Fingering patterns on an expanding miscible drop in a rotating Hele‐Shaw cell

We perform a detailed numerical study for the evolution of an expanding miscible drop in a rotating Hele‐Shaw cell. Two mathematical formulations applied to model the coating layer expansion during practical spin‐coating process, such as thinning of the layer by cell pressing and drop spreading outward due to injection, are investigated. Including miscible interfacial stresses, we focus on the investigation of dynamical and morphological influences of two different stabilizing parameters: the gap width parameter for the pressing cell and the injecting strength. In the case of a pressing cell, the fingering features of the expanding miscible drop, such as the critical radius, are distinct from those ones in the experiments of spin coating due to the different distributions of the inherent radial velocity. On the other hand, the global interfacial evolutions of an expanding drop with an additional injection bear remarkable resemblances to their immiscible counterparts. The better agreement for an injecting model suggests its appropriateness when we simulate the emerging fingering instabilities in the spin‐coating process. Moreover, we investigate the effects of Coriolis force at higher miscible Bond numbers. Coriolis force affects significantly the onset of fingering instability and the tilting angles of fingers. These stable effects are in line with the results from the previous studies for miscible and immiscible flow fields. Copyright © 2007 John Wiley & Sons, Ltd.     

On the possibility of using linear current generators for geophysical research

A procedure was developed to obtain analytical estimates of the operating modes of linear electric generators (LEGs) of the railgun type with current supply to active inductive loads. An analysis of the solutions obtained shows a low effectiveness of using generators of this type to power geophysical dipoles in electrical prospecting (the efficiency does not exceed 20%). Induction coil-type LEGs provide for much higher efficiency. Numerical modeling was performed of the operating modes of one design of such generators. Versions of nonlinear-inductance LEG circuits that produce quasirectangular load current pulses and are of interest for solving electrical prospecting problems were proposed and studied. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 175–184, September–October, 2006.     

Liquid vorticity computation in non‐spherical bubble dynamics

The purpose of this work is to compare efficiency of a number of numerical techniques of computation of liquid vorticity from non‐spherical bubble oscillations. The techniques based on the finite‐difference method (FDM), the collocation method (one with differentiating (CMd) the integral boundary condition and another without it (CM)) and the Galerkin method (GM) have been considered. The central‐difference approximations are used in FDM. Sinus functions are chosen as the basis in GM. Problems of decaying a small distortion of the spherical shape of a bubble and dynamics of a bubble under harmonic liquid pressure variation with various parameters are used for comparison. The FDM technique has been found to be most efficient in all the cases. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical Simulation of Magnetoresonance Absorption for Multilayered Co/Cu Nanostructures in the Millimeter Waveband

The numerical simulation of the Ferromagnetic Resonance (FMR) absorption in the millimeter waveband has been performed for magnetic multilayered nanostructures. The model taking into account a random bilinear exchange interaction between layers was applied for the theoretical approach elaboration. Results of the simulation are in good agreement with experimental data registered for samples of nanostructure Co/Cu with buffer layer Fe in the frequency band 20-37 GHz. The explanation of the origin of unexpectedly broad magnetoresonance peaks formed in specimen under study is suggested.     

On the formation of sharp transition layers in two-dimensional reaction-diffusion models

For a singularly perturbed parabolic equation in two dimensions, the formation of a solution with a sharp transition layer from a sufficiently general initial function is considered. An asymptotic analysis is used to estimate the time required for the formation of a contrast structure. Numerical results are presented.     

Non smooth dynamics of mechanical systems with history term

Models involving maximal monotone term and history (delay) term are considered in a mathematical and numerical point of view.     

Numerical study of the HP version of mixed discontinuous finite element methods for reaction‐diffusion problems: The 1D case

This article studies the stability and convergence of the hp version of the three families of mixed discontinuous finite element (MDFE) methods for the numerical solution of reaction‐diffusion problems. The focus of this article is on these problems for one space dimension. Error estimates are obtained explicitly in the grid size h, the polynomial degree p, and the solution regularity; arbitrary space grids and polynomial degree are allowed. These estimates are asymptotically optimal in both h and p for some of these methods. Extensive numerical results to show convergence rates in h and p of the MDFE methods are presented. Theoretical and numerical comparisons between the three families of MDFE methods are described. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 525–553, 2003     

Fast Construction of the Fejér and Clenshaw–Curtis Quadrature Rules

We present an elegant algorithm for stably and quickly generating the weights of Fejér’s quadrature rules and of the Clenshaw–Curtis rule. The weights for an arbitrary number of nodes are obtained as the discrete Fourier transform of an explicitly defined vector of rational or algebraic numbers. Since these rules have the capability of forming nested families, some of them have gained renewed interest in connection with quadrature over multi-dimensional regions. AMS subject classification (2000) 65D32, 65T20, 65Y20     

High‐order compact numerical schemes for non‐hydrostatic free surface flows

The development of a numerical scheme for non‐hydrostatic free surface flows is described with the objective of improving the resolution characteristics of existing solution methods. The model uses a high‐order compact finite difference method for spatial discretization on a collocated grid and the standard, explicit, single step, four‐stage, fourth‐order Runge–Kutta method for temporal discretization. The Cartesian coordinate system was used. The model requires the solution of two Poisson equations at each time‐step and tridiagonal matrices for each derivative at each of the four stages in a time‐step. Third‐ and fourth‐order accurate boundaries for the flow variables have been developed including the top non‐hydrostatic pressure boundary. The results demonstrate that numerical dissipation which has been a problem with many similar models that are second‐order accurate is practically eliminated. A high accuracy is obtained for the flow variables including the non‐hydrostatic pressure. The accuracy of the model has been tested in numerical experiments. In all cases where analytical solutions are available, both phase errors and amplitude errors are very small. Copyright © 2006 John Wiley & Sons, Ltd.     

Description of Finite-amplitude Standing Acoustic Waves Using Convection-diffusion Equations

This paper deals with problems of finite-amplitude standing waves in acoustical resonators of variable cross-section. Set of two one-dimensional partial differential equations in the third approximation, formulated in conservative form, is derived from the fundamental equations of gas dynamics. The model equations which takes into account external driving force, gas dynamic nonlinearities and thermoviscous dissipation are solved numerically in time domain using a central scheme developed for convection-diffusion equations integration. In this paper numerical results for closed air-filled acoustic resonators are presented.