Fourier’s law has been used to successfully describe heat transport for over 200 years. However, it is known that it leads to inaccuracies at extremely short time scales or at very small length scales. For example, experiments of laser heating of ultrathin layers or simulations of heat transport in solids using molecular dynamics show dramatic discrepancies with respect to classical laws.
One example which illustrates the failure of the classical equations is the fact that the thermal conductivity, a key parameter for describing heat transport and originally introduced by Fourier as an intrinsic property of the medium, has been shown to become size-dependent at small length scales.
The Guyer-Krumhansl equation is a popular extension to the classical Fourier’s law that includes memory and non-local effects, which are assumed to become important as the time or length scales decrease. In addition, the similarity between this equation and the Navier-Stokes equations exhaustively studied in fluid dynamics allows us to drive important analogies between heat and fluid flow.
With the aim of obtaining a predictive model for the effective thermal conductivity, which agrees with the available experimental data, we use the Guyer-Krumhansl equation to describe the heat conduction across a circular nanowire which is held at different temperatures at both of its extrema. The simple geometry allows us to find an analytical solution for the flux and an expression for the effective thermal conductivity which shows excellent agreement with experimental data. Experimental and theoretical results are compared in the figure below for nanowires of different diameters (37 nm (red), 56 nm (blue), 115 nm (green)) across a wide range of temperatures. Further details may be found in [1,2].
By Marc Calvo-Schwarzwälder , PhD student, Centre de Recerca Matematica.
[1] M Calvo-Schwarzwälder, MG Hennessy, P Torres, TG Myers, FX Alvarez. A slip-based model for the size-dependent effective thermal conductivity of nanowires. International Communications in Heat and Mass Transfer 91, 57-63, 2018.
[2] M Calvo-Schwarzwälder, MG Hennessy, P Torres, TG Myers, FX Alvarez. Effective thermal conductivity of rectangular nanowires based on phonon hydrodynamics. International Journal of Heat and Mass Transfer 126, 1120-1128.