Exact analytical solutions for the piston effect in supercritical fluids under post-acoustic approximation -- Short-time asymptotics, thermal penetration depth and comparison with the Spacelab D-2 experiments
Near the liquid-vapor critical point, fluids become highly compressible, giving rise to a special, strongly coupled thermo-mechanical process: the piston effect. In this phenomenon, a thin thermal boundary layer develops near a heated wall; owing to strong thermal expansion, this layer acts like a piston, compressing the bulk fluid adiabatically and resulting in a seemingly accelerated thermal response. Although the piston effect is a thermo-acoustic process, the characteristic time scale of the boundary perturbation is typically orders of magnitude larger than the acoustic time scale of the setup. Consequently, rapid acoustic propagation can be neglected, justifying a post-acoustic approximation with a spatially uniform but time-dependent bulk pressure. Within the linear regime, the temporal evolution of pressure can be directly connected to the heat flux entering through the boundaries. As a result, the problem reduces to a diffusion equation governed by a spatially homogeneous source term that depends explicitly on the boundary conditions. Exact, closed-form analytical solutions are derived for effectively one-dimensional problems in both Cartesian and spherical coordinates, considering boundary conditions of the first and second kinds. Short-time asymptotic behavior and thermal penetration depth are analyzed for all four cases. By incorporating the heat capacity of a container via a homogeneous model, an effective boundary condition coupling the wall heat flux and the time derivative of the wall temperature is derived, allowing for a direct comparison with experimental data from the Spacelab D-2 mission. The analytical predictions show good agreement with the experimental results without relying on any numerical simulations.