球形膠體粒子在同心球形孔隙中之磁流力運動

Abstract

本論文探討在擬穩態與低雷諾數的情況下，一球形膠體粒子在一充滿導電流體之球形孔隙中心受磁流力效應作用所進行的各種運動。勞倫茲力 (Lorentz force)為磁場與電流交互作用下所產生的作用力，將廣義互等定理 (generalized reciprocal theorem) 應用於含有勞倫茲力的Stokes方程式並考慮粒子表面上的馬克斯威爾應力 (Maxwell stress)，可求得粒子在任意粒子/球孔半徑比值之移動與轉動速度。一般來說，粒子受到磁流力作用進行運動之邊界效應，與粒子受到重力進行沉降之邊界效應大小相當，但也存在明顯差異。 在第二章中，吾人探討球形帶電膠體粒子於一任意無邊界電解質溶液中並施以任意流場與均勻磁場，因磁流力效應所產生之移動與轉動，帶電粒子周圍的電雙層厚度可為任意值。透過微擾法與廣義互等定理可處理含有靜電力與勞倫茲力效應的Stokes方程式，並利用求解線性Poisson-Boltzmann方程式所得之平衡時電雙層電位分佈，可求得球形粒子在一階微擾展開下因磁流力效應所產生的移動與轉動速度。此移動與轉動速率與無因次電動力半徑ka為一單調遞增函數之關係，其中ka為德拜 (Debye) 屏蔽參數，a為粒子半徑。在電雙層很厚的情況下，電解質在純轉動的Stokes流場下不具磁流力效應。此外，電解質溶液在純形變流動情況下的磁流力效應可使粒子轉動，但不會造成粒子移動。 在第三章中，吾人探討球形帶電膠體粒子於充滿任意電解質溶液之帶電球孔中心並施以均勻磁場下，因磁流力效應所產生之移動與轉動，帶電粒子周圍及球孔表面的電雙層厚度相較粒子與球孔半徑可為任意值。透過與第二章相同的分析方法，亦可求得球形粒子在任意粒子/球孔半徑比a/b及任意ka值下因磁流力效應所產生的移動與轉動速度，其中b為球孔半徑。一般來說，邊界接近時會使磁流效應之移動速率減慢，但會加強轉動速率。 第四章探討一球形膠體粒子在充滿導電流體球孔中心之電磁泳運動。在施加均勻電場與磁場的情況下，可求得電流密度與磁通密度在任意電導度與磁透度下之解析解。將廣義互等定理應用於含有勞倫茲力的Stokes方程式並計算相對重要的馬克斯威爾應力在粒子上之作用，可求得粒子在任意a/b值下的移動速度。除了粒子相對於流體具有高導電性與低磁透性的情況下，粒子速率一般隨a/b值遞增而遞減。本章亦分別探討邊界效應在低與高a/b值的情況下，電磁作用力與粒子可動度的漸近現象。 在第五章中，吾人探討具表面自發電化學反應之球形膠體粒子，在充滿電解質溶液之球孔中心因磁場誘導所產生之運動。粒子表面之電位可為任意分佈，但電雙層厚度遠小於粒子半徑或粒子與球孔間距。求得電流與磁通密度分佈在任意電導度與磁透度下之解析解後，將廣義互等定理應用於含有勞倫茲力的Stokes方程式，可求得粒子在任意a/b值下的移動與轉動速度。粒子的移動與轉動分別與表面電位分佈的偶極 (dipole) 與四極 (quadrupole) 有關。此磁場誘發之運動相當重要，且在物理上之意義不同於現有之電磁泳動及其他泳動效應。粒子運動速率亦隨a/b值增加而遞減。具表面自發電化學反應之粒子，因磁流力誘導所產生運動之邊界效應較其他泳動現象為強。

