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The Boltzmann transport equation for phonons is recast directly in terms ofthe heat-flux by means of iteration followed by truncation at the second orderin the spherical harmonic expansion of the distribution function. Thisprocedure displays the heat-flux in an explicitly coordinate-invariant form,and leads to a natural decomposition into two components, namely the solenoidalcomponent in addition to the usual irrotational component. The solenoidalheat-flux is explicitly shown to arise by applying the heat-flux equation to aright-circular cylinder. These findings are important in the context of phononresonators that utilize the strong quasi-ballistic thermal transport reportedrecently in silicon membranes at room temperature.

On the solenoidal heat-flux in quasi-ballistic thermal conduction