Long-range memory effects in flows involving abrupt changes in geometry: Part I: flows associated with I-shaped and T-shaped geometries

Abstract

This is a theoretical paper which attempts to study for the first time the effect of high elasticity in flow situations involving elastico-viscous liquids and abrupt changes in geometry. It is argued that implicit rheological models are essential in this exercise and, accordingly, the numerical method of solution is forced to recognise the equations of continuity, the stress equations of motion and the rheological equations as separate equations involving velocity, pressure and stress variables with appropriate boundary conditions on these variables. The present paper is concerned with L-shaped and T-shaped geometries, and the effect of elasticity is assessed by comparing the numerical predictions for an elastic liquid with those for an inelastic liquid with the same “viscosity” behaviour. This comparison is facilitated by a simple limiting procedure outlined in Section 2. The main conclusions from the work are that, in general terms, elasticity works against inertia, reducing the pressure drop caused by the abrupt change in geometry and reducing the area of influence of the bend (for finite Reynolds numbers). So far as the stress fields are concerned most interest centres on the corner region, as one would expect, but there is also a region of normal-stress activity, which is generated by “stretching” rather than “shearing”. In an appendix, some consideration is given to the entry-length and exit-length problems. It is concluded that the overall problem is a complex one, since it depends to a large measure on the criterion one uses for “fully-developed” flow. If a fairly crude criterion is used, fluid elasticity is found to decrease the entry-length and increase the exit-length.