Vorticity and compressibility hydrodynamics in electron and phonon fluids

Recent experiments have provided compelling evidence for the theoretical predictions of the emergence of phonon hydrodynamics in graphene and in other fast thermal conductors. At variance with diffusive heat transport, this regime, similar to that of viscous fluids, is dominated by normal phonon-phonons scattering. Here, we use the viscous heat equations (VHE) - that are a thermal analogue of the Navier-Stokes equations (NSE) in the laminar regime - to study a 2D device, where we can decouple the steady-state VHE in Fourier space and determine the interplay between thermal vorticity and compressibility as a distinctive hallmark of viscous transport. We also establish a clear link between thermal viscosity and a measurable negative nonlocal macroscopic thermal resistance, or heat backflow from cooler to warmer regions, proposing an experimental apparatus for its detection. Last, we identify the NSE for the electron fluid as the incompressible limit of the compressible phonon VHE.