On the objective rate of heat and stress fluxes. Connection with micro/nano-scale heat convection

In this paper, the derivation of the convected derivatives for the heat flux and stress tensor is revisited. A kinematic approach is adopted based on material invariance. These upper-convected derivatives are used in the literature to generalize Newton's law of viscosity and Fourier's heat law of heat. The former constitutive law represents the behaviour of a viscoelastic fluid of the Boger type obeying the Oldroyd-B model, and the latter represents fluids obeying the Maxwell-Cattaneo's heat equation. The invariance of the derivatives under orthogonal transformation is also shown. Although the presentation here is limited to the derivatives of vector and second-rank tensor fluxes, the formulation can be generalized to generate the convected derivative of a tensor flux of arbitrary rank. Finally, the connection with micro- or nano-channel flow is noted.