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

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