ON THE DIRICHLET-NEUMANN BOUNDARY PROBLEM FOR SCALAR CONSERVATION LAWS

We consider a Dirichlet-Neumann boundary problem in a bounded domain for scalar conservation laws. We construct an approximate solution to the problem via an elliptic approximation for which, under appropriate assumptions, we prove that the corresponding limit satisfies the considered equation in the interior of the domain. The basic tool is the compensated compactness method. We also provide numerical examples.

Reynolds-number measurements for low-Prandtl-number turbulent convection of large-aspect-ratio samples

We present experimental results for the Reynolds number ${\mathit{Re}}_{U} $ based on the horizontal mean-flow velocity $U$ and for ${\mathit{Re}}_{V} $ based on the root-mean-square horizontal fluctuation velocity $V$ for turbulent Rayleigh–Bénard convection in a cylindrical sample of aspect ratio $\Gamma = 10. 9$ over the Prandtl number range $0. 18\leq \mathit{Pr}\leq 0. 88$. The results were derived from space–time cross-correlation functions of shadowgraph images, using the elliptic approximation of He & Zhang (Phys. Rev. E, vol. 73, 2006, 055303). The data cover the Rayleigh number range from $3\times 1{0}^{5} $ to $2\times 1{0}^{7} $. We find that ${\mathit{Re}}_{U} $ is nearly two orders of magnitude smaller than the values given by the Grossmann–Lohse (GL) model (Grossmann & Lohse, Phys. Rev. E, vol. 66, 2002, 016305) for $\Gamma = 1. 00$ and attribute this difference to averaging caused by lateral random diffusion of the large-scale circulation cells in large-$\Gamma $ samples. For the fluctuations we found ${\mathit{Re}}_{V} = {\tilde {R} }_{0} {\mathit{Pr}}^{\alpha } {\mathit{Ra}}^{\eta } $, with ${\tilde {R} }_{0} = 0. 31$, $\alpha = - 0. 53\pm 0. 11$ and $\eta = 0. 45\pm 0. 03$. That result agrees well with the GL model. The close agreement of the coefficient ${\tilde {R} }_{0} $ must be regarded as a coincidence because the GL model was for $\Gamma = 1. 00$ and for a mean-flow velocity $U$.

Intermediate Phases in LixFePO4

Rietveld analysis for the time of flight powder neutron diffraction profile for LiFePO4 at room temperature was performed. Refined tensor elements of the unisotropic thermal factor under the elliptic approximation showed the principal axis of the lithium vibration is toward the face shared vacant tetrahedral space and is consistent with the theoretical prediction; lithium ions diffuse along curved one-dementional chain along b-axis. Impact of temperature on the phase diagram of LixFePO4 with > 200nm particle size was slight under the unmixing line around 200 C. While the reduction in particle size down to <100 nm seems to have significant effect to the room temperature miscibility gap. The thermodynamic concepts for the extended solution in smaller particles are discussed, followed by a demonstration of very high rate capability observed for the small spherical particles < 80 nm.