Applied Maths Seminars at Courant
During the 2021-2022 academic year, I was co-organizing the Applied Maths Seminars (AMS) at the Courant Institute. For record purposes, the schedule of the seminars (with invited speakers and abstracts) can be found on this page.
Schedule
Could convection in porous media explain the geysers of Enceladus?
September 17, 2021 -- Samuel Boury (New York University, Courant Institute of Mathematical Sciences)
Motivated by Enceladus cryovolcanism and possible shear heating along its south pole fractures, we perform a theoretical and numerical study of boundary-layer convection along a vertical heated wall in a bounded ideal mushy region. By focusing on this simple model, we demonstrate the existence of four regions with different regimes and scalings that are studied asymptotically, showing a good agreement between the theory and the numerical simulations. Close to the heated wall, the convection in the mushy layer is similar to a rising buoyant plume abruptly stopped at the top, leading to increased pressure and temperature in the upper region, that could be the ignition of Enceladus' geysers.
Singular measures and information capacity of turbulent cascades
September 24, 2021 -- Gregory Falkovich (Weizmann Institute)
How weak is the weak turbulence? We analyze turbulence of weakly interacting waves using the tools of information theory. It offers a unique perspective for comparing thermal equilibrium and turbulence. The mutual information between resonant modes in a finite box is shown to be stationary and small in thermal equilibrium, yet to grow with time in weak turbulence. We trace this growth to the concentration of probability on the resonance surfaces, which can go all the way to a singular measure. The surprising conclusion is that no matter how small is the nonlinearity and how close to Gaussian is the statistics of any single amplitude, a stationary phase-space measure is far from Gaussian, as manifested by a large relative entropy. At the end I shall describe a new class of the so-called Fibonacci models that describe resonantly interacting waves and allow turbulence close to thermal equilibrium. (0.1103/PhysRevLett.125.104501)
Surface gravity waves enhance the settling and dispersion of finite-size particles
October 1, 2021 -- Michelle DiBenedetto (University of Washington)
Surface gravity waves transport mass through a process referred to as Stokes drift. Finite-sized particles of arbitrary shape exhibit additional Stokes drift-like phenomena. In this talk, I explore the effects of particle inertia and shape on particle behavior in waves. I demonstrate both an angular analogue of Stokes drift for non-spherical particles and a vertical Stokes drift for settling particles. Experimental observations are compared with these analytical results, and these behaviors are discussed in the context of microplastic transport in the ocean.
Untangling in time: designing time varying applied fields to reveal interior structure
October 15, 2021 -- Graeme Milton (University of Utah)
In two phase materials, each phase having a non-local response in time, we were surprised to discover that for appropriate driving fields the response somehow untangles at specific times, allowing one to directly infer useful information about the geometry of the material, such as the volume fractions of the phases. This rests on the existence of approximate, measure independent, linear relations between the values that Markov functions take at a given set of possibly complex points, not belonging to the interval [-1,1] where the measure is supported. The problem is reduced to simply one of polynomial approximation of a given function on the interval [-1,1]. In the context of the motivating problem, the analysis also yields bounds on the response at any particular time for any driving field, and allows one to estimate the response at a given frequency using an appropriately designed driving field that effectively is turned on only for a fixed interval of time. The approximation extends directly to Markov-type functions with a positive semi-definite operator valued measure, and this has applications to determining the shape of an inclusion in a body from boundary flux measurements at a specific time, when the time-dependent boundary potentials are suitably tailored. This is joint work with Ornella Mattei and Mihai Putinar.
Dragonflies and Fruit flies: From equations to behavior
October 29, 2021 -- Jane Wang (Cornell University)
Jane has been filming dragonflies this summer in the Applied Math Lab and has also started filming flies this semester; she will explain why.
