scholarly journals A Björling representation for Jacobi fields on minimal surfaces and soap film instabilities

2020 ◽  
Vol 476 (2238) ◽  
pp. 20190903
Author(s):  
Gareth P. Alexander ◽  
Thomas Machon
Keyword(s):  
Minimal Surfaces ◽  
General Framework ◽  
Soap Film ◽  
Jacobi Field ◽  
Soap Films ◽  
Jacobi Fields ◽  
One Dimensional ◽  
Stable Domain ◽  
Transverse Gradient ◽  
Geometric Nature

We develop a general framework for the description of instabilities on soap films using the Björling representation of minimal surfaces. The construction is naturally geometric and the instability has the interpretation as being specified by its amplitude and transverse gradient along any curve lying in the minimal surface. When the amplitude vanishes, the curve forms part of the boundary to a critically stable domain, while when the gradient vanishes the Jacobi field is maximal along the curve. In the latter case, we show that the Jacobi field is maximally localized if its amplitude is taken to be the lowest eigenfunction of a one-dimensional Schrödinger operator. We present examples for the helicoid, catenoid, circular helicoids and planar Enneper minimal surfaces, and emphasize that the geometric nature of the Björling representation allows direct connection with instabilities observed in soap films.

scholarly journals On the Study of Jacobi Fields in Riemannian Manifolds

1970 ◽  
Vol 43 (4) ◽  
pp. 521-528
Author(s):  
Khondokar M Ahmed
Keyword(s):  
Field Equation ◽  
Key Words ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Jacobi Equation ◽  
Standard Form ◽  
Jacobi Field ◽  
Jacobi Fields ◽  
New Approach

A new approach of finding a Jacobi field equation with the relation between curvature and geodesics of a Riemanian manifold M has been derived. Using this derivation we have made an attempt to find a standard form of this equation involving sectional curvature K and other related objects. Key words: Riemanign curvature, Sectional curvature, Jacobi equation, Jacobifield.    doi: 10.3329/bjsir.v43i4.2242 Bangladesh J. Sci. Ind. Res. 43(4), 521-528, 2008

scholarly journals A comparison theorem on magnetic jacobi fields

1997 ◽  
Vol 40 (2) ◽  
pp. 293-308 ◽  
Author(s):  
Toshiaki Adachi
Keyword(s):  
Magnetic Field ◽  
Comparison Theorem ◽  
Charged Particles ◽  
Kähler Manifold ◽  
Jacobi Field ◽  
Jacobi Fields ◽  
Kahler Manifold ◽  
Scalar Multiple ◽  
Kähler Form

A scalar multiple of the Kähler form of a Kähler manifold is called a Kähler magnetic field. We are focused on trajectories of charged particles under this action. As a variation of trajectories we define a magnetic Jacobi field. In this paper we discuss a comparison theorem on magnetic Jacobi fields, which corresponds to the Rauch's comparison theorem.

scholarly journals The dynamics of a viscous soap film with soluble surfactant

2001 ◽  
Vol 442 ◽  
pp. 387-409 ◽  
Author(s):  
JEAN-MARC CHOMAZ
Keyword(s):  
Film Thickness ◽  
Elastic Wave ◽  
Surfactant Concentration ◽  
Stokes Equations ◽  
Film Surface ◽  
Soap Film ◽  
Fluid Viscosity ◽  
Two Dimensional ◽  
Leading Order ◽  
Soap Films

Nearly two decades ago, Couder (1981) and Gharib & Derango (1989) used soap films to perform classical hydrodynamics experiments on two-dimensional flows. Recently soap films have received renewed interest and experimental investigations published in the past few years call for a proper analysis of soap film dynamics. In the present paper, we derive the leading-order approximation for the dynamics of a flat soap film under the sole assumption that the typical length scale of the flow parallel to the film surface is large compared to the film thickness. The evolution equations governing the leading-order film thickness, two-dimensional velocities (locally averaged across the film thickness), average surfactant concentration in the interstitial liquid, and surface surfactant concentration are given and compared to similar results from the literature. Then we show that a sufficient condition for the film velocity distribution to comply with the Navier–Stokes equations is that the typical flow velocity be small compared to the Marangoni elastic wave velocity. In that case the thickness variations are slaved to the velocity field in a very specific way that seems consistent with recent experimental observations. When fluid velocities are of the order of the elastic wave speed, we show that the dynamics are generally very specific to a soap film except if the fluid viscosity and the surfactant solubility are neglected. In that case, the compressible Euler equations are recovered and the soap film behaves like a two-dimensional gas with an unusual ratio of specific heat capacities equal to unity.

