Ring waves as a mass transport mechanism in air-driven core-annular flows

Air-driven core-annular fluid flows occur in many situations, from lung airways to engineering applications. Here we study, experimentally and theoretically, flows where a viscous liquid film lining the inside of a tube is forced upwards against gravity by turbulent airflow up the center of the tube. We present results on the thickness and mean speed of the film and properties of the interfacial waves that develop from an instability of the air-liquid interface. We derive a long-wave asymptotic model and compare properties of its solutions with those of the experiments. Traveling wave solutions of this long-wave model exhibit evidence of different mass transport regimes: Past a certain threshold, sufficiently large-amplitude waves begin to trap cores of fluid which propagate upward at wave speeds. This theoretical result is then confirmed by a second set of experiments that show evidence of ring waves of annular fluid propagating over the underlying creeping flow. By tuning the parameters of the experiments, the strength of this phenomenon can be adjusted in a way that is predicted qualitatively by the model

Traveling waves for a model of gravity-driven film flows in cylindrical domains

Traveling wave solutions are studied for a recently-derived model of a falling viscous film on the interior of a vertical rigid tube. By identifying a Hopf bifurcation and using numerical continuation software, families of non-trivial traveling wave solutions may be traced out in parameter space. These families all contain a single solution at a ‘turnaround point’ with larger film thickness than all others in the family. In an earlier paper, it was conjectured that this turnaround point may represent a critical thickness separating two distinct flow regimes observed in physical experiments as well as two distinct types of behavior in transient solutions to the model. Here, these hypotheses are verified over a range of parameter values using a combination of numerical and analytical techniques. The linear stability of these solutions is also discussed; both large- and small-amplitude solutions are shown to be unstable, though the instability mechanisms are different for each wave type. Specifically, for small-amplitude waves, the region of relatively flat film away from the localized wave crest is subject to the same instability that makes the trivial flat-film solution unstable; for large-amplitude waves, this mechanism is present but dwarfed by a much stronger tendency to relax to a regime close to that followed by small-amplitude waves

Tropical intraseasonal variability and the stochastic skeleton method

In this text, modern applied mathematics and physical insight are used to construct the simplest and first nonlinear dynamical model for the Madden-Julian oscillation (MJO), i.e. the stochastic skeleton model. This model captures the fundamental features of the MJO and offers a theoretical prediction of its structure, leading to new detailed methods to identify it in observational data. The text contributes to understanding and predicting intraseasonal variability, which remains a challenging task in contemporary climate, atmospheric, and oceanic science. In the tropics, the Madden-Julian oscillation (MJO) is the dominant component of intraseasonal variability. One of the strengths of this text is demonstrating how a blend of modern applied mathematical tools, including linear and nonlinear partial differential equations (PDEs), simple stochastic modeling, and numerical algorithms, have been used in conjunction with physical insight to create the model. These tools are also applied in developing several extensions of the model in order to capture additional features of the MJO, including its refined vertical structure and its interactions with the extratropics. This book is of interest to graduate students, postdocs, and senior researchers in pure and applied mathematics, physics, engineering, and climate, atmospheric, and oceanic science interested in turbulent dynamical systems as well as other complex systems