Data-driven methods for identifying models of complex biological systems
November 5, 2021 -- Niall Mangan (Northwestern University)
Inferring the structure and dynamical interactions of complex biological systems is critical to understanding and controlling their behavior. I am interested in discovering mechanistic and informative models, assuming I have time-series data of important state variables and knowledge of the possible types of interactions between state variables. The problem is then selecting which interactions, or model terms, are most likely responsible for the observed dynamics. Several challenges make model selection difficult including nonlinearities and unmeasured state variables. I will discuss methods for reframing these problems so that sparse model selection is possible. I will discuss preliminary results on parameter estimation, model selection, and experimental design to characterize a spatially organized metabolism pathway in bacteria and generic chaotic systems. Parameter estimation and model selection are challenging in these cases because only some of the metabolite pools or state variables can be measured and the other variables are hidden or latent. We use a combination of data assimilations techniques and sparse optimization to perform model selection. Experimental design is enabled through sensitivity analysis of the model manifold.
Asymmetric Rectified Electric Field (AREF)
November 12, 2021 -- Aref Hashemi Amrei (New York University, Courant Institute of Mathematical Sciences)
We demonstrate that application of an oscillatory electric field to a liquid yields a long-range steady field, provided the ions present have unequal mobilities. The main physics is illustrated by a two-ion harmonic oscillator, yielding an asymmetric rectified field whose time average scales as the square of the applied field strength. Computations of the fully nonlinear electrokinetic model corroborate the two-ion model and further demonstrate that steady fields extend over large distances between two electrodes. The heretofore unsuspected existence of a long-range steady field helps explain several long-standing questions regarding the behavior of particles and electrically induced fluid flows in response to oscillatory potentials.
The Biomechanics and Evolutionary Origins of Flight in Animals
November 19, 2021 -- Robert Dudley (Berkeley)
What is the aerodynamic use of half a wing? More generally, how do novel structures and behaviors evolve? Controlled aerial behaviors in ant workers and other insects of the tropical rain forest canopy demonstrate directed gliding in the complete absence of wings. Importantly, tree-dwelling bristletails (the sister group to the winged insects) also exhibit righting responses and directed descent while falling; our recent work on wingless stick insect larvae elucidates the mechanisms of such aerial behavior. Ontogenetic, biomechanical, paleontological, and phylogenetic analyses furthermore suggest that controlled aerial behaviors preceded the origin of wings in birds, indicating functional aerodynamics for only partially feathered structures and for rudimentary flapping kinematics. Our recent use of a robotic Archaeopteryx similarly demonstrates biomechanical functionality of the intermediate-winged condition, consistent with arboreal and gravitationally assisted origins of flight in all volant taxa. I will also present some ongoing work on aerial maneuverability in hummingbirds, describing a variety of experimental perturbations to elicit extreme examples of flight control (e.g., flight through apertures).
Life in a Tight Spot: How Bacteria Swim, Disperse, and Grow in Crowded Spaces
December 3, 2021 -- Sujit Datta (Princeton University)
Bacterial motility and growth play central roles in agriculture, the environment, and medicine. While bacterial behavior is typically studied in bulk liquid or on flat surfaces, many bacterial habitats --e.g., soils, sediments, and biological gels/tissues-- are complex and crowded spaces. In this talk, I will describe my group's work using tools from soft matter physics to address this gap in knowledge. In particular, using studies of E. coli in transparent 3D porous media, we demonstrate how confinement in a crowded medium fundamentally alters bacterial behavior. In particular, we show how the paradigm of run-and-tumble motility is dramatically altered by pore-scale confinement, both for cells performing undirected motion and those performing chemotaxis, directed motion in response to a chemical stimulus. Our porous media also enable precisely structured multi-cellular communities to be 3D printed. Using this capability, we show how spatial variations in the ability of cells to perform chemotaxis enable populations to autonomously stabilize large-scale perturbations in their overall morphology. Finally, we show how when the pores are small enough to prevent cells from swimming through the pore space, expansion of a community via cellular growth and division gives rise to distinct, highly-complex, large-scale community morphologies. Together, our work thus reveals new principles to predict and control the organization of bacteria, and active matter in general, in complex and crowded environments.
