fluid dynamics - Stokes law in 2-dimensions

Stokes' law states that force on slow moving sphere (i.e. $Re\ll1$) in liquid is $$ F_d = 6 \pi \mu R V $$

In two dimensions we are in trouble (flow around disk in 2d or around cylinder in 3d), because there is no solution to the Stokes' problem (known as Stokes' paradox), but from dimensional analysis we can still conclude that

$$ F_d = C \mu V $$

I did some numerical tests of Navier-Stokes equations for small Reynolds numbers and found that $F_d$ really does not depend on $R$ and $C\approx 4\pi$.

I find it quite counter-intuitive that the force in 2D does not depend on the disk radius. Have I done something wrong? Or it really does not depend on radius of the disk?

Only thing which depends on the disk radius is the admissible range of input velocities. If you increase $R$ than you have to lower the max $V$ to ensure the condition $Re \ll 1$.