The Asymptotic Behaviour at a Re-Entrant Corner for a PTT Fluid in the Limit of Small kappa

Jonathan D Evans, David N Sibley

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

We consider the Upper Convected Maxwell (UCM) limit of the Phan-Thien-Tanner (PTT) equations for steady planar flow around re-entrant comers. The PTT equations give the UCM equations in the limit of vanishing model parameter kappa, this dimensionless parameter being associated with the quadratic stress terms in the PTT model. The critical length scale local to the comer is r = O(kappa (1/2(1-alpha))) as kappa -> 0, where pi/alpha a is the re-entrant comer angle with alpha is an element of [1/2, 1) and r the radial distance. On distances far smaller than this we obtain the PTT kappa = 1 problem, whilst on distances greater (but still small) we obtain the UCM problem kappa = 0. This critical length scale is that on which intermediate behaviour of the PTT model is obtained where both linear and quadratic stress terms are present in the wall boundary layer equations. The double limit kappa -> 0, r -> 0 thus yields a nine region local asymptotic structure.
LanguageEnglish
Title of host publicationNumerical Analysis and Applied Mathematics, Vols I-III
EditorsT E Simos, G Psihoyios, C Tsitouras
PublisherAmerican Institute of Physics
Pages1680-1683
Number of pages4
Volume1281
ISBN (Print)978-0-7354-0834-0
DOIs
StatusPublished - 2010

Publication series

NameAIP Conference Proceedings
PublisherAmerican Institute of Physics

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Asymptotic Behavior
Fluid
Length Scale
Term
Pi
Maxwell's equations
Dimensionless
Boundary Layer
Model
Angle

Cite this

Evans, J. D., & Sibley, D. N. (2010). The Asymptotic Behaviour at a Re-Entrant Corner for a PTT Fluid in the Limit of Small kappa. In T. E. Simos, G. Psihoyios, & C. Tsitouras (Eds.), Numerical Analysis and Applied Mathematics, Vols I-III (Vol. 1281, pp. 1680-1683). (AIP Conference Proceedings). American Institute of Physics. https://doi.org/10.1063/1.3498165

The Asymptotic Behaviour at a Re-Entrant Corner for a PTT Fluid in the Limit of Small kappa. / Evans, Jonathan D; Sibley, David N.

Numerical Analysis and Applied Mathematics, Vols I-III. ed. / T E Simos; G Psihoyios; C Tsitouras. Vol. 1281 American Institute of Physics, 2010. p. 1680-1683 (AIP Conference Proceedings).

Research output: Chapter in Book/Report/Conference proceedingChapter

Evans, JD & Sibley, DN 2010, The Asymptotic Behaviour at a Re-Entrant Corner for a PTT Fluid in the Limit of Small kappa. in TE Simos, G Psihoyios & C Tsitouras (eds), Numerical Analysis and Applied Mathematics, Vols I-III. vol. 1281, AIP Conference Proceedings, American Institute of Physics, pp. 1680-1683. https://doi.org/10.1063/1.3498165
Evans JD, Sibley DN. The Asymptotic Behaviour at a Re-Entrant Corner for a PTT Fluid in the Limit of Small kappa. In Simos TE, Psihoyios G, Tsitouras C, editors, Numerical Analysis and Applied Mathematics, Vols I-III. Vol. 1281. American Institute of Physics. 2010. p. 1680-1683. (AIP Conference Proceedings). https://doi.org/10.1063/1.3498165
Evans, Jonathan D ; Sibley, David N. / The Asymptotic Behaviour at a Re-Entrant Corner for a PTT Fluid in the Limit of Small kappa. Numerical Analysis and Applied Mathematics, Vols I-III. editor / T E Simos ; G Psihoyios ; C Tsitouras. Vol. 1281 American Institute of Physics, 2010. pp. 1680-1683 (AIP Conference Proceedings).
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N2 - We consider the Upper Convected Maxwell (UCM) limit of the Phan-Thien-Tanner (PTT) equations for steady planar flow around re-entrant comers. The PTT equations give the UCM equations in the limit of vanishing model parameter kappa, this dimensionless parameter being associated with the quadratic stress terms in the PTT model. The critical length scale local to the comer is r = O(kappa (1/2(1-alpha))) as kappa -> 0, where pi/alpha a is the re-entrant comer angle with alpha is an element of [1/2, 1) and r the radial distance. On distances far smaller than this we obtain the PTT kappa = 1 problem, whilst on distances greater (but still small) we obtain the UCM problem kappa = 0. This critical length scale is that on which intermediate behaviour of the PTT model is obtained where both linear and quadratic stress terms are present in the wall boundary layer equations. The double limit kappa -> 0, r -> 0 thus yields a nine region local asymptotic structure.

AB - We consider the Upper Convected Maxwell (UCM) limit of the Phan-Thien-Tanner (PTT) equations for steady planar flow around re-entrant comers. The PTT equations give the UCM equations in the limit of vanishing model parameter kappa, this dimensionless parameter being associated with the quadratic stress terms in the PTT model. The critical length scale local to the comer is r = O(kappa (1/2(1-alpha))) as kappa -> 0, where pi/alpha a is the re-entrant comer angle with alpha is an element of [1/2, 1) and r the radial distance. On distances far smaller than this we obtain the PTT kappa = 1 problem, whilst on distances greater (but still small) we obtain the UCM problem kappa = 0. This critical length scale is that on which intermediate behaviour of the PTT model is obtained where both linear and quadratic stress terms are present in the wall boundary layer equations. The double limit kappa -> 0, r -> 0 thus yields a nine region local asymptotic structure.

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