# Fluid Mechanics and Waves Seminar - Fall 2022

Seminars are held on Mondays from 2:30 - 3:30PM in CULM 611, unless otherwise noted.

Seminars are held on Mondays from 2:30 - 3:30PM in CULM 611, unless otherwise noted.

For questions about the seminar schedule, please contact Travis Askham.

To join the Fluid Mechanics and Waves seminar mailing list visit https://groups.google.com/a/njit.edu/forum/#!forum/math-fmws/join (Google Profile required). To join the mailing list without a Google Profile, submit the seminar request form.

**Jose Alvarado**, University of Texas at Austin

**Location: **Webex

**Elastoviscous Effects of Soft, Passive Hair Beds**

We are ‘hairy’ on the inside: beds of passive fibers anchored to a surface and immersed in fluids are prevalent in many biological systems, including intestines, tongues, and blood vessels. These hairs are soft enough to deform in response to stresses from fluid flows. Yet fluid stresses are in turn affected by hair deformation, leading to a coupled elastoviscous problem. Here I present measurements of an experimental model system of soft elastomer hairs subject to Stokes flows. We find that hair bending introduces a geometrical nonlinearity, which results in effects such as drag-reduction, rectification, and mixing. This work helps to understand the biological function of biological hair beds, and can be used to develop novel engineering applications.

**Dan Fortunato**, Flatiron Institute

**Location: **CULM 111

**A High-Order Fast Direct Solver for Surface PDEs**

We introduce a fast direct solver for variable-coefficient elliptic partial differential equations on surfaces based on the hierarchical Poincaré–Steklov method. The method takes as input a high-order quadrilateral mesh of a surface and discretizes surface differential operators on each element using a high-order spectral collocation scheme. Elemental solution operators and Dirichlet-to-Neumann maps tangent to the surface are precomputed and merged in a pairwise fashion to yield a hierarchy of solution operators that may be applied in $\mathcal{O}(N \log N)$ operations for a mesh with $N$ elements. The resulting fast direct solver may be used to accelerate implicit time-stepping schemes, as the precomputed operators can be reused for fast elliptic solves on surfaces. We apply the method to a range of problems on both smooth surfaces and surfaces with sharp corners and edges, including the static Laplace–Beltrami problem, the Hodge decomposition of a tangential vector field, and some time-dependent reaction--diffusion systems.

**Fruzsina Agocs**, Flatiron Institute

**Location: **CULM 611

**A Fast and Arbitrarily High-Order Solver for Highly Oscillatory ODEs**

Oscillatory systems are ubiquitous in physics: they arise in celestial and quantum mechanics, electrical circuits, molecular dynamics, and beyond. Yet even in the simplest case, when the frequency of oscillations changes slowly but is large, the vast majority of numerical methods struggle to solve such equations. Methods based on approximating the solution with polynomials are forced to take $\mathcal{O}(k)$ timesteps, where $k$ is the characteristic frequency of oscillations. This scaling can generate unacceptable computational costs when the ODE in question needs to be solved billions of times, e.g.\ as the forward modelling step of Bayesian parameter estimation.

In this talk I will introduce an efficient method for solving 2nd order, linear ODEs with highly oscillatory solutions.

The solver employs two methods: in regions where the solution varies slowly, it uses a spectral method based on Chebyshev nodes and with an adaptive stepsize, but in the highly oscillatory phase it automatically switches over to an asymptotic method. The asymptotic method constructs a nonoscillatory phase function solution of the Riccati equation associated with the ODE. In the talk I will present how the method fits in the landscape of oscillatory solvers, the theoretical underpinnings of the asymptotic solver, a summary of the switching and stepsize-update algorithms, results from numerical experiments, and a brief error analysis.

**Yassine Tissaoui**, NJIT

**Location: **CULM 611

**Efficient Lower-Atmospheric Simulations Using Unstructured Grids and Spectral Elements: Added Complexity and Possible Solutions Featuring Non-Column Based Rain**

Beyond the Navier-Stokes equations, a significant component of atmospheric simulations is modeling the physical interactions taking place at scales beyond the current resolution of most atmospheric models. This involves trying to accurately model microphysical processes such as cloud generation and precipitation, as well unresolved dynamical processes such as turbulence. To this end, the atmospheric modeling community has done a great deal of work to develop new models and improve older ones and physical parametrizations are more accurate than ever. However, independent as they are from the numerical method being used in the model, not all physical parametrizations are easily implementable across the variety of existing numerical methods available. This can be attributed to factors such as the lack of locality to algorithmic limitation forced by the requirement of certain data structures. I will explore these limitations, present a commonly used existing solution within the atmospheric modeling community, and also present a simple but novel approach towards adapting an existing microphysics model to function using spectral elements on an unstructured grid.

**Amir Sagiv**, Columbia University

**Location: **CULM 611

**Floquet Hamiltonians - Effective Gaps and Resonant Decay**

Floquet topological insulators are an emerging category of materials whose properties are transformed by time-periodic forcing. Can their properties be understood from their first-principles continuum PDE models?

Experimentally, graphene is known to transform into an insulator under a time-periodic driving. A spectral gap, however, is conjectured to not form. How do we reconcile these two facts? We show that the original Schrodinger equation has an “effective gap” - a new and physically-relevant relaxation of a spectral gap.

Next, we challenge the prevailing notion of "Floquet edge modes"; due to resonance, localized modes in periodically-forced media are only metasable. Sufficiently rapid forcing couples the localized mode to the bulk, and so energy eventually leaks away from the localized edge/defect, in the spirit of the Fermi Golden Rule.

**Aminur Rahman**, University of Washington

**Location: **WebEx

**Bouncing Droplets as a Damped-Driven System**

Damped-driven systems are ubiquitous in science, however the damping and driving mechanisms are often quite convoluted. This talk presents a fluidic droplet on a vertically vibrating fluid bath as a damped-driven system. Fortunately, the damping and driving in the present system are relatively segregated. By separating the two mechanisms, we show that the droplet exhibits similar bifurcations present in other more complex damped-driven systems. In this investigation we study a fluidic droplet in an annular cavity with the fluid bath forced above the Faraday wave threshold. We model the droplet as a kinematic point particle in air and as inelastic collisions during impact with the bath. In both experiments and the model the droplet is observed to chaotically change velocity with a Gaussian distribution leading to diffusion-like behavior. In addition, the forcing above the Faraday wave threshold on the fluid bath simplifies the wave dynamics to that of a standing wave, which allows us to explicitly segregate the damping and driving mechanisms. The energy gain comes from the kinematics of the droplet between impacts, and the energy loss comes from the hydrodynamic damping at impact. We show that this energy gain-loss formulation exhibits dynamical behavior canonical to a wide range of damped-driven systems. The bifurcations and route to chaos present in the droplet system reveals analogies with other well-studied systems from optics, detonation, and electronics.This also hints at analogies with many more systems that have yet to be analyzed in detail, and paves a framework with which other systems can be analyzed.

*Updated November 28, 2022*