表面滑移球形粒子在非同心球形孔洞中之緩流運動

Abstract

本文以半解析的方法，研究一球形液滴或一表面滑移之剛性球形粒子於一球形孔洞所包覆的黏性流體中，沿著連接它們中心的軸在擬穩態和Reynolds數很小情況下之緩流運動。為求解主導流場之Stokes方程式，需要建立一個包括粒子和孔洞相關兩個球座標系統之通解。以邊界取點法使通解滿足粒子表面和孔壁之邊界條件，在各種不同的粒子與孔洞半徑比值、粒子的相對位置、粒子的相對黏度或滑移係數，以及不同的孔壁滑移係數情形下，計算流體施加於粒子的阻力。當粒子與孔洞同心時，或粒子接近一曲度很小之孔壁時，我們得到的阻力結果分別與文獻中一球形粒子在同心的孔洞以及球形粒子靠近一平面的結果近乎相同。 如預期的，粒子所受到的阻力在所有情況下都是隨著粒子與孔洞半徑比值呈單調遞增的情況，而在粒子接觸孔壁時阻力會趨近於無窮大。一般而言，粒子所受之阻力會隨著其相對黏度的增加而增加或隨著表面滑移係數增加而遞減，不過在粒子與孔壁半徑比較大時有例外的情況發生。

A semianalytical study for the creeping flow caused by a spherical fluid or solid particle with a slip surface translating in a viscous fluid within a spherical cavity along the line connecting their centers is presented in the quasisteady limit of small Reynolds number. To solve the Stokes equations for the flow field, a general solution is constructed from the superposition of the fundamental solutions in the two spherical coordinate systems based on the particle and cavity respectively. The boundary conditions on the particle surface and cavity wall are satisfied by a collocation technique. Numerical results for the hydrodynamic drag force exerted on the particle are obtained with good convergence for various values of the ratio of particle-to-cavity radii, the relative location of the particle, the relative viscosity or slip coefficient of the particle, and the slip coefficient of the cavity wall. In the limits of the motions of a spherical particle in a concentric cavity and near a cavity wall with a small curvature, our drag results are in good agreement with the available solutions in the literature. As expected, the boundary-corrected drag force exerted on the particle for all cases is a monotonic increasing function of the ratio of particle-to-cavity radii, and becomes infinite in the touching limit. For a specified ratio of particle-to-cavity radii, the drag force is minimal when the particle is situated at the cavity center and increases monotonically to infinity in the limit when it is located extremely away from the cavity center. The drag force acting on the particle in general increases with an increase in its relative viscosity or with a decrease in its slip coefficient for a given configuration, but there are exceptions when the ratio of particle-to-cavity radii is large.