Higher-order lubrication analytic solutions of viscoelastic flows in slowly varying slits

A regular perturbation method in which the small parameter is taken as the characteristic frequency of variation of a two-dimensional slit wall height is applied to the creeping flow of a viscoelastic fluid. The Upper-Convected Maxwell model has been chosen as the constitutive equation. The equations of motion are expressed in terms of the stream function as the only unknown. The stream function is expressed then as a regular perturbation expansion up to the fourth order in the small parameter. When these expansions are written for slits with slowly varying walls of any shape, curvature and higher-order derivatives appear in the expansion solution. From the stream function solution, we distinguish two different types of viscoelastic higher-order effects: one is symmetric and the other is a loss of fore-and-aft flow symmetry due to the convected derivatives appearing in the constitutive equation. The expansion is formally validated with a ratio between two consecutive higher-order terms. It shows that the accuracy of the perturbation will depend on four factors: the small parameter ϵ, the Weissenberg number We0, the local depth h and the shape of the slit. In some instances, Padé approximants can restore a diverging perturbation expansion and extend the range of viscoelastic levels for which the expansion is usable. Results for diverging and for converging flows in a wedge, as well as flows in corrugated slits, are checked against Finite Elements Method calculations. At the same level of viscoelasticity, the stress profiles are completely different in a diverging flow from a converging flow. In particular, the shear stress component of the stress tensor becomes non-monotonic in divergent flows.