# Fluid Mechanics and Waves Seminar - Spring 2023

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.

**February 6**

**Pedro Jordan**, U.S. Naval Research Laboratory

**Please note, this talk will be held virtually via WebEx**

**Nonlinear Acoustics: Fundamental Concepts and Shock Applications**

I will review the fundamental concepts and equations of nonlinear acoustics theory, derive equations of motion in terms of the scalar velocity potential, and present a number of example problems involving shock phenomena.

**February 20**

**Fredrik Fryklund**, NYU

**An Integral Equation Method for the Advection-Diffusion Equation on Time-Dependent Domains in the Plan**

Boundary integral methods are attractive for solving homogeneous elliptic partial differential equations on complicated geometries, since they can offer accurate solutions with a computational cost that is linear or close to linear in the number of discretization points on the boundary of the domain. However, these numerical methods are not straightforward to apply to time-dependent equations, which often arise in science and engineering. We address this problem with an integral equation-based solver for the advection-diffusion equation on moving and deforming geometries in two space dimensions. In this method, an adaptive high-order accurate time-stepping scheme based on semi-implicit spectral deferred correction is applied. One time-step then involves solving a sequence of non-homogeneous modified Helmholtz equations, a method known as elliptic marching. Our solution methodology utilizes several recently developed methods, including special purpose quadrature, a function extension technique and a spectral Ewald method for the modified Helmholtz kernel. Special care is also taken to handle the time-dependent geometries. The numerical method is tested through several numerical examples to demonstrate robustness, flexibility and accuracy.

**March 6**

**Pablo Ravazzoli**, The Center for Research in Physics and Engineering of the Center of the Province of Buenos Aires (CIFICEN)

**Equilibrium Solutions of 3 and 4 Phase Systems**

We study the equilibrium shape of liquid lens (two liquids and a gas phase) and liquid bridges (two liquids, gas and solid phase) considering the surface tension forces in nonwetting situations (negative

spreading factor). For the lenses, we obtain analytical expressions for the drop shape when gravity can be neglected and when gravity is include in the analysis, we find two different families of equilibrium solutions for the same set of

physical parameters. For the bridges, we did not consider the gravity effects, but we also found more than one solution for the same set of physical parameters. In both cases we determine the family that has the smallest energy, and therefore is the most probable to be found in nature.

**April 3**

**Zewen Shen**, University of Toronto

**Is Polynomial Interpolation in the Monomial Basis Unstable?**

In this talk, we will show that the monomial basis is generally as good as a well-conditioned polynomial basis for interpolation, provided that the condition number of the Vandermonde matrix is smaller than the reciprocal of machine epsilon. We will also show that the monomial basis is more advantageous than other polynomial bases in a number of applications.

**April 17**

**Heather Wilber**, UT Austin

**Please note, this talk will be held virtually**

**Hierarchical Solvers for Special Linear Systems **

We introduce a superfast method for solving special linear systems that involve matrices with displacement structure, such as Toeplitz, Cauchy and Vandermonde matrices. These matrices appear pervasively across a range of important applications in computational mathematics and signal processing, and as we show, they can be transformed into matrices that have hierarchical low rank structures. We analyze the rank structure of these matrices and develop fast direct solvers based on the structures. This includes solvers for the overdetermined case. In particular, we highlight the application of our method as a means for computing the inverse nonuniform discrete Fourier transform in settings where signal samples are far from equally-spaced.

**May 1**

**Rodolfo Brandão Macena Lira**, Princeton

**Elastic Filaments in Low-Reynolds-Number Flows**

Hydrodynamic couplings of elastic filaments with external flows are crucial for the functioning of various biological and synthetic processes, such as the locomotion of flagellated and ciliated organisms. In this talk, we will analyze the steady-state deformation of an elastic filament in unidirectional flows at low Reynolds numbers, assuming that the filament is clamped at one end or tethered at its midpoint. We will begin by discussing the basic equations for the problem, which consist of the nonlinear Euler-Bernoulli equations coupled with a local slender-body description of the hydrodynamic forces. In dimensionless form, the problem depends on a single compliance parameter. By employing singular perturbation techniques, we will derive asymptotic approximations in the limits of small and large compliances. Our approximations are in closed form and show excellent agreement with numerical simulations of the full problem. Lastly, we will comment on a generalization of our analysis to fluid filaments fixed at both ends and subject to external flows—which follows from the "Stokes-Rayleigh analogy". In this case, our analysis not only predicts the shape of the fluid filament but also offers insight into its breakup.

*Updated: April 28, 2023*