Falling liquid films in narrow tubes: occlusion scenarios

Linear stability and nonlinear dynamics in a long-wave model of film flows inside a tube in the presence of surfactant

“…They found that travelling wave solution branches could be readily found for thin films; as the film thickness increased, however, a turning point in the family of solutions was reached, and beyond this point no solutions could be found. These turning points have been shown to correspond well with the critical film thickness separating plug formation from no plugs in both experiments and model simulations in a variety of models (Camassa et al 2014(Camassa et al , 2016Ding et al 2019;Dietze, Lavalle & Ruyer-Quil 2020).…”

“…(2019) or Dietze et al. (2020)) that the maximum film thickness for which travelling wave solutions can be found decreases with increasing period size, with the maximum film thickness appearing to approach some limiting value as . This can be understood by noting that turning point solutions for different period size consist of identical wave profiles, but with different lengths of substrate surrounding the wave; figure 7 in Camassa et al.…”

“…The reason for this choice has to do with the sensitivity of turning point to period size. It has been shown (see, e.g., figure 6 of Camassa et al (2016), figure 7 of Ding et al (2019) or Dietze et al (2020)) that the maximum film thickness for which travelling wave solutions can be found decreases with increasing period size, with the maximum film thickness appearing to approach some limiting value as T → ∞. This can be understood by noting that turning point solutions for different period size consist of identical wave profiles, but with different lengths of substrate surrounding the wave; figure 7 in Camassa et al (2016) shows this for two different period size turning point solutions.…”

Superconfined falling liquid films: linear versus nonlinear dynamics

“…We perform linear stability calculations by solving the dispersion equation , obtained by linearizing (2.3) around , for at a given : We also compute nonlinear TWS, which remain unaltered in a reference frame moving at the wave speed , through numerical continuation based on (2.3), using Auto07P (Doedel 2008). Our code allows us to track TWS at the linearly most-amplified angular frequency , via the following constraints (Dietze, Lavalle & Ruyer-Quil 2020): Finally, we check the stability of nonlinear TWS via transient computations based on (2.3), using either periodic or inlet/outlet boundary conditions (Lavalle et al. 2020).…”

A long-wave model for film flow inside a tube with slip

“…Long-wave models contain more complicated nonlinear terms, arising from the cylindrical geometry of the tube, that can improve the quantitative (and in some cases qualitative) agreement between model and experiments; examples of such models for flow along the interior or exterior of a cylinder include those of Lin & Liu (1975), Craster & Matar (2006) and Camassa et al (2012). Integral boundary layer models are able to successfully model flows at moderate Reynolds numbers; see, for example, Dietze & Ruyer-Quil (2015) and Dietze, Lavalle & Ruyer-Quil (2020) for models of flow inside a tube. The current study is focused on the flow of highly viscous films and is primarily concerned with long-wave models, with discussion of a thin-film counterpart model as well.…”