Fluids Mechanics and Waves Seminar - Spring 2024

For questions about the seminar schedule, please contact Thi Phong Nguyen.

February 19

Xinyu Zhao, McMaster University

Systemic Search for Singularities and Instabilities in Euler Flows

The Euler equations are used to describe fluid flow in regimes where viscosity can be neglected. It remains one of the central open questions in mathematical fluid mechanics whether solutions of the three-dimensional incompressible Euler equations can develop finite-time singularities from smooth initial conditions, i.e., certain norms of the solutions blow up in finite time. In this talk, we present a numerical approach to this problem where we develop PDE-constrained optimization methods to systematically search for initial data that may lead to a potential singularity. In two dimensions, the Euler equations are globally well-posed but the stability of even very simple equilibrium solutions has not been fully understood due to the subtle interplay between the point spectrum and the essential spectrum of the corresponding linear operator. We adopt a PDE-constrained optimization approach to discover the growth of non-modal perturbations associated with the essential spectrum.

March 4

Ross Parker, Center for Communications Research, Princeton

Multi-modal solitary wave solutions to nonlinear wave equations

While complex patterns in linear systems can be constructed by superposition, this is not possible in nonlinear systems. Nonetheless, we will show that we can “glue together” simple structures in certain nonlinear wave equations to create more complex structures. Specifically, we will consider solitary waves, which are localized disturbances in a medium that move at constant speed. As a prototypical model, we will use a fourth-order variant of the nonlinear Schrodinger equation. We will first show that we can construct multi-modal solitary waves, as long as certain geometric constraints are satisfied. We can prove, however, that all such solutions are unstable. We then consider so-called “dark soliton” solutions. By contrast, numerical results suggest that some multi-modal dark solitons are in fact stable.

March 18

Liet Vo, University of Illinois at Chicago

Mixed finite element methods for the Stochastic Navier-Stokes equations

Navier-Stokes equation is one of the most well-known equations in fluid mechanics because of its broad applications. In this talk, I will introduce the stochastic version of the equation. It is well-known that the stochastic Navier-Stokes equation is used for a better understanding of turbulence and thermodynamic fluctuations present in fluid flows. We will focus on the Euler-Maruyama-mixed finite element methods for solving the stochastic Navier-Stokes equation with additive noise. For the error analysis of the approximate solutions, the interplay between the stochastic noise term and the Navier-Stokes nonlinear term will make the stochastic version completely differ from the deterministic counterpart and will cause many issues in obtaining a strong error estimate. So, we present some exponential stability estimates of the numerical solutions to overcome the issue and obtain strong error estimates in both the L^2-norm and the energy norm. Numerical experiments are also presented to validate the theoretical results.

Homepage: https://sites.google.com/view/liet-vo/home?authuser=0

April 1

Binan Gu, Worcester Polytechnic Institute

Dynamic Pore Network Modeling of Nonlinear Species, Thermal and Surface Reactive Transport with Far-field Forcings

This talk has two connected components:

1. Predicting the fluid, thermal, and species transport in an evolving complex network of pores requires a fundamental description of the transport processes and their coupling to the underlying reaction chemistry. To better understand these processes at different spatial and temporal scales, we focus on the transport dynamics in a single slender pore, where aqueous reactions occur on shorter time scales than pore surface reactions, and derive an axisymmetric model (a system of PDEs) that captures the effects of advection, diffusion, surface reaction and pore evolution in the Stokes flow limit. We explore the reaction landscape for calcite-based carbonaceous chemical reaction as a function of temperature and species concentration near reaction equilibrium under constant far-field forcings, to understand the long-time effect on net thermal and species transport.

2. The solutions to the PDE model formulated in part 1 form the foundation for adaptations for dynamic pore network models (PNM). We present two approaches that derive a reduced system of ODEs: one is based on taking spatial averages of the PDE and then assuming spatial independence, while the other is based on linear stability theory. We demonstrate their strength and weaknesses in terms of fidelity to the PDE solution and computational complexity. Regardless of which approach we take, in a network of coupled pores, the reduced ODEs act as reasonable approximations to the pore dynamics in the edges while being computationally inexpensive. The coupling between the pores/edges is enforced via the conservation of fluid, thermal, and species flux at pore junctions. We demonstrate a worked example of a general approach to such PNMs using weighted graph Laplacians.

