Mathematical modeling of local non-equilibrium hydrodynamics in moving liquids

Abstract

By using a modified formula of Newton's Law, in which the velocities and accelerations of the tangential stress and velocity gradient are taken into account, the locally non-equilibrium Navier-Stokes equation, which describes the fluid flow with respect to relaxation phenomena, is obtained. The derived equation allows for the elimination of the infinite rate of momentum transfer problem described by the classical equations of motion. This can be explained by the fact that using the modified Newton's Law formula makes it possible to avoid the instantaneous change in the tangential stress (described by the classical formula) at a change in the velocity gradient because, in this case, they appear to be time dependent. Analysis of a numerical solution to the boundary problem for velocity distribution in a flat channel allowed us to conclude that the formation of the velocity profile in time in locally non-equilibrium conditions occurs with some lag, which was estimated by the relaxation coefficients of the liquid. It was also shown that, at any point of the channel cross section, there are velocity oscillations, the frequency and amplitude of which are determined by the relaxation properties of the liquid.