Bifurcation and Stability Theory of Periodic Solutions For Integrodifferential Equations

This paper is concerned with a nonlinear integrodifferential equation (the delay logistic equation) governing the growth dynamics of a single species N(t) for time t > 0. This equation contains a positive parameter λ. Suppose that there exists a positive equilibrium solution N = c which is stable for all small values of λ. Assume also that this solution loses stability as λ is increased past a critical value λ*. This will correspond to a simple pure imaginary conjugate pair of roots of a characteristic equation associated with the linearized stability of N = c at λ = λ*. Then we will construct a unique bifurcating time periodic solution of the equation as a Taylor series in a parameter ε. Furthermore this solution exists either for supercritical values of the parameter (λ > λ*) or for subcritical values (λ < λ*). The stability behavior of this small periodic solution can be characterized according to whether the bifurcation is supercritical or subcritical-supercritical solutions are stable, but subcritical solutions are unstable. Therefore these results are analogous to Hoprs bifurcation theorem for autonomous systems of differential equations.     

Hydrodynamics of a particle impact on a wall

The problem of a particle impacting on a wall, a common phenomenon in particle-laden flows in the minerals and process industries, is investigated computationally using a spectral-element method with the grid adjusting to the movement of the particle towards the wall. Remeshing is required at regular intervals to avoid problems associated with mesh distortion and the constantly reducing maximum time-step associated with integration of the non-linear convective terms of the Navier–Stokes equations. Accurate interpolation between meshes is achieved using the same high-order interpolation employed by the spectral-element flow solver. This approach allows the full flow evolution to be followed from the initial approach, through impact and afterwards as the flow relaxes. The method is applied to the generic two-dimensional and three-dimensional bluff body geometries, the circular cylinder and the sphere. The principal case reported here is that of a particle colliding normally with a wall and sticking. For the circular cylinder, non-normal collisions are also considered. The impacts are studied for moderate Reynolds numbers up to approximately 1200. A cylindrical body impacting on a wall produces two vortices from its wake that convect away from the cylinder along the wall before stalling while lifting induced wall vorticity into the main flow. The situation for a sphere impact is similar, except in this case a vortex ring is formed from the wake vorticity. Again, secondary vorticity from the wall and particle plays a role. At higher Reynolds number, the secondary vorticity tends to form a semi-annular structure encircling the primary vortex core. At even higher Reynolds numbers, the secondary annular structure fragments into semi-discrete structures, which again encircle and orbit the primary core. Vorticity fields and passive tracer particles are used to characterize the interaction of the vortical structures. The evolution of the pressure and viscous drag coefficients during a collision are provided for a typical sphere impact. For a Reynolds number greater than approximately 1000 for a sphere and 400 for a cylinder, the primary vortex core produced by the impacting body undergoes a short-wavelength instability in the azimuthal/spanwise direction. Experimental visualisation using dye carried out in water is presented to validate the predictions.     

Enthalpies of reaction of bis(triphenylphosphine)(trans-stilbene)platinum(0) with diphenylcyclopropenone and benzocyclobutenedione

Enthalpies, ΔH(1) ?94.8 ± 6.0 and ΔH(6) ?57.1 ± 5.1 kJ mol^?1, of the following reactions have been measured calorimetrically [Pt(trans-stilbene)(PPh₃)₂](s) + dpcp(g) → (PPh₃)₂Pt(dpcb)(s) + trans-stilbene(g) (1) [Pt(trans-stilbene)(PPh₃)₂](s) + bcbd(g) → (PPh₃)₂Pt(bcpd)(s) + trans-stilbene (g) (6) where dpcp is diphenylcyclopropenone, (PPh₃)₂Pt(dpcb) is (1,1-bistriphenylphosphine)platinadiphenylcyclobutenone, (PPh₃)₂$\text{PtC(Ph)C(Ph)C}$O, bcbd is benzocyclobutene-1,2-dione and (PPh₃)₂Pt(bcpd) is (1,1-bistriphenylphosphine)platinabenzocyclopentanedione,

. It is concluded that the five-membered platinacyclo ring system in (PPh₃)₂Pt(dpcb) is not heavily strained.     

Synthesis of high-purity α- and β-PbO and possible applications to synthesis and processing of other lead oxide materials

Litharge, the red tetragonal form of lead oxide α-PbO and massicot, the yellow orthorhombic form β-PbO, are synthesized from lead(II) salts in aqueous media at elevated temperatures. Scanning electron microscopy (SEM) and X-ray diffraction (XRD) were used to characterize the size, morphology, and crystallographic structural forms of the products. The role of impurities in the experimental synthesis of the materials and microstructural variations in the final products are described. The implications of these observations regarding the synthesis of different conducting lead oxides and other related materials are discussed. PACS 71.20.Ps; 72.80.Jc; 74.62.Bf; 74.62.Dh; 75.50.Tt     

Cryopreservation of protonemata of Ditrichum cornubicum (paton) comparing the effectiveness of four cryoprotectant pretreatments

If an ex situ programme for the conservation of rare and threatened bryophytes is to succeed, a method of long-term, basal storage is required. The aim of this study was to determine the effectiveness of sucrose, ABA and sucrose/ABA combined, as cryoprotectant pretreatments, for the long-term storage of protonemata of the endangered Cornish path moss, Ditrichum cornubicum. This work is a prelude to the in vitro conservation of rare and endangered UK bryophytes. The addition of sucrose and/or ABA in the pre-culture medium, reduced protonemal growth rate. However, pretreatment afforded a high level of protection against tissue damage during dehydration and freezing, with 100 % regeneration of pretreated plants after thawing, compared to 53 % regeneration of controls. Sucrose combined with ABA in the pre-culture medium gave the most effective cryoprotection, with protonemata having fewer necrotic tissues and rapid recovery of growth.     

Trimethylphosphine complexes of tungsten(O) and (IV). X-ray crystal structures of trimethylphosphine tris (phenylacetylene) tungsten(O); bis(trimethylphosphine) tetrakis (methylisocyanide) tungsten(O), and oxodichlorotris (trimethylphosphine) tungsten (IV)

The reduction of WCl₄(PMe₃)₃ by sodium amalgam in presence of phenylacetylene gives W(PMe₃)(PhCCH)₃ (A). Reduction in presence of methylisocyanide gives W(PMe₃)₂(MeNC)₄ (B), while in presence of excess PMe₃ in tetrahydrofuran under hydrogen, WH₂Cl₂(PMe₃)₄ (C) is formed. The reaction of WCl₂(PMe₃)₄ with methanol in tetrahydrofuran gives mixtures of WH₂Cl₂(PMe₃)₄ and WOC1₂(PMe₃)₃ (D).The structures of A, B, and D have been determined by X-ray diffraction.     

Non-isothermal developing flow of a power-law fluid in a converging slit

A numerical solution of the developing non-isothermal flow of a generalised power-law fluid in a slightly converging slit is presented, a problem which is relevance to some polymer processing operations.The results are presented in graphical form. They indicate that inertia forces and kinetic energy increases along the slit and these factors affect the development of the velocity and stress distributions. The effect of other parameters on the process is also brought out.     

Heat transfer to flowing polymer melts

A theoretical and experimental study of heat transfer to polymer melts flowing through ducts is presented. The mathematical model provides for shear heating and expansion cooling effects for various wall boundary conditions.Experimental results, obtained in tube flow, show reproducible temperature and velocity profiles and the data confirm the predictions of the magnitude of the shear heating and expansion cooling effects.It is concluded that the method can be justified for predicting heat transfer in more complex geometries and some preliminary results are presented.