球形軟質粒子垂直兩平板之緩慢轉動

Abstract

本文探討一個軟質球形粒子於充滿不可壓縮的牛頓流體之兩平行平板間任意位置，以其垂直於平板之直徑為轉軸，所進行的穩態低雷諾數轉動。粒子外部的流體速度透過Stokes方程式在圓柱坐標系統和球坐標系統的個別通解相加獲得流速通解，而粒子多孔表面層內的流體速度則透過 Brinkman 方程式在球坐標系統下獲得流速通解。首先帶入平板的邊界條件並透過 Hankel 轉換法解析計算，接著帶入粒子表面的邊界條件，最後使用邊界取點法數值計算，獲得流體施加於粒子之力矩。吾人探討正規化力矩與各相關無因次參數（硬質核心半徑與粒子半徑之比值、粒子半徑與單板間距之比值、粒子在平板間的相對位置、粒子半徑與流體於多孔層穿透長度之比值）之關係。平板對粒子轉動之邊界效應影響相當明顯，當固定粒子直徑與平板間距之比值時，粒子位於兩平板中間的正規化力矩最小，正規化力矩會隨粒子與任一平板的相對間距減小而增加（當粒子靠近任一平板時正規化力矩上升），即使粒子碰觸平板時正規化力矩仍然是有限的。在保持其他參數不變的情況下，軟質粒子的正規化力矩小於實心粒子（或具有較小厚度或穿透長度的多孔層之軟質粒子）。

The creeping flow of a viscous fluid around a soft colloidal sphere rotating about a diameter normal to two planar walls at an arbitrary position between them is theoretically investigated in the steady limit of small Reynolds numbers. The fluid velocity outside the particle consists of the general solutions of the Stokes equation in circular cylindrical and spherical coordinates, while the fluid velocity inside the porous surface layer of the particle is expressed by the general solution of the Brinkman equation in spherical coordinates. The boundary conditions are implemented first on the planar walls by means of the Hankel transforms and then at the particle and hard-core surfaces by a collocation technique. The torque exerted on the particle by the fluid is calculated as a function of the ratio of the core-to-particle radii, ratio of the particle radius to the flow penetration length of the porous layer, and relative particle-to-wall spacings over the entire range. The wall effect on the rotating soft particle can be significant. The hydrodynamic torque exerted on the confined soft sphere increases as the relative particle-to-wall spacings decrease and stays finite even when the soft sphere contacts the plane walls. It is smaller than the torque on a hard sphere (or soft one with a reduced thickness or penetration length of the porous layer), holding the other parameters constant. For a given relative wall-to-wall spacing, this torque is minimal when the particle is situated midway between the walls and rises as it locates closer to either wall.