Entry-flow calculations for the Oldroyd-B and FENE equations

Abstract

Entry-flow calculations are presented for the Oldroyd-B and FENE equations in the limit of high contraction ratio in both planar and axisym- metric geometries. In addition to abrupt entry flow a cone-shaped entry flow of half-angle π/4 is also considered. A decoupled finite-difference scheme is used with time stepping to converge to the solution of the non-linear equations. For the solution of the stress equation a streamline integration technique is developed. For the FENE equation it is found that a reordering of the elliptic operator for the stream function enables larger time steps to be taken at high We when the polymers are near full extension. These calculations show that the FENE equation can predict much larger vortices than the Oldroyd-B equation when in the non-linear regime of the FENE spring. No vortices are observed for the Oldroyd-B equation in planar flow and only weak vortices when the flow is axisymmetric. For the cone entry flow, larger vortices are again predicted for the FENE equation but they remain localised near the entry, in a manner similar to lip vortices. Pressure drops are also considered and it is argued that at high We the non-dimensional decrease in pressure drop compared with the Newtonian pressure drop should be linear in We for the Oldroyd-B equation. An estimate of this pressure drop is made in the high We, but low concentration, limits.