The purpose of this chapter9.1 is to discuss the formulation of momentum friction used in MOM. Maintaining certain symmetry properties of the frictional stress tensor guarantees that the resulting friction vector, which is constructed as the covariant divergence of the stress tensor, dissipates total kinetic energy without introducing internal sources of angular momentum. Providing a representation of these properties using curvilinear coordinates is facilitated with some of the tools of tensor analysis.
The central reference for this chapter is the review article by Smagorinsky (1993) as well as some unpublished notes of his from 1963. The related papers by Williams (1972) and Wajsowicz (1993) are also of use. These papers are generally complete, yet the derivations lack some of the tools of modern tensor analysis. Consequently, it has been found useful to rederive the results in this chapter using such tools in hopes of adding some clarity and generality to arbitrary orthogonal coordinates. Smagorinsky based much of his ideas on works from elasticity theory (e.g., Love 1944, Landau and Lifshitz 1986, Synge and Schild 1949, and Segel 1987). The fluid mechanics books by Aris (1962) and Landau and Lifshitz (1987) also consider many of the matters dealt with in the following. The mathematical formalism of Aris, which is consistent with the tensor analysis used by many mathematical physicists today, is closely followed in this chapter. Those more familiar with general relativity will also find such books as Weinberg (1972) useful.