Nonlinear Dynamics of a Bubble Contrast Agent Oscillating near an Elastic Wall

Contrast agent microbubbles, which are encapsulated gas bubbles, are widely used to enhance ultrasound imaging. There are also several new promising applications of the contrast agents such as targeted drug delivery and noninvasive therapy. Here we study three models of the microbubble dynamics: a nonencapsulated bubble oscillating close to an elastic wall, a simple coated bubble and a coated bubble near an elastic wall.We demonstrate that complex dynamics can occur in these models. We are particularly interested in the multistability phenomenon of bubble dynamics. We show that coexisting attractors appear in all of these models, but for higher acoustic pressures for the models of an encapsulated bubble.We demonstrate how several tools can be used to localize the coexisting attractors. We provide some considerations why the multistability can be undesirable for applications.     

Numerical Investigation of Cavitation Interacting with Pressure Wave

A computational fluid dynamics solver based on homogeneous cavitation model is employed to compute the two-phase cavitating flow. The model treats the two-phase regime as the homogeneous mixture of liquid and vapour which are locally assumed to be under both kinetic and thermodynamic equilibrium. As our focus is on pressure wave formation, propagation and its impact on cavitation bubble, the compressibility effects of liquid water have to be accounted for and hence the flow is considered to be compressible. The cavitating flow disturbed by the introduced pressure wave is simulated to investigate the unsteady features of cavitation due to the external perturbations. It is observed that the cavity becomes unstable, locally experiencing deformation or collapse, which depends on the shock wave intensity and freestream flow speed.     

A curvilinear finite-volume method to solve compressible gas dynamics in semi-Lagrangian coordinates

We present a new finite-volume method to solve compressible gas dynamics in semi-Lagrangian coordinates on curvilinear grids. The approach relies on a weak formulation to compute the mesh velocity using an acoustic Riemann solver approximation. We prove this method to be both conservative and entropic.     

The effect of magnetic field on dynamics of gas bubbles in visco-elastic fluids

The effect of magnetic field on nonlinear oscillations of a spherical, acoustically forced gas bubble in nonlinear visco-elastic media is studied. The constitutive equation UCM used for modeling the rheological behaviors of the fluid. By starting from the momentum equations for bubbles considering the magnetic force and considering some simplifying assumptions, the modified bubble dynamics equation (the modified Rayleigh–Plesset equation) has been achieved. Assumptions concerning the trace of the stress tensor are addressed in light of the incorporation of visco-elastic constitutive equations into modified bubble dynamics equations. The governing equations are non-dimesionalized and numerically solved by using 4th order Runge–Kutta method. The accuracy of the calculations and the formulation is compared with the previous works done for models without the presence of magnetic field. Furthermore, the bubble size variations due to acoustic motivations and stress tensor components variations in presence of different magnitudes of magnetic fields are studied. Also, the bubble size dependence on fluid conductivity variations is declared. The relevance and importance of this approach to biomedical ultrasound applications are highlighted. Preliminary results indicate that magnetic field may be an important consideration for the risk assessment of potential cavitations and also it could be possible to damp the bubble oscillations by using magnetic fields or in opposite case amplify the oscillations which could result in higher level light emissions in sonoluminescence approach.     

Weak oscillations of a gas bubble in a spherical volume of compressible liquid

The following spherically symmetric problem is considered: a single gas bubble at the centre of a spherical flask filled with a compressible liquid is oscillating in response to forced radial excitation of the flask walls. In the long-wave approximation at low Mach numbers, one obtains a system of differential-difference equations generalizing the Rayleigh-Lamb-Plesseth equation. This system takes into account the compressibility of the liquid and is suitable for describing both free and forced oscillations of the bubble. It includes an ordinary differential equation analogous to the Herring-Flinn-Gilmore equation describing the evolution of the bubble radius, and a delay equation relating the pressure at the flask walls to the variation of the bubble radius. The solutions of this system of differential-difference equations are analysed in the linear approximation and numerical analysis is used to study various modes of weak but non-linear oscillations of the bubble, for different laws governing the variation of the pressure or velocity of the liquid at the flask wall. These solutions are compared with numerical solutions of the complete system of partial differential equations for the radial motion of the compressible liquid around the bubble.     

Hybrid computational aeroacoustics based on compressible flow data at low Mach numbers

In some practical applications (e.g. cavity with a lip), even at low Mach numbers, acoustic feedback mechanisms excite flow structures. The compressible flow simulation cannot distinguish between a pure fluid dynamic part and acoustic phenomena. With this in mind, we propose a workflow based on Helmholtz-Hodge decomposition, to extract pure source terms of the compressible flow simulation, to model the sound radiation. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Simulation of Bubbles in Shear Flow