Abrupt Circulation Changes in the Tropical Atmosphere
December 10, 2021 -- Corentin Herbert (École Normale Suérieure de Lyon)
A crucial point in the public debate about global warming is the existence of "tipping points", i.e. bifurcations potentially leading to abrupt climate change. Paleoclimate records suggest that such events have occurred in Earth's past, on timescales which do not exceed a decade. Yet, it remains virtually unknown whether the large-scale atmospheric circulation (the fastest component of the climate system) may undergo such transitions. In this talk I will discuss the possibility of abrupt changes in the circulation of the tropical atmosphere: specifically, the reversal of the mean zonal winds ("superrotation") and the collapse of the meridional circulation (Hadley cell). I will discuss theoretical mechanisms, based in particular on Rossby wave resonance, and their relevance for the Earth using numerical simulations across a hierarchy of models of increasing complexity.
Twisted topological tangles or: the knot theory of knitting
February 4, 2022 -- Sabetta Matsumoto (Georgia Institute of Technology)
Imagine a 1D curve, then use it to fill a 2D manifold that covers an arbitrary 3D object -- this computationally intensive materials challenge has been realized in the ancient technology known as knitting. This process for making functional materials 2D materials from 1D portable cloth dates back to prehistory, with the oldest known examples dating from the 11th century CE. Knitted textiles are ubiquitous as they are easy and cheap to create, lightweight, portable, flexible and stretchy. As with many functional materials, the key to knitting's extraordinary properties lies in its microstructure. At the 1D level, knits are composed of an interlocking series of slip knots. At the most basic level there is only one manipulation that creates a knitted stitch -- pulling a loop of yarn through another loop. However, there exist hundreds of books with thousands of patterns of stitches with seemingly unbounded complexity. The topology of knitted stitches has a profound impact on the geometry and elasticity of the resulting fabric. This puts a new spin on additive manufacturing – not only can stitch pattern control the local and global geometry of a textile, but the creation process encodes mechanical properties within the material itself. Unlike standard additive manufacturing techniques, the innate properties of the yarn and the stitch microstructure has a direct effect on the global geometric and mechanical outcome of knitted fabrics.
Coarse-Grained Models out of Equilibrium
February 25, 2022 -- Tanya Schilling (Institute of Physics, University of Freiburg)
Active matter, responsive materials and materials under time-dependent load are systems out of thermal equilibrium. To construct coarse-grained models for such systems, one needs to integrate out a distribution of microstates that evolves in time. This is a challenging task. In this talk, we we review recent developments in theoretical approaches to the non-equilibrium coarse-graining problem, in particular, time-dependent projection operator formalisms and numerical schemes to construct explicitly time-dependent memory kernels.
Different aspects of swimming and schooling in fish
March 11, 2022 -- Benjamin Thiria (ESPCI, Université Paris-Cité)
In this talk, I will show three different experiments on live fish swimming and schooling. I will first try to bring some answers to the old fundamental question raised by fluid dynamicists: is there a hydrodynamic advantage of swimming in schools? I will then give some details about the intermittent dynamics that characterize the swimming of small and medium size fish and discuss the dependence of the global cost of transport to the parameters of the swimming. At last, I will show the role of vision in the formation of spontaneous global behavior in a large population of fish.
A thermomechanical model for frozen sediments
March 25, 2022 -- Colin Meyer (Dartmouth College)
Ice-infiltrated sediment, known as a frozen fringe, leads to phenomena such as frost heave, ice lenses, and meters of debris-rich ice under glaciers. Understanding the dynamics of frozen fringe development is important as frost heave is responsible for damaging infrastructure at high latitudes; frozen sediments at the base of glaciers can modulate glacier flow, influencing the rate of global sea level rise; and frozen water ice exists within the sediments of the top several meters on Mars and in places on the Moon. Here we study the fluid physics of interstitial freezing water in sediments and focus on the conditions relevant for subglacial and planetary environments. We describe the thermomechanics of liquid water flow through and freezing in ice-saturated frozen sediments. The force balance that governs the frozen fringe thickness depends on the weight of the overlying material, the thermomolecular force between ice and sediments across premelted films of liquid, and the water pressure within liquid films that is required by flow according to Darcy's law. Our model accounts for premelting at ice-sediment contacts, partial ice saturation of the pore space, water flow through the fringe, the thermodynamics of the ice-water-sediment interface, and vertical force balance. We explicitly account for the formation of ice lenses, regions of pure ice that cleave the fringe at the depth where the interparticle force vanishes.
