Mode selection in nearly-self-sustained explosive crystallization

We study a model of a driven thermally activated, exothermic process, in the context of explosive crystallization of a thin amorphous film. In the limit in which self-sustained crystallization at a speed close to the laser scan speed is almost possible, the model reduces to a single autonomous second-order differential equation for the amorphous-crystalline interface position. The steady state of this equation represents a straight-line interface advancing into the amorphous region at a speed equal to the laser scan speed. As a control parameter is increased, this steady state of the model becomes linearly unstable, undergoing a Hopf bifurcation which gives rise to oscillations of the interface velocity. For parameter ranges in which the interface remains a straight line, the amplitude, period, and form of these oscillations is calculated as a function of the control parameter. A similar equation, valid when the interface does not necessarily remain straight but may take on a long-wavelength shape, is also derived; it is harder to analyze.     

Critical behavior of a heavy particle in a granular fluid

A second order phase transition is observed for the homogeneous cooling state of a heavy impurity particle in a granular fluid. The order parameter straight phi is the ratio of impurity mean square velocity to that of the fluid, with a conjugate field h proportional to the mass ratio. A parameter beta, measuring the fluid cooling rate relative to the impurity-fluid collision rate, is the analog of the inverse temperature. For beta<1 the fluid is "normal" with straight phi = 0 at h = 0, as in the case of elastic collisions. For beta>1 an "ordered" state with straight phi not equal0 occurs at h = 0, representing an extreme breakdown of equipartition. Critical slowing and qualitative changes in the velocity distribution function near the transition are noted.     

Field-induced phase separation in one dimension

Phase separation is induced in the one-dimensional Ising chain (or lattice-gas model of a fluid) by means of an external field that changes sign in the middle of the chain. The magnetization profile (or density profile of the analogous fluid) is obtained analytically. It is found to decay exponentially rapidly to the bulk-phase magnetizations (or densities), the exponential decay parameter being the correlation length in the bulk phases in the presence of the field. This is in accord with earlier theoretical ideas. The interfacial tension is also obtained analytically. In an appropriately defined limit of large neighboring-site spin-spin interactions and small external field the interface becomes infinitely broad while the amplitude of the profile and the interfacial tension both vanish, in close imitation of the approach to a critical point in a real fluid. In this asymptotic limit the interfacial tension is related to the amplitude of the profile in the way that is predicted by earlier theories of interfaces near critical points, with critical-point exponents now those appropriate to one dimension. The exact interfacial profile and tension are used to test several approximations, including a corrected form of the barometric law and local (square-gradient) and nonlocal forms of the van der Waals theory.     

Interfacial properties of a driven diffusive system

A study is made of the low-temperature interfacial properties of a driven system with a single conserved density whose bulk properties were first analyzed, using computer simulations, by Katz, Lebowitz, and Spohn in 1983. The system corresponds to a nearest neighbor interacting lattice gas of charged particles (henceconserved order parameter), which are acted upon by a uniform, constant external electric fieldE. Starting from a bulk kinetic equation, an integral equation for the interface is derived. Nonlocal coupling between different parts of the interface arises from local particle conservation. The interface at any angle is shown to be stable against small deformations ofall wavelengths that are large compared to the interfacial width. However, the relaxation rate(k) for the interface exhibits a strongorientational dependence, which can be understood in terms of the modification of nonlocality byE. The wandering of the interface is considered. Also, the possible stabilizing effect of periodic boundary conditions on the orientation toward the direction ofE is discussed.     

Dynamics of the critical liquid-vapor interface close to a solid wall

We discuss the dynamics of an interface separating two phases of a fluid close to its critical point and in the presence of a third, inert phase regarded as a solid wall. The dispersion relation for the interface undulations is derived and compared with the predictions of the theories of capillary waves and interfacial motion for a conserved order parameter.     

Apparent slip at a polymer-polymer interface

We consider a planar interface between strongly-segregated homopolymers subjected to steady shear in the plane of the interface. We develop a constitutive equation for stress relaxation in an inhomogeneous system for chains obeying Rouse dynamics. Using this equation, the interfacial viscosity for a symmetric blend is found to be in agreement with a scaling prediction due to de Gennes, where is the bead friction coefficient, b is the segment length, is the segment volume and is the Flory-Huggins interaction parameter driving the phase separation. We generalize our results to asymmetric blends and describe a phenomenological extension to entangled melts. Received: 18 August 1997 / Received in final form: 1 December 1997 / Accepted: 2 December 1997     

A velocity decomposition approach for moving interfaces in viscous fluids

We present a second-order accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness. The fluid flow is described by the Navier–Stokes equations, with a singular force due to the stretching of the moving interface. We decompose the velocity into a “Stokes” part and a “regular” part. The first part is determined by the Stokes equations and the singular interfacial force. The Stokes solution is obtained using the immersed interface method, which gives second-order accurate values by incorporating known jumps for the solution and its derivatives into a finite difference method. The regular part of the velocity is given by the Navier–Stokes equations with a body force resulting from the Stokes part. The regular velocity is obtained using a time-stepping method that combines the semi-Lagrangian method with the backward difference formula. Because the body force is continuous, jump conditions are not necessary. For problems with stiff boundary forces, the decomposition approach can be combined with fractional time-stepping, using a smaller time step to advance the interface quickly by Stokes flow, with the velocity computed using boundary integrals. The small time steps maintain numerical stability, while the overall solution is updated on a larger time step to reduce computational cost.     

