Modeling Rayleigh–Taylor instability in viscoelastic liquid film flow

This study presents a first-order weighted-residual model for viscoelastic thin-film flow on inverted substrates, incorporating van der Waals interactions. The model is derived via systematic scaling analysis, boundary-layer approximations, and a Galerkin weighted-residual method based on the Oldroyd-B constitutive framework. It achieves a balance between computational efficiency and accurate representation of viscoelastic effects. A notable feature is the ability to independently adjust the Deborah number (De) and the retardation ratio (r), addressing the limitation of Benney-type models that depend only on the combined parameter M=(1−r)De. The model remains accurate for large De. Analysis of the principal dimensionless parameters (De, S, r, A, Ga) clarifies the stability behavior: increasing De enhances elastic instability, whereas a larger r weakens viscoelastic effects. The cutoff wavenumber k_c depends on gravity (Ga), surface tension (S), and van der Waals forces (A), but not on De or r. Validation against linearized Navier–Stokes (LNS) solutions and direct numerical simulations (DNS) shows closer agreement than the Benney-type model, particularly at high De. The model also reproduces ultrathin-film rupture via cusp formation induced by van der Waals forces and predicts the scaling law h_min∝(tr−t)^1/5 near rupture.