The UCM limit of the PIT equations at a re-entrant corner

Jonathan D Evans, David N Sibley

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We consider the Upper Convected Maxwell (UCM) limit of the Phan-Thien-Tanner (PTT) equations for steady planar flow around re-entrant corners. 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. We show that the critical length scale local to the corner is r = O (kappa(1/2(1-alpha))) as kappa -> 0, where pi/alpha is the re-entrant corner angle with alpha is an element of [1/2, 1) and r the radial distance. On distances far smaller than this we obtain the PIT K = 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.
Original languageEnglish
Pages (from-to)1543-1549
Number of pages7
JournalJournal of Non-Newtonian Fluid Mechanics
Volume165
Issue number21-22
DOIs
Publication statusPublished - Nov 2010

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pulsed inductive thrusters
Length Scale
boundary layer equations
Maxwell equations
Term
Pi
Maxwell's equations
Dimensionless
Maxwell equation
Boundary Layer
Boundary layers
Model
Angle

Keywords

  • re-entrant corner
  • Phan-Thien-Tanner
  • stress singularity
  • upper convected Maxwell
  • elastic boundary layers

Cite this

The UCM limit of the PIT equations at a re-entrant corner. / Evans, Jonathan D; Sibley, David N.

In: Journal of Non-Newtonian Fluid Mechanics, Vol. 165, No. 21-22, 11.2010, p. 1543-1549.

Research output: Contribution to journalArticle

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AB - We consider the Upper Convected Maxwell (UCM) limit of the Phan-Thien-Tanner (PTT) equations for steady planar flow around re-entrant corners. 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. We show that the critical length scale local to the corner is r = O (kappa(1/2(1-alpha))) as kappa -> 0, where pi/alpha is the re-entrant corner angle with alpha is an element of [1/2, 1) and r the radial distance. On distances far smaller than this we obtain the PIT K = 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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