Instabilities and bifurcations: The good, the bad and the beauty
Author: Prof. Dr. Karin Jacobs
"What are you researching?" - "Instabilities and bifurcations!" - "Instabilities? You mean risk analyses?" Small talk like this is something many of us may have had before. Dealing with "instabilities" sounds weak, fragile, and a bit pessimistic, too. "Bifurcations," on contrast, sounds stronger, more promising (because unknown?), and more optimistic. Yet in our research work, neither of these attributes are there. As an experimentalist, I come more in contact with "instabilities", and having been in this "business" for more than two decades, I suffer from a "déformation professionelle" and see instabilities everywhere. Is that a good thing or a bad thing? Depends on to whom you are talking…
However, instabilities in experiments or simulations often carry an irresistible beauty. That's what happened to me when I saw "my first instability": a dewetting thin polymer film. The analysis took some time, but the result is still important to my research today! Multiple instabilities are for instance also involved in bacterial adhesion, which in my group is characterized by atomic force microscopy in single-cell force spectroscopy mode. So I’ll take
you on a journey through the good, the bad, and the beauty of instabilities!
Image of an instability of a thin, dewetting film that is not yet fully understood…
Author: Prof. Dr. Andrew Hazel
The behaviour of fluids is beautiful, complex and difficult to predict. If a viscous liquid containing gas bubbles is driven through a confined geometry, it exhibits complex nonlinear behaviour, even in the absence of fluid inertia. The behaviour is driven by interaction between the geometry, capillary forces at the liquid-gas interface, and viscous forces within the bulk liquid. The system is relatively simple, yet it exhibits a remarkable variety of nonlinear phenomena, making it ideally suited to studies of bifurcations and instabilities in fluid dynamics. At the Manchester Centre for Nonlinear Dynamics, we use a combined experimental and theoretical approach to study such flows through Hele-Shaw channels with small axially-uniform variations in channel depth. From a theoretical point of view, the system is attractive because it can be accurately described by depth-averaged equations. Moreover, the state of the system is characterised by the bubbles' shapes and relative locations within the channel, which are easy to access experimentally. An unusual feature of the system is its frequent topology changes as bubbles break up into smaller bubbles that can later recombine or not. Topology changes alter the entire solution structure, meaning that a single bifurcation diagram cannot capture the system's dynamics. For a fixed set of control parameters (e.g. flow rate and total bubble volume) the solution structure can change several times over the course of an experiment. In addition, the system shows heightened sensitivity to perturbations in regimes where the bubble is likely to break up. We find multiple steadily propagating and oscillatory states, regions of multi-stability and practical unpredictability, where the final outcome is extremely sensitive to minor perturbations. We use numerical bifurcation techniques to determine the structures that organise the dynamics of the system, including key roles played by unstable periodic orbits and weakly unstable steadily propagating solutions. We find a strong correspondence between single and multiple bubble solutions which allows us to predict and construct non-trivial stable multi-bubble states. Although our interest in these systems is motivated by the desire to understand complex nonlinear dynamics, there are applications to microfluidics. In this talk, I will present a survey of what we know about the system so far, it's connection to other interfacial flows and present some of the open challenges.
Bifurcation Analysis of Ocean Flows
Author: Prof. dr. ir. Henk A. Dijkstra
The global ocean circulation is a complex three-dimensional flow, but it can roughly be described by a mostly wind-driven surface
flow and a density affected overturning flow. Each of these components is susceptible to large-scale instabilities due to the existence of several feedbacks.
In this talk, I will give an overview of these instability phenomena and the numerical techniques to determine the associated bifurcations, with a focus on the western boundary currents (e.g. the Gulf Stream) and the Atlantic Meridional Overturning Circulation.