The flow of an Oldroyd fluid around a sharp corner

Abstract

A similarity solution is constructed for the flow of an Oldroyd-B fluid around a 270" re-entrant corner. The velocity is found to vanish like r5/9 and the stress to be singular like Y-~/~. A simple expression is found for the streamfunction.

Plane sudden expansion flows of viscoelastic liquids

“…All three stresses exhibit the same asymptotic variation as they approach the corner, varying as r −2/3 (represented as the straight lines of slope −2/3 in the log-log plot of Fig. 8), in agreement with the theoretical predictions of Hinch [23].…”

Development of an optimal hybrid finite volume/element method for viscoelastic flows

“…In this manner, Alves et al were able to reach a Weissenberg number of ÿve on their third ÿnest mesh, and three on their fourth, most ÿne mesh. Relevant issues that arise in Reference [14] can be summarized as follows: diminishing lip-vortex with mesh reÿnement as reported by Matallah et al [1], and Xue et al [16]; lip-vortex growth (see References [1,13] to a lesser extent), and diminishing salient-corner vortex with increasing elasticity; agreement with the asymptotic behaviour of velocities and stresses near the re-entrant corner, as predicted theoretically by Hinch [17] for an Oldroyd-B uid (see also, Renardy [18,19] for an UCM uid). The present article develops these themes somewhat further.…”

“…The singular asymptotic behaviour of the stress at a re-entrant corner has been studied theoretically by a number of researchers, using an (r,Â) polar coordinate system of reference, centred at the corner [17][18][19]47]. For the ow of an Oldroyd-B uid past a sharp corner and for a given angle Â, Hinch [17] established that velocity components asymptote towards the corner like r 5=9 .…”

Mixed finite element methods for viscoelastic flow analysis: a review

“…Moreover, the theoretical results of Hinch [98] with respect to the order of the corner singularity for the axi-symmetric four-to-one contraction are recovered numerically. Furthermore, comparison of computed corner vortex intensities with results obtained by the finite volume method of Sasmal [104] and the EEME/SUPG results of Coates et al [102], indicates the low order convergence behaviour of the discretization chosen by Sasmal.…”

“…Nevertheless, it is fair to state that significant progress has been made over the past decade. In three recent papers, Davies and Devlin [97], Hinch [98] and Renardy [99], seek local solutions near the singular point. Renardy assumed a Newtonian velo- For the flow of an Oldroyd-B fluid through an axi-symmetric 4:1 contraction and using the 4 × 4 SU formulation, Marchal and Crochet [20] achieved solutions upto a Weissenberg beyond 60.…”