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

Abstract

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