Modelling falling viscoelastic liquid films at moderate Weissenberg number and Reynolds number

It is well-known that viscoelasticity plays a stabilising role in thin liquid film flow at moderate Reynolds numbers, which, however, cannot be resolved by current long-wave models. All these long-wave models can only predict the destabilising effect of viscoelasticity. This paper proposes a novel integral boundary layer model for a two-dimensional Oldroyd-B liquid film flow down a vertical plane. It consists of four dynamical variables – the film thickness h h h, local flow rate q q q, depth-integral normal viscoelastic stress upper A A A, and depth-integral shear viscoelastic stress upper B B B – which successfully capture the destabilising mechanism of viscoelasticity at small Reynolds numbers and the stabilising mechanism of viscoelasticity at moderate Reynolds numbers by a linear stability analysis. Energy budget analysis reveals that the overall effect of viscoelasticity results from competition between destabilising surface shear stress and stabilising total polymeric stress work. Nonlinear travelling wave analysis reveals that wave speed is enhanced by weak viscoelasticity but suppressed by strong viscoelasticity, while the maximum film thickness grows monotonically with viscoelasticity at low Reynolds numbers, displaying a non-monotonic response characterised by an initial increase followed by a subsequent decrease under moderate Reynolds number conditions. Direct numerical simulations show that our integral boundary layer model is remarkably accurate when the Reynolds number is moderate, which predicts the nonlinear wave speed within 2 percent sign 2 % 2\, \% errors against the full Navier–Stokes equations.