For gas flows, the Navier-Stokes (NS) equations are established by
mathematically expressing conservations of mass, momentum and energy. The
advantage of the NS equations over the Euler equations is that the NS equations
have taken into account the viscous stress caused by the thermal motion of
molecules. The viscous stress arises from applying Isaac Newton's second law to
fluid motion, together with the assumption that the stress is proportional to
the gradient of velocity1. Thus, the assumption is the only empirical element
in the NS equations, and this is actually the reason why the NS equations
perform poorly under special circumstances. For example, the NS equations
cannot describe rarefied gas flows and shock structure. This work proposed a
correction to the NS equations with an argument that the viscous stress is
proportional to the gradient of momentum when the flow is under compression,
with zero additional empirical parameters. For the first time, the NS equations
have been capable of accurately solving shock structure and rarefied gas flows.
In addition, even for perfect gas, the accuracy of the prediction of heat flux
rate is greatly improved. The corrected NS equations can readily be used to
improve the accuracy in the computation of flows with density variations which
is common in nature.Comment: 13 pages, 7 figure