Geometric approach to stationary shapes in rotating Hele-Shaw flows

We consider the geometry of a special family of curves associated with the motion of a two-fluid interface in a rotating Hele-Shaw cell. This family of stationary exact solutions with surface tension consists of interface shapes which balance exactly the competing capillary and centrifugal forces. The result is the formation of patterned structures presenting fingers that eventually assume a teardrop-like shape, and tend to be detached from the main body of the rotating fluid (occurrence of a pinch-off event). By using the vortex-sheet formalism, we approach the problem analytically, and show that the curvature of these particular curves can be very simply expressed as a function of the radial distance to the rotation axis. Motivated by this fact, and through a simple geometric interpretation, we show that the exact solutions for the rotating Hele-Shaw problem satisfy a first-order ordinary differential equation which is readily solved, so that the shape solutions are determined up to quadratures. The ability to probe a number of key morphological features of such solutions analytically is demonstrated, and a gallery of plots is provided to highlight these findings. The possibility of accessing a criterion to predict the occurrence of pinch-off is also discussed. Finally, we prove that the general solutions are dense, and use this fact to guarantee the very existence of the neat, highly symmetric physical patterns.     

10.1007/s10955-006-9040-z

We derive hydrodynamic equations describing the evolution of a binary fluid segregated into two regions, each rich in one species,which are separated (on the macroscopic scale) by a sharp interface. Our starting point is a Vlasov-Boltzmann (VB) equation describing the evolution of the one particle position and velocity distributions, f_i (x, v, t), i = 1, 2. The solution of the VB equation is developed in a Hilbert expansion appropriate for this system. This yields incompressible Navier-Stokes equations for the velocity field u and a jump boundary condition for the pressure across the interface. The interface, in turn, moves with a velocity given by the normal component of u.     

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.     

Lattice Boltzmann Model for Free Surface Flow for Modeling Foaming

We present a 2D- and 3D-lattice Boltzmann model for the treatment of free surface flows including gas diffusion. Interface advection and related boundary conditions are based on the idea of the lattice Boltzmann equation. The fluid dynamic boundary conditions are approximated by using the mass and momentum fluxes across the interface, which do not require explicit calculation of gradients. A similar procedure is applied to fulfill the diffusion boundary condition. Simple verification tests demonstrate the correctness of the algorithms. 2D- and 3D-foam evolution examples demonstrate the potential of the method.     

A composite grid solver for conjugate heat transfer in fluid–structure systems

We describe a numerical method for modeling temperature-dependent fluid flow coupled to heat transfer in solids. This approach to conjugate heat transfer can be used to compute transient and steady state solutions to a wide range of fluid–solid systems in complex two- and three-dimensional geometry. Fluids are modeled with the temperature-dependent incompressible Navier–Stokes equations using the Boussinesq approximation. Solids with heat transfer are modeled with the heat equation. Appropriate interface equations are applied to couple the solutions across different domains. The computational region is divided into a number of sub-domains corresponding to fluid domains and solid domains. There may be multiple fluid domains and multiple solid domains. Each fluid or solid sub-domain is discretized with an overlapping grid. The entire region is associated with a composite grid which is the union of the overlapping grids for the sub-domains. A different physics solver (fluid solver or solid solver) is associated with each sub-domain. A higher-level multi-domain solver manages the entire solution process.     

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.     

A hybrid numerical method for interfacial fluid flow with soluble surfactant

We address a significant difficulty in the numerical computation of fluid interfaces with soluble surfactant that occurs in the physically representative limit of large bulk Peclet number Pe. At the high values of Pe in typical fluid-surfactant systems, there is a transition layer near the interface in which the surfactant concentration varies rapidly, and large gradients at the interface must be resolved accurately to evaluate the exchange of surfactant between the interface and bulk flow. We use the slenderness of the layer to develop a fast and accurate ‘hybrid’ numerical method that incorporates a separate, singular perturbation analysis of the dynamics in the transition layer into a full numerical solution of the interfacial free boundary problem. The accuracy and efficiency of the method is assessed by comparison with a more ‘traditional’ numerical approach that uses finite differences on a curvilinear coordinate system exterior to the bubble, without the separate transition layer reduction. The traditional method implemented here features a novel fast calculation of fluid velocity off the interface.     

