Re-entrant corner flows of PTT fluids in the Cartesian stress basis

Abstract

We consider the planar flow of Phan-Thien–Tanner (PTT) fluids around a re-entrant corner of angle π / α where α ∈ [ 1 / 2 , 1 ) . 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 votex. The local asymptotic solution structure is similar to that for the upper convected Maxwell (UCM) model and is shown to comprise a core flow (outer) region in which the fluid behaves elastically, together with wall boundary layers (inner regions) of similar thickness as those in the UCM model. In the core flow, the stress singularity is that for UCM, namely O ( r − 2 ( 1 − α ) ) where r is the radial distance from the corner, although the stream function vanishes at the slower rate O ( r α ( 1 + α ) ) compared to O ( r α ( 3 − α ) ) for UCM—this latter feature being a consequence of the shear thinning property of the PTT model. The amplitudes of the velocity and stress fields are determined and are seen to be independent from this local analysis, any link between them appearing to require global flow information away from the corner. The analysis is performed here in the Cartesian stress formulation of the problem, allowing the description of a similarity solution for the core flow and upstream boundary layer. The analysis remains to be completed by a solution for the downstream boundary layer which requires the use of the natural stress basis.