scholarly journals Soap Films as 1D waveguides

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
Olivier Emile ◽  
Janine Emile
Keyword(s):  
Film Thickness ◽  
Laser Light ◽  
Soap Film ◽  
Mode Propagation ◽  
Free Standing ◽  
One Dimensional ◽  
Strong Selection ◽  
Self Selection ◽  
Optical Modes ◽  
Selection Of

AbstractLaser light is injected in a free standing horizontal draining soap film through the glass frame sustaining the film. Two propagation regimes are clearly identified depending on the film thickness. At the beginning of the drainage, the soap film behaves as a multimode-one dimensional optofiuidic waveguide. In particular, we observe that the injected light creates a bottleneck in the film and part of the injected light is refracted leading to whiskers. At the end of the drainage where the film thickness is below 1μm, there is a strong selection among the various possible optical modes in the film, and part of the light is defiected. This leads to a self selection of the mode propagation inside the film.

scholarly journals Second order homogenization of quasi-periodic structures

2018 ◽  
Vol 40 (4) ◽  
pp. 325-348
Author(s):  
Duc Trung Le ◽  
Jean-Jacques Marigo
Keyword(s):  
General Framework ◽  
Volume Fraction ◽  
Effective Properties ◽  
Periodic Structures ◽  
Expansion Method ◽  
Second Order ◽  
Unit Cell ◽  
One Dimensional ◽  
Minimization Problems ◽  
Asymptotic Expansion Method

The paper develops a general framework to derive the effective properties of quasi-periodic elastic medium. By using the asymptotic expansion method, the solution is expanded to the second order by solving a sequence of minimization problems. The effective stiffness tensors fields entering in the expression of the macroscopic energy are obtained by solving several families of microscopic problems posed on the unit cell and which bring into play only the microstructure. As an illustrative example, we consider an anti-plane elastic case of a heterogeneous cylinder made of a bi-layer laminate and submitted to the gravity. The unit cell being one-dimensional, all the associated elementary problems can be solved in a closed form and one shows that the effective energy of the medium expanded up to the second order depends not only on the strain gradient, but also on the gradient of the volume fraction \(\theta\) characterizing the repartition of the two materials in the laminate.

Quasi-steady vortical structures in vertically vibrating soap films

1998 ◽  
Vol 372 ◽  
pp. 213-230 ◽  
Author(s):  
JOSÉ M. VEGA ◽  
F. J. HIGUERA ◽  
P. D. WEIDMAN
Keyword(s):  
Numerical Simulations ◽  
Liquid Flow ◽  
Excitation Frequency ◽  
Soap Film ◽  
Soap Films ◽  
Vortical Structures ◽  
Steady Streaming ◽  
Streaming Motion

An analysis of the quasi-steady streaming of the liquid in a vertically vibrated horizontal soap film is reported. The air around the soap film is seen to play a variety of roles: it transmits normal and tangential oscillatory stresses to the film, damps out Marangoni waves, and forces non-oscillatory deflection of the film and tangential motion of the liquid. Non-oscillatory volume forcing originating inside the liquid is also analysed. This forcing dominates the quasi-steady streaming when the excitation frequency is close to the eigenfrequency of a Marangoni mode of the soap film, while both volume forcing in the liquid and surface forcing of the gas on the liquid are important when no Marangoni mode resonates. Different manners by which the combined forcings can induce quasi-steady streaming motion are discussed and some numerical simulations of the quasi-steady liquid flow are presented.

Jacobi Fields in Optimal Control: One-dimensional Variations

2020 ◽  
Vol 26 (4) ◽  
pp. 685-732
Author(s):  
Andrei Agrachev ◽  
Ivan Beschastnyi
Keyword(s):  
Optimal Control ◽  
Jacobi Fields ◽  
One Dimensional
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