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Minimal surfaces on mirror-symmetric frames: a fluid dynamics analogy

Published online by Cambridge University Press:  19 June 2020

Mars M. Alimov Open the ORCID record for Mars M. Alimov [Opens in a new window] ,
Alexander V. Bazilevsky  and
Konstantin G. Kornev Open the ORCID record for Konstantin G. Kornev [Opens in a new window]
Mars M. Alimov
Affiliation:
Kazan Federal University, Kazan420008, Russia
Alexander V. Bazilevsky
Affiliation:
Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow 119526, Russia
Konstantin G. Kornev*
Affiliation:
Department of Materials Science and Engineering, Clemson University, Clemson, SC 29634-0971, USA
*
†Email address for correspondence: [email protected]
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Abstract

Chaplygin’s hodograph method of classical fluid mechanics is applied to explicitly solve the Plateau problem of finding minimal surfaces. The minimal surfaces are formed between two mirror-symmetric polygonal frames having a common axis of symmetry. Two classes of minimal surfaces are found: the class of regular surfaces continuously connecting the supporting frames forming a tube with complex shape; and the class of singular surfaces which have a partitioning film closing the tube in between. As an illustration of the general solution, minimal surfaces supported by triangular frames are fully described. The theory is experimentally validated using soap films. The general solution is compared with the known particular solutions obtained by the Weierstrass inverse method.

JFM classification

Complex Fluids: Foams Interfacial Flows (free surface): Thin films Mathematical Foundations: General fluid mechanics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 897 , 25 August 2020 , A36
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Alimov et al. supplementary movie

Three dimensional configuration of a periodic cell of the Schwarz' H triply periodic minimal surface obtained using Chaplygin's hodograph method.

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