Viscoelastic Flows of PTT Fluid

Abstract

We consider the planar flow of Phan-Thien-Tanner (PTT) fluids in geometries where a singular stress is encountered. The main problem considered is flow around a sharp corner, with also preliminary results for sink flow in a wedge. Two distinct cases arise for the corner geometry, where the corner angle is denoted by π/α. For 1/2 ≤ α < 1 we have a re-entrant corner, whilst for 1 < α < ∞ a so called salient corner occurs. These two regimes have markedly different flow behaviour. The model is considered in the absence of a solvent viscosity and the flow situation assumes complete flow around the corner with the absence of a lip vortex.

For the re-entrant corner problem a class of self-similar solutions has been identified with stress singularities of O(r−2(1−α)) and stream function behaviour O(r(1+α)α) (r being the radial distance from the corner). These behaviours arise in a core flow region away from the walls and are shown to be solutions of the incompressible Euler equations. This region is reconciled with boundary layers at the upstream and downstream walls using the method of matched asymptotic expansions. The analysis benefits from the representation of the stress in both Cartesian and natural stress formulations, and is performed when the Weissenberg number (the dimensionless relaxation time) and the PTT model parameter κ (a dimensionless mobility factor) are both O(1). The analysis is then extended for the various limits of these two parameters.

For the salient corner case the mathematically simpler Newtonian balance for the flow and stress fields are shown to dominate away from the walls. This gives a stream function behaviour of O(r1+λ0 ) and stress behaviour O(rλ0−1), where λ0 is the Newtonian problem eigenvalue. This behaviour is again reconciled with boundary layers at the walls which recover viscometric behaviour.

Date of Award	1 Mar 2010
Original language	English
Awarding Institution	University of Bath
Supervisor	Jonathan Evans (Supervisor)