On icicle ripples

IF 1.4 4区工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY Journal of Engineering Mathematics Pub Date : 2024-03-28 DOI:10.1007/s10665-024-10336-4

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Abstract

Natural icicles have an overall conical shape modulated by surface ripples. It has been noted from many observations of icicles formed in nature and in the laboratory that the wavelength of the ripples has a very narrow spectrum between about 8 and 12 mm and that, as time evolves, the phase of the ripples migrates upwards. In this pedagogical review, I explore some of the physical mechanisms that can cause and mediate the formation and migration of ripples on icicles using simple mathematical models. To keep the mathematics more straightforward and transparent, I confine attention to two dimensions. A key physical parameter is the surface tension between the film of water that coats an icicle and the air that surrounds it, which causes a phase shift between the film–air interface and the ice–film interface. I show that the wavelength of ripples is dominantly proportional to the cube root of the square of the gravity-capillary length times the thickness of the water film. At high film-flow rates, advection-dominated heat transfer coupled with the interfacial phase shift becomes the dominant driver of instability. Gibbs–Thomson undercooling provides an unexpectedly large stabilisation of small wavelengths at these large flow rates, sufficient to maintain wavelength selection at millimetre scales.

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冰柱涟漪上

摘要天然冰柱整体呈圆锥形，由表面波纹调制。通过对自然界和实验室中形成的冰柱的多次观察发现，波纹的波长在大约 8 至 12 毫米之间有一个非常窄的频谱，而且随着时间的推移，波纹的相位会向上移动。在这篇教学综述中，我将利用简单的数学模型探讨一些可能导致和介导冰柱上波纹的形成和迁移的物理机制。为了使数学知识更加简单明了，我将注意力限制在两个维度上。一个关键的物理参数是包裹冰柱的水膜与环绕冰柱的空气之间的表面张力，它导致了水膜-空气界面和冰膜界面之间的相移。我的研究表明，波纹的波长主要与重力-毛细管长度乘以水膜厚度的平方的立方根成正比。在水膜流速较高时，平流主导的传热与界面相移耦合成为不稳定的主要驱动力。吉布斯-汤姆森过冷度在这些大流速下为小波长提供了意想不到的巨大稳定性，足以维持毫米尺度的波长选择。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Engineering Mathematics 工程技术-工程：综合

CiteScore

2.10

自引率

7.70%

发文量

审稿时长

6 months

期刊介绍： The aim of this journal is to promote the application of mathematics to problems from engineering and the applied sciences. It also aims to emphasize the intrinsic unity, through mathematics, of the fundamental problems of applied and engineering science. The scope of the journal includes the following: • Mathematics: Ordinary and partial differential equations, Integral equations, Asymptotics, Variational and functional−analytic methods, Numerical analysis, Computational methods. • Applied Fields: Continuum mechanics, Stability theory, Wave propagation, Diffusion, Heat and mass transfer, Free−boundary problems; Fluid mechanics: Aero− and hydrodynamics, Boundary layers, Shock waves, Fluid machinery, Fluid−structure interactions, Convection, Combustion, Acoustics, Multi−phase flows, Transition and turbulence, Creeping flow, Rheology, Porous−media flows, Ocean engineering, Atmospheric engineering, Non-Newtonian flows, Ship hydrodynamics; Solid mechanics: Elasticity, Classical mechanics, Nonlinear mechanics, Vibrations, Plates and shells, Fracture mechanics; Biomedical engineering, Geophysical engineering, Reaction−diffusion problems; and related areas. The Journal also publishes occasional invited ''Perspectives'' articles by distinguished researchers reviewing and bringing their authoritative overview to recent developments in topics of current interest in their area of expertise. Authors wishing to suggest topics for such articles should contact the Editors-in-Chief directly. Prospective authors are encouraged to consult recent issues of the journal in order to judge whether or not their manuscript is consistent with the style and content of published papers.