  In a viscous incompressible fluid, the hydrodynamic forces acting on two
close-to-touch rigid particles in relative motion always become arbitrarily
large, as the interparticle distance parameter $\varepsilon$ goes to zero. In
this paper we obtain asymptotic formulas of the hydrodynamic forces and torque
in $2\mathrm{D}$ model and establish the optimal upper and lower bound
estimates in $3\mathrm{D}$, which sharply characterizes the singular behavior
of hydrodynamic forces. These results reveal the effect of the relative
convexity between particles, denoted by index $m$, on the blow-up rates of
hydrodynamic forces. Further, when $m$ degenerates to infinity, we consider the
particles with partially flat boundary and capture that the largest blow-up
rate of the hydrodynamic forces is $\varepsilon^{-3}$ both in 2D and 3D. We
also clarify the singularities arising from linear motion and rotational
motion, and find that the largest blow-up rate induced by rotation appears in
all directions of the forces.
