Spatiotemporal instability and topological bifurcations of a weakly elastic film flowing down a vertical fibre

This paper investigates the temporal, spatiotemporal and nonlinear instability of a weakly viscoelastic liquid film flowing down a vertical cylindrical fibre. The fluid is described by the Walters’ liquid B model, and the analysis combines long-wave asymptotic modelling, a long-wave-weighted Orr–Sommerfeld stability analysis and nonlinear numerical simulations. Linear stability analysis reveals that while viscoelasticity is temporally destabilising, it significantly expands the domain of absolute instability in parameter space, facilitating the upstream propagation of disturbances. In the nonlinear regime, viscoelasticity exhibits a distinct dual effect: it monotonically suppresses the saturation amplitude r Subscript italic max r max rₘₐₓ with increasing elasticity upper M M M , whilst accelerating the onset of finite-time singularity during rupture dominated by absolute instability at larger elasticity parameters. The rupture dynamics obeys a self-similar scaling h Subscript italic min Baseline tilde left parenthesis t Subscript c Baseline minus t right parenthesis Superscript m h min ∼ ( t c − t ) m hₘᵢₙ ~ (tc - t)ᵐ . Driven by gravity-induced axial stretching, the system deviates from classical pure-capillary pinch-off, shifting the Newtonian baseline to a friction-dominated drainage regime with m almost equals 0.50 m ≈ 0.50 m ≈ 0.50 and an asymmetric spatial collapse. Based on this gravitationally modified baseline, increased elasticity further reduces the temporal exponent m m m , indicating that elastic effects accelerate the local thinning dynamics while preserving the main upstream similarity structure. Travelling-wave solutions further reveal elasticity-induced changes in the phase-space topology of the saturated states: in the solitary-wave regime, increasing elasticity promotes a transition from near-homoclinic trajectories to stable limit cycles, whereas in the parasitic-wave regime it produces more complex homoclinic-like trajectories associated with satellite droplets in physical space.