To gain a basic understanding of foam flow, as it can be found e.g. in transport of aerated food, simulation tools can help to provide better insight. Shearing of the bubbles appears in different flow geometries and is for a bubble assembly not captured analytically. Also experimentally, those flow fields are hard to observe so that simulations are the method of choice. Our method to simulate foams uses a volume of fluid approach that is based on the free surface algorithm by Körner et al. [1]. Different from classical multiphase methods, only the liquid phase is simulated and special boundary conditions at the liquid-gas interface account for the gas phase. With this approach high density ratios, e.g. in water-air systems, are easier to realize than in other methods. High density ratios are even necessary to physically justify the model, where the dynamics of the lighter phase are partially neglected. This method is integrated in the Lattice Boltzmann software framework waLBerla [3] (widely applicable Lattice Boltzmann solver from Erlangen†) that can be used on massively parallel computers and thus allows to simulate even large bubble assemblies. As first validation, single bubbles are sheared with different capillary numbers and the simulation results are compared to literature [2] and show good agreement. The next step is shearing a bubble assembly which is arranged like a dense sphere packing. In order to investigate the geometrical configuration of the assembly and its impact on the behavior during a shear deformation, the bubble assembly is rotated with different angles with respect to the shear direction. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

梢涡流场中气泡轨迹及形态数值模拟

针对气泡在舰船尾迹涡流场运动特性,根据其是否为尾涡所捕获,将数值模拟过程分为两个阶段:准球状运动阶段和非球状运动阶段．分别应用单向耦合质点粒子追踪法（PTM）和边界元法（BEM）模拟这两个阶段,将第1阶段结束的物理量作为第2阶段的初始条件,从而完成整个数值模拟过程．在已有数值研究结果和实验数据基础上,探讨空化发生条件,追踪尾迹空泡运动轨迹,模拟尾迹气泡的运动、变形、溃灭等,以及被尾涡捕获后的撕裂等运动特性,旨在为优化设计尾流场提供参考．     

Convergence of the compressible magnetohydrodynamic equations to incompressible magnetohydrodynamic equations

In this paper, we study the incompressible limit of the three-dimensional compressible magnetohydrodynamic equations, which models the dynamics of compressible quasi-neutrally ionized fluids under the influence of electromagnetic fields. Based on the convergence-stability principle, we show that, when the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient are sufficiently small, the initial-value problem of the model has a unique smooth solution in the time interval where the ideal incompressible magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient go to zero. Moreover, we obtain the convergence of smooth solutions for the model forwards those for the ideal incompressible magnetohydrodynamic equations with a sharp convergence rate.     

Local Decay of Acoustic Waves in the Low Mach Number Limits on General Unbounded Domains Under Slip Boundary Conditions

We discuss several aspects of the problem of propagation and dispersion of acoustic waves arising in the low Mach number asymptotic limits of compressible fluid systems. A general approach is proposed based on analysis of the spectral measures associated to the corresponding wave propagator. In particular, the local decay estimates based on a result of Tosio Kato and on RAGE theorem are obtained as limit cases. The approach is applied to problems on domains their shape may vary with the Mach number.     

狭窄流域内气泡破裂现象数值模拟

基于边界积分法建立狭窄流域内气泡破裂数值模型,开发相应的计算程序,分别模拟对称破裂与非对称破裂两类典型工况并与相应实验结果进行对比,计算值与实验值吻合很好,表明三维数值模型的有效性．从狭窄流域内气泡运动的基本现象入手,基于开发的程序系统地研究气泡的对称破裂与非对称破裂,在已有数值研究成果和实验数据基础上,提出气泡破裂的可行性准则,研究分裂后子气泡的动力学特性,并分析距离参数对气泡破裂特性及子气泡动力学行为的影响,总结相应规律,旨在为相关气泡破裂特性研究提供参考．     

Low Mach Number Limit of Viscous Polytropic Flows: Formal Asymptotics in the Periodic Case

This article is devoted to the low Mach number limit of weak solutions to the compressible Navier–Stokes equations for polytropic fluids with periodic boundary conditions and ill‐prepared data. We derive formally the equation satisfied by the mean value of the velocity and the equations governing the dynamics of the nonlinear acoustic waves in dimension d= 2 or 3.     

Lower Branch Modes of the Compressible Boundary Layer Flow Due to a Rotating-Disk

In this article, a theoretical study is pursued to investigate the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the compressible boundary layer flow due to a rotating-disk. Special attention is focused on to the short-wavelength stationary/nonstationary compressible crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic framework introduced in [ 1 ] for the incompressible stationary modes, it is demonstrated here that the compressible modes having sufficiently long time scale can also be described by an asymptotic expansion procedure based on the triple-deck approach. Making use of this rational asymptotic technique, which rigorously takes into account the nonparallel effects, the asymptotic structure of the nonstationary modes is shown to be adjusted by a balance between viscous and Coriolis forces, and resulted from the fact of vanishing shear stress at the disk surface, as in the incompressible Von Karman's flow. As a consequence of matching successive regions in the asymptotic procedure, it is found that the wavenumber and the orientation of the compressible lower branch modes are governed by an eigenrelation, which is akin to the one obtained previously in [ 1 ] for the incompressible stationary mode and in [ 2 ] for the compressible stationary modes. The nonparallel influences are toward destabilizing all the modes, though the wall insulation and heating are relatively stabilizing for the modes in the vicinity of the stationary mode, unlike the wall cooling. The asymptotic compressible data obtained at high Reynolds number limit compares fairly well with the numerical results generated directly solving the linearized compressible system with usual parallel flow approximation.     