A multirate time integrator for regularized Stokeslets

The method of regularized Stokeslets is a numerical approach to approximating solutions of fluid–structure interaction problems in the Stokes regime. Regularized Stokeslets are fundamental solutions to the Stokes equations with a regularized point-force term that are used to represent forces generated by a rigid or elastic object interacting with the fluid. Due to the linearity of the Stokes equations, the velocity at any point in the fluid can be computed by summing the contributions of regularized Stokeslets, and the time evolution of positions can be computed using standard methods for ordinary differential equations. Rigid or elastic objects in the flow are usually treated as immersed boundaries represented by a collection of regularized Stokeslets coupled together by virtual springs which determine the forces exerted by the boundary in the fluid. For problems with boundaries modeled by springs with large spring constants, the resulting ordinary differential equations become stiff, and hence the time step for explicit time integration methods is severely constrained. Unfortunately, the use of standard implicit time integration methods for the method of regularized Stokeslets requires the solution of dense nonlinear systems of equations for many relevant problems. Here, an alternate strategy using an explicit multirate time integration scheme based on spectral deferred corrections is incorporated that in many cases can significantly decrease the computational cost of the method. The multirate methods are higher-order methods that treat different portions of the ODE explicitly with different time steps depending on the stiffness of each component. Numerical examples on two nontrivial three-dimensional problems demonstrate the increased efficiency of the multi-explicit approach with no significant increase in numerical error.     

Analytical solutions of the lattice Boltzmann BGK model

Analytical solutions of the two-dimensional triangular and square lattice Boltzmann BGK models have been obtained for the plane Poiseuille flow and the plane Couette flow. The analytical solutions are written in terms of the characteristic velocity of the flow, the single relaxation time , and the lattice spacing. The analytic solutions are the exact representation of these two flows without any approximation. Using the analytical solution, it is shown that in Poiseuille flow the bounce-back boundary condition introduces an error of first order in the lattice spacing. The boundary condition used by Kadanoffet al. in lattice gas automata to simulate Poiseuille flow is also considered for the triangular lattice Boltzmann BGK model. An analytical solution is obtained and used to show that the boundary condition introduces an error of second order in the lattice spacing.     

Stability of core-annular flow of power-law fluids in the presence of interfacial surfactant

The shear-thinning influence on the core-annular flow stability of two immiscible power-law fluids is considered by making a linear stability analysis.The flow is driven by an axial pressure gradient in a straight pipe with the interface between the two fluids occupied by an insoluble surfactant.Given the basic flow for this core-annular arrangement,the analytical solution is obtained with respect to the power-law fluid model.The linearized equations for the evolution of infinitesimal disturbances are deriv...     

Density fluctuations in lattice-Boltzmann simulations of multiphase fluids in a closed system

A two-dimensional single component two-phase lattice Boltzmann model was used to simulate the Rayleigh–Taylor instability in a closed system. Spatiotemporally variable densities were generated by gravity acting on the fluid density. The density fluctuations were triggered by rapid changes in the fluid velocity induced by changes in the interface geometry and impact of the dense fluid on the rigid lower boundary of the computational domain. The ratio of the maximum density fluctuations to the maximum fluid velocity increased more rapidly at low velocities than at high velocities. The ratio of the maximum density fluctuations in the dense phase to its maximum velocity was on the order of the inverse of the sound speed. The solution became unstable when the density-based maximum local Knudsen number exceeded 0.13.     

