Pressure-driven wrinkling of soft inner-lined tubes

A simple equation modelling an inextensible elastic lining of an inner-lined tube subject to an imposed pressure difference is derived from a consideration of the idealised elastic properties of the lining and the pressure and soft-substrate forces. Two cases are considered in detail, one with prominent wrinkling and a second one in which wrinkling is absent and only buckling remains. Bifurcation diagrams are computed via numerical continuation for both cases. Wrinkling, buckling, folding, and mixed-mode solutions are found and organised according to system-response measures including tension, in-plane compression, maximum curvature and energy. Approximate wrinkle solutions are constructed using weakly nonlinear theory, in excellent agreement with numerics. Our approach explains how the wavelength of the wrinkles is selected as a function of the parameters in compressed wrinkling systems and shows how localised folds and mixed-mode states form in secondary bifurcations from wrinkled states. Our model aims to capture the wrinkling response of arterial endothelium to blood pressure changes but applies much more broadly.

Periodicity disruption of a model quasi-biennial oscillation of equatorial winds

<p>In the Earth's atmosphere, fast propagating equatorial waves generate slow reversals of the large scale stratospheric winds with a  period of about 28 months. This quasi-biennial oscillation is a spectacular manifestation of wave-mean flow interactions in stratified fluids, with analogues in other planetary atmospheres and laboratory experiments. Recent observations of a disruption of this periodic behavior have been attributed to external perturbations, but the mechanism explaining the disrupted response has remained elusive. We show the existence of secondary bifurcations and a quasiperiodic route to chaos in simplified models of the equatorial atmosphere ranging from the classical Holton-Lindzen-Plumb model to fully nonlinear simulations of stratified fluids. Perturbations of the slow oscillations are widely amplified in the proximity of the secondary bifurcation point. This suggests that intrinsic dynamics may be equally influential as external variability in explaining disruptions of regular wind reversals  [1].</p><p>[1] Renaud, A., Nadeau, L. P., & Venaille, A. (2019). Periodicity Disruption of a Model Quasibiennial Oscillation of Equatorial Winds. Physical Review Letters, 122(21), 214504.<br> </p>

Minimal Energy Configurations of Finite Molecular Arrays

In this paper, we consider the problem of characterizing the minimum energy configurations of a finite system of particles interacting between them due to attractive or repulsive forces given by a certain intermolecular potential. We limit ourselves to the cases of three particles arranged in a triangular array and that of four particles in a tetrahedral array. The minimization is constrained to a fixed area in the case of the triangular array, and to a fixed volume in the tetrahedral case. For a general class of intermolecular potentials we give conditions for the homogeneous configuration (either an equilateral triangle or a regular tetrahedron) of the array to be stable that is, a minimizer of the potential energy of the system. To determine whether or not there exist other stable states, the system of first-order necessary conditions for a minimum is treated as a bifurcation problem with the area or volume variable as the bifurcation parameter. Because of the symmetries present in our problem, we can apply the techniques of equivariant bifurcation theory to show that there exist branches of non-homogeneous solutions bifurcating from the trivial branch of homogeneous solutions at precisely the values of the parameter of area or volume for which the homogeneous configuration changes stability. For the triangular array, we construct numerically the bifurcation diagrams for both a Lennard–Jones and Buckingham potentials. The numerics show that there exist non-homogeneous stable states, multiple stable states for intervals of values of the area parameter, and secondary bifurcations as well.

Bifurcations and convective instabilities of steady flows in a constricted channel

Stability and nonlinear analyses were employed to study symmetric and asymmetric steady flows through a straight channel with a smooth constriction with 50 % occlusion. Linear stability analysis was carried out to determine the unstable regions and the critical Reynolds numbers for the primary and secondary global instabilities. The primary bifurcation demonstrated an intricate aspect: the three-dimensional modes transfer their energy to the two-dimensional mode, which causes a symmetry breaking of the flow. This behaviour could be observed for Reynolds number lower than the critical, showing that this primary bifurcation is hysteretical. The secondary bifurcation also presented subcritical behaviour, exhibiting a pitchfork diagram with a large hysteretic curve. Given the subcritical character of both bifurcations, the relevance of non-normal growth in these flows were assessed, so that their convective mechanisms were exhaustively investigated. In addition, we could show that, for the secondary instability, optimal initial disturbances with large enough initial energy were able to promote a subcritical nonlinear saturation for a Reynolds number lower than the critical. The physical mechanism behind the transition process occurred by nonlinear interaction between the two- and three-dimensional modes, which established oscillatory behaviour, moreover, this energy exchange between the modes led the flow to the nonlinear saturated state. Therefore, the two-dimensional modes play a key role in the primary and secondary bifurcations of this system.

Formation of high aspect ratio wrinkles and ridges on elastic bilayers with small thickness contrast

An elastic bilayer composed of a stiff film bonded to a soft substrate forms wrinkles under compression. Experiments and finite element simulations reveal that at small thickness contrast, secondary bifurcations such as period doubling are delayed, providing access to high aspect ratio wrinkles. For high modulus contrast, the periodic wrinkles can evolve into a regular pattern of ridges with even higher aspect ratio.

