Bulk and shear viscosities of the two-dimensional electron liquid in a doped graphene sheet

Hydrodynamic flow occurs in an electron liquid when the mean free path for electron-electron collisions is the shortest length scale in the problem. In this regime, transport is described by the Navier-Stokes equation, which contains two fundamental parameters, the bulk and shear viscosities. In this paper, we present extensive results for these transport coefficients in the case of the two-dimensional massless Dirac fermion liquid in a doped graphene sheet. Our approach relies on microscopic calculations of the viscosities up to second order in the strength of electron-electron interactions and in the high-frequency limit, where perturbation theory is applicable. We then use simple interpolation formulas that allow to reach the low-frequency hydrodynamic regime where perturbation theory is no longer directly applicable. The key ingredient for the interpolation formulas is the "viscosity transport time" ?v, which we calculate in this paper. The transverse nature of the excitations contributing to ?v leads to the suppression of scattering events with small momentum transfer, which are inherently longitudinal. Therefore, contrary to the quasiparticle lifetime, which goes as -1/[T2ln(T/TF)], in the low-temperature limit we find ?v~1/T2.