Conservation form of the Navier-Stokes equations in general nonsteady coordinates

Introduction R interest in the generation of general bodyoriented curvilinear coordinate systems," for the purpose of solving the complete Navier-Stokes system of equations for subsonic and supersonic flows, has given rise to many forms of presentations of the equations both in conservative and nonconservative formulations. Based on the available solutions of the gasdynamic equations (e.g., Ref. 5) the conservation-law form of the equations seems definitely preferable, particularly when shocks are present. Although the above statement cannot be repeated in a definitive sense for the case of viscous subsonic and supersonic flow prediction through the Navier-Stokes equations, nevertheless, it is expected that the conservation-law form may eventually be more acceptable for numerical purposes. The purpose of this paper is to derive the conservation-law form of the Navier-Stokes equations in general nonsteady coordinate systems in a simple and direct fashion. Previous work on this subject has been done by McVittie, Viviand, and Vinokur. It will be shown in this Note that the equations in the conservation-law form can be obtained simply by a little manipulation of some standard vector and tensor formulas.