Second-sound beyond Maxwell–Cattaneo: Nonlocal effects in hyperbolic heat transfer at the nanoscale

We investigate a recently proposed model of second-sound at the nanoscale—One that goes beyond the usual Maxwell–Catteneo theory of hyperbolic heat transfer. The model in question is based on a dynamical non-equilibrium temperature, denoted herein by β, that takes into account the effects of relaxation and nonlocality, both of which play an important role in heat-transport phenomena at the nanoscale. Employing a combination of analytical and numerical methodologies, we examine the propagation of both traveling waves and temperature-rate waves in a class of (thermally conducting) rigid solids; the impact of nonlocality on these waveforms is illustrated via comparisons with results predicted by Maxwell–Cattaneo theory. In particular, we show that nonlocality can: alter the speed at which second-sound propagates; lead to bounded but increasing, semi-compact, traveling wave profiles; and impact the evolution of temperature-rate wave amplitudes. Lastly, we review connections between our findings and known results in other fields, possible follow-on studies are noted, and a paradox revealed by our traveling wave analysis is discussed.