  Inspired by some experimental (numerical) works on fractional diffusion PDEs,
we develop a rigorous framework to prove that solutions to the fractional
Navier-Stokes equations, which involve the fractional Laplacian operator
$(-\Delta)^{\frac{\alpha}{2}}$ with $\alpha<2$, converge to a solution of the
classical case, with $-\Delta$, when $\alpha$ goes to $2$. Precisely, in the
setting of mild solutions, we prove uniform convergence in both the time and
spatial variables and derive a precise convergence rate, revealing some
phenomenological effects. Finally, our results are also generalized to the
coupled setting of the Magnetic-hydrodynamic (MHD) system.
