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

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 onto 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 into jump conditions at the interface between the internal and external media. The equation can be solved numerically, and an excellent agreement is found with a fullycomputational approach we developed in parallel. Comparisons are also shown with the results available in the scientific literature for drops, and a model based on the concept of embarked fluid is presented, which allows for a good representation of the results and a consistent interpretation of the underlying physics.

An unstructured solver for simulations of deformable particles in flows at arbitrary Reynolds numbers

“…Small yet larger differences are observed for the damping rate (8% and 5% for modes 2 and 3, respectively). A more detailed comparison of the results is presented by Martins Afonso et al [67]. This very good comparison confirms the correct behavior of the present method in the test case of damped oscillations of an elastic capsule.…”

“…Depending on the elastic modulus of the membrane and the fluid dynamic viscosity μ, the membrane oscillates or not before reaching the equilibrium state. More precisely, for low viscosity values, the capsule oscillates and the period of the oscillations is shown to be related with an elastic characteristic time t e = (ρR 0 4 /E e (R f − R 0 )) 1 2 [67,73]. The damping rate of the oscillation is related to the viscous time t D = ρR 0 2 /μ.…”

“…In order to further analyze the oscillatory relaxation of 2-D capsules, Martins Afonso et al [67] used normal-mode analysis to obtain semi-analytical frequency and damping rate of the oscillations of capsules. The analysis stands for small perturbations of the different azimuthal modes (the case shown in Fig.…”

Fluid‐structure interaction of a pulsatile flow with an aortic valve model: A combined experimental and numerical study

“…In particular, the so‐called Sigma subgrid model is well adapted to configurations with weak turbulence, as it has been designed not to add subgrid viscosity in a number of laminar canonical cases. The present fluid solver was validated on the benchmark of idealized cardiovascular device proposed by the US Food and Drug Administration (FDA), as well as on a variety of test cases …”

“…The present fluid solver was validated on the benchmark of idealized cardiovascular device proposed by the US Food and Drug Administration (FDA), 45 as well as on a variety of test cases. 35,46,47…”