Hydrodynamic flow in the 3D electron Fermi liquid: Non-Newtonian viscosity and arguments for a general critical Reynolds number

Abstract

Macroscopic hydrodynamic flow in a 3D Fermi liquid within high-purity bulk metals manifests as stationary current jets and vortices—extending up to 10 cm—featuring backflow against the applied electric field. Experiments were conducted on high-purity and tungsten-doped molybdenum single crystals (residual resistivity ratio = 1700– 240 000, diameters up to 6 mm, and lengths up to 20 cm) at low temperatures (T = 1.9, 4.2, and 14 K), focusing on current flow from point-like injection and the influence of magnetic fields up to B = 1 T, and partially up to B = 16 T. Contrary to prevailing assumptions, momentum-conserving normal electron–electron scattering is not responsible for the observed hydrodynamic behavior. Instead, the dominant contribution stems from momentum-relaxing scattering mechanisms in the bulk material, as demonstrated through variations in temperature, current density, crystal defects, and magnetic field. These processes give rise to a non-Newtonian, shear-thickening electron viscosity that facilitates diffuse momentum transport perpendicular to the main current flow. This form of viscosity may have broad relevance, ranging from electromigration phenomena to degenerate plasma jets. An observed anomaly in electron jetting allows the inference of a critical Reynolds number, defined in terms of the maximum ratio of laminar (parallel) to diffuse (perpendicular) momentum transport, yielding Recrit ≈ 4000. A theoretical derivation from a Landau cylinder model for conduction electrons in magnetic fields supports this value: Recrit = (2π3)2 ≈ 4000 which aligns with the classical hydrodynamic critical value when the maximum (central) velocity in Poiseuille flow is used instead of the average flow velocity. Thus, it is proposed that this critical Reynolds number may be universally applicable to particle flows—whether molecular or electronic.