Fully-developed pipe and planar flows of multimode viscoelastic fluids

Two solutions are presented for fully-developed pipe and planar flows of multimode viscoelastic models. The fluids have a Newtonian solvent contribution and the polymer modes are described by the Phan-Thien-Tanner (PTT), the FENE-P or the Giesekus equation. The first solution is exact and can handle any number of modes, but is only semi-analytical. The second solution, which is presented only for the PTT model with a linear stress coefficient and the FENE-P model, can also handle any number of modes. It is based on a truncated series expansion and is completely analytical, but provides only an approximated solution. The complexity of the multimode solutions is investigated first with the exact semi-analytical method and it is shown that at high Deborah number flows the high-order stresses can become as important as the stress of the first mode. It is also under these conditions that the approximated analytical solution deviates from the exact semi-analytical solution. A criterion for the accurate use of the approximated solution is presented. Fortran codes are provided to obtain these solutions at the internet address at the end.