The forces acting on an element of a continuous media are of two kinds. External or body forces, such as gravitation, Coriolis, or electromagnetic forces, act throughout the media. Internal or contact forces, such as pressure forces, act on an element of volume through its bounding surface. The balance between these forces and acceleration leads, through Newton's second law, to the equations of motion. Furthermore, if all torques acting on the fluid arise from macroscopic forces, which is the case in typical Newtonian fluids, then fluid elements respect an angular momentum conservation law. As MOM assumes the fluid it simulates to be Newtonian, ideally its solutions conserve angular momentum for closed systems.
The stresses acting within a continuous media can be organized into a
second order stress tensor with generally
independent
elements. The divergence of these stresses gives rise to the internal
forces acting in the media. As seen in the following discussion, a
proper account of the angular momentum budget implies that the stress
tensor is symmetric, which brings the number of independent stress
elements down to six. It is useful to note that symmetry of the
stress tensor is equivalent to Cauchy's reciprocal theorem
(section 5.13 of Aris), which says that each of two stresses at a
point has an equal projection on the normal to the surface on which
the other acts.
For simplicity, the analysis in this section assumes Cartesian coordinates. Generalizations are straightforward (e.g., Chapters 7 and 8 of Aris).