Remarks on complex fluid models

P. Constantin

doi:10.1017/CBO9781139235792.005

4 - Remarks on complex fluid models

Published online by Cambridge University Press: 05 November 2012

Edited by

José L. Rodrigo and

P. Constantin: Affiliation:
Princeton University
James C. Robinson: Affiliation:
University of Warwick
José L. Rodrigo: Affiliation:
University of Warwick
Witold Sadowski: Affiliation:
Uniwersytet Warszawski, Poland

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Summary

Abstract We review recent global existence and uniqueness results of solutions for models of complex fluids in ℝd. We describe results concerning the Oldroyd-B and related models.

Introduction

Complex fluids are ubiquitous in nature and manifest a rather large number of different behaviours. There is no single accepted general model for all these, and the presence of a large array of complicated models is an indication of the difficulties encountered at a fundamental level. In this article I will describe some of the mathematical issues. A complex fluid is a mixture between a solvent, which is treated as a normal fluid, and particulate matter in it. The particles are sufficiently many and sufficiently small compared to the characteristic scales of the motions of the solvent, so that one may hope for a description that does not have to resolve the fluid mechanical problem of flow past the particles. The particles themselves are treated in a simplified manner as objects m ∈ M, where M is a finite-degrees-of-freedom configuration space accounting for the salient features of the particles. For instance M can be a subset of ℝN or a more complicated metric space. Models have been devised to deal with microscopic elastic thread-like objects such as polymers (Doi & Edwards, 1998, Öttinger, 1996). The complicated hydrodynamic interactions are simplified using the separation of scales, replacing the many degrees of freedom due to them by a few representative ones.

Type: Chapter
Information: Mathematical Aspects of Fluid Mechanics , pp. 70 - 87

DOI: https://doi.org/10.1017/CBO9781139235792.005 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2012

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