Active Matter for Self Assembly
April 1, 2022 -- Jérémie Palacci (IST Austria)
Biological systems show remarkable and robust self-assembly: bacteria form colonies, cells reshape and muscle fibers collectively contract… Those phenomena stem from the non-equilibrium nature of living matter, a prototypical example of active matter in which self-driven units convert an energy source into useful motion and work. Inspired by the biological world, we will show how we can build and control man-made materials powered from within. As a first example, we will discuss the effect of an active bacterial bath on the aggregation of attractive microbeads. We will notably show that we can control the morphology of the aggregates, possibly programming the mechanical response of a soft material.
Colloidal Robotics: Engineering Autonomous Behaviors of Self-Propelled Particles
April 15, 2022 -- Kyle Bishop (Columbia University)
Mobile robots combine sensory information with mechanical actuation to move autonomously through complex environments and perform specific tasks (e.g., a robot vacuum cleaner). The miniaturization of such robots to the size of living cells (ca. 2-40 um) is actively pursued for applications in biomedicine, materials science, and environmental sustainability. In pursuit of these "microrobots", we seek to understand the many mechanisms underlying the self-propulsion of colloidal particles through viscous fluids. Building on this understanding, we seek to design active particles capable of autonomous behaviors such as navigation of structured environments. In this talk, I discuss two recent efforts – on Quincke oscillators and magnetic topotaxis, respectively – that highlight these complementary aims to understand and design active colloids. In part one, I explain how static electric fields drive the oscillatory motion of micron-scale particles commensurate with the thickness of a field-induced boundary layer in nonpolar electrolytes. In part two, I describe how spatially uniform, time-periodic magnetic fields can be designed to power and direct the migration of ferromagnetic spheres up local gradients in surface topography.
Mathematical Models of Organelle Size Control and Scaling
April 22, 2022 -- Thomas Fai (Brandeis University)
To address the fundamental question of how cells maintain the size of their organelles despite the constant turnover of proteins and biomolecules, we analyze mathematical models of organelle size control rooted in the physicochemical principles of transport, chemical kinetics, and force balance. In particular, we consider how a model based on osmotic force balance predicts the stable nuclear-to-cell size ratio observed experimentally in yeast. In addition, we consider flagellar length control in the unicellular green algae Chlamydomonas reinhardtii and develop a minimal model in which diffusion gives rise to a length-dependent concentration of depolymerase at the flagellar tip. By studying the mathematical symmetries of competing models, we arrive at general principles of organelle size control.
Soft Boundaries in Two-Phase Flows
May 6, 2022 -- Emilie Dressaire (University of California San Barbara)
Fluid flows can generate stresses sufficiently large to destabilize interfaces, deform and fracture solid substrates. While the interactions between fluid flow and surface deformation are typically studied with a single fluid, two-phase flows are common in agricultural, manufacturing, and energy processes. The two immiscible liquids are separated by an interface, which results in capillary stresses. In this talk, I will present two recent studies on multiphase flows confined by deformable, soft boundaries. First, we will revisit the classical displacement flow problem in a growing fracture. The successive injection of two fluids in a brittle elastic material drives the formation and growth of a penny-shape fracture. Using model experiments, we characterize the fracture dynamics. The viscous dissipation in the fluid, the elastic stresses and toughness of the matrix control the flow in the fluid-filled fracture. We demonstrate that the role of the two fluids depends on the regime of fracture propagation, fluid-controlled vs matrix-controlled. Then, I will present results on the atomization of oil in water emulsions. We study the impact of a drop of emulsion on a small target which lead to the formation and atomization of a liquid sheet. Here the dispersed phase is confined between air/liquid interfaces. The oil droplets modify the dynamics of an aqueous liquid sheet and its break-up into droplets. As the viscosity of the oil phase decreases, the liquid sheet becomes more unstable, and holes become primarily responsible for the atomization of the sheet. These studies illustrate the complex couplings between two-phase flows and substrate deformation.