Research problems

This section appears from time to time. Contributions are invited, and should be submitted to R. C. Thompson, Mathematics Department, University of California, Santa Barbara, CA 93106.     

Numerical study of simultaneous natural convection heat transfer from both surfaces of a uniformly heated thin plate with arbitrary inclination

 Numerical studies were conducted to investigate the natural convection heat transfer around a uniformly heated thin plate with arbitrary inclination in an infinite space. The numerical approach was based on the finite volume technique with a nonstaggered grid arrangement. For handling the pressure–velocity coupling the SIMPLE-algorithm was used. QUICK scheme and first order upwind scheme were employed for discretization of the momentum and energy convective terms respectively. Plate width and heating rate were used to vary the modified Rayleigh number over the range of 4.8×10⁶ to 1.87×10⁸. Local and average heat transfer characteristics were compared with regarding to the inclination angle. The empirical expressions for local and average Nusselt number were correlated. It has been found that for inclination angle less than 10^∘, the flow and heat transfer characteristics are complicated and the average Nusselt number can not be correlated by one equation while for inclination angle larger than 10^∘, the average Nusselt number can be correlated into an elegant correlation. Received on 18 April 2001 / Published online: 29 November 2001     

Finite element method in the solution of the Euler and Navier-Stokes equations for internal flow

Finite element solutions of the Euler and Navier-Stokes equations are presented, using a simple dissipation model. The discretization is based on the weak-Galerkin weighted residual method and equal interpolation functions for all the unknowns are permitted. The nonlinearity is iterated upon using a Newton method and at each iteration the linear algebraic system is solved by a direct solver with all unknowns fully coupled. Results are presented for two-dimensional transonic inviscid flows and two- and three-dimensional incompressible viscous flows. Convergence of the algorithm is shown to be quadratic, reaching machine accuracy in very few iterations. The inviscid results demonstrate the existence of nonunique numerical solutions to the steady Euler equations.     

Interpretation of the polyelectrolyte and antipolyelectrolyte effects of poly(N‐vinylimidazole‐co‐sodium styrenesulfonate) hydrogels

Hydrogels of N‐vinylimidazole (VI) and sodium styrenesulfonate (SSS) were synthesized in aqueous solution by radical crosslinking copolymerization with N,N′‐methylene‐bis(acrylamide) as crosslinker. Swelling in several saline solutions was measured for hydrogel samples synthesized with different comonomer concentrations (C_T = 10, 25, or 40%) and with SSS mole fractions covering a broad range (f_SSS = 0–0.7), while the crosslinker ratio was 2 wt % in all cases. The degree of swelling in aqueous solution with a specific ionic strength (μ), plotted versus the SSS composition of the feed, shows a minimum for any set of samples synthesized with a fixed C_T. The dependence of swelling on μ shows both polyelectrolyte (f_SSS beyond the minimum) and antipolyelectrolyte behaviors (in the low f_SSS limit). It was found that the nonGaussian factor of the crosslinking density and the polymer‐solvent interaction parameter increase with f_SSS for any C_T. Moreover, in the low f_SSS limit, the osmotic swelling pressure is governed not only by the ionic contribution, but also by the polymer‐solvent mixing and, the concentration of mobile counterions inside the gel is not proportional to the net fixed charge but to the addition of cationic and anionic side groups, what discards the formation of ionic pairs. The antipolyelectrolyte effect is interpreted as due to the increasing protonation of VI as μ goes up. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 1683–1693, 2007     

On 2-ciass field towers of some imaginary quadratic number fields

We construct an infinite family of imaginary quadratic number fields with 2-class groups of type (2, 2, 2) whose Hilbert 2-class fields are finite.     

A note on dephasing and pair-breaking

It is shown that the dephasing which suppresses the weak localization correction to the conductivity has the same physical origin than pair-breaking in superconductors and thus-following de Gennes-may be expressed in terms of the correlation function of the operator for time reversal.     

A globally convergent multi-grid algorithm for moving boundary problems of two-phase Stefan type

In this paper we deal with the numerical solution of movingboundary problems of two-phase Stefan type. Based on an implicitdiscretization in time and the use of continuous, piecewiselinear finite elements in the space variables with respect tothe weak formulation of the problem, a globally convergent multi-gridalgorithm is developed. That algorithm strongly relies on thevariational characterization of the fully discretized problemas the unconstrained minimization of a subdifferentiable convexobjective functional. Numerical results indicate a significantimprovement in efficiency compared with previous multi-gridapproaches.     

Matrix effects in the determination of butyltin compounds in environmental samples by GC AA after hydride generation

Experiments for the determination of mono-, di and tri-butyltin (MBT, DBT and TBT) by hydride generation/gas chromatography/atomic absorption spectrometry in various matrices (sediment, suspended matter, mussel, algae and water) have revealed that poor butyltin recoveries are obtained in sediments displaying high sulphur and hydrocarbon contents; very poor recoveries were also observed for TBT in sediments with high chlorophyll pigment contents as well as in algal samples. It was however not clear whether the hydride generatin was inhibited by these infering compounds, as was previously assumed in the case of hydrocarbons, or whether interferences affected the atomization rate. Further studies were performed to solve this problem in order to validate this method in the case of analyses of, for example, oil-contaminated sediment and algae. This paper presents the results obtained. It is concluded here that the poor recoveries were due to an inhibition of hydride generation rather than to interference at the atomization stage.     

X-ray diffraction study of 'side-on fixed' homopolysiloxanes from nematic to smectic C phases

The X-ray patterns for the nematic phase in a series of side-on fixed polysiloxanes show different kinds of diffuse elements which imply complex short range ordering. A systematic structural study of the evolution of the patterns versus two molecular parameters, namely the spacer length and the length of the terminal aliphatic tails, suggests the possible occurrence of a smectic C phase for certain values of these parameters. Taking into account these tendencies, new synthesis led to a nematic-smectic C polymorphism observed for the first time in side-on fixed polysiloxanes.