General evaluation of the pressure gradient for lubrication flows in varying channels with applications to linear and hyperbolic contractions

Under the classic lubrication approximation, we develop a unified framework for evaluating the pressure gradient of an incompressible, isothermal viscoelastic fluid in a symmetric channel with slowly varying geometry, including inertia. Exploiting the independence of the pressure gradient from the wall-normal coordinate—a property absent in general 2D planar or 3D axisymmetric flows—we derive multiple integral expressions for the pressure gradient and the corresponding average pressure drop required to maintain a constant flow rate. The derivations use the momentum balance formulated via the extra-stress tensor, providing a flexible, formal, and rigorous procedure, and the physical significance of each expression is discussed.

To bypass choosing among these expressions, we introduce a new set of lubrication equations based on a streamfunction, mapped coordinates, and transformed polymer extra-stress components. This formulation automatically satisfies the continuity equation, the constraints due to fluid incompressibility, the boundary conditions, and the flow symmetries, allowing the pressure gradient to be determined a posteriori and providing a tool for consistency and accuracy checks.

The equivalence of the integral expressions is illustrated in two representative cases: (i) Newtonian inertial flow in a linearly contracting channel, and (ii) viscoelastic inertialess flow in a hyperbolic contraction. In both cases, the predicted average pressure drop agrees very well with high-order asymptotic solutions post-processed via Padé approximants, high-accuracy spectral simulations, and DNS results from the literature. The framework provides a rigorous, general, and computationally robust tool for analyzing lubrication flows of viscoelastic fluids and can be easily extended to other complex fluids and broader flow conditions.