Tensor analysis applied to the equations of contiuum mechanics I

The Navier-Stokes equations, the Euler equations, the equations of elasticity and expressions derived from those, are in most cases treated in Cartesian coordinates. In some cases it can necessary to handle those equations in other coordinate systems. In the cases of cylindrical coordinates, for example in the description of the flow around acoustic antennas, it is natural to use cylinder coordinates. In this report, we present the formalism necessary to handle the mentioned equations and related expressions in generalized coordinates. The formalism include tensor analysis, developed during 1850-1900 by Gregorio Ricci Kurbastro, Tullio Levi-Civita, Sophus Lie and others. Albert Einstein used tensor analysis as the mathematical basis for the General Theory of Relativity. In this report we will limit our self to describe the classical fluid equations in generalized coordinates. The tensor-theory can appear to be difficult and one can ask if it is necessary to go through all these complicated calculations. Can’t they be found at the web or in standard collections of formulas? We have looked for expressions, for example ∇ · (∇(ρT)), where T is the momentum flux density tensor that appears in Lighthill’s equation. We could not find this derived in cylinder coordinates and it was necessary to calculate it by hand to achieve our goals. In the analysis of flow around an acoustic antenna, various tensors appear, for example the strain rate tensor, structural tensors and tensorial expressions involved in the RANS equations, it was necessary to follow the formalism of tensor analysis in detail. With data given in cylinder coordinates, it is natural to do the analysis also in cylinder coordinates. Physical components of both vectors and tensors are used in the physical interpretations of the data. Although the treatment in cylinder coordinates addressed in this report only is directly applicable to a limited number of applications, the concept of tensor analysis is fundamental in practically all applications of continuum mechanics.