Reduction of the effective thermal conductivity by circulation of the   quasi-ballistic heat-flux
  For a thermal conductivity that is spatially uniform and independent of temperature, Fourier's law of heat transfer predicts a curl-free heat-flux. In the quasi-ballistic phonon transport regime, where the Fourier law breaks down, it has been proven starting from the Boltzmann transport equation that the constitutive relation for the heat-flux contains a solenoidal (circulatory) term with zero divergence and non-zero curl. In this paper we show how to reduce the effective thermal conductivity of a material with dimensions on the same scale as the phonon mean-free path, by exploiting the solenoidal term in the constitutive relation. 
Ashok T. Ramu, Carl D. Meinhart and John E. Bowers
cond-mat.mes-hall