A novel implementation of lubrication approximation for viscoelastic internal flows with extensional characteristics

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.