牛顿流体中中性浮性刚性粒子(内含物)的非胶态(可忽略布朗运动)浓悬浮的数学模型在流体力学、地质学、化学工程、复合材料的制造和生物学等领域有着广泛的应用. 本学位论文主要研究牛顿流体中的中性浮性刚性粒子的非胶态的浓悬浮液, 其数学模型由~Stokes~方程给出, 每个内含物的边界上都给定了刚性粒子的速度. 当两个相对运动的刚性粒子间距~$\varepsilon$~充分小时, 作用在粒子上的流体动力~$\mathbf{F}$~和流体力矩~$\mathbf{T}$~会非常大. 本文主要围绕任意~$m$-凸内含物的流体动力和力矩关于~$\varepsilon$~ 的渐近展式问题取得了一些进展.


一方面, 我们利用润滑理论证明得到流体动力和力矩关于~$m$~和~$\varepsilon$~的渐近展式. 具体地, 在二维空间中, ~$\mathbf{F}$~在两个坐标轴方向上的爆破速度分别为~$\frac{1}{\varepsilon^{1-\frac{1}{m}}}$~和~$\frac{1}{\varepsilon^{3-\frac{3}{m}}}$, 见定理~\ref{thm3}; 在三维中, ~$\mathbf{F}$~在三个坐标轴方向上的爆破速度分别为~$\frac{1}{\varepsilon^{1-\frac{2}{m}}}$,~$\frac{1}{\varepsilon^{1-\frac{2}{m}}}$~
和~$\frac{1}{\varepsilon^{3-\frac{4}{m}}}$, 见定理~\ref{thm1}. 从中可以发现流体动力~$\mathbf{F}$~的奇异性随着粒子表面相对凸性的减弱而增大. 另一方面, 我们也证明得到了带有部分“平坦”边界的粒子的渐近等式, 见定理~\ref{thm2}~和定理~\ref{thm4}, 发现此时二维和三维中流体动力~$\mathbf{F}$~在三个坐标轴方向中最大的爆破速率都是~$\varepsilon^{-3}$.

The mathematical model of non colloidal (negligible Brownian motion) concentrated suspension of neutral buoyant rigid particles (inclusions) in Newtonian fluid is widely used in the fields of hydrodynamics, geology, chemical engineering, manufacturing of composite materials and biology. In this dissertation, we mainly study the non colloidal concentrated suspension of the neutral floating rigid particles in Newtonian fluid. The mathematical model is given by Stokes equation, and the velocity of the rigid particles is given on the boundary of each inclusion. When the distance $\varepsilon$ between the two relatively moving rigid particles is sufficiently small, the hydrodynamic forces $\mathbf{F}$ and the hydrodynamic torque $\mathbf{T}$ acting on particles are very large. In this paper, some progress has been made on the asymptotic expansion of the hydrodynamic forces and the hydrodynamic torque for arbitrary $m$-convex inclusions.


On the one hand, we use lubrication theory to prove the asymptotic formula of the hydrodynamic force and the hydrodynamic torque about $m$ and $\varepsilon$. Specifically, in two-dimensional space, the blow-up rate of $\mathbf{F}$ in two coordinate axes is $\frac{1}{\varepsilon^{1-\frac{1}{m}}}$ and $\frac{1}{\varepsilon^{3-\frac{3}{m}}}$ respectively, see Theorem \ref{thm3}; in three-dimensional space, the blow-up rate of $\mathbf{F}$ in three coordinate axes is $\frac{1}{\varepsilon^{1-\frac{2}{m}}}$, $\frac{1}{\varepsilon^{1-\frac{2}{m}}}$ and $\frac{1}{\varepsilon^{3-\frac{4}{m}}}$ respectively, see Theorem \ref{thm1}. It can be found that the singularity of hydrodynamic force $\mathbf{F}$ increases with the decrease of the relative convexity of particle surface. On the other hand, we also prove the asymptotic equation of particles with partial "flat" boundary, see theorems \ref{thm2} and \ref{thm4}, and it is found that the largest blow-up rate of the hydrodynamic force $\mathbf{F}$ in three coordinate axes in two-dimensional and three-dimensional is both $\varepsilon^{-3}$.