Multiscale theorems

Abstract

We present 71 multiscale theorems that transform various derivatives of a function from one scale to another and contain all 128 such theorems in the literature. These theorems are grouped into integration theorems and averaging theorems. The former refers to those which change any or all spatial scales of a derivative from the microscale (or any continuum scale) to the megascale by integration. The integration domain is allowed to translate and deform with time, and can be a volume, a surface or a curve. The latter is those which change any or all spatial scales of a derivative from microscale to macroscale by averaging. An averaging volume may be located in space and integration is performed over volumes, surfaces, or curves contained within the averaging volume. These theorems not only provide simple tools to model multiscale phenomena at various scales and to interchange among those scales, but they also form generalized and universal integration theorems. As with all mathematical theorems, the tools used to prove each theorem and the logic behind the proof itself also hold significant value. Furthermore, these theorems are endowed with important information regarding geometrical and topological structures of and interactions among various entities such as phases, interfaces, common curves, and common points. Therefore, they also provide critical information for resolving the closure problems that are routinely encountered in multiscale science. In Section 1, we present some examples of multiscale issues in materials science, physics, hydrology, chemical engineering, transport phenomena in biological systems, and the universe. We then discuss scales and scaling and introduce the multiscale theorems as powerful tools for model-scaling. Section 2 covers the derivatives of a function, including various gradients, divergences, curls, and partial time derivatives. In particular, we denote all derivatives using three orthonormal vectors and develop relations among them. In Section 3, we extend the familiar Heaviside step function to form indicator functions for identifying portions of a curve, a surface, or a space. We then develop some useful identities involving indicator functions and their derivatives. Finally, we present two major applications of indicator functions in developing the multiscale theorems: the selection and interchange of integration domains. Section 4 gives the detailed proofs of the 38 integration theorems that convert integrals of the gradient, divergence, curl, and partial time derivatives of a microscale function to some combination of derivatives of integrals of the function and integrals over domains of reduced dimensionality. These theorems are powerful tools for scaling between the microscale (or any continuum scale) and the megascale. In Sectiion 5, we develop 33 averaging theorems that relate the integral of a derivative of a microscale function to the derivative of that integral. These theorems are powerful tools for scaling among microscales, macroscales, and megascales. Section 6 covers applications of the multiscale theorems for modeling single-phase turbulent flow, heat conduction in two-phase systems, transport in porous and multiphase systems and for developing the thermodynamically constrained averaging theory (TCAT) approach for modeling flow and transport phenomena in multiscale porous-medium systems. These applications lead to transport-phenomena models, which contain important information for developing closure, and some new results such as the scale-by-scale energy budget equations in real space for turbulent flows, the energy flux rates among various groups of eddies and the intrinsic equivalence between dual-phase-lagging heat conduction and Fourier heat conduction in two-phase systems that are subject to a lack of local thermal equilibrium. In the last section, we conclude with some remarks regarding advances in multiscale science and multiscale theorems (their features; tools and logic behind their proof; their applications). © 2008 Elsevier Inc.