Lubrication Approximation of Pressure-Driven Viscoelastic Flow in a Hyperbolic Channel

We investigate theoretically the steady incompressible viscoelastic flow in a hyperbolic contracting channel. The fluid viscoelasticity is modeled using the upper convected Maxwell (UCM), Oldroyd-B, Phan-Thien and Tanner (PTT), Giesekus, and the finite elasticity non-linear elastic dumbbell with the Peterlin approximation (FENE-P) models. We first develop the general governing equations for flow within a non-deformable channel whose cross section varies with the distance from the inlet. We then exploit the classic lubrication approximation, assuming a small aspect ratio of the channel to simplify the original governing equations. The final equations, which we formulate in terms of the stream unction, are then solved analytically using a high-order asymptotic scheme in terms of the Deborah number, De, and the formulas for the average pressure drop are derived up to eight orders in De. The accuracy of the original perturbation solution is enhanced and extended over a wide range of parameters by implementing a convergence acceleration method for truncated series. Furthermore, convergence of the transformed solutions for the average pressure drop is demonstrated. The validity and accuracy of the theoretical results is independently confirmed through comparison with numerical results from simulations performed using high-order finite differences and pseudospectral methods. The results reveal the decrease in the average pressure drop with increasing the Deborah number, the polymer viscosity ratio, and the ratio of the inlet to the outlet height. We also show that the fundamental UCM and Oldroyd-B models can predict the major viscoelastic phenomena for this type of internal and confined lubrication flows, while the effect of the rheological parameters of the PTT, Giesekus, and FENE-P models on the results is minor.