A transform method for linear evolution PDEs on a finite interval

We study initial boundary value problems for linear scalar evolutionpartial differential equations, with spatial derivatives ofarbitrary order, posed on the domain {t > 0, 0 < x <L}. We show that the solution can be expressed as an integralin the complex k-plane. This integral is defined in terms ofan x-transform of the initial condition and a t-transform ofthe boundary conditions. The derivation of this integral representationrelies on the analysis of the global relation, which is an algebraicrelation defined in the complex k-plane coupling all boundaryvalues of the solution. For particular cases, such as the case of periodic boundaryconditions, or the case of boundary value problems for even-orderPDEs, it is possible to obtain directly from the global relationan alternative representation for the solution, in the formof an infinite series. We stress, however, that there existinitial boundary value problems for which the only representationis an integral which cannot be written as an infinite series.An example of such a problem is provided by the linearized versionof the KdV equation. Similarly, in general the solution of odd-orderlinear initial boundary value problems on a finite intervalcannot be expressed in terms of an infinite series.     

Numerical simulations of a filament in a flowing soap film

Experiments concerning the properties of soap films have recently been carried out and these systems have been proposed as experimental versions of theoretical two‐dimensional liquids. A silk filament introduced into a flowing soap film, was seen to demonstrate various stable modes, and these were, namely, a mode in which the filament oscillates and one in which the filament is stationary and aligns with the flow of the liquid. The system could be forced from the oscillatory mode into the non‐ oscillatory mode by varying the length of the filament. In this article we use numerical and computational techniques in order to simulate the strongly coupled behaviour of the filament and the fluid. Preliminary results are presented for the specific case in which the filament is seen to oscillate continuously for the duration of our simulation. We also find that the filament oscillations are strongly suppressed when we reduce the effective length of the filament. We believe that these results are reminiscent of the different oscillatory and non‐oscillatory modes observed in experiment. The numerical solutions show that, in contrast to experiment, vortices are created at the leading edge of the filament and are preferentially grown in the curvature of the filament and are eventually released from the trailing edge of the filament. In a similar manner to oscillating hydrofoils, it seems that the oscillating filaments are in a minimal energy state, extracting sufficient energy from the fluid to oscillate. In comparing numerical and experimental results it is possible that the soap film does have an effect on the fluid flow especially in the boundary layer where surface tension forces are large. Copyright © 2004 John Wiley & Sons, Ltd.     

LES of turbulent heat transfer: proper convection numerical schemes for temperature transport

Large eddy simulations of two basic configurations (decay of isotropic turbulence, and the academic plane channel flow) with heat transfer have been performed comparing several convection numerical schemes, in order to discuss their ability to evaluate temperature fluctuations properly. Results are compared with the available incompressible heat transfer direct numerical simulation data. It is shown that the use of regularizing schemes (such as high order upwind type schemes) for the temperature transport equation in combination with centered schemes for momentum transport equation gives better results than the use of centred schemes for both equations. Copyright © 2004 John Wiley & Sons, Ltd.     

High‐order accurate numerical solutions of incompressible flows with the artificial compressibility method

A high‐order accurate, finite‐difference method for the numerical solution of incompressible flows is presented. This method is based on the artificial compressibility formulation of the incompressible Navier–Stokes equations. Fourth‐ or sixth‐order accurate discretizations of the metric terms and the convective fluxes are obtained using compact, centred schemes. The viscous terms are also discretized using fourth‐order accurate, centred finite differences. Implicit time marching is performed for both steady‐state and time‐accurate numerical solutions. High‐order, spectral‐type, low‐pass, compact filters are used to regularize the numerical solution and remove spurious modes arising from unresolved scales, non‐linearities, and inaccuracies in the application of boundary conditions. The accuracy and efficiency of the proposed method is demonstrated for test problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Sulfonated poly(aryl ether ketone)s containing naphthalene moieties obtained by direct copolymerization as novel polymers for proton exchange membranes

A series of sulfonated poly(aryl ether ketone)s (SPAEKs) were prepared by aromatic nucleophilic polycondensation of 2,6‐dihydroxynaphthalene with 5,5′‐carbonyl‐bis(2‐fluorobenzenesulfonate) and 4,4′‐difluorobenzophenone. The structure and degree of sulfonation (DS) of the SPAEKs were characterized using ¹H NMR spectroscopy. The experimentally observed DS values were close to the expected values derived from the starting material ratios. The thermal stabilities of the SPAEKs were characterized by thermogravimetric analysis, which showed that in acid and sodium salt forms they were thermally stable in air up to about 240 and 380 °C, respectively. Transparent membranes cast from the directly polymerized SPAEKs exhibited good mechanical properties in both dry and hydrated states. The dependence of water uptake and of membrane swelling on the DS at different temperatures was studied. SPAEK membranes with a DS from 0.72 to 1.60 maintained adequate mechanical properties after immersion in water at 80 °C for 24 h. The proton conductivity of SPAEK membranes with different degrees of sulfonation was measured as a function of temperature. The proton conductivity of the SPAEK films increased with increased DS, and the highest room temperature conductivity (4.2 × 10^?2 S/cm) was recorded for a SPAEK membrane with a DS of 1.60, which further increased to 1.1 × 10^?1 S/cm at 80 °C. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 2866–2876, 2004     

Photoinitiators with functional groups. VII. Covalently bonded camphorquinone—amines

Camphorquinone (CQ), a widely used photoinitiator (PI) in dental applications, was covalently bonded to aromatic amines to enhance the rate of electron and proton transfer effect due to the close vicinity of the diketone and the amine group. 10‐bromocamphorquinone and 10‐bromomethylcamphorquinone were selected as suitable precursors for esterification with the carboxyl group containing aromatic amines based on 4‐dimethylaminobenzoic acid. Properties of the new photoinitiating systems were investigated by UV spectroscopy and differential scanning photocalorimetry in lauryl acrylate. Compared to physical mixtures, in all cases similar or even better performance was obtained. Surprisingly, 10‐acetyl derivatives 7 – 9 and 18 especially, were found to be highly reactive. Compared to CQ/ethyl 4‐dimethylaminobenzoate, the rate of photopolymerization was increased by a factor of up to 2. Intramolecular reaction was confirmed by photo‐differential scanning calorimetry experiments with varying PI concentrations. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 4948–4963, 2004