Exact maximal height distribution of fluctuating interfaces

We present an exact solution for the distribution P(h(m),L) of the maximal height h(m) (measured with respect to the average spatial height) in the steady state of a fluctuating Edwards-Wilkinson interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(h(m),L)=L(-1/2)f(h(m)L(-1/2)) for all L>0, where the function f(x) is the Airy distribution function that describes the probability density of the area under a Brownian excursion over a unit interval. For the free boundary case, the same scaling holds, but the scaling function is different from that of the periodic case. Numerical simulations are in excellent agreement with our analytical results. Our results provide an exactly solvable case for the distribution of extremum of a set of strongly correlated random variables.     

An immersed interface method for Stokes flows with fixed/moving interfaces and rigid boundaries

We present an immersed interface method for solving the incompressible steady Stokes equations involving fixed/moving interfaces and rigid boundaries (irregular domains). The fixed/moving interfaces and rigid boundaries are represented by a number of Lagrangian control points. In order to enforce the prescribed velocity at the rigid boundaries, singular forces are applied on the fluid at these boundaries. The strength of singular forces at the rigid boundary is determined by solving a small system of equations. For the deformable interfaces, the forces that the interface exerts on the fluid are calculated from the configuration (position) of the deformed interface. The jumps in the pressure and the jumps in the derivatives of both pressure and velocity are related to the forces at the fixed/moving interfaces and rigid boundaries. These forces are interpolated using cubic splines and applied to the fluid through the jump conditions. The positions of the deformable interfaces are updated implicitly using a quasi-Newton method (BFGS) within each time step. In the proposed method, the Stokes equations are discretized via the finite difference method on a staggered Cartesian grid with the incorporation of jump contributions and solved by the conjugate gradient Uzawa-type method. Numerical results demonstrate the accuracy and ability of the proposed method to simulate incompressible Stokes flows with fixed/moving interfaces on irregular domains.     

基于求解Riemann问题的界面处理方法

通过在界面处构造Riemann问题,根据流体的法向速度和压力在界面(接触间断)处连续的特性,利用Riemann问题的解不仅定义了ghost流体的值,而且对真实流体中邻近界面的点值进行了更新,使得在界面处的流体的状态满足接触间断的性质,给出了更加精确的界面边界条件,守恒误差分析表明该方法在界面计算过程中引入较小的误差.数值试验表明该方法能准确地捕捉界面和激波的位置.     

Nonlinear dynamics of an interface between shear bands

We study numerically the nonlinear dynamics of a shear banding interface in two-dimensional planar shear flow, within the nonlocal Johnson-Segalman model. Consistent with a recent linear stability analysis, we find that an initially flat interface is unstable with respect to small undulations for a sufficiently small ratio of the interfacial width l to cell length L(x). The instability saturates in finite amplitude interfacial fluctuations. For decreasing l/L(x) these undergo a nonequilibrium transition from simple traveling interfacial waves with constant average wall stress, to periodically rippling waves with a periodic stress response. When multiple shear bands are present we find erratic interfacial dynamics and a stress response suggesting low dimensional chaos.     

超声波振动辅助下填充填缝工艺的数值研究

本文基于数值方法研究了超声波振动辅助倒装芯片成型下填充填缝工艺。采用了一种基于紧耦合的流固耦合算法,对结构和流场之间的耦合运动进行了数值模拟,其中流场部分采用有限体积法对任意拉格朗日-欧拉方法的不可压缩N-S方程进行离散计算,而结构部分采用有限元法对拉格朗日坐标下的弹性动力学方程进行离散计算。在一个时间步内,结构与流场的计算区域的交界面上进行了多次的数值传递和插值,以保证满足耦合面边界条件。计算了超声波振动作为边界条件作用于结构场,而引起的流场中流体流速的变化规律,并通过实验验证了本方法的正确性。同时,监测了流场入口体积流量的变化规律,探究了超声波振幅、频率以及流体粘度等因素对流体流速的影响,为超声波振动辅助倒装芯片成型下填充工艺的应用提供了理论依据。     

微通道内反应流体粘性指进的数值研究

采用格子Boltzmann方法数值模拟化学反应中混溶流体在微通道中的粘性指进现象.模拟采用单浓度变量的双稳态化学反应模型,重点研究指进的形态位置随化学反应速率和稳态浓度参数（即无化学反应发生的界面浓度）的变化.结果表明：随着反应速率的增加,指进界面变薄;而稳态浓度参数的变化则影响反应区的分布以及反应速率,导致指进形态以及位置的改变,甚至出现指尖液滴分离.     

