Friction tensor for a pair of Brownian particles: Spurious finite-size effects and molecular dynamics estimates

In this work, we show thatin any finite system, the binary friction tensor for two Brownian particlescannot be directly estimated from an evaluation of the microscopic Green-Kubo formula, involving the time integral of force-force autocorrelation functions. This pitfall is associated with a subtle inversion of the thermodynamic and long-time limits and leads to spurious results for the estimates of the friction matrix based on molecular dynamics simulations. Starting from a careful analysis of the coupled Langevin equations for two interacting Brownian particles, we derive a method to circumvent these effects and extract the binary friction tensor from the correlation function matrix of the instantaneous forces exerted by the bath particles on the fixed Brownian particles, and from the relaxation of the total momentum of the bath in afinite system. The general methodology is applied to the case of two hard or soft Brownian spheres in a bath of light particles. Numerical estimates of the relevant correlation functions and of the resulting self and mutual components of the matrix of friction tensors are obtained by molecular dynamics simulations for various spacings between the Brownian particles. This paper is dedicated to B. Jancovici on the occassion of his 65th birthday.     

Further improvement of weakly nonlinear theory of hydrodynamic stability

The problems of the weakly nonlinear theory of hydrodynamic stability has been re-addressed; namely, (i) why is the radius of convergence of its solution too small? (ii) it is not appropriate to use it to deal with the evolution problem of disturbances; the main reason is that the way of calculating the mean flow distortion is inappropriate, (iii) it cannot be used to deal with the initial value problem. Ways of its improvement were proposed. Its correctness was confirmed by its comparison with the results of numerical simulations.     

Sedimentation of homogeneous suspensions in finite vessels

The question of a possible container shape dependence of the sedimentation velocity in a homogeneous suspension is reexamined. To this end we develop a statistical theory of suspensions based on low-Reynolds-number hydrodynamics of spherical particles in a container. It is shown, to first order in the volume fraction, that in an arbitrary vessel the relative sedimentation velocity is shape independent, but that at the same time shape-dependent convection occurs. The theory forms a bridge between earlier calculations for special geometries by Beenakker and Mazur and a phenomenological theory recently proposed by Nozières.This paper is dedicated to N. G. van Kampen.     

Hydrodynamics of dielectric fluid films

We introduce a general hydrodynamic model to study the stability of lipid films against thermal fluctuations. As one novel aspect the model accounts before all for a complete intrinsic surface rheology of the film interfaces. Thus the rheological behaviour of the surface adsorbed lipids is modelled which screen the hydrophobic film interior against the aqueous exterior. For coloured films we demonstrate first the influence of electrical forces on the dynamics and film stability. For that we perform a linear stability analysis on a simplified mechanically symmetric film with i) symmetric surface charge distribution and ii) linear electric potential drop across the film. Based on the complete film model we then categorize the complete set of solutions of the linearized equations of motion and we study the growth rates of unstable film modes. Finally we discuss the stability properties of a black film after introducing a repulsive mechanism due to the steric hindrance of the interfacial lipids.     

Anisotropic flow at RHIC

STAR’s measurement of directed flow for pions,kaons(K 0 S ),protons and anti-protons,for Au+Au collisions at 200 GeV obtained in Run7 are presented,as well as elliptic flow for identified particles measured in Au+Au(Run7) and Cu+Cu(Run5) collisions.It is found that the slope of proton v 1 (y) at midrapidity is extremely small.Elliptic flow results are compared to Hydro calculation and the discrepancy is discussed.     

