Stress blow-up analysis when a suspending rigid particle approaches the boundary in Stokes flow: 2D case

It is an interesting and important topic to study the motion of small particles in a viscous liquid in current applied research. In this paper we assume the particles are convex with arbitrary shapes and mainly investigate the interaction between the rigid particles and the domain boundary when the distance tends to zero. In fact, even though the domain and the prescribed boundary data are both smooth, it is possible to cause a definite increase of the blow-up rate of the stress. This problem has the free boundary value feature due to the rigidity assumption on the particle. We find that the prescribed local boundary data directly affects on the free boundary value on the particle. Two kinds of boundary data are considered: locally constant boundary data and locally polynomial boundary data. For the former we prove the free boundary value is close to the prescribed constant, while for the latter we show the influence on the blow-up rate from the order of growth of the prescribed polynomial. Based on pointwise upper bounds in the neck region and lower bounds at the midpoint of the shortest line between the particle and the domain boundary, we show that these blow-up rates obtained in this paper are optimal. These precise estimates will help us understand the underlying mechanism of the hydrodynamic interactions in fluid particle model.