The Scaling Limit of Polymer Pinning Dynamics and a One Dimensional Stefan Freezing Problem

We consider the stochastic evolution of a 1 + 1-dimensional interface (or polymer) in the presence of a substrate. This stochastic process is a dynamical version of the homogeneous pinning model. We start from a configuration far from equilibrium: a polymer with a non-trivial macroscopic height profile, and look at the evolution of a space-time rescaled interface. In two cases, we prove that this rescaled interface has a scaling limit on the diffusive scale (space rescaled by L in both dimensions and time rescaled by L ² where L denotes the length of the interface) which we describe. When the interaction with the substrate is such that the system is unpinned at equilibrium, then the scaling limit of the height profile is given by the solution of the heat equation with Dirichlet boundary condition ; when the attraction to the substrate is infinite, the scaling limit is given by a free-boundary problem which belongs to the class of Stefan problems with contracting boundary, also referred to as Stefan freezing problems. In addition, we prove the existence and regularity of the solution to this problem until a maximal time, where the boundaries collide. Our result provides a new rigorous link between Stefan problems and Statistical Mechanics.     

激波作用不同椭圆氦气柱过程中流动混合研究

在激波与气柱相互作用问题中,压力与密度间断不平行产生的斜压涡量会引起流动的不稳定性,从而促进物质间的混合.本文基于双通量模型,结合五阶加权基本无振荡(WENO)格式,求解多组分二维Navier-Stokes方程,分析激波作用面积相同结构不同的椭圆气柱所致的流动和混合.数值结果清晰地显示了激波诱导Richtmyer-Meshkov不稳定性引起的气柱界面变形和波系演化.同时定量地从界面运动、界面结构参数变化(长度和高度)、气柱体积压缩率、环量及混合率等角度分析激波诱导的流动混合机制,研究椭圆几何构型对氦气混合过程的影响.结果表明,界面及相关参数的演化与气柱初始形状密切相关.当激波沿椭圆长轴作用于气柱时,气柱前端出现空气射流结构,且射流不断增长并渗透到下游界面,致使气柱分离成两个独立涡团,离心率越大,射流发展越快;同时激波作用气柱后在界面处产生不规则反射现象.圆形气柱界面演化与这种作用情形类似.当激波沿椭圆短轴作用于气柱时,界面上游出现类平面结构,随后平面上下缘处产生涡旋,主导流动发展,激波在界面作用产生规则反射,离心率越大,这些现象越明显.界面高度、长度、体积压缩率也因此有所差异.对界面演化、环量和混合率的综合分析表明,激波沿长轴作用于气柱且离心率较大时,流动发展较快,不稳定性导致的流动越复杂,越有利于氦气与环境介质的混合.     

A HOLOGRAPHIC INTERFEROMETRY STUDY OF THE DOUBLE-DIFFUSIVE LAYERED SYSTEM

This study investigates double-diffusive convection in a two-layer, salt-stratified solution destablized by lateral Keating and cooling. The two-wavelength holographic interferometry method was used to measure the transient temperature, concentration, and density distributions. The evolution of the two-layered system can be divided into three stages. In the first stage, the thermally driven convective layers form rapidly, and the existing diffusion layer adjusts itself into a thin interface by convection motion. In the second stage, a quasi-steady state is attained. The temperature distribution is S-type and the temperature difference across the diffusion region does not change much. The concentration distribution is uniform in the two fluid layers, but the concentration and density differences decrease linearly with time. When the interface becomes very thin, unstable finger-type convection appears. Finally, the interface is destroyed by the boundary layers at the side walls in the third stage. The interfacial Nusselt number and Sherwood number art found to increase with the thermal Rayleigh number, and the effect of the solutal Rayleigh number seems to be less significant. The dimemumless mixing time is found to correlate well with thermal and solutal Rayleigh numbers. Results from numerical simulation are demonstrated and compared with the experiments.     

