Runge-Kutta stability on a Floquet problem

This work examines the stability of explicit Runge-Kutta methods applied to a certain linear ordinary differential equation with periodic coefficients. On this problem naïve use of the eigenvalues of the Jacobian results in misleading conclusions about stable behaviour. It is shown, however, that a valid analogue of the classical absolute stability theory can be developed. Further, using a suitable generalisation of the equilibrium theory of Hall [ACM Trans. on Math. Soft. 11 (1985), pp. 289–301], accurate predictions are made about the performance of modern, adaptive algorithms.Supported by the University of Dundee Research Initiatives Fund.     

On the analytic model of a class of semi-hyponormal operators

The analytic model of a class of semi-hyponormal operators is derived using three kernal functions. In addition, explicit forms of the kernal functions are given and the Pincus principal functions of the operators are calculated.     

Time resolved measurements of the flow generated by suction feeding fish

The majority of aquatic vertebrates are suction feeders: by rapidly expanding the mouth cavity they generate a fluid flow outside of their head in order to draw prey into their mouth. In addition to the biological relevance, the generated flow field is interesting fluid mechanically as it incorporates high velocities, is localized in front of the mouth, and is unsteady, typically lasting between 10 and 50 ms. Using manometry and high-speed particle image velocimetry, this is the first study to quantify pressure within and outside the mouth of a feeding fish while simultaneously measuring the velocity field outside the mouth. Measurements with a high temporal (2 ms) and spatial (<1 mm) resolution were made for several feeding events of a single largemouth bass (Micropterus salmoides). General properties of the flow were evaluated, including the transient velocity field, its relationship to pressure within the mouth and pressure at the prey. We find that throughout the feeding event a relationship exists for the magnitude of fluid speed as a function of distance from the predator mouth that is based on scaling the velocity field according to the size of the mouth opening and the magnitude of fluid speed at the mouth. The velocity field is concentrated within an area extending approximately one mouth diameter from the fish and the generated pressure field is even more local to the mouth aperture. Although peak suction pressures measured inside the mouth were slightly larger than those that were predicted using the equations of motion, we find that these equations give a very accurate prediction of the timing of peak pressure, so long as the unsteady nature of the flow is included.     

An inviscid model for vortex shedding from a deforming body

An inviscid vortex sheet model is developed in order to study the unsteady separated flow past a two-dimensional deforming body which moves with a prescribed motion in an otherwise quiescent fluid. Following Jones (J Fluid Mech 496, 405–441, 2003) the flow is assumed to comprise of a bound vortex sheet attached to the body and two separate vortex sheets originating at the edges. The complex conjugate velocity potential is expressed explicitly in terms of the bound vortex sheet strength and the edge circulations through a boundary integral representation. It is shown that Kelvin’s circulation theorem, along with the conditions of continuity of the normal velocity across the body and the boundedness of the velocity field, yields a coupled system of equations for the unknown bound vortex sheet strength and the edge circulations. A general numerical treatment is developed for the singular principal value integrals arising in the solution procedure. The model is validated against the results of Jones (J Fluid Mech 496, 405–441, 2003) for computations involving a rigid flat plate and is subsequently applied to the flapping foil experiments of Heathcote et al. (AIAA J, 42, 2196–2204, 2004) in order to predict the thrust coefficient. The utility of the model in simulating aquatic locomotion is also demonstrated, with vortex shedding suppressed at the leading edge of the swimming body.      

Assessing the exchange coupling in binuclear lanthanide(iii) complexes and the slow relaxation of the magnetization in the antiferromagnetically coupled Dy2 derivative

We report here the synthesis and the investigation of the magnetic properties of a series of binuclear lanthanide complexes belonging to the metallacrown family. The isostructural complexes have a core structure with the general formula [Ga₄Ln₂(shi^3–)₄(Hshi^2–)₂(H₂shi^–)₂(C₅H₅N)₄(CH₃OH)_x(H₂O)_x]·xC₅H₅N·xCH₃OH·xH₂O (where H₃shi = salicylhydroxamic acid and Ln = Gd^III1; Tb^III2; Dy^III3; Er^III4; Y^III5; Y^III_0.9Dy^III_0.16). Apart from the Er-containing complex, all complexes exhibit an antiferromagnetic exchange coupling leading to a diamagnetic ground state. Magnetic studies, below 2 K, on a single crystal of 3 using a micro-squid array reveal an opening of the magnetic hysteresis cycle at zero field. The dynamic susceptibility studies of 3 and of the diluted DyY 6 complexes reveal the presence of two relaxation processes for 3 that are due to the excited ferromagnetic state and to the uncoupled Dy^III ions. The antiferromagnetic coupling in 3 was shown to be mainly due to an exchange mechanism, which accounts for about 2/3 of the energy gap between the antiferro- and the ferromagnetic states. The overlap integrals between the Natural Spin Orbitals (NSOs) of the mononuclear fragments, which are related to the magnitude of the antiferromagnetic exchange, are one order of magnitude larger for the Dy₂ than for the Er₂ complex.     

