The classical Navier-Stokes equation (NSE) is the fundamental partial differential equation that describes the flow of fluids, but in certain cases, like high local density and temperature gradient, it is inconsistent with the experimental results. Some extended Navier-Stokes equations with diffusion terms taken into consideration have been proposed. However, a consensus conclusion on the specific expression of the additional diffusion term has not been reached in the academic circle. The models adopt the form of the generalized Newtonian constitutive relation by substituting the convection velocity with a new term, or by using some analogy. In this study, a new constitutive relation for momentum transport and a momentum balance equation are obtained based on the molecular kinetic theory. The new constitutive relation preserves the symmetry of the deviation stress, and the momentum balance equation satisfies Galilean invariance. The results show that for Poiseuille flow in a circular micro-tube, self-diffusion in micro-flow needs considering even if the local density gradient is very low.