Effective damping of proton transfer in the H-bond caused by asymmetry of the reservoir

When studying some relaxation phenomena, the thermal reservoir is usually taken as a system of harmonic oscillators. In the model used here this type of the reservoir is coupled to the proton in such a way, that it distinguishes whether the proton is in the left or in the right well of the potential. Due to this asymmetry it may be expected that the steady state of the proton will not be symmetric. Surprisingly, in the Born approximation the reservoir does not change the previous symmetry of the steady state of the proton but changes its character of relaxation.I would like to thank to Dr. P. Baacký and to Dr. V. ápek for many useful discussions.     

Influence of Non-linear Radiation Heat Flux on Rotating Maxwell Fluid over a Deformable Surface: A Numerical Study

Mathematical model for Maxwell fluid flow in rotating frame induced by an isothermal stretching wall is explored numerically. Scale analysis based boundary layer approximations are applied to simplify the conservation relations which are later converted to similar forms via appropriate substitutions. A numerical approach is utilized to derive similarity solutions for broad range of Deborah number. The results predict that velocity distributions are inversely proportional to the stress relaxation time. This outcome is different from that observed for the elastic parameter of second grade fluid. Unlike non-rotating frame, the solution curves are oscillatory decaying functions of similarity variable. As angular velocity enlarges, temperature rises and significant drop in the heat transfer coefficient occurs. We note that the wall slope of temperature has an asymptotically decaying profile against the wall to ambient ratio parameter. From the qualitative view point, temperature ratio parameter and radiation parameter have similar effect on the thermal boundary layer. Furthermore, radiation parameter has a definite role in improving the cooling process of the stretching boundary.A comparative study of current numerical computations and those from the existing studies is also presented in a limiting case. To our knowledge, the phenomenon of non-linear radiation in rotating viscoelastic flow due to linearly stretched plate is just modeled here.     

Crack propagation as a free boundary problem

A sharp interface model of crack propagation as a phase transition process is discussed. We develop a multipole expansion technique to solve this free boundary problem numerically. We obtain steady state solutions with a self-consistently selected propagation velocity and shape of the crack, provided that elastodynamic effects are taken into account. Also, we find a saturation of the steady state crack velocity below the Rayleigh speed, tip blunting with increasing driving force, and a tip splitting instability above a critical driving force.     

Crystallinity and dielectric relaxations in semi-crystalline poly(ether ether ketone)

The Maxwell-Wagner-Sillars (MWS) relaxation is studied for semi-crystalline polymers poly (ether ether ketone) (PEEK), in the range 20 Hz-1 MHz and temperature varying from 80 to 330 °C. The parameter is the crystallization condition in the case of PEEK, which is a semi-crystalline polymer considered as a particulate composite. The relaxation found in the semi-crystalline polymers above the α relaxation of the PEEK is ascribed to the trapping of conductive carriers at the interface between crystalline lamellae and the amorphous matrix. The study of PEEK microstructure is based on differential calorimetry and X-rays diffraction. Two lamellae populations have been detected, that depends on the crystallization temperature and its duration. The crystallinity rate is increasing with crystallization temperature and duration. In dielectric studies, the use of the electric modulus instead of permittivity allows us to minimize the ionic conduction and then leads to the appearance of the interfacial relaxation. According to our measurements, the crystallinity rate is not the main factor of the interfacial relaxation intensity, which also depends on the nature and degree of perfection of the lamellae.     

Selection of steady states in planar Darcy convection

The planar natural convection of an incompressible fluid in a porous medium is considered. We study the selection of steady states under temperature perturbations on the boundary. A selection map is introduced in order to analyze the selection of a steady state from a continuous family of equilibria which exists under zero boundary conditions. The results of finite-difference modeling for a rectangular enclosure are presented.     

A boundary condition capturing immersed interface method for 3D rigid objects in a flow

To simulate the flow around an object, we can replace the object with the fluid enclosed by a singular force. We can then simulate the flow on a fixed domain with a fluid–fluid interface supporting the singular force. In this paper, we present a boundary condition capturing approach to determine the singular force for a 3D rigid object. We apply a discontinuous body force to enforce the rigid motion of the fluid replacing the object and compute the singular force based on the kinematics of the object. Due to the singular force and the body force, the flow is not smooth across the interface. We solve the flow using the immersed interface method. Our boundary condition capturing immersed interface method is very efficient and stable, and its accuracy based on the infinity norm is near second order for the velocity and above first order for the pressure.