Stability and nonlinear dynamics of three-layer viscous films inside a vertical cylindrical tube

Abstract

We investigate the dynamics and stability of three immiscible viscous liquid layers coating the interior of a vertical cylindrical tube, a configuration relevant to stratified core--annular transport processes. A long-wave asymptotic analysis yields a coupled system of nonlinear evolution equations governing the motion of the three interfaces. Linear stability analysis predicts a persistent long-wave instability, the capillary (Rayleigh--Plateau) instability of the air--core interface, together with secondary finite-wavenumber instability bands that emerge from interfacial coupling in certain parameter regimes. These stability characteristics depend sensitively on the layer thicknesses, viscosity ratios, and surface tension parameters, and include mode-switching associated with competing maxima in the dispersion relation. Nonlinear simulations reveal three distinct dynamical outcomes: saturation to finite-amplitude travelling waves, air-core closure through plug formation, and rupture of the intermediate liquid layer while the air core remains open. The intermediate-layer rupture mechanism is unique to the three-layer configuration which has no analogue in one- or two-interface cylindrical film flows. Numerical continuation is used to compute branches of travelling-wave solutions and their associated limit points. Comparison with time-dependent simulations shows that travelling-wave branches successfully predict the transition from saturated waves to plug formation, but do not capture the distinct rupture mechanism associated with collapse of the intermediate layer.

36 pages, 21 figures. Submitted to the Journal of Fluid Mechanics