Single theoretical model for breakup of viscous thread with and without a fiber

In this study, we introduce a comprehensive theoretical model for viscous liquid systems exhibiting Rayleigh-Plateau instability, accommodating cases both with and without a solid fiber. Employing the lubrication approach and implementing the hydrodynamic interaction at the solid-liquid interface, we formulate one-dimensional evolution equations for the breakup of viscous liquid threads and films on a fiber. Through several validations, we showed that our model exhibits a good agreement with experimental results in comparison to numerical simulations. Finally, our model, which incorporates the flow effect from the inner boundary condition by reconsidering the ansatz of a conventional long-wave approximation, provides a necessary condition for satellite droplet formation and determines the most unstable mode proportional to ${k^{*}}^2$, where $k^{*}$ is the most unstable wavenumber. In addition, we observed that the volume of the satellite droplets exponentially decays depending on the wavenumber. Moreover, our single model integrates the findings of Goren's liquid film on a fiber and Rayleigh's viscous liquid thread, demonstrating its versatility and relevance to a wide range of systems.