We develop a novel implementation of the lubrication approximation for viscoelastic internal flows exhibiting both shear and strong extensional characteristics. We illustrate its application to two-dimensional symmetric channels with rigid walls. The approach is based on a single perturbation scheme in terms of the channel aspect ratio and is applied directly to the original dimensionless governing equations, without rescaling the polymer-related variables (extra-stress or conformation tensors). The implementation enables higher-order corrections to be computed more efficiently than in previous approaches and captures strain-hardening behavior in purely extensional regions of the flow. Analytical expressions for the integrated (over the cross section) pressure drop required to drive the flow at a constant volumetric rate reveal higher-order effects not previously reported in the literature. For a two-dimensional symmetric hyperbolic channel, these additional contributions increase the driving pressure, unlike the purely viscoelastic terms that can also be obtained using standard lubrication formulation. Excellent agreement with full two-dimensional simulations Mahapatra et al., J. Fluid Mech. 1009, A12 (2025) across the valid parameter range demonstrates the accuracy, robustness, and practical utility of the framework for analyzing confined flows in slender geometries with combined shear and extensional characteristics.
Housiadas et al. (Wed,) studied this question.