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Womersley flow of generalized Newtonian liquid

Part of: Foundations, constitutive equations, rheology Incompressible viscous fluids

Published online by Cambridge University Press:  22 March 2016

Giovanni P. Galdi  and
Carlo R. Grisanti
Giovanni P. Galdi
Affiliation:
University of Pittsburgh, 630 Benedum Engineering Hall, Pittsburgh, PA 15621, USA ([email protected])
Carlo R. Grisanti
Affiliation:
Dipartimento di Matematica, Università di Pisa, Via Buonarroti 1/c, 56127 Pisa, Italy ([email protected])
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Extract

We show that in an infinite straight pipe of arbitrary (sufficiently smooth) cross-section, a generalized non-Newtonian liquid admits one and only one fully developed time-periodic flow (Womersley flow) when either the flow rate (problem 1) or the axial pressure gradient (problem 2) is prescribed in analogous time-periodic fashion. In addition, we show that the relevant solution depends continuously upon the data in appropriate norms. As is well known from the Newtonian counterpart of the problem, the latter is pivotal for the analysis of flow in a general unbounded pipe system with cylindrical outlets (Leray's problem). It is also worth remarking that problem 1 possesses an intrinsic interest from both mathematical and physical viewpoints, in that it constitutes a (nonlinear) inverse problem with a significant bearing on several applications, including blood flow modelling in large arteries.

Keywords

pipe flowtime-periodicgeneralized Newtonian76A0576D03

MSC classification

Secondary: 76A05: Non-Newtonian fluids 76D03: Existence, uniqueness, and regularity theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 4 , August 2016 , pp. 671 - 692
Copyright
Copyright © Royal Society of Edinburgh 2016 

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