Viscoelastic sink flow in a wedge for the UCM and Oldroyd-B models

Abstract

The steady planar sink flow through wedges of angle π / α with α ≥ 1 / 2 of the upper convected Maxwell (UCM) and Oldroyd-B fluids is considered. The local asymptotic structure near the wedge apex is shown to comprise an outer core flow region together with thin elastic boundary layers at the wedge walls. A class of similarity solutions is described for the outer core flow in which the streamlines are straight lines giving stress and velocity singularities of O ( r − 2 ) and O ( r − 1 ) , respectively, where r ≪ 1 is the distance from the wedge apex. These solutions are matched to wall boundary layer equations which recover viscometric behaviour and are subsequently also solved using a similarity solution. The boundary layers are shown to be of thickness O ( r 2 ) , their size being independent of the wedge angle. The parametric solution of this structure is determined numerically in terms of the volume flux Q and the pressure coefficient p 0 , both of which are assumed furnished by the flow away from the wedge apex in the r = O ( 1 ) region. The solutions as described are sufficiently general to accommodate a wide variety of external flows from the far-field r = O ( 1 ) region. Recirculating regions are implicitly assumed to be absent.