The deformation and stability of an elastic cell in a uniform flow

Adam Yorkston, Mark Blyth, Emilian Parau

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The deformation and stability of a two-dimensional inextensible elastic cell in an inviscid uniform stream are investigated using a conformal mapping method. At low flow speeds equilibrium solutions are obtained using an asymptotic expansion, and the sequence of critical dimensionless pressures identified by Flaherty et al. (1972) for a circular cell exposed to a uniform transmural pressure is shown to play a crucial role.
Below the smallest critical pressure a circular cell in a weak flow deforms into a near-elliptical shape with its major axis perpendicular to the flow, and above this critical pressure its major axis is aligned with the flow. At each subsequent critical pressure the bifurcations produce in alternating sequence cells with either one or two axes of symmetry. In the former case cells with left-right symmetry and cells with top-bottom symmetry are found. Equilibria for general flow speeds are calculated numerically, and their linear stability is analysed.
Cells with two degrees of rotational symmetry whose longest chord is perpendicular to the uniform stream are found to be always stable.
Other configurations are found to be stable only for certain parameter values.
The nonlinear evolution of unstable cells subject to a small perturbation are computed numerically, and parameter values are located for which the cell falls into one of two distinct regular motions, either flipping over in alternating directions or bulging out to the side, while being intermittently propelled downstream with the flow.
Original languageEnglish
Pages (from-to)71–94
Number of pages24
JournalSIAM Journal on Applied Mathematics (SIAP)
Issue number1
Early online date8 Jan 2020
Publication statusPublished - Jan 2020


  • Conformal mapping
  • Hydroelasticity
  • Potential flow

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