Abstract: This paper ls intended to analyze the mathematical physics of the non-Fourier heat transfer in fluid flows. To give the basis for analytical and numerical approaches,the zones of dependence and influence, With their physical implications, of the governing equations are ex-pounded via the viewpoint of characteristics and sub-characteristics.The validitv of the constitutive relation is justified by a fixed-heat-source problem in the flow,and thermal shocks are shown to exist in the flow field when the flow velocity is equal to or faster than the thermal propagation speed of the fluid.

Key words: heat flux, heat transfer, fluid flow