Abstract
We consider the planar flow of Oldroyd-B fluids around sharp corners. 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 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(3−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 elastic 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) is O(1). These results hold for all values of the retardation parameter β ∈ [0,1), but breakdownin the Newtonian limit β → 1−. This latter singular limit is considered along with the other singular regimes of low and high Weissenberg number, in order to extend the parameter dependence of the solution.
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. These boundary layers are markedly different from those of the re-entrant corner case.
| Date of Award | 1 Dec 2010 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Jonathan Evans (Supervisor) |
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