Abstract
The deformation and stability of a twodimensional 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 nearelliptical 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 leftright symmetry and cells with topbottom 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.
Below the smallest critical pressure a circular cell in a weak flow deforms into a nearelliptical 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 leftright symmetry and cells with topbottom 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 language  English 

Pages (fromto)  71–94 
Number of pages  24 
Journal  SIAM Journal on Applied Mathematics (SIAP) 
Volume  80 
Issue number  1 
Early online date  8 Jan 2020 
DOIs  
Publication status  Published  Jan 2020 
Keywords
 Conformal mapping
 Hydroelasticity
 Potential flow
Profiles

Mark Blyth
 School of Mathematics  Professor of Applied Mathematics
 Fluid and Solid Mechanics  Member
Person: Research Group Member, Academic, Teaching & Research

Emilian Parau
 School of Mathematics  Professor of Applied Mathematics & Head of School
 Fluid and Solid Mechanics  Member
Person: Research Group Member, Academic, Teaching & Research