  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.
