Center for Fluid Mechanics, Division of Applied Mathematics Fluids and Thermal Systems, School of Engineering Joint Seminar Series
Abstract: Across many scientific fields, imaging is an integral tool that leads to the discovery and description of new phenomena. On the other hand, the science of imaging itself has advanced to push the boundaries of experimental techniques. This talk will present research along each of these fronts as we discuss the physics of water skipping elastic spheres and novel three-dimensional (3D) light field imaging for fluid flows (and beyond). In the former, we make use of high speed imaging to discover the physical mechanisms responsible for the magnificent water skipping properties of certain highly deformable spheres. It will be shown that the highly compliant nature of the spheres is responsible both for the enhanced skipping behavior and an unexpected fluid-structure interaction mode involving material waves that propagate in the spheres. The latter research relates to the development of a novel 3D imaging method and its application to fluid flows densely packed with bubbles, droplets and/or particles. The technique enables measurements that are difficult or impossible to achieve with most existing quantitative imaging methods. Finally, preliminary results from the extension of the 3D imaging method to field study of the collective behavior of animal groups are presented.
ICERM / Applied Math Talk
Abstract: CFSE analysis of proliferating cell populations is a tool of growing popularity for the study of cell division and division-linked changes in cell behavior. Partial differential equation (PDE) models are presented to describe lymphocyte dynamics in a CFSE proliferation assay. Previously unknown physical mechanisms accounting for the exact degree of dye dilution by division are explained in the context of cellular auto fluorescence. The rate at which label decays/diffuses out of the cell is also quantified using a Gompertz decay process. A new class of division-dependent compartmental models allows one to separate proliferation and death rates from intracellular label dynamics. By fitting the new models to the commonly used histogram representation of the data, it is shown that these improvements result in models with a strong physical basis which are still fully capable of replicating the behavior observed in in vitro data. Some mathematical aspects of the corresponding inverse problems are discussed. The new models provide quantitative techniques that are useful for the comparison of CFSE proliferation assay data across different data sets and experimental conditions.
Scientific Computing Seminar
Abstract: Development of efficient algorithms with rigorous analysis for partial differential equations (PDE) on domains and surfaces requires knowledge of the regularity of solutions of the PDE. In this talk, for the Navier-Stokes equations on rotating spheres we discuss (i) the global regularity for real and complexified time; (ii) a high-order algorithm with stability and convergence analysis; (iii) long time simulation of a benchmark atmospheric flow model and justification of a turbulence theory.
Abstract: We study parabolic equations which consist on a drift term (a vector field times the gradient) plus a diffusion term (either the laplacian or the fractional Laplacian). We analyze what assumptions on the drift would assure that the solution remains continuous for positive time. A particularly interesting case is when the drift is a divergence free vector field, since these appear frequently in equations from fluid mechanics. Assuming that the drift is bounded respect to some norm which is invariant by the scaling of the equation gives Holder continuity estimates in many cases. We will prove that when this scaling condition is violated, discontinuities can form in finite time, even if the drift is divergence free. A notable exception is for an equation with classical diffusion (with the usual Laplacian), in 2 space dimensions, and a drift which is independent of time. For that case a modulus of continuity is obtained for any divergence free drift in $L^1$ (which is highly supercritical).
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