表面滑移長形粒子在圓柱形孔隙中之軸對稱緩流運動

Abstract

本論文探討在黏性流體中一軸對稱長形粒子於一圓柱形孔隙中之軸對稱緩流運動，所考慮之系統為穩定狀態及雷諾數極小，且流體在固體表面上可以具有滑移現象。吾人利用半解析半數值方法來求解系統中流場之主導方程式，先以球形奇點分佈法沿粒子內對稱軸上分佈一組球形奇點以求得流速場通解，再利用邊界取點法滿足在粒子表面上之邊界條件以解出通解中之未知係數。分別以長形橢球與卡西尼卵形體為例，在不同的形狀參數、粒子於孔隙中相對位置、粒子滑移係數，以及孔壁滑移係數之情形下，所計算出之流體施加於粒子的阻力，皆可獲得良好之收斂性。 當一長形橢球粒子在圓柱形孔隙中之軸對稱運動於表面不滑移時，以及當一球形粒子在圓柱形孔隙中之軸對稱運動於表面滑移時，我們所得到的流體施加於粒子的阻力結果與文獻中存在的相關解析及數值解一致。在固定粒子於孔隙中相對位置情況下，粒子所受到的正規化流體阻力一般隨著粒子長短軸比增加而遞增，然而在粒子表面較為滑移的情形下，會有例外的情形發生。當粒子接近孔壁時，粒子運動所受到之孔壁邊界效應影響相當顯著。

A theoretical study of the fluid flow caused by an axially symmetric prolate particle translating along its axis of revolution in a coaxial circular cylindrical pore at low Reynolds numbers is presented, where the viscous fluid may slip at the solid surfaces. A method of distribution of a set of spherical singularities along the axis of revolution within the particle is used to find the general solution of the fluid velocity field that satisfies the boundary conditions at the pore wall and at infinity. The slip condition on the particle surface is then satisfied by applying a boundary collocation technique to this general solution to determine the unknown constants. The drag force exerted on the particle by the ambient fluid is calculated with good convergence behavior for the cases of a prolate spheroid and a prolate Cassini oval (whose surface can be partly concave) with broad ranges of their aspect ratio (shape parameter) and distance from the pore wall. For the axially symmetric translations of a spheroid in a cylindrical pore with nonslip surfaces and of a sphere in a cylindrical pore with slip surfaces, our drag results agree with the relevant solutions available in the literature. For a fixed particle-wall separation parameter, the normalized drag force in general increases with an increase in the axial-to-radial aspect ratio of the particle, but there are exceptions when the particle is highly slippery. The boundary effect on the motion of the particle can be very significant when it gets close to the pore wall.