Under insonation contrast agents are known to perform nonlinear pulsations and deform statically, in the form of buckling, or dynamically via parametric mode excitation, and often exhibit jetting and break-up like bubbles without coating. Boundary element simulations are performed in the context of axisymmetry in order to establish the nonlinear evolution of these patterns. The viscoelastic stresses that develop on the coating form the dominant force balance tangentially to the shell-liquid interface, whereas the dynamic overpressure across the shell balances viscoelastic stresses in the normal direction. Strain softening and strain hardening behavior is studied in the presence of shape instabilities for various initial conditions. Simulations recover the pattern of static buckling, subharmonic/harmonic excitation, and dynamic buckling predicted by linear stability. Preferential mode excitation during compression is obtained supercritically for strain softening phospholipid shells while the shell regains its sphericity at expansion. It is a result of energy transfer between the emerging unstable modes and the radial mode, eventually leading to saturated oscillations of shape modes accompanied by asymmetric radial pulsations in favor of compression. Strain softening shells are more prone to sustain saturated pulsations due to the mechanical behavior of the shell. As the sound amplitude increases and before the onset of dynamic buckling, both types of shells exhibit transient break-up via unbalanced growth of a number of unstable shape modes. The effect of pre-stress in lowering the amplitude threshold for shape mode excitation is captured numerically and compared against the predictions of linear stability analysis. The amplitude interval for which sustained shape oscillations are obtained is extended, in the presence of pre-stress, by switching from a strain softening constitutive law to a strain hardening one once the shell curvature increases beyond a certain level. This type of mechanical behavior models the formation of lipid bilayer structures on the shell beyond a certain level of bending, as a result of a lipid monolayer folding transition. In this context a compression only type behavior is obtained in the simulations, which is accompanied by preferential shape deformation during compression at relatively small sound amplitudes in a manner that bears significance on the interpretation of available experimental observations exhibiting similar dynamic behavior.

The time-asymptotic linear stability of pulsatile flow in a channel with compliant walls is studied together with the evaluation of modal transient growth within the pulsation period of the basic flow as well as non-modal transient growth. Both one (vertical-displacement) and two (vertical and axial) degrees-of-freedom compliant-wall models are implemented. Two approaches are developed to study the dynamics of the coupled fluid–structure system, the first being a Floquet analysis in which disturbances are decomposed into a product of exponential growth and a sum of harmonics, while the second is a time-stepping technique for the evolution of the fundamental solution (monodromy) matrix. A parametric study of stability in the non-dimensional parameter space, principally defined by Reynolds number ($Re$), Womersley number ($Wo$) and amplitude of the applied pressure modulation ($\unicode[STIX]{x1D6EC}$), is then conducted for compliant walls of fixed geometric and material properties. The flow through a rigid channel is shown to be destabilized by pulsation for low $Wo$, stabilized due to Stokes-layer effects at intermediate $Wo$, while the critical $Re$ approaches the steady Poiseuille-flow result at high $Wo$, and that these effects are made more pronounced by increasing $\unicode[STIX]{x1D6EC}$. Wall flexibility is shown to be stabilizing throughout the $Wo$ range but, for the relatively stiff wall used, is more effective at high $Wo$. Axial displacements are shown to have negligible effect on the results based upon only vertical deformation of the compliant wall. The effect of structural damping in the compliant-wall dynamics is destabilizing, thereby suggesting that the dominant inflectional (Rayleigh) instability is of the Class A (negative-energy) type. It is shown that very high levels of modal transient growth can occur at low $Wo$, and this mechanism could therefore be more important than asymptotic amplification in causing transition to turbulent flow for two-dimensional disturbances. Wall flexibility is shown to ameliorate mildly this phenomenon. As $Wo$ is increased, modal transient growth becomes progressively less important and the non-modal mechanism can cause similar levels of transient growth. We also show that oblique waves having non-zero transverse wavenumbers are stable to higher values of critical $Re$ than their two-dimensional counterparts. Finally, we identify an additional instability branch at high $Re$ that corresponds to wall-based travelling-wave flutter. We show that this is stabilized by the inclusion of structural damping, thereby confirming that it is of the Class B (positive-energy) instability type.

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We study the fluid–structure interaction (FSI) of a compliant panel with developing Blasius boundary-layer flow. The linearised Navier–Stokes equations in velocity–vorticity form are solved using a Helmholtz decomposition coupled with the dynamics of a plate-spring compliant panel couched in finite-difference form. The FSI system is written as an eigenvalue problem and the various flow- and wall-based instabilities are analysed. It is shown that global temporal instability can occur through the interaction of travelling wave flutter (TWF) with a structural mode or as a resonance between Tollmien–Schlichting wave (TSW) instability and discrete structural modes of the compliant panel. The former is independent of compliant panel length and upstream inflow disturbances while the specific behaviour arising from the latter phenomenon is dependent upon the frequency of a disturbance introduced upstream of the compliant panel. The inclusion of axial displacements in the wall model does not lead to any further global instabilities. The dependence of instability-onset Reynolds numbers with structural stiffness and damping for the global modes is quantified. It is also shown that the TWF-based global instability is stabilised as the boundary layer progresses downstream while the TSW-based global instability exhibits discrete resonance-type behaviour as Reynolds number increases. At sufficiently high Reynolds numbers, a globally unstable divergence instability is identified when the wavelength of its wall-based mode is longer than that of the least stable TSW mode. Finally, a non-modal analysis reveals a high level of transient growth when the flow interacts with a compliant panel which has structural properties capable of reducing TSW growth but which is prone to global instability through wall-based modes.

Abstract. We develop a hybrid of computational and theoretical approaches suited to study the fluid-structure interaction (FSI) of a compliant panel, flush between rigid upstream and downstream wall sections, with a Blasius boundary-layer flow. The ensuing linear-stability analysis is focused upon global instability and transient growth of disturbances. The flow solution is developed using a combination of vortex and source boundary-element sheets on a computational grid while the dynamics of a plate-spring compliant wall are couched in finite-difference form. The fully-coupled FSI system is then written as an eigenvalue problem and the eigenvalues of the various flow-and wall-based instabilities are analysed. It is shown that coalescence or resonance of a structural eigenmode with either a flow-based Tollmien-Schlichting Wave (TSW) or wall-based travelling-wave flutter (TWF) modes can occur. This can render the nature of these well-known convective instabilities to become global for a finite compliant wall giving temporal growth of system disturbances. Finally, a non-modal analysis, based on the linear superposition of the extracted temporal modes is presented. This reveals a high level of transient growth when the flow interacts with a compliant panel that has structural properties which render the FSI system prone to global instability. Thus, to design stable finite compliant panels for applications such as boundary-layer transition postponement, both global instabilities and transient growth must be taken into account.