An exact Riemann solver based solution for regular shock refraction

We study the classical problem of planar shock refraction at an oblique density discontinuity, separating two gases at rest. When the shock impinges on the density discontinuity, it refracts and in the hydrodynamical case 3 signals arise. Regular refraction means that these signals meet at a single point, called the triple point. After reflection from the top wall, the contact discontinuity becomes unstable due to local Kelvin-Helmholtz instability, causing the contact surface to roll up and develop the Richtmyer-Meshkov instability. We present an exact Riemann solver based solution strategy to describe the initial self similar refraction phase, by which we can quantify the vorticity deposited on the contact interface. We investigate the effect of a perpendicular magnetic field and quantify how addition of a perpendicular magnetic field increases the deposition of vorticity on the contact interface slightly under constant Atwood number. We predict wave pattern transitions, in agreement with experiments, von Neumann shock refraction theory, and numerical simulations performed with the grid-adaptive code AMRVAC. These simulations also describe the later phase of the Richtmyer-Meshkov instability.Comment: 21 pages, 17 figures in 41 ps-files, accepted by J. Fluid Mec

Graphs whose minimal rank is two : the finite fields case

Let F be a finite field, G = (V,E) be an undirected graph on n vertices, and let S(F,G) be the set of all symmetric n × n matrices over F whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let mr(F,G) be the minimum rank of all matrices in S(F,G). If F is a finite field with pt elements, p ??= 2, it is shown that mr(F,G) = 2 if and only if the complement of G is the join of a complete graph with either the union of at most (pt+1)/2 nonempty complete bipartite graphs or the union of at most two nonempty complete graphs and of at most (pt - 1)/2 nonempty complete bipartite graphs. These graphs are also characterized as those for which 9 specific graphs do not occur as induced subgraphs. If F is a finite field with 2t elements, then mr(F,G) = 2 if and only if the complement of G is the join of a complete graph with either the union of at most 2t +1 nonempty complete graphs or the union of at most one nonempty complete graph and of at most 2t-1 nonempty complete bipartite graphs. A list of subgraphs that do not occur as induced subgraphs is provided for this case as well

Graphs whose minimal rank is two

Let F be a field, G = (V,E) be an undirected graph on n vertices, and let S(F,G) be the set of all symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. For example, if G is a path, S(F,G) c onsists of the symmetric irreducible tridiagonal matrices. Let mr(F,G) be the minimum rank over all matrices in S(F,G). Then mr(F,G) = 1 if and only if G is the union of a clique with at least 2 vertices and an independent set. If F is an infinite field such that char F ??= 2, then mr(F,G) = 2 if and only if the complement of G is the join of a clique and a graph that is the union of at most two cliques and any number of complete bipartite graphs. A similar result is obtained in the case that F is an infinite field with char F = 2. Furthermore, in each case, such graphs are characterized as those for which 6 specific graphs do not occur as induced subgraphs. The number of forbidden subgraphs is reduced to 4 if the graph is connected. Finally, similar criteria is obtained for the minimum rank of a Hermitian matrix to be less than or equal to two. The complement is the join of a clique and a graph that is the union of any number of cliques and any number of complete bipartite graphs. The number of forbidden subgraphs is now 5, or in the connected case, 3

One-dimensional conduction in Charge-Density Wave nanowires

We report a systematic study of the transport properties of coupled one-dimensional metallic chains as a function of the number of parallel chains. When the number of parallel chains is less than 2000, the transport properties show power-law behavior on temperature and voltage, characteristic for one-dimensional systems.Comment: 4 pages, 5 figures, submitted to Phys. Rev. Let

Transverse stability of relativistic two-component jets

Context: Astrophysical jets from various sources seem to be stratified, with a fast inner jet and a slower outer jet. As it is likely that the launching mechanism for each component is different, their interface will develop differential rotation, while the outer jet radius represents a second interface where disruptions may occur. Aims: We explore the stability of stratified, rotating, relativistic two-component jets, in turn embedded in static interstellar medium. Methods: In a grid-adaptive relativistic hydrodynamic simulation with the AMRVAC code, the non-linear azimuthal stability of two-component relativistic jets is investigated. We simulate until multiple inner jet rotations have been completed. Results: We find evidence for the development of an extended shear flow layer between the two jet components, resulting from the growth of a body mode in the inner jet, Kelvin-Helmholtz surface modes at their original interface, and their nonlinear interaction. Both wave modes are excited by acoustic waves which are reflected between the symmetry axis and the interface of the two jet components. Their interaction induces the growth of near stationary, counterrotating vortices at the outer edge of the shear flow layer. The presence of a heavy external jet allows to slow down their further development, and maintain a collimated flow. At the outer jet boundary, small-scale Rayleigh-Taylor instabilities develop, without disrupting the jet configuration. Conclusion: We demonstrate that the cross-section of two-component relativistic jets, with a heavy, cold outer jet, is non-linearly stable.Comment: Accepted in A&A 24/09/200