On unifying the concepts of scale, instrumentation, and stochastics in the development of multiphase transport theory

A generalized theory of multiphase transport is presented which combines the concepts of scale, instrumentation, stochastics, and time series with the development of transport equations. By defining the filtering process as a convolution of a measure P with a field property ψ, we are able to exploit the Fourier transform to place plausible restrictions on an instrument in frequency space so as to make its measurement relevant to its physical environment. An ideal instrument is defined which filters out high-frequency noise (corresponding to short distances) and yet does not alter the structure of low frequencies. Using ideal instruments we successively filter out lower frequency noise in a multiscale, multiphase environment. Formulas are developed to relate the autocorrelation of a field property on one scale of motion to that on any other scale while taking into account the types of instruments used in the measuring process. An equation relating the integral scale on one scale of motion to the integral scale on any other scale of motion is developed. Power spectra are developed which relate spectra on different scales to the measuring instrument used. By successively applying filtering theorems, a hierarchy of multiscale transport equations is developed. Filtered properties in the transport equations are mass averages. Different properties are allowed to be measured by different instruments and different instruments are allowed on different scales of motion and for different phases. The concept of a wide sense stationary, ergodic process is introduced to develop mass average autocorrelations and spectra over scales of motion as a function of measuring devices.

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