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 (VHE), that resemble the Navier-Stokes equations (NSE) in the laminar regime. Here, we show that the VHE 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 VHE are analyzed, showing how the latter mirrors the NSE for the electron fluid. This work 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 work provides novel analytical tools to design hydrodynamic phonon flow, highlights its generalization for electron hydrodynamics, and promotes additional avenues to explore experimentally such fascinating phenomena.

💡 Research Summary

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The paper addresses the burgeoning interest in phonon hydrodynamics by providing a rigorous analytical framework for the viscous heat equations (VHE), which govern heat transport when momentum‑conserving phonon‑phonon collisions dominate. Starting from the steady‑state, isotropic VHE that couple temperature T and phonon drift velocity u, the authors introduce two scalar fields: the thermal compressibility Φ = ∇·u and the thermal vorticity W = ∇×u. By eliminating the vector field u, they decouple the VHE into two modified Helmholtz equations (Eq. 3). The compressibility equation contains a term proportional to the volume viscosity μ = ζ + η/3, while the vorticity equation involves only the shear viscosity η and the momentum‑relaxation rate γ (Umklapp and boundary scattering).

To solve these equations analytically, the authors employ a Helmholtz decomposition of the velocity field, u = −∇ϕ + ∇×ψ, where ϕ is a scalar potential (associated with compressibility) and ψ is the out‑of‑plane component of the stream‑function vector (associated with vorticity). Substituting this decomposition yields two fourth‑order “modified biharmonic” equations for ϕ and ψ (Eq. 5). In the limit γ → 0 the vorticity equation reduces to the classic biharmonic equation of incompressible Stokes flow, confirming consistency with known fluid‑mechanics results.

The authors then apply this formalism to a concrete geometry: an infinitely long two‑dimensional graphite strip of width h, with point‑like contacts that inject both a temperature gradient ΔT and a phonon drift velocity U at the lower edge and extract them at the upper edge. Boundary conditions enforce no‑slip for the tangential velocity, a prescribed normal velocity u_y = U δ(x), and temperature jumps T = \bar T ± ΔT δ(x) at the contacts. By Fourier transforming in the longitudinal direction, they obtain analytic expressions for ψ(k,y) and ϕ(k,y) (Eq. 9), where the decay constants q_ψ and q_ϕ contain the viscous and compressibility parameters.

A key result is the decomposition of the temperature field into two contributions, T(x,y) = T_ϕ(x,y) + T_ψ(x,y), where T_ϕ stems from compressibility (∇·u) and T_ψ from vorticity (∇×u). This decomposition is universal and not limited to the strip geometry; it provides a direct way to experimentally separate the two mechanisms by measuring spatial temperature profiles.

To quantify deviations from Fourier’s law, the authors introduce the Fourier Deviation Number (FDN) = 1/(ε + ξ), where ε = γh²/η measures the strength of momentum‑relaxing scattering and ξ = καΔT h/U measures the relative importance of the injected drift velocity. Small ε and ξ (i.e., weak Umklapp scattering and strong drift) give large FDN, indicating a strong hydrodynamic regime.

In the diffusive limit (large ε, ξ) the temperature reduces to the standard Fourier solution (Eq. 13). In the opposite, ideal hydrodynamic limit (ε → 0, ξ → 0) the temperature exhibits an inverse‑square dependence on the distance from the contacts and develops nodal lines along y = ±x, reflecting the presence of vortices. The analytical expression (Eq. 14‑16) predicts a negative temperature difference across the strip near the contacts, i.e., a negative thermal resistance or heat backflow from the colder to the hotter region. This backflow originates from the dominance of the viscous term (∝ ∇²u) at short distances, which overwhelms the diffusive contribution.

Numerical evaluation (Fig. 2) shows that as ξ is reduced, the temperature difference T(x,0) − T(x,h) changes sign: positive in the far field (diffusive regime) and negative near the contacts (viscous regime). The effect of varying ε (Umklapp scattering) is comparatively modest, confirming that a sufficiently low ξ—i.e., a large drift velocity U—is essential for observing thermal backflow. For a realistic ΔT = 1 K, the authors estimate that U ≈ 2 × 10⁴ m s⁻¹ is required to produce a measurable backflow of ~0.1 K; isotopically purified graphite (¹²C 99.95 %) can raise this to ~0.4 K.

Finally, the paper discusses the streamlines obtained from the complex potential χ = iψ − ϕ. Because both ϕ and ψ contribute, the flow is neither purely incompressible nor irrotational; compressibility and vorticity jointly shape the streamlines, leading to a central channel flanked by counter‑rotating vortices. This contrasts with electron fluids, which are often approximated as incompressible, and highlights a fundamental difference between phonon and electron hydrodynamics.

In summary, the work provides (i) a complete analytical decoupling of the VHE into compressibility and vorticity sectors, (ii) explicit solutions for a prototypical device geometry, (iii) a clear physical picture of how thermal vortices generate negative thermal resistance, and (iv) quantitative guidelines (material parameters, drift velocities, isotopic purity) for experimental observation of phonon‑hydrodynamic phenomena. The methodology opens the door to designing devices that exploit phonon vortices and backflow for novel thermal management applications.