Prandtl number

The Prandtl number (Pr) is a dimensionless quantity in fluid mechanics defined as the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. It quantifies the relative ease with which momentum and heat diffuse through a fluid. When Pr ≈ 1, as in most gases, momentum and heat diffuse at comparable rates, and the Reynolds analogy between skin friction and heat transfer is approximately valid. When Pr >> 1, as in oils and water, momentum diffuses much faster than heat, and the thermal boundary layer is much thinner than the velocity boundary layer. When Pr << 1, as in liquid metals, heat diffuses faster than momentum, and the reverse is true.

The Prandtl number is not merely a parameter for engineering correlations. It is a fundamental property that reveals the microscopic mechanism of energy transport in a fluid. In gases, both momentum and heat are transported by molecular collisions, and the Prandtl number is determined by the ratio of collision cross-sections for momentum and energy exchange. In liquids, the mechanisms differ: momentum transport is dominated by the short-range ordering of molecules, while heat transport is dominated by the propagation of phonons and the diffusion of molecular kinetic energy. The Prandtl number therefore encodes the difference between mechanical and thermal transport at the molecular scale.

The Prandtl number also governs the stability of convective flows. In Rayleigh-Bénard convection — the paradigmatic system for studying thermal instability — the critical Rayleigh number for the onset of convection depends on the Prandtl number. At low Prandtl numbers, the instability is oscillatory; at high Prandtl numbers, it is stationary. This Prandtl-number dependence has been observed in experiments ranging from liquid helium to molten metals, and it is one of the most robust results in thermal instability theory. The Nusselt number — the dimensionless heat transfer coefficient — is a function of both the Rayleigh number and the Prandtl number, and their interplay determines the efficiency of convective heat transport in planetary interiors, stellar atmospheres, and industrial heat exchangers.