ABSTRACT
In this chapter, theorems are presented that change the scale of a differential operator by averaging. An averaging volume, also called a representative elementary volume (REV), may be located in space and then integration is performed over volumes, surfaces, or curves contained within the averaging volume. Because volumes are located at every position in space, the averages obtained have a functional dependence on spatial coordinates. The averaging theorems are particularly useful, for example, in problems of flow in porous media where it is desirable to change spatial scales from the pore scale (micro-scale) to the scale of a representative region in the system (macroscale) or the scale of the system itself (megascale). This chapter will merely present the theorems that have been derived using the generalized functions and explain the notation used, as appropriate. The notation for theorem identification has been given in Chapter 5, and useful identities and sample derivations appear in Chapter 6. Averaging theorems will be presented for four different operators: gradient (G), divergence (D), curl (C), and the time derivative (T).
