球形複合粒子在非同心球形孔洞中之緩慢移動及轉動

Abstract

本研究以半解析方式探討一個複合球形粒子(由一實心硬核外部包覆可穿透之多孔層所組成)在一個充滿黏性流體之球形孔洞中，於非同心位置沿其垂直連心方向，所進行的擬穩態低雷諾數移動及轉動。多孔層內外之流體速度分布分別由Brinkman方程式及Stokes方程式主導，其中粒子外部流體的速度表示式為以粒子中心及孔洞中心為原點的球座標系統之通解組合而成。對於滿足孔洞表面及多孔層內外表面的邊界條件所得之方程組，本研究將透過邊界取點法數值求解，計算出流體施加於粒子之阻力及力矩，且數值解在不同參數組合下均呈現良好收斂性。從計算結果可得複合粒子移動、轉動時所受拖曳力及力矩與粒子結構(如多孔層的厚度及流體穿透度)、粒子在孔洞中的相對位置及大小之關係。流體施加於複合粒子之拖曳力和力矩會隨多孔層穿透度下降、實心硬核對粒子半徑比值增加以及粒子對孔洞半徑比值增加而呈現單調遞增。粒子所受拖曳力和力矩大致上亦會隨其偏心程度增加而遞增。此外，孔洞對於複合粒子移動時的阻礙影響會相較於相同粒子轉動時所受影響更為顯著。複合粒子在孔洞中移動伴隨轉動之耦合效應較為複雜，且並非為粒子對孔洞半徑比值之單調函數。

A semi-analytical study of coupled translation and rotation of a composite spherical particle (a hard sphere core coated with a permeable porous layer) in a viscous fluid inside an eccentric spherical cavity normal to their common diameter is presented in the quasi-steady limit of low Reynolds number. To solve the Stokes and Brinkman equations for the flow fields outside and inside the porous layer, respectively, a general solution is constructed from the fundamental solutions in the two spherical coordinate systems based on both the composite particle and the cavity. The boundary conditions at the cavity wall and inner and outer surfaces of the porous layer are satisfied by a collocation method. Numerical results for the force and torque exerted on the particle by the fluid are obtained with good convergence for various values of the relevant parameters in practical applications. For the translation and rotation of a composite sphere inside a concentric cavity, our force and torque results agree well with the available solutions in the literature. The force and torque on a translating and rotating particle increase monotonically with increases in the ratios of particle radius to porous layer permeation length, core-to-particle radii, and particle-to-cavity radii. In general, they also increase with an increase in the relative distance between the particle and cavity centers. The boundary effect of the cavity on the translation of the particle is much more pronounced than that on the rotation. The coupling effect in the simultaneous translation and rotation inside an eccentric spherical cavity is complicated and not a monotonic function of the particle-to-cavity radius ratio.