Corner transport upwind lattice Boltzmann model for bubble cavitation

Aiming to study the bubble cavitation problem in quiescent and sheared liquids, a third-order isothermal lattice Boltzmann (LB) model that describes a two-dimensional ($2D$) fluid obeying the van der Waals equation of state, is introduced. The evolution equations for the distribution functions in this off-lattice model with 16 velocities are solved using the corner transport upwind (CTU) numerical scheme on large square lattices (up to $6144 \times 6144$ nodes). The numerical viscosity and the regularization of the model are discussed for first and second order CTU schemes finding that the latter choice allows to obtain a very accurate phase diagram of a nonideal fluid. In a quiescent liquid, the present model allows to recover the solution of the $2D$ Rayleigh-Plesset equation for a growing vapor bubble. In a sheared liquid, we investigated the evolution of the total bubble area, the bubble deformation and the bubble tilt angle, for various values of the shear rate. A linear relation between the dimensionless deformation coefficient $D$ and the capillary number $Ca$ is found at small $Ca$ but with a different factor than in equilibrium liquids. A non-linear regime is observed for $Ca \gtrsim 0.2$.Comment: Accepted for publication in Phys. Rev.

Solitary and compact-like shear waves in the bulk of solids

We show that a model proposed by Rubin, Rosenau, and Gottlieb [J. Appl. Phys. 77 (1995) 4054], for dispersion caused by an inherent material characteristic length, belongs to the class of simple materials. Therefore, it is possible to generalize the idea of Rubin, Rosenau, and Gottlieb to include a wide range of material models, from nonlinear elasticity to turbulence. Using this insight, we are able to fine-tune nonlinear and dispersive effects in the theory of nonlinear elasticity in order to generate pulse solitary waves and also bulk travelling waves with compact support

Modular symbols in Iwasawa theory

This survey paper is focused on a connection between the geometry of $\mathrm{GL}_d$ and the arithmetic of $\mathrm{GL}_{d-1}$ over global fields, for integers $d \ge 2$. For $d = 2$ over $\mathbb{Q}$, there is an explicit conjecture of the third author relating the geometry of modular curves and the arithmetic of cyclotomic fields, and it is proven in many instances by the work of the first two authors. The paper is divided into three parts: in the first, we explain the conjecture of the third author and the main result of the first two authors on it. In the second, we explain an analogous conjecture and result for $d = 2$ over $\mathbb{F}_q(t)$. In the third, we pose questions for general $d$ over the rationals, imaginary quadratic fields, and global function fields.Comment: 43 page

Global generalized solutions for Maxwell-alpha and Euler-alpha equations

We study initial-boundary value problems for the Lagrangian averaged alpha models for the equations of motion for the corotational Maxwell and inviscid fluids in 2D and 3D. We show existence of (global in time) dissipative solutions to these problems. We also discuss the idea of dissipative solution in an abstract Hilbert space framework.Comment: 27 pages, to appear in Nonlinearit

Velocity distribution function of spontaneously evaporating atoms

Numerical solutions of the Enskog-Vlasov (EV) equation are used to determine the velocity distribution function of atoms spontaneously evaporating into near-vacuum conditions. It is found that an accurate approximation is provided by a half-Maxwellian including a drift velocity combined with different characteristic temperatures for the velocity components normal and parallel to the liquid-vapor interface. The drift velocity and the temperature anisotropy reduce as the liquid bulk temperature decreases but persist for relatively low temperatures corresponding to a vapor behaviour which is only slightly non-ideal. Deviations from the undrifted isotropic half-Maxwellian are shown to be consequences of collisions in the liquid-vapor interface which preferentially backscatter atoms with lower normal-velocity component

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final publication is available at http://www.springerlink.co

Bjorken flow attractors with transverse dynamics

In the context of the longitudinally boost-invariant Bjorken flow with transverse expansion, we use three different numerical methods to analyze the emergence of attractor solutions in an ideal gas of massless particles exhibiting constant shear viscosity to entropy density ratio $\eta / s$. The fluid energy density is initialized using a Gaussian profile in the transverse plane, while the ratio $\chi = \mathcal{P}_L / \mathcal{P}_T$ between the longitudinal and transverse pressures is set at initial time $\tau_0$ to a constant value $\chi_0$ throughout the system employing the Romatschke-Strickland distribution. We introduce the hydrodynamization time $\delta \tau_H = (\tau_H - \tau_0)/ \tau_0$ based on the time $\tau_H$ when the standard deviation $\sigma(\chi)$ of a family of solutions with different $\chi_0$ reaches a minimum value at the point of maximum convergence of the solutions. In the $0+1{\rm D}$ setup, $\delta \tau_H$ exhibits scale invariance, being a function only of $(\eta / s) / (\tau_0 T_0)$. With transverse expansion, we find a similar $\delta \tau_H$ computed with respect to the local initial temperature, $T_0(r)$. We highlight the transition between the regimes where the longitudinal and transverse expansions dominate. We find that the hydrodynamization time required for the attractor solution to be reached increases with the distance from the origin, as expected based on the properties of the $0+1{\rm D}$ system defined by the local initial conditions. We argue that hydrodynamization is predominantly the effect of the longitudinal expansion, being significantly influenced by the transverse dynamics only for small systems or for large values of $\eta / s$.Comment: Accepted version. 20 pages, 11 figures, 1 tabl

OPTICAL AND PHOTOCATALYTIC PROPERTIES OF ELECTROSPUN ZnO FIBERS

ZnO nanofibers were obtained by electrospinning a solution of zinc acetate dihydrate and polyvinylpyrrolidone in N,N-dimethylformamide, followed by calcination at 500, 650 or 800 °C for 1 h. X-ray diffraction, selected area electron diffraction, scanning electron microscopy, transmission electron microscopy, reflectance spectroscopy and photoluminescence spectroscopy were used for the characterization of the resulting fibers. The thermally treated samples exhibit ZnO single phase with polycrystalline hexagonal structure. The morphological investigation revealed an accentuated contraction process during calcination, as well as the increase of the crystallite size and the appearance of a breaking tendency with the calcination temperature enhancement. Both UV and Visible emissions under excitation at 350 nm were showed by the optical studies, which also led to band gap values slightly lower than those reported for similar one-dimensional nanostructures. In order to assess the photocatalytic activity of ZnO fibers, the photodegradation of methylene blue in aqueous medium (10 -3 M) under UV irradiation (368 nm) was analyzed