表面滑移球形粒子在圓柱形細孔中之緩慢運動

Abstract

本文以半解析半數值的方法，研究一剛性球形粒子於黏性流體中，沿著一無限長圓柱形細孔之中心軸移動所造成的緩流運動。所考慮之流體可為稍微稀薄之氣體，並可以在粒子的表面產生滑移現象。為求解主導流場之Stokes方程式，需要建立其在圓柱座標及球座標系統之通解。首先以Fourier轉換，使此通解滿足孔壁上的邊界條件，再以邊界取點法使通解滿足球形粒子表面上的邊界條件。在各種不同的粒子半徑與圓孔半徑之比值，與不同的粒子表面滑移係數的情形下，分別計算流體施加於粒子之阻力，可得到良好之收斂數值結果。對於粒子沿著圓柱形細孔中心軸運動，在粒子表面為完全不滑移與完全滑移的情況下，所計算出的粒子所受阻力值，皆與現有文獻中之數值相吻合。而在任意固定的粒子半徑與圓孔半徑之比值下，粒子所受之阻力值，大致隨著粒子表面滑移係數之增加而有逐漸遞減的情形。我們比較表面滑移球形粒子在不同形狀的孔洞內做軸向運動可以發現，當粒子在圓柱形細孔中運動時所受到的阻力值，小於其在球形孔洞內運動之值而大於其在狹縫形孔洞內運動之阻力值。

A combined analytical-numerical study for the creeping flow caused by a rigid spherical particle translating in a viscous fluid along the centerline of a circular cylindrical pore is presented. The fluid, which may be a slightly rarefied gas, is allowed to slip at the surface of the particle. To solve the axisymmetric Stokes equation for the fluid velocity field, a general solution is constructed from the superposition of the fundamental solutions in both circular cylindrical and spherical coordinate systems. Boundary conditions are enforced first at the pore wall by the Fourier transforms and then on the particle surface by a collocation technique. Numerical results for the hydrodynamic drag force acting on the particle are obtained with good convergence for various values of the slip coefficient of the particle and of the relative separation distance between the particle and the pore wall. For the motions of a no-slip sphere and a perfectly-slip sphere along the centerline of a cylindrical pore, our drag results are in good agreement with the available solutions in the literature. The boundary-corrected drag force exerted on the particle in general decreases with an increase in the slip coefficient for a given ratio of particle-to-pore radii, but there are exceptions. In a comparison for the pore shape effect on the axial translation of a slip sphere, it is found that the particle in a circular cylindrical pore acquires a lower hydrodynamic drag than in a spherical cavity but a higher drag force than in a slit pore.