8 - Time Averaging and Systems with Changing Material Content

Summary

Preamble

Time averaging is shown here to pervade the microscopic foundations of macroscopic modelling, whether in appreciating the nature of intermolecular forces in terms of subatomic dynamics (recall Section 5.4), establishing links between continuum field values and actual measurements (recall Section 3.10), deriving balance relations for systems with changing material content, or exploring the conceptual foundations of statistical mechanics.

After recalling the notion of a Δ-time average introduced in Section 5.4, such averaging is applied directly to the continuity equation and balances of linearmomentum and energy established in Chapters 4, 5, and 6. Field values in the resulting relations are thereby identified as local averages of molecular quantities computed jointly in both space and time. Such relations apply to material systems which do not change with time. Any system M⁺ whose material content does change with time is regarded to be a subset of an unchanging material system M. Relations for M⁺ are established by introducing a membership function e_i for each element P_i of M: e_i(t) = 1 (or 0) according to whether P_i ∊ M⁺ (or P_i ∉ M⁺) at time t. An equation which prescribes the time evolution of a single global representative velocity for M⁺ is derived utilising time averaging: this serves to establish the methodology to be used later in deriving local balances of mass, linear momentum, and energy, and also to discuss the fundamentals of rocket and jet propulsion.