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Article: Multiscale theorems
Title  Multiscale theorems 

Authors  
Keywords  Averaging theorems Complexity Homogenization Integration theorems Mixture theory Modeldownscaling Modelscaling Modelupscaling Multiscale modeling Multiscale phenomena Multiscale science Multiscale theorems Scales Scaling 
Issue Date  2008 
Publisher  Academic Press. 
Citation  Advances In Chemical Engineering, 2008, v. 34, p. 175468 How to Cite? 
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 modelscaling. 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 singlephase turbulent flow, heat conduction in twophase 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 porousmedium systems. These applications lead to transportphenomena models, which contain important information for developing closure, and some new results such as the scalebyscale energy budget equations in real space for turbulent flows, the energy flux rates among various groups of eddies and the intrinsic equivalence between dualphaselagging heat conduction and Fourier heat conduction in twophase 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. 
Persistent Identifier  http://hdl.handle.net/10722/64496 
ISSN  2015 SCImago Journal Rankings: 0.183 
References 
DC Field  Value  Language 

dc.contributor.author  Wang, L  en_HK 
dc.contributor.author  Xu, M  en_HK 
dc.contributor.author  Wei, X  en_HK 
dc.date.accessioned  20100713T04:52:15Z   
dc.date.available  20100713T04:52:15Z   
dc.date.issued  2008  en_HK 
dc.identifier.citation  Advances In Chemical Engineering, 2008, v. 34, p. 175468  en_HK 
dc.identifier.issn  00652377  en_HK 
dc.identifier.uri  http://hdl.handle.net/10722/64496   
dc.description.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 modelscaling. 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 singlephase turbulent flow, heat conduction in twophase 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 porousmedium systems. These applications lead to transportphenomena models, which contain important information for developing closure, and some new results such as the scalebyscale energy budget equations in real space for turbulent flows, the energy flux rates among various groups of eddies and the intrinsic equivalence between dualphaselagging heat conduction and Fourier heat conduction in twophase 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.  en_HK 
dc.language  eng  en_HK 
dc.publisher  Academic Press.  en_HK 
dc.relation.ispartof  Advances in Chemical Engineering  en_HK 
dc.subject  Averaging theorems  en_HK 
dc.subject  Complexity  en_HK 
dc.subject  Homogenization  en_HK 
dc.subject  Integration theorems  en_HK 
dc.subject  Mixture theory  en_HK 
dc.subject  Modeldownscaling  en_HK 
dc.subject  Modelscaling  en_HK 
dc.subject  Modelupscaling  en_HK 
dc.subject  Multiscale modeling  en_HK 
dc.subject  Multiscale phenomena  en_HK 
dc.subject  Multiscale science  en_HK 
dc.subject  Multiscale theorems  en_HK 
dc.subject  Scales  en_HK 
dc.subject  Scaling  en_HK 
dc.title  Multiscale theorems  en_HK 
dc.type  Article  en_HK 
dc.identifier.openurl  http://library.hku.hk:4550/resserv?sid=HKU:IR&issn=00652377&volume=34&spage=175&epage=468&date=2008&atitle=Multiscale+theorems  en_HK 
dc.identifier.email  Wang, L:lqwang@hkucc.hku.hk  en_HK 
dc.identifier.authority  Wang, L=rp00184  en_HK 
dc.description.nature  link_to_subscribed_fulltext   
dc.identifier.doi  10.1016/S00652377(08)000045  en_HK 
dc.identifier.scopus  eid_2s2.055449110990  en_HK 
dc.identifier.hkuros  155005  en_HK 
dc.relation.references  http://www.scopus.com/mlt/select.url?eid=2s2.055449110990&selection=ref&src=s&origin=recordpage  en_HK 
dc.identifier.volume  34  en_HK 
dc.identifier.spage  175  en_HK 
dc.identifier.epage  468  en_HK 
dc.publisher.place  United States  en_HK 
dc.identifier.scopusauthorid  Wang, L=35235288500  en_HK 
dc.identifier.scopusauthorid  Xu, M=7403607587  en_HK 
dc.identifier.scopusauthorid  Wei, X=23669842200  en_HK 