheroidal particles. The results agree quite well with the available solutions in the literature and in the previous chapters. In Chapter 5, the thermophoresis and translation of a particle of revolution with fore-and-aft symmetry perpendicular to the axis of revolution are explored using the same method of the distribution of spherical singularities combined with the boundary-collocation technique. The thermophoretic velocity of the particle and the drag force acting on the particle by the fluid are calculated with good convergence behavior for various cases, including the spherical and spheroidal particles. The results show excellent agreement with the relevant analytical solutions in the literature and in Chapter 2 for a spheroid whose shape deviates slightly from that of a sphere. It is found that the thermophoretic mobility of the spheroid normalized by the corresponding value for a sphere with equal equatorial radius in general is a monotonic function of the aspect ratio of the spheroid, but there are some exceptions. For most practical cases of a spheroid with a specified aspect ratio, the thermophoretic mobility of the particle is not a monotonic function of its relative jump/slip coefficients and thermal conductivity. Depending on the value of the slip parameter, the hydrodynamic drag force and torque acting on a moving spheroid normalized by the corresponding values for a no-slip sphere with equal equatorial radius are not necessarily monotonic functions of the aspect ratio of the spheroid. For a moving spheroid with a fixed aspect ratio, its normalized hydrodynamic drag force and torque decrease monotonically with an increase in the slip capability of the particle.