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{{Over-quotation|date=August 2019}}
[[File:Holec2016P40.svg|thumb|Modeling approaches and their scales]]
'''Multiscale modeling''' or '''multiscale mathematics''' is the [[Branches of science|field]] of solving problems that have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids,<ref>{{Cite journal|last1=Chen|first1=Shiyi|last2=Doolen|first2=Gary D.|date=1998-01-01|title=Lattice Boltzmann Method for Fluid Flows|journal=Annual Review of Fluid Mechanics|volume=30|issue=1|pages=329–364|doi=10.1146/annurev.fluid.30.1.329|bibcode=1998AnRFM..30..329C}}</ref><ref name="Steinhauser 20082">{{cite book|title=Multiscale Modeling of Fluids and Solids - Theory and Applications|year=2017|isbn=978-3662532225|first1=M. O.|last1=Steinhauser|publisher=Springer }}</ref><ref>{{Cite journal |
An example of such problems involve the [[Navier–Stokes equations]] for incompressible fluid flow.
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\\ \nabla\cdot\mathbf{u}=0. \end{array}</math>
In a wide variety of applications, the stress tensor <math>\tau</math> is given as a linear function of the gradient <math>\nabla u</math>. Such a choice for <math>\tau</math> has been proven to be sufficient for describing the dynamics of a broad range of fluids. However, its use for more complex fluids such as polymers is dubious. In such a case, it may be necessary to use multiscale modeling to accurately model the system such that the stress tensor can be extracted without requiring the computational cost of a full microscale simulation.<ref>{{Cite book |last=E |first=Weinan
==History==
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<references>
<ref name="Horstemeyer 2009">
{{cite book |first1=M. F. |last1=Horstemeyer |year=2009 |chapter=Multiscale Modeling: A Review |chapter-url=https://books.google.com/books?id=esOANcsz5w8C&pg=PA87 |pages=87–135 |editor1-first=Jerzy |editor1-last=Leszczyński |editor2-first=Manoj K. |editor2-last=Shukla |title=Practical Aspects of Computational Chemistry: Methods, Concepts and Applications |publisher=Springer |isbn=978-90-481-2687-3}}
</ref>
<ref name="Horstemeyer 2012">
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==Further reading==
*{{cite journal |pmid=19136256 |year=2009 |last1=Hosseini |first1=SA |last2=Shah |first2=N |title=Multiscale modelling of hydrothermal biomass pretreatment for chip size optimization |volume=100 |issue=9 |pages=2621–8 |doi=10.1016/j.biortech.2008.11.030 |journal=Bioresource Technology|bibcode=2009BiTec.100.2621H }}
*{{cite journal |bibcode=2009BAMS...90..515T |title=A Multiscale Modeling System: Developments, Applications, and Critical Issues |last1=Tao |first1=Wei-Kuo |last2=Chern |first2=Jiun-Dar |last3=Atlas |first3=Robert |last4=Randall |first4=David |last5=Khairoutdinov |first5=Marat |last6=Li |first6=Jui-Lin |last7=Waliser |first7=Duane E. |last8=Hou |first8=Arthur |last9=Lin |first9=Xin |last10=Jiang |first10=Jonathan |last11=Hou |first11=Arthur |last12=Lin |first12=Xin |last13=Peters-Lidard |first13=Christa |volume=90 |year=2009 |pages=515–534 |journal=Bulletin of the American Meteorological Society |doi=10.1175/2008BAMS2542.1 |issue=4|display-authors=8 |hdl=2060/20080039624 |hdl-access=free }}
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