球形複合粒子平行單一或兩平板之緩慢移動與轉動

Abstract

本論文以半解析的方式探討一個球形複合粒子(構造為一個流體不可滲透的硬質核心固體，外圍包覆著一層流體可滲透的多孔物質)在黏性流體中，平行於一個或兩個平板邊界，所進行的穩態低雷諾數耦合移動與轉動。多孔層內外之流體速度分別由Brinkman方程式及Stokes方程式主導。研究中，分別利用球坐標與直角坐標下的基本解建立通解。首先代入平板的邊界條件並透過傅立葉轉換法解析求解，接著代入多孔層內外球面的邊界條件並使用邊界取點法數值計算，獲得流體施加於粒子之拖曳力與力矩。數值計算顯示，在廣泛的物理參數範圍內，所得的阻力與力矩均具有良好的收斂性。為驗證數據合理性，針對不可滲透硬質固體球平行於單一或兩平板的結果，與既有文獻中的對應結果進行比對，吻合良好。流體施加於複合粒子的拖曳力與力矩隨著粒子半徑與流體於多孔層穿透長度之比值、固體核半徑與粒子半徑之比值、粒子半徑與較近平板間距之比值增加而單調遞增。結果亦發現，平板對粒子移動的影響明顯大於對其轉動的影響。比較平行於平板與垂直於平板的運動時，平面邊界在粒子平行運動時對粒子產生的流體拖曳力較小，但所施加的力矩則較大。此外，複合球在平行於平板運動時，移動與轉動的耦合作用表現出複雜的行為，其變化並不隨系統參數的改變而單調變化。

A semi-analytical investigation is conducted to examine the coupled translational and rotational motions of a composite spherical particle (consisting of an impermeable hard core surrounded by a permeable porous shell) immersed in a viscous fluid parallel to one or two planar boundaries under the steady condition of a low Reynolds number. The fluid flow is described using the Stokes equations outside the porous shell and the Brinkman equation within it. A general solution is formulated by employing fundamental solutions in both spherical and Cartesian coordinate systems. The boundary conditions on the planar walls are implemented using the Fourier transform method, while those on the inner and outer boundaries of the porous shell are applied via a collocation technique. Numerical calculations yield hydrodynamic force and torque results with good convergence across a broad range of physical parameters. For validation, the results corresponding to an impermeable hard sphere parallel to one or two planar walls are shown to be in close agreement with established solutions from the literature. The hydrodynamic drag force and torque experienced by the composite particle increase steadily with larger values of the ratio of the particle radius to the porous shell’s permeation length, the ratio of the core radius to the total particle radius, and the separations between the particle and the walls. It has been observed that the influence of the walls on translational motion is significantly stronger than that on rotational motion. When comparing motions parallel versus normal to the walls, the planar boundaries impose weaker hydrodynamic forces but stronger torques during parallel motions. The coupling between the translation and rotation of the composite sphere parallel to the walls exhibits complex behavior that does not vary monotonically with changes in system parameters.