基于近似Riemann解的有限体积ALE方法

研究二维平面坐标系和二维轴对称坐标系中四边形网格上可压缩流体力学的有限体积ALE(Arbitrary Lagrangian Eulerian)方法.数值方法采用节点中心有限体积法,数值通量采用适用于任意状态方程的HLLC(Harten-Lax-Van Leer-Collela)通量.空间二阶精度通过用WENO(weighted essentially non-oscillatory)方法对原始变量进行重构获得,时间离散采用两步显式Runge-Kutta格式.数值例子显示,方法具有良好的激波分辨能力和高精度的数值逼近能力.     

Stokes–Darcy boundary integral solutions using preconditioners

In a system where a free fluid flow is coupled to flow in a porous medium, different PDEs are solved simultaneously in two subdomains. We consider steady Stokes equations in the free region, coupled across a fixed interface to Darcy equations in the porous substrate. Recently, the numerical solution of this system was obtained using the boundary integral formulation combined with a regularization-correction procedure. The correction process also results in the improvement of the condition number of the linear system. In this paper, an appropriate preconditioner based on the singular part of corrections is introduced to improve the convergence of a Krylov subspace method applied to solve the integral formulation.     

A nonlinear PSE method for two-fluid shear flows with complex interfacial topology

A nonlinear stability method is developed for laminar two-fluid shear flows which undergo changes in the interface topology. The method is based on the nonlinear parabolized stability equations (PSE) and incorporates a scalar-based interface capturing (IC) scheme in order to track complex deformations of the fluid interface. In doing so, the formulation retains the flexibility and physical insight of instability-wave based methods, while providing hydrodynamic modeling capabilities similar to direct numerical calculations: the new formulation, referred to as the IC-PSE, can capture the nonlinear physical mechanisms responsible for generating large-scale, two-fluid structures, without incurring heavy computational costs. This approach is valid for spatially developing, laminar two-fluid shear flows which are convectively unstable, and can naturally account for the growth of finite amplitude interfacial waves, along with changes to the interfacial topology. We demonstrate the accuracy of the IC-PSE against direct Navier–Stokes calculations for two-fluid mixing layers with density and viscosity stratification. The comparisons show that the IC-PSE can predict the dynamics of the instability waves and capture the formation of Kelvin–Helmholtz vortex rolls and large scale liquid structures, at an order of magnitude less computational cost than direct calculations. The role of surface tension in the IC-PSE formulation is shown to be valid for flows in which Re/We ? 1, and the method accurately predicts the formation and non-linear evolution of flow structures in this limit. This is demonstrated for spatially developing mixing layers which lead to vortex roll-up and ligaments, prior to droplet formation. The pinch-off process itself is a high surface tension phenomenon and in not considered herein. The method also accurately captures the effect of interfacial waves on the mean flow, and the topology changes during the non-linear evolution of the two-fluid structures.     

Role of the channel geometry on the bubble pinch-off in flow-focusing devices

The formation of bubbles by flow focusing of a gas and a liquid in a rectangular channel is shown to depend strongly on the channel aspect ratio. Bubble breakup consists in a slow linear 2D collapse of the gas thread, ending in a fast 3D pinch-off. The 2D collapse is predicted to be stable against perturbations of the gas-liquid interface, whereas the 3D pinch-off is unstable, causing bubble polydispersity. During 3D pinch-off, a scaling w_(m) approximately tau(1/3) between the neck width w_(m) and the time tau before breakup indicates that breakup is driven by the inertia of both gas and liquid, not by capillarity.     

Perfect fluid spheres in general relativity

Spherically symmetric perfect fluid distributions in general relativity have been investigated under the assumptions of (i) uniform expansion or contraction and (ii) the validity of an equation of state of the formp=p(ρ) with nonuniform density. An exact solution which is equivalent to a solution found earlier by Wyman is obtained and it is shown that the solution isunique. The boundary conditions at the interface of fluid distribution and the exterior vacuum are discussed and as a consequence the following theorem is established:Uniform expansion or contraction of a perfect fluid sphere obeying an equation of state with nonuniform density is not admitted by the field equations. It is further shown that the Wyman metric is not suitable on physical grounds to represent a cosmological solution.