  It is vital important in material sciences and fluid mechanics to study the
field enhancements in the narrow region between two inclusions. Complex fluids
including particle suspensions usually result in complicated flow behavior. In
this paper we establish the pointwise upper bounds of the gradient and the
second-order partial derivatives for the Stokes flow when two rigid particles
are closely spaced suspending in an open bounded domain and away from the
boundary in dimension two. Moreover, the lower bounds of the gradient estimates
at the narrowest place of the neck region show the optimality of the blow-up
rate. These results are valid for inclusions with arbitrary shape.
