一表面滑移球體垂直於兩表面滑移平板之緩慢轉動

Abstract

本論文探討一個具有滑移表面的球形粒子於不可壓縮之牛頓流體中，以其垂直於具有滑移表面的單一平板或兩個平行平板之直徑為轉軸，所進行的穩定緩慢轉動。吾人以Stokes方程式在球坐標系統與圓柱坐標系統的個別流速通解組合成整體流速通解，先代入平板之邊界條件，使用Hankel轉換法解析計算，再代入粒子表面之邊界條件，使用邊界取點法數值計算，獲得流體施加於粒子之力矩。 流體施加於粒子之正規化力矩解在廣泛不同的幾何及滑移參數下皆可獲得甚佳的收斂值，且具有滑移表面的平板對轉動之滑移粒子所造成的影響很有趣。正規化力矩會隨著平板表面之黏滯程度的增加，從完全滑移至非滑移表面而單調遞增。當平板之黏滯參數大於一特定值時，粒子所受力矩會比在無邊界存在情況下轉動之粒子所受到之力矩大，且會隨粒子表面之黏滯參數及球半徑與平板間距之比值的增加，呈現單調遞增關係。相對地，當平板之黏滯參數小於此特定值時，粒子所受力矩會比在無邊界存在情況下轉動之粒子所受到之力矩小，且會隨粒子表面之黏滯參數及球半徑與平板間距之比值的增加，呈現單調遞減關係。

The steady creeping flow of an incompressible Newtonian fluid around a slip spherical particle rotating about its diameter perpendicular to one or two slip plane walls is analyzed in this thesis. To satisfy the Stokes equation for fluid velocity, the general solution consists of the sum of the essential solutions in spherical and cylindrical coordinates. Boundary conditions are implemented first on the plane walls by means of the Hankel transforms and then on the particle surface through a collocation method. The hydrodynamic torque exerted on the particle is obtained with excellent convergence for various values of the pertinent geometrical and stick-slip parameters, and the effect of the slip planes on the rotational motion of the slip particle is interesting. The torque increases with an increase in the stickiness of the walls from the limit of full slip to the limit of no slip. When the stick parameters of the plane walls are larger than some critical values, the hydrodynamic torque is more than that on an identical particle in the unbounded fluid and an increasing function of the stickiness of the particle surface and ratio of the particle radius to distance from the walls. When the stick parameters of the plane walls are smaller than the critical values, on the contrary, the torque is less than that on the particle in the unbounded fluid and a decreasing function of the surface stickiness and relative radius of the particle.