Dynamic Scaling in Miscible Viscous Fingering

We consider dynamic scaling in gravity driven miscible viscous fingering. We prove rigorous one-sided bounds on bulk transport and coarsening in regimes of physical interest. The analysis relies on comparison with solutions to one-dimensional conservation laws, and new scale-invariant estimates. Our bounds on the size of the mixing layer are of two kinds: a naive bound that is sharp in the absence of diffusion, and a more careful bound that accounts for diffusion as a selection criterion in the limit of vanishingly small diffusion. The naive bound is simple and robust, but does not yield the experimental speed of transport. In a reduced model derived by Wooding [20], we prove a sharp upper bound on the size of the mixing layer in accordance with his experiments. Woodings model also provides an example of a scalar conservation law where the entropy condition is not the physically appropriate selection criterion.     

2001 Publication Programme

Publishers Note

2001 Publication Programme     

On the ratio Σ0/Λ in electroproduction off protons

Recent experimental results on electroproduction off protons show that the ratio σ(ep → eK⁺Σ⁰)/σ(ep → eK⁺Λ) decreases strongly with increasing Q². A simple argument is given in the framework of the quark parton model which could provide a qualitative understanding of this fact. The decrease of the Σ⁰/Λ ratio is related to the decrease of the ratio $\text{F}{}_1{}^{\mathrm{<mtext>\gamma </mtext><mtext>n</mtext>}}\text{/F}{}_1{}^{\mathrm{<mtext>\gamma </mtext><mtext>p</mtext>}}$ as Q² increases, where $\text{F}{}_1{}^{\mathrm{<mtext>\gamma </mtext><mtext>p</mtext>}}\text{and F}{}_1{}^{\mathrm{<mtext>\gamma </mtext><mtext>n</mtext>}}$ are the usual structure functions for deep inelastic electron-nucleon scattering.     

Head-to-head polymers. XXIV. Synthesis of head-to-head polyisobutylene by Grignard coupling reaction

Head-to-head polyisobutylene of molecular weight 3000 to 10,000 was prepared in about 30% yield by Grignard coupling polymerization of 2,2,3,3-tetramethyl-1,4-dibromobutane with copper(I) tris(triphenylphosphino)bromide as the catalyst. Head-to-head polyisobutylene is crystalline, with a T_m of 187°C and a T_g of 87°C and still has bromine end groups left. The polymers have been characterized by their infrared, ¹H-, and ¹³C-NMR spectra and compared to the corresponding head-to-tail polymers of similar molecular weight. A substantial portion of the reaction product of the Grignard coupling polymerization is not polymeric but was identified as cyclized product, 1,1,2,2-tetramethylcyclobutane.