Vortices and Backflow in Hydrodynamic Heat Transport

Recent experiments have provided compelling and renewed interest in phonon hydrodynamics. At variance with ordinary diffusive heat transport, this regime is primarily governed by momentum-conserving phonon collisions. At the mesoscopic scale it can be described by the viscous heat equations (VHEs), that resemble the Navier-Stokes equations (NSEs) in the laminar regime. Here, we show that the VHEs can be separated and recast as modified biharmonic equations in the velocity potential and stream function-solvable analytically. These two can be merged into a complex potential defining the flow streamlines, and give rise to two distinct temperature contributions, ultimately related to thermal compressibility and vorticity. The irrotational and incompressible limits of the phonon VHEs are analyzed, showing how the latter mirror the NSEs for the electron fluid. This Letter also extends to the electron compressible regime that arises when drift velocities can be higher than plasmonic velocities. Finally, by examining thermal flow within a 2D graphite strip device, we explore the boundary conditions and transport coefficients needed to observe thermal vortices and negative thermal resistance, or heat backflow from cooler to warmer regions. This Letter provides analytical tools to design hydrodynamic phonon and electron flows, offering a foundational framework to understand and guide experiments on nonstandard transport, plasmonics, and collective excitations in microscale systems.