For two nearly touching rigid particles in a steady Navier-Stokes fluid, the velocity gradient blows up at rate 1/(epsilon log(1/epsilon)) in three dimensions and 1/sqrt(epsilon) in two dimensions, and these rates are optimal.
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Stress concentration between two adjacent rigid particles in Navier-Stokes flow
For two nearly touching rigid particles in a steady Navier-Stokes fluid, the velocity gradient blows up at rate 1/(epsilon log(1/epsilon)) in three dimensions and 1/sqrt(epsilon) in two dimensions, and these rates are optimal.