The quasi-steady motions of a spherical colloidal particle inside a concentric spherical cavity filled with a conducting fluid induced by the magnetohydrodynamic (MHD) effect are analyzed at low Reynolds number. Through the use of a generalized reciprocal theorem to the Stokes equations modified with the Lorentz force density resulting from the interaction of an applied magnetic field with the existing electric current and the consideration of the Maxwell stress to the force exerted on the particle, the translational and angular velocities of the particle under various conditions are obtained in closed forms valid for an arbitrary value of the particle-to-cavity radius ratio. The boundary effects on the motions of the particle caused by the MHD force are generally equivalent to (yet different from) that in sedimentation. In Chapter 2, an analytical study is presented for the MHD effects on a translating and rotating charged sphere in an arbitrary unbounded electrolyte solution prescribed with a general flow field and a uniform magnetic field. The electric double layer surrounding the charged particle may have an arbitrary thickness relative to the particle radius. Through the use of a simple perturbation method, the Stokes equations modified with an electric force term, including the Lorentz force contribution, are dealt with using a generalized reciprocal theorem. Using the equilibrium double-layer potential distribution from solving the linearized Poisson-Boltzmann equation, we obtain closed-form formulas for the translational and angular velocities of the spherical particle induced by the MHD effects to the leading order. It is found that the MHD effects on the particle movement associated with the translation and rotation of the particle and the ambient fluid are monotonically increasing functions of ka , where k is the Debye screening parameter and a is the particle radius. Any pure rotational Stokes flow of the electrolyte solution in the presence of the magnetic field exerts no MHD effect on the particle directly in the case of a very thick double layer. The MHD effect caused by the pure straining flow of the electrolyte solution can drive the particle to rotate, but it makes no contribution to the translation of the particle. In Chapter 3, the MHD effects on the translation and rotation of a charged sphere situated at the center of a charged spherical cavity filled with an arbitrary electrolyte solution when a constant magnetic field is imposed are analyzed. The electric double layers adjacent to the solid surfaces may have an arbitrary thickness relative to the particle and cavity radii. Through the same method of analysis in Chapter 2, we obtain explicit formulas for the translational and angular velocities of the colloidal sphere produced by the MHD effects valid for all values of the particle-to-cavity size ratio. The boundary effect on the MHD motion of the spherical particle is a qualitatively and quantitatively sensible function of the parameters a/b and ka , where b is the radius of the cavity. In general, the proximity of the cavity wall reduces the MHD migration but intensifies the MHD rotation of the particle. In Chapter 4, the electromagnetophoretic (EMP) motion of a spherical colloidal particle positioned at the center of a spherical cavity filled with a conducting fluid is analyzed. Under uniformly applied electric and magnetic fields, the electric current and magnetic flux density distributions are solved for the particle and fluid phases of arbitrary electric conductivities and magnetic permeabilities. Applying a generalized reciprocal theorem to the Stokes equations modified with the resulted Lorentz force density, we obtain a closed-form formula for the migration velocity of the particle valid for an arbitrary value of the particle-to-cavity radius ratio. The particle velocity in general decreases monotonically with an increase in this radius ratio, with an exception for the case of a particle with high electric conductivity and low magnetic permeability relative to the suspending fluid. The asymptotic behaviors of the boundary effect on the EMP force and mobility of the confined particle at small and large radius ratios are discussed. In Chapter 5, an analytical study is presented for the magnetic-field-induced motion of a colloidal sphere with spontaneous electrochemical reactions on its surface situated at the center of a spherical cavity filled with an electrolyte solution. The zeta potential associated with the particle surface may have an arbitrary distribution, whereas the electric double layers adjoining the particle and cavity surfaces are taken to be thin relative to the particle size and the spacing between the solid surfaces. The electric current and magnetic flux density distributions are solved for the particle and fluid phases of arbitrary electric conductivities and magnetic permeabilities. Applying a generalized reciprocal theorem to the Stokes equations with the resulted Lorentz force term, we obtain explicit formulas for the translational and angular velocities of the colloidal sphere valid for all values of the particle-to-cavity size ratio. The dipole and quadrupole moments of the zeta potential distribution over the particle surface cause the particle translation and rotation, respectively. The induced velocities of the particle are unexpectedly significant, and their dependence on the characteristics of the particle-fluid system is physically different from that for EMP particles or phoretic swimmers. The particle velocities decrease monotonically with an increase in the particle-to-cavity size ratio. The boundary effect on the movement of the particle with interfacial self-electrochemical reactions induced by the MHD force is much stronger than that in phoretic swimming.