开裂孔隙材料渗透率的细观力学模型研究

采用细观力学方法对含随机裂纹网络的孔隙材料渗透性进行研究.开裂孔隙材料渗透性的影响因素包括裂纹网络的密度、连通度、裂纹的开度以及孔隙材料基体渗透性.对于不连通的裂纹网络,该文采用已有的相互作用直推法(interaction direct derivative, IDD)的理论框架,引入裂纹的密度ρ和裂纹开度比b,提出了裂纹夹杂-基体两相复合材料渗透率的IDD理论解.对于部分连通裂纹网络,考虑局部裂纹团内部各个裂纹对有效渗透率的相互放大作用,引入裂纹网络的连通度f,定义与连通度相关的水平裂纹密度ρ~h,按照增量法将表征连通特征的水平裂纹嵌入有效基体中,以此方式来考虑裂纹夹杂间的相互搭接,提出了考虑裂纹连通特征的扩展IDD理论解,分别考虑了基体材料渗透率K_m、裂纹密度ρ、裂纹开度比b以及与连通度f相关的ρ~h.最后通过对有限区域内含随机裂纹网络孔隙材料渗透过程的有限元模拟分别验证了不连通和部分连通裂纹网络扩展IDD模型的适用性:(1)当裂纹不连通时,由于基体对流体渗透的阻隔作用,裂纹的开度对有效渗透率影响不大;(2)当裂纹部分连通时,裂纹密度分别小于1.1 (无关联裂纹网络,分形维数为2.0)、1.2 (关联裂纹网络,分形维数为1.75)时,扩展IDD模型能够很好地估计开裂孔隙材料的有效渗透率,但是随着裂纹进一步扩展,最大裂纹团主导作用凸显,扩展IDD模型不再适用.     

复合材料水翼水动力与结构强度特性数值研究

针对复合材料水翼存在的流固耦合求解问题,结合其自身特有属性,对复合材料水翼结构变形特性进行了数值仿真计算研究.研究建立了复合材料水翼流固耦合数值计算模型,并将数值计算结果与Zarruk等的实验结果进行对比,验证模型的正确性,得出复合材料水翼尖端扭转角随雷诺数的增加而增加的研究结论.基于数值计算模型,系统地研究了不同铺层角对复合材料水翼水动力特性及强度特性的影响,结果表明:不同铺层角复合材料水翼的尖端扭转角,随铺层角的增大而减小,而其尖端位移量随铺层角的增大先减小后增大.为了削弱工程常数的影响对复合材料水翼变形的影响,研究提出了无量纲扭转角和无量纲位移量,进一步探究复合材料水翼结构的弯扭耦合作用对其变形特性的影响.最后利用蔡$\!$-$\!$-$\!$吴失效准则进行复合材料水翼强度特性的判断和分析,结果表明:不同铺层角复合材料水翼的蔡$\!$-$\!$-$\!$吴系数,随铺层角的增大呈现先减小后增大的趋势,其中0$^\circ$铺层时的复合材料水翼蔡$\!$-$\!$-$\!$吴系数最小,50$^\circ$铺层时的复合材料水翼的蔡$\!$-$\!$-$\!$吴系数最大.      

定面射孔新工艺对套管强度的影响研究1)

用于非常规储层的定面射孔工艺可有效地降低水力裂缝的起裂压力,提高近井筒的泄油面积;但该工艺对套管强度的影响研究较少。本文基于流固耦合理论,建立了地层$\!$-$\!$-$\!$水泥环$\!$-$\!$-$\!$射孔套管的应力耦合模型,退化为裸套管与API标准进行比对,效验了模型,并发现地层约束条件下套管承受内压的能力提高了1.48$\sim$2.16倍,在此基础上,给出了套管尺寸、射孔直径及射孔相位角对套管承压能力的变化规律。计算表明：螺旋射孔方位角为45$^\circ$时,定面射孔间夹角为30$^\circ$时,套管内径越小,套管壁厚越厚,则套管的承压能力越高;相同参数下,定面射孔工艺比螺旋射孔工艺引起的套管承压能力略低。     

16MnR钢缺口试样解理断裂行为研究

研究了低合金热轧钢16MnR缺口试样在$-196\,{^\circ}$C和$-130\,{^\circ}$C的解理断裂机 理. 拉伸试验、单、双缺口四点弯曲实验、断口形貌观察以及有限元分析结果表明, 缺口试 样发生解理断裂时均起裂于夹杂物粒子, 一种位于缺口根部前端(IC型), 另一种位于距缺口 根部较远的条形裂纹前端(SIC型); 且随温度升高, 起裂源的类型从$-196\,{^\circ}$C下的IC 型转变为$-130\,{^\circ}$C下的SIC型. 微裂纹均形核于夹杂物, 最终的断裂由