ABSTRACT
We examine the buckling shape and critical compression of confined inhomogeneous composite sheets lying on a liquid foundation. The buckling modes are controlled by the bending stiffness of the sheet, the density of the substrate, and the size and the spatially dependent elastic coefficients of the sheet. We solve the beam equation describing the mechanical equilibrium of a sheet when its bending stiffness varies parallel to the direction of confinement. The case of a homogeneous bending stiffness exhibits a degeneracy of wrinkled states for certain lengths of the confined sheet; we explain this degeneracy using an asymptotic analysis valid for long sheets, and show that it corresponds to the switching of the sheet between symmetric and antisymmetric buckling modes. This degeneracy disappears for spatially dependent elastic coefficients. Medium length sheets buckle similarly to their homogeneous counterparts, whereas the wrinkled states in large length sheets concentrate the bending energy towards the soft regions of the sheet.
ABSTRACT
The Bénard-von Kármán vortex shedding instability in the wake of a cylinder is perhaps the best known example of a supercritical Hopf bifurcation in fluid dynamics. However, a simplified physical description that accurately accounts for the saturation amplitude of the instability is still missing. Here, we present a simple self-consistent model that provides a clear description of the saturation mechanism and quantitatively predicts the saturated amplitude and flow fields. The model is formally constructed by a set of coupled equations governing the mean flow together with its most unstable eigenmode with finite size. The saturation amplitude is determined by requiring the mean flow to be neutrally stable. Without requiring any input from numerical or experimental data, the resolution of the model provides a good prediction of the amplitude and frequency of the vortex shedding as well as the spatial structure of the mean flow and the Reynolds stress.
ABSTRACT
Mixed monolayers of the ganglioside G(M1) and the lipid dipalmitoylphosphatidlycholine (DPPC) at air-water and solid-air interfaces were investigated using various biophysical techniques to ascertain the location and phase behavior of the ganglioside molecules in a mixed membrane. The effects induced by G(M1) on the mean molecular area of the binary mixtures and the phase behavior of DPPC were followed for G(M1) concentrations ranging from 5 to 70 mol %. Surface pressure isotherms and fluorescence microscopy imaging of domain formation indicate that at low concentrations of G(M1) (<25 mol %), the monolayer becomes continually more condensed than DPPC upon further addition of ganglioside. At higher G(M1) concentrations (>25 mol %), the mixed monolayer becomes more expanded or fluid-like. After deposition onto a solid substrate, atomic force microscopy imaging of these lipid monolayers showed that G(M1) and DPPC pack cooperatively in the condensed phase domain to form geometrically packed complexes that are more ordered than either individual component as evidenced by a more extended total height of the complex arising from a well-packed hydrocarbon tail region. Grazing incidence x-ray diffraction on the DPPC/G(M1) binary mixture provides evidence that ordering can emerge when two otherwise fluid components are mixed together. The addition of G(M1) to DPPC gives rise to a unit cell that differs from that of a pure DPPC monolayer. To determine the region of the G(M1) molecule that interacts with the DPPC molecule and causes condensation and subsequent expansion of the monolayer, surface pressure isotherms were obtained with molecules modeling the backbone or headgroup portions of the G(M1) molecule. The observed concentration-dependent condensing and fluidizing effects are specific to the rigid, sugar headgroup portion of the G(M1) molecule.
