Preturbulence in momentum-relaxing Navier-Stokes hydrodynamics

In a hydrodynamic regime of electronic fluids, the electron scattering on impurities and phonons inhibits the development of oscillating preturbulent phenomena (such as the K\'arm\'an vortex shedding) and enforces a laminar flow. Working in a local generalization of the Navier-Stokes hydrodynamics of incompressible two-dimensional fluids, we show that the critical relaxation time separating these two regimes is described by a surprisingly simple fractional power function of the Reynolds number. The critical exponent, $\alpha \simeq 4/3$, sets particular experimental conditions for the observation of preturbulent effects in electronic fluids.