All-acrylic film-forming colloidal polymer/silica nanocomposite particles prepared by aqueous emulsion polymerization

The efficient synthesis of all-acrylic, film-forming, core-shell colloidal nanocomposite particles via in situ aqueous emulsion copolymerization of methyl methacrylate with n-butyl acrylate in the presence of a glycerol-functionalized ultrafine silica sol using a cationic azo initiator at 60 °C is reported. It is shown that relatively monodisperse nanocomposite particles can be produced with typical mean weight-average diameters of 140-330 nm and silica contents of up to 39 wt %. The importance of surface functionalization of the silica sol is highlighted, and it is demonstrated that systematic variation of parameters such as the initial silica sol concentration and initiator concentration affect both the mean particle diameter and the silica aggregation efficiency. The nanocomposite morphology comprises a copolymer core and a particulate silica shell, as determined by aqueous electrophoresis, X-ray photoelectron spectroscopy, and electron microscopy. Moreover, it is shown that films cast from n-butyl acrylate-rich copolymer/silica nanocomposite dispersions are significantly more transparent than those prepared from the poly(styrene-co-n-butyl acrylate)/silica nanocomposite particles reported previously. In the case of the aqueous emulsion homopolymerization of methyl methacrylate in the presence of ultrafine silica, a particle formation mechanism is proposed to account for the various experimental observations made when periodically sampling such nanocomposite syntheses at intermediate comonomer conversions.     

Adsorption of sophorolipid biosurfactants on their own and mixed with sodium dodecyl benzene sulfonate, at the air/water interface

The adsorption of the lactonic (LS) and acidic (AS) forms of sophorolipid and their mixtures with the anionic surfactant sodium dodecyl benzene sulfonate (LAS) has been measured at the air/water interface by neutron reflectivity, NR. The AS and LS sophorolipids adsorb with Langmuir-like adsorption isotherms. The more hydrophobic LS is more surface active than the AS, with a lower critical micellar concentration, CMC, and stronger surface adsorption, with an area/molecule ～70 ?(2) compared with 85 ?(2) for the AS. The acidic sophorolipid shows a maximum in its adsorption at the CMC which appears to be associated with a mixture of different isomeric forms. The binary LS/AS and LS/LAS mixtures show a strong surface partitioning in favor of the more surface active and hydrophobic LS component but are nevertheless consistent with ideal mixing at the interface. In contrast, the surface composition of the AS/LAS mixture is much closer to the solution composition, but the surface mixing is nonideal and can be accounted for by regular solution theory, RST. In the AS/LS/LAS ternary mixtures, the surface adsorption is dominated by the sophorolipid, and especially the LS component, in a way that is not consistent with the observations for the binary mixtures. The extreme partitioning in favor of the sophorolipid for the LAS/LS/AS (1:2) mixtures is attributed to a reduction in the packing constraints at the surface due to the AS component. Measurements of the surface structure reveal a compact monolayer for LS and a narrow solvent region for LS, LS/AS, and LS/LAS mixtures, consistent with the more hydrophobic nature of the LS component. The results highlight the importance of the relative packing constraints on the adsorption of multicomponent mixtures, and the impact of the lactonic form of the sophorolipid on the adsorption of the sophorolipid/LAS mixtures.     

Cyclization of protected N-acylhydroxyguanidine to 3-amino-1,2,4-oxadiazole

The acid mediated cyclization of a protected N-acylhydroxyguanidine into the corresponding 3-amino-1,2,4-oxadiazole and confirmation of its structure by single crystal X-ray crystallography is reported herein. The yield of the cyclization is comparable to literature reports utilizing alternative procedures. Importantly, these new conditions provide complementary chemoselectivity to current synthetic procedures which may be useful for the synthesis of 3-amino-1,2,4-oxadiazoles in general.