Eldonioids with associated trace fossils from the lower Cambrian Emu Bay Shale Konservat-Lagerstätte of South Australia

Rare specimens of eldonioids recovered from the lower Cambrian (Series 2, Stage 4) Emu Bay Shale (EBS) Konservat-Lagerstätte represent the first record of the group for the Cambrian of East Gondwana. The disc-shaped body of the EBS taxon bears fine concentric corrugations on the dorsal surface and, ventrally, a series of internal lobes that have primary and secondary bifurcations, as well as a coiled sac. It appears to be most similar toRotadiscusandPararotadiscusof the Cambrian Chengjiang and Kaili biotas of South China, respectively. While the structure of the internal lobes would indicate that this occurrence in the EBS represents a new taxon within the Rotadiscidae, lack of detail regarding the precise number of internal lobes and the condition of the circumoral tentacles warrants a more conservative approach in leaving the genus and species under open nomenclature. The EBS specimens also host trace fossils, including the remains of a burrow, which are generally lacking in the body-fossil-bearing layers of the Konservat-Lagerstätte interval. These traces appear to have been made by small organisms and are similar to traces associated with the discs ofPararotadiscus guizhouensis(Zhao and Zhu, 1994) from the Kaili Biota. The available taphonomic, paleoenvironmental, and ichnological evidence indicates that the EBS eldonioids are most likely vagrants that were transported or settled into the ‘preservational trap’ and subsequently exposed on the substrate for a brief period before burial, thereby allowing organisms to exploit their carcasses for nutrients or other purposes.

RETRACTED—Eldonioids with associated trace fossils from the lower Cambrian Emu Bay Shale Konservat-Lagerstätte of South Australia

Rare specimens of eldonioids recovered from the lower Cambrian (Series 2, Stage 4) Emu Bay Shale (EBS) Konservat-Lagerstätte represent the first record of the group for the Cambrian of East Gondwana. The disc-shaped body of the EBS taxon bears fine concentric corrugations on the dorsal surface and, ventrally, a series of internal lobes that have primary and secondary bifurcations, as well as a coiled sac. It appears to be most similar to Rotadiscus and Pararotadiscus of the Cambrian Chengjiang and Kaili biotas of South China, respectively. While the structure of the internal lobes would indicate that this occurrence in the EBS represents a new taxon within the Rotadiscidae, lack of detail regarding the precise number of internal lobes and the condition of the circumoral tentacles warrants a more conservative approach in leaving the genus and species under open nomenclature. The EBS specimens also host trace fossils, including the remains of a burrow, which are generally lacking in the body-fossil-bearing layers of the Konservat-Lagerstätte interval. These traces appear to have been made by small organisms and are similar to traces associated with the discs of Pararotadiscus guizhouensis Zhao and Zhu, 1994 from the Kaili Biota. The available taphonomic, paleoenvironmental, and ichnological evidence indicates that the EBS eldonioids are most likely vagrants that were transported or settled into the ‘preservational trap’ and subsequently exposed on the substrate for a brief period before burial, thereby allowing organisms to exploit their carcasses for nutrients or other purposes.

Instabilities of Elastic and Spinning Systems: Concepts and Phenomena

Bifurcational instabilities of increasing complexity are described, with instructive and intriguing applications. Multiple folds in the gravitational collapse of a star triggers a discussion about possible sequencing of folds. Columns, rings and a quirky floating log illustrate safe and dangerous pitchforks, with self-weight models as lecture demonstrations. Localized buckling is employed by biologists using magnetic tweezers to twist a DNA molecule and evaluate a cutting and de-knotting enzyme. Unfolding of complex bifurcations into “structurally-stable” forms by a number (codimension) of perturbations illuminates the imperfection-sensitivity of shells, and the interactive buckling of a guyed mast at a hyperbolic-umbilic catastrophe. Spinning of primitive planets introduces secondary bifurcations, while Kirchhoff’s analogy is displayed for rods and spinning tops. Examples of gyroscopic stabilization and levitation mimic the “traps” that are used in modern experimental physics to hold in place individual electrons and neutrons.

Wrinkling instabilities in soft bilayered systems

Wrinkling phenomena control the surface morphology of many technical and biological systems. While primary wrinkling has been extensively studied, experimentally, analytically and computationally, higher-order instabilities remain insufficiently understood, especially in systems with stiffness contrasts well below 100. Here, we use the model system of an elastomeric bilayer to experimentally characterize primary and secondary wrinkling at moderate stiffness contrasts. We systematically vary the film thickness and substrate prestretch to explore which parameters modulate the emergence of secondary instabilities, including period-doubling, period-tripling and wrinkle-to-fold transitions. Our experiments suggest that period-doubling is the favourable secondary instability mode and that period-tripling can emerge under disturbed boundary conditions. High substrate prestretch can suppress period-doubling and primary wrinkles immediately transform into folds. We combine analytical models with computational simulations to predict the onset of primary wrinkling, the post-buckling behaviour, secondary bifurcations and the wrinkle-to-fold transition. Understanding the mechanisms of pattern selection and identifying the critical control parameters of wrinkling will allow us to fabricate smart surfaces with tunable properties and to control undesired surface patterns like in the asthmatic airway. This article is part of the themed issue ‘Patterning through instabilities in complex media: theory and applications.’

Transition in the asymptotic suction boundary layer over a heated plate

The asymptotic suction boundary layer (ASBL) is a parallel shear flow that becomes turbulent in a bypass transition in parameter regions where the laminar profile is stable. We here add a temperature gradient perpendicular to the plate and explore the interaction between convection and shear in determining the transition. We find that the laminar state becomes unstable in a subcritical bifurcation and that the critical Rayleigh number and wavenumber depend strongly on the Prandtl number. We also track several secondary bifurcations and identify states that are localized in two directions, showing different symmetries. In the subcritical regime, transient turbulent states which are connected to exact coherent states and follow the same transition scenario as found in linearly stable shear flows are identified and analysed. The study extends the bypass transition scenario from shear flows to thermal boundary layers and highlights the intricate interactions between thermal and shear forces.