Stability problems with generalized Navier–Stokes–Voigt theories

Abstract

We investigate some hydrodynamic stability problems in the context of the Navier–Stokes–Voigt equations. It is pointed out that one should regard the usual set of equations known as the Navier–Stokes–Voigt equations as being the Navier–Stokes equations to which a regularizing term has been added. We investigate other models which have features very similar to the Navier–Stokes–Voigt equations, but which arise from proper continuum thermodynamic approaches, including employing an objective time derivative rather than simply the Laplacian of the partial time derivative of the velocity field. It is shown that in some cases, particularly those connected to straightforward thermal convection studies, the linear theory of the more physically based models reduces to that of the classical Navier–Stokes–Voigt theory. However, these are special problems and we also display other problems where the generalized theories based on continuum mechanics principles lead to very different results from what one finds with traditional Navier–Stokes–Voigt theory. Finally, two further models pertaining to Navier–Stokes–Voigt theory which were introduced by Oskolkov are investigated.