Slow viscous flow near a superhydrophobic surface with trapped gas bubbles

The Boundary Element Method (BEM) is used to solve the problem of Stokes flow of a viscous fluid over a periodic striped texture of a superhydrophobic surface (SHS), partially filled with frictionless gas bubbles. The shape of the bubble surfaces and the position of the meniscus pinning points relative to the cavity walls are taken into account in the study. Two kinds of flows important for practical applications are considered: a pressure-driven flow in a thin channel with a bottom superhydrophobic wall and a shear-driven flow over a periodic texture. We study the flow pattern in the fluid over a single cavity containing a bubble with a curved phase interface shifted into the cavity. A parametric numerical study of the averaged slip length of the SHS is performed as a function of the geometric parameters of the texture. It is shown that the curvature of the phase interface and/or its shift into the cavity both result in the decrease in the average slip length. It is demonstrated that the BEM can be an efficient tool for studying Stokes flows over textured superhydrophobic surfaces with different geometries of microcavities and phase interfaces. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New Applications of the Boundary Element Method in Dynamics of Solids

The boundary element method (BEM) is developed and applied in new fields of dynamic fracture mechanics, dynamics of composite, elasto–plastic and piezoelectric materials. The BEM results are compared with solutions computed by the finite element method (FEM) showing high accuracy of the BEM. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Acoustic limit for the Boltzmann equation in optimal scaling

Based on a recent L² ? L^∞ framework, we establish the acoustic limit of the Boltzmann equation for general collision kernels. The scaling of the fluctuations with respect to the Knudsen number is optimal. Our approach is based on a new analysis of the compressible Euler limit of the Boltzmann equation, as well as refined estimates of Euler and acoustic solutions. © 2009 Wiley Periodicals, Inc.     

Nonuniform Approach to Terminal Velocity for Single Mode Rayleigh-Taylor Instability

Abstract The temporal development of a single mode Rayleigh-Taylor instability consists of three stages:the linear, free fall and terminal velocity regimes. The purpose of this paper is to report on new phenomenaobserved in the approach to terminal velocity. Our numerical study shows an unexpected nonuniform approachto terminal velocity. The nonuniformity applies especially to the spikes, which are fingers of heavy fluid fallinginto the light fluid, but it also applies to the rising bubbles of light fluid. For spikes especially our results callinto question the meaningfulness of a terminal velocity for moderate values of the Atwood number A. After ashort time period of pseudo-terminal plateau, the spike velocity increases to a significantly higher maximum,followed by a decrease. This phenomena appears to be due to a slow evolution in the shape of the spike andbubble. We find a relation between the spike (bubble) acceleration and the tip curvature. In correlation with anincrease in the spike velocity, th     

Application of the program package TURBO problem solver for some fluid dynamics problems

A technology for building parallel applications for numerical simulation based on hyperbolic partial differential equations is described. A formalization of problems and methods that makes it possible to describe new problems and methods for their solution by configuring the universal technology for specific cases is proposed. Results of numerical simulation of spatial flows in shear layers of a compressible inviscid perfect medium and of the Rayleigh–Taylor instability are presented.     

壁面效应对剪切稀化流体内气泡上浮特性的影响

数值研究了壁面效应对剪切稀化流体内气泡上浮运动特性的影响,气液两相的界面捕捉采用流体体积(VOF)法,剪切稀化流体流变特性和气液相间表面张力的计算分别采用Carreau模型和连续表面张力模型.详细研究了不同流变指数下,壁面效应对气泡形状、液相流场和气泡终端速度的影响.结果表明,强的壁面效应或弱的剪切稀化程度会限制气泡的变形和尾涡的形成,使气泡的终端速度减小;气泡终端速度最易受壁面效应的影响;强的壁面效应和强的剪切稀化程度会导致高剪切速率区域出现在壁面附近,引起壁面附近液相表观黏度大幅度的下降.     

On a structural acoustic model which incorporates shear and thermal effects in the structural component

In this paper we consider a structural acoustic model which takes account of thermal effects over and above displacement, rotational inertia and shear effects in the flat flexible structural component of the model. Thus the structural medium is a Reissner-Mindlin plate into which an additional degree of freedom, viz. temperature variation in the plate, has been introduced and the constitutive equations for the structural acoustic model couple parabolic dynamics with hyperbolic dynamics. We show unique solvability of the mathematical model and investigate the effect of the presence of thermal effects on the mechanical dissipation devices needed to attain uniform stabilization of the two-dimensional model in which the structural component is a Timoshenko beam. It turns out that, as in linear structural acoustic models which use the Euler-Bernoulli equation or the Kirchoff equation to describe the deflections of the thermo-elastic structural medium, uniform stabilization of the energy associated with the model can be attained without introducing mechanical dissipation at the free edge of the beam. Open problems with regard to the stabilization of the three-dimensional model are outlined.