Homepage: https://users.wpi.edu/~bgu/?_gl=1*d8jrlp*_ga*MTE5Nzc3MzU3NS4xNzExNDU4ODM...

April 15

Jay Meegoda, NJIT Civil and Environmental Engineering

Mysteries of Nanobubbles

Nanobubbles are gas-filled cavities in an aqueous solution size smaller than 200nm and formed by any gas. They are generated by shear flow, nucleation, cavitation, shockwaves, or a combination. The existence of stable nanobubbles that last months is still in debate, as it contradicts accepted physics theories based on Laplace pressure. However, research has confirmed the existence of nanobubbles, and commonly, bubble characterization uses dynamic laser scattering techniques and nanoparticle tracking analysis methods. An extensive research was proformed at NJIT, including experimental investigation, theoretical analysis, and numerical simulations to address the unusual features of nanobubbles. The experimental program showed: 1. Reactive gases produced larger and highly negatively charged nanobubbles; 2. Low pH solutions produced larger and low negatively charged nanobubbles, and the surface charge became positive after one week; 3. High-temperature solutions produced larger and low negatively charged nanobubbles, and 4. High salt solitons produced larger and low negatively charged nanobubbles. The above results were similar to the diffused double layer theory of clay-water electrolyte system. Hence a theoretical analysis was performed by applying the diffused double layer theory to nanobubbles. Bubble size, surface charge density, and the number of negative charges on the bubble surface increased with increased NaCl concentration, while the magnitude of zeta/surface potential, double layer thickness, internal pressure, and electrostatic repulsion force decreased. The lower NaCl concentration of 0.001 M showed stable bubbles with a 6.99x10 -20  J energy barrier, which prevented bubble coalescence. Ion profiles revealed high concentrations of cations absorption at the bubble interface confirming negatively charged nanobubbles. To investigate the long-term stability of nanobubbles with high internal gas pressure, the molecular dynamic simulation was performed simulating high inner density O 2 gas nanobubble. The gas and liquid profiles showed that the gas diffusion occurred only out of the bubble. Density profiles also showed the thin gas-liquid interface around the bubble, and the back-calculated surface tension values shown were smaller than the typical values. Further, the calculated diffusion coefficient values were smaller than the typical values attributed to gas supersaturation, confirming the longevity of the bubble. Elevated bubble temperatures caused faster declination of the internal gas count, faster gas diffusion, and higher internal bubble pressure, leading to unstable conditions. We are now staring to theoretically investigate the bubble generation and bubble implosion due to ultrasound.

Dr. Meegoda is the program director of Geotechnical group and a Distinguished Professor of Civil and Environmental Engineering at New Jersey Institute of Technology. He received his BS (Honors) from University of Sri Lanka and his M.S. and his Ph.D. from the University of California at Davis. He has been working as educator, consultant and researcher in engineering for over 49 years. He utilizes scientific concepts and engineering technologies in his research to provide solutions to real world problems. Dr. Meegoda has worked with state and local governments, and foreign governments to provide technical input for a broad range of problems.

Homepage: https://people.njit.edu/profile/meegoda

April 29

Emad Masroor, Swarthmore

Vortex Dynamics in the Wake

Wakes form behind any solid object immersed in a fluid that is moving at an appreciable speed. Although viscous, dissipative forces at the fluid-solid interface are largely responsible for the production of a wake, it is often possible to model the wake itself using the techniques of inviscid fluid dynamics when the Reynolds number is moderately large. This technique was first used by Theodore von Karman in a series of papers in 1911 that modeled the wake behind a bluff body as a double row of alternately-signed point vortices, yielding a formula for the drag force on the object as a function of the vortex street characteristics.

In this talk, we will explore how vortex dynamics in an ideal fluid can illuminate our understanding of wakes and provide useful quantitative predictions. We will first present a generalization of von Karman’s drag law for periodic vortex-street wakes with N vortices per period in a two-dimensional wake. We will then consider periodic arrays of axisymmetric thin-cored vortex rings as a model for three-dimensional wakes. Lastly, we will present a novel numerical method to model the wake behind a flexible swimmer, improving upon the method of Silas Alben (2009, J. Comp. Physics). We will apply this method to a simple model of carangiform swimming.

Homepage: https://emadmasroor.github.io