Hydrodynamic assist and the dynamic contact angle in the coalescence of liquid drops

^** Email: decentsp{at}for.mat.bham.ac.uk The coalescence of two viscous liquid drops in an inviscid gasor in a vacuum is studied using the interface formation model.In the very early stages of coalescence during the formationof the ‘liquid-bridge’ connecting the two drops,this model predicts a moving contact line and a dynamic contactangle. This paper examines the dynamic evolution of this contactangle, and for small Reynolds number and small Capillary number,relevant particularly in micro-fluidics, a non-linear differentialequation is derived for the contact angle and solved computationally.It is found that the contact angle evolution can only be evaluatedby determining information about the flow away from the contactline. This is a manifestation of so-called hydrodynamic assist,studied experimentally in the context of curtain coating byBlake et al. (1999 Experimental evidence of non-local hydrodynamicinfluence on the dynamic contact angle. Phys. Fluids, 11, 1995–2007).For small Capillary number and small Reynolds number, the free-surfaceevolution is determined for the coalescence of two cylindersof equal radius. Finally, some comments are made on experimentsin coalescence, as well as on issues arising in a computationalsolution of the full model described here.     

A stable boundary elements method for magnetohydrodynamic channel flows at high Hartmann numbers

The article is devoted to extension of boundary element method (BEM) for solving coupled equations in velocity and induced magnetic field for time dependent magnetohydrodynamic (MHD) flows through a rectangular pipe. The BEM is equipped with finite difference approach to solve MHD problem at high Hartmann numbers up to 10⁶. In fact, the finite difference approach is used to approximate partial derivatives of unknown functions at boundary points respect to outward normal vector. It yields a numerical method with no singular boundary integrals. Besides, a new approach is suggested in this article where transforms 2D singular BEM's integrals to 1D nonsingular ones. The new approach reduces computational cost, significantly. Note that the stability of the numerical scheme is proved mathematically when computational domain is discretized uniformly and Hartmann number is 40 times bigger than length of boundary elements. Numerical examples show behavior of velocity and induced magnetic field across the sections.     

Extension criterion on regularity for weak solutions to the 3D MHD equations

In this paper, we consider the regularity criteria for weak solutions to the 3D incompressible magnetohydrodynamic equations and prove some regularity criteria which are related only with u+B or u?B. This is an improvement of the result given by He and Wang (J. Differential Equations 2007; 238:1–17; Math. Meth. Appl. Sci. 2008; 31:1667–1684) and He and Xin (J. Differential Equations 2005; 213(2):235–254). Copyright © 2010 John Wiley & Sons, Ltd.     

Analytical modeling of active magnetic bearing geometry

The aim of this paper is to present a new graphical approach to the shape design of the active magnetic bearing (AMB) stator. The AMB is a tool to levitate the rotor without contact. The standard design method uses a computer-aided design (CAD) software in the modeling process. Therefore the designed AMB shape consists of graphical primitives like lines and arcs with fixed properties. For the advanced interdisciplinary analysis of the AMB construction the shape generation and modifications ought to be done automatically. The proposed method is based on mathematical analysis and representation of the AMB stator by curves. Second and third order Bézier curves given in polynomial and rational form are compared to the circle and arc based arcs. The fitting quality is considered for the selection of the appropriate arc representation. The obtained shapes are ready to be used in the magnetic field analysis and optimization procedures to find an optimal form of the AMB construction. The author’s experience in modeling and vector graphics was a motivation to look at the AMB construction from mathematical and programming point of view. The AMB components are modeled with parametric curves under constraints defined by the AMB static and dynamic properties. Such a described 2D or 3D model can be generated automatically in a programming way for a wide range of AMB configurations in further research. Selected configurations are presented to show features of the proposed method and realized algorithm. The selected features of the proposed solution as well as feedback from industry are discussed.     

Nonlinear study of a rotor–AMB system under simultaneous primary-internal resonance

A rotor–active magnetic bearing (AMB) system subjected to a periodically time-varying stiffness with quadratic and cubic non-linearities under multi-parametric excitations is studied and solved. The method of multiple scales is applied to analyze the response of two modes of a rotor–AMB system with multi-parametric excitations and time-varying stiffness near the simultaneous primary and internal resonance. The stability of the steady state solution for that resonance is determined and studied using Rung–Kutta method of fourth order. It is shown that the system exhibits many typical non-linear behaviors including multiple-valued solutions, jump phenomenon, hardening and softening non-linearities and chaos in the second mode of the system. The effects of the different parameters on the steady state solutions are investigated and discussed also. A comparison to published work is reported.