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On the damped oscillations of an elastic quasi-circular membrane in a two-dimensional incompressible fluid

Published online by Cambridge University Press:  01 April 2014

Marco Martins Afonso Open the ORCID record for Marco Martins Afonso [Opens in a new window] ,
Simon Mendez  and
Franck Nicoud
Marco Martins Afonso*
Affiliation:
Institut de Mathématiques et de Modélisation de Montpellier, CNRS UMR 5149, Université Montpellier 2, CC 051, 34095 Montpellier CEDEX 5, France Laboratoire de Mécanique, Modélisation et Procédés Propres, CNRS UMR 7340, Aix-Marseille Université, Ecole Centrale Marseille, 13451 Marseille CEDEX 13, France
Simon Mendez
Affiliation:
Institut de Mathématiques et de Modélisation de Montpellier, CNRS UMR 5149, Université Montpellier 2, CC 051, 34095 Montpellier CEDEX 5, France
Franck Nicoud
Affiliation:
Institut de Mathématiques et de Modélisation de Montpellier, CNRS UMR 5149, Université Montpellier 2, CC 051, 34095 Montpellier CEDEX 5, France
*
†Email address for correspondence: [email protected]
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Abstract

We propose a procedure – partly analytical and partly numerical – to find the frequency and the damping rate of the small-amplitude oscillations of a massless elastic capsule immersed in a two-dimensional viscous incompressible fluid. The unsteady Stokes equations for the stream function are decomposed into normal modes for the angular and temporal variables, leading to a fourth-order linear ordinary differential equation in the radial variable. The forcing terms are dictated by the properties of the membrane and result in jump conditions at the interface between the internal and external media. The equation can be solved numerically, and excellent agreement is found with a fully computational approach that we have developed in parallel. Comparisons are also shown with results available in the scientific literature for drops, and a model based on the concept of entrained fluid is presented, which allows for a good representation of the present results and a consistent interpretation of the underlying physics.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Biological Fluid Dynamics: Capsule/cell dynamics Biological Fluid Dynamics: Membranes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 746 , 10 May 2014 , pp. 300 - 331
Copyright
© 2014 Cambridge University Press 

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