We have investigated the thermal conductivity tenser of strongly shear
ed atomic and molecular liquids by computer simulation methods. Accord
ing to linear nonequilibrium thermodynamics, heat and transverse momen
tum transport are uncoupled in the linear regime. We also expect the t
hermal conductivity to be independent of the zero-wavevector strain ra
te in the linear regime. Away from the linear regime, the situation is
different. Although even a large zero-wavevector strain rate cannot i
nduce a heat flux, the thermal conductivity can become strain-rate dep
endent. Furthermore, the thermal conductivity becomes a tenser because
a strong velocity profile can destroy the isotropy of a fluid. These
effects are only apparent at extremely high strain rates for atomic li
quids, but are experimentally observable for polymeric liquids.