arXiv subjects

Ray Treinen

Publications and source records attributed to Ray Treinen.

6 recordsLinked to original sources

arXiv2026

Comparison of the potential energy for different equilibrium configurations of symmetric and asymmetric floating drops

We provide a numerical method for computing solutions to a free boundary problem arising from the equilibrium state of a floating drop. This numerical method is based on a Newton's method for the underlying nonlinear boundary value problems, and at each iterative step a Chebyshev spectral collocation method is employed. The problems considered here are those that can be described by using generating curves, and include problems in $\mathbb{R}^2$ and $\mathbb{R}^3$. The resulting nine-dimensional space of physical parameters is explored, and examples are given that highlight the potential energy of centrally located drops, wall-bound drops, and asymmetrical configurations in $\mathbb{R}^2$. Non-uniqueness of solutions to the corresponding Euler-Lagrange equations is displayed, and also strong evidence of non-uniqueness of energy minimizers is given.

physics.flu-dyn↗

arXiv2024

Spectral Prescribed Mean Curvature

We consider prescribed mean curvature equations whose solutions are minimal surfaces, constant mean curvature surfaces, or capillary surfaces. We consider both Dirichlet boundary conditions for Plateau problems and nonlinear Neumann boundary conditions for capillary problems and we consider domains in $\mathbf{R}^2$ to be rectangles, disks, or annuli. We present spectral methods for approximating solutions of the associated boundary value problems. These are either based on Chebyshev or Chebyshev-Fourier methods depending on the geometry of the domain. The non-linearity in the prescribed mean curvature equations is treated with a Newton method. The algorithms are designed to be adaptive; if the prescribed tolerances are not met then the resolution of the solution is increased until the tolerances are achieved. 22

math.NA↗

arXiv2023

Multi-scale spectral methods for bounded radially symmetric capillary surfaces

We consider radially symmetric capillary surfaces that are described by bounded generating curves. We use the arc-length representation of the differential equations for these surfaces to allow for vertical points and inflection points along the generating curve. These considerations admit capillary tubes, sessile drops, and fluids in annular tubes as well as other examples. We present a multi-scale pseudo-spectral method for approximating solutions of the associated boundary value problems based on interpolation by Chebyshev polynomials. The multi-scale approach is based on a domain decomposition with adaptive refinements within each sub-domain.

math.NA↗

arXiv2022

Discussion of a uniqueness result in "Equilibrium Configurations for a Floating Drop"

We analyze a uniqueness result presented by Elcrat, Neel, and Siegel \cite{ENS} for unbounded liquid bridges, and show that the proof they presented is incorrect. We add a choice of three hypothesis to their stated theorem and show that their result holds under this condition. Then we use Chebyshev spectral methods to build a numerical method to approximate solutions to a related boundary value problems that show one of the three hypothesis holds.

math.CA↗

arXiv2022

Spectral Methods for capillary surfaces described by bounded generating curves

We consider capillary surfaces that are constructed by bounded generating curves. This class of surfaces includes radially symmetric and lower dimensional fluid-fluid interfaces. We use the arc-length representation of the differential equations for these surfaces to allow for vertical points and inflection points along the generating curve. These considerations admit capillary tubes, sessile drops, and fluids in annular tubes as well as other examples. We present a pseudo-spectral method for approximating solutions to the associated boundary value problems based on interpolation by Chebyshev polynomials. This method is observably more stable than the traditional shooting method and it is computationally lean and fast. The algorithm is also adaptive, but does not use the adaptive automation in Chebfun.

math.NA↗

arXiv2016

The geometry of the triple junction between three fluids in equilibrium

We conduct an analysis of the blow up at the triple junction of three fluids in equilibrium. Energy minimizers have been shown to exist in the class of functions of bounded variations, and the classical theory implies that an interface between two fluids is an analytic surface. We prove two monotonicity formulas at the triple junction for the three-fluid configuration, and show that blow up limits exist and are always cones. We discuss some of the geometric consequences of our results.

math.AP↗