An official website of the United States government
Abstract We discuss the problem of unique determination of the finite free discrete Schrödinger operator from its spectrum, also known as the Ambarzumian problem, with various boundary conditions, namely any real constant boundary condition at zero and Floquet boundary conditions of any angle. Then we prove the following Ambarzumian-type mixed inverse spectral problem: diagonal entries except the first and second ones and a set of two consecutive eigenvalues uniquely determine the finite free discrete Schrödinger operator.
more »
« less
Density-constrained Chemotaxis and Hele-Shaw flow
We consider a model of congestion dynamics with chemotaxis, where the density of cells follows the chemical signal it generates, while observing an incompressibility constraint (incompressible parabolic-elliptic Patlak-Keller-Segel model). We show that when the chemical diffuses slowly and attracts the cells strongly, then the dynamics of the congested cells is well approximated by a surface-tension driven free boundary problem. More precisely, we rigorously establish the convergence of the solution to the characteristic function of a set whose evolution is determined by the classical Hele-Shaw free boundary problem with surface tension. The problem is set in a bounded domain, which leads to an interesting analysis on the limiting boundary conditions. Namely, we prove that the assumption of Robin boundary conditions for the chemical potential leads to a contact angle condition for the free interface (in particular Neumann boundary conditions lead to an orthogonal contact angle condition, while Dirichlet boundary conditions lead to a tangential contact angle condition).
more »
« less
- PAR ID:
- 10504974
- Publisher / Repository:
- Transactions of the Amer. Math. Soc.
- Date Published:
- Journal Name:
- Transactions of the American Mathematical Society
- ISSN:
- 0002-9947
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
In an effort to study the stability of contact lines in fluids, we consider the dynamics of an incompressible viscous Stokes fluid evolving in a two-dimensional open-top vessel under the influence of gravity. This is a free boundary problem: the interface between the fluid in the vessel and the air above (modeled by a trivial fluid) is free to move and experiences capillary forces. The three-phase interface where the fluid, air, and solid vessel wall meet is known as a contact point, and the angle formed between the free interface and the vessel is called the contact angle. We consider a model of this problem that allows for fully dynamic contact points and angles. We develop a scheme of a priori estimates for the model, which then allow us to show that for initial data sufficiently close to equilibrium, the model admits global solutions that decay to equilibrium exponentially quickly.more » « less
-
In this work, we use the immersed boundary method with four extensions to simulate a moving liquid–gas interface on a solid surface. We first define a moving contact line model and implements a static-dynamic friction condition at the immersed solid boundary. The dynamic contact angle is endogenous instead of prescribed, and the solid boundary can be non-stationary with respect to time. Second, we simulate both a surface tension force and a Young's force with one general equation that does not involve estimating local curvature. In the third extension, we splice liquid–gas interfaces to handle topological changes, such as the coalescence and separation of liquid droplets or gas bubbles. Finally, we re-sample liquid–gas interface markers to ensure a near-uniform distribution without exerting artificial forces. We demonstrate empirical convergence of our methods on non-trivial examples and apply them to several benchmark cases, including a slipping droplet on a wall and a rising bubble.more » « less
-
Recent observations of plasma-activated water (PAW)’s surfactant behavior suggest that the activation of water with non-equilibrium plasma can decrease the surface tension of the water. This suggested change to the surface tension also indicates that the addition of plasma can lead to changes in the physical properties of the water, knowledge of which can expand existing PAW applications and open new ones. While the chemical behavior of PAW has been extensively analyzed, to the best of our knowledge the physical properties of PAW have not been investigated. This study focuses on the need for experimental determination of PAW’s physical properties—namely, surface tension, viscosity, and contact angle. The experimental results of this study show that the addition of plasma lowers the surface tension of water at room temperature, increases the viscosity of water at high temperatures, and lowers the contact angle of droplets on glass surfaces at room temperatures. Potential factors influencing these changes include plasma alteration of the mesoscopic structure of water at low temperatures and plasma additives acting as foreign particles in water at higher temperatures. Ultimately, this investigation demonstrates that the physical properties of water change due to plasma activation, which could lead to potential industrial applications of PAW as a surfactant or as a washing-out and cleaning agent.more » « less
-
Abstract It is widely accepted that solid‐state membranes are indispensable media for the graphene process, particularly transfer procedures. But these membranes inevitably bring contaminations and residues to the transferred graphene and consequently compromise the material quality. This study reports a newly observed free‐standing graphene‐water membrane structure, which replaces the conventional solid‐state supporting media with liquid film to sustain the graphene integrity and continuity. Experimental observation, theoretical model, and molecular dynamics simulations consistently indicate that the high surface tension of pure water and its large contact angle with graphene are essential factors for forming such a membrane structure. More interestingly, water surface tension ensures the flatness of graphene layers and renders high transfer quality on many types of target substrates. This report enriches the understanding of the interactions on reduced dimensional material while rendering an alternative approach for scalable layered material processing with ensured quality for advanced manufacturing.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/tran/8934
