Presure driven flow of a nonlinear viscoelastic fluid in a plane channel

The pressure-driven flow of a non-linear viscoelastic fluid characterized by a tensor internal parameter in a plane channel is studied. A full set of exact analytical solutions to this problem obtained in a parametric form is presented. The solutions have been analyzed to identify physically inadmissible solutions. The distributions of anisotropy tensor components, the flow velocity and the velocity gradient throughout the height of the channel have been obtained for different parameters of the rheological model. It is shown that the pressure drops exceeding critical values lead to ambiguity of the solutions, which manifests itself as discontinuities in the profiles of the components of the anisotropy tensor. The same problem has been solved for a two-dimensional case using the finite element method. A comparison of the numerical and analytical results has shown that under subcritical conditions the results agree well. Under supercritical conditions, the analytical solution has discontinuities, whereas the numerical solution is continuous for normal components of the anisotropy tensor and gives underestimated values of the longitudinal velocity in the regime of active loading and overestimated values in the case of unloading. The flow rate - pressure drop characteristics are of a hysteresis character.