Analytical solution of steady 2D wall-free extensional flows of UCM fluids

Abstract

The general analytical solution for the two-dimensional steady planar extensional flow with wall-free stagnation point is obtained for viscoelastic fluids described by the upper convected Maxwell model providing the stress and pressure fields. The two normal stress fields contain terms that are unbounded for vertical bar a vertical bar De < 1/2, vertical bar a vertical bar De > 1/2 and even for any vertical bar a vertical bar De, where De denotes the Deborah number and vertical bar a vertical bar De denotes the Weissenberg number, but the pressure field is only unbounded for vertical bar a vertical bar De < 1/2. Properties of the first invariant of the stress tensor impose relations between the various stress and pressure coefficients and also require that they are odd functions of vertical bar a vertical bar De. The solution is such that no stress singularities exist if the stress boundary conditions are equal to the stress particular solutions. For vertical bar a vertical bar De < 1/2 the only way for the pressure to be bounded is for the stresses to be constant in the whole extensional flow domain and equal to those particular stresses, in which case the loss of stress smoothness, reported previously in the literature, does not exist. For vertical bar a vertical bar De > 1/2, however, the pressure remains bounded even in the presence of stress singularities. In all flow cases studied, the stress and pressure fields are contained by the general solution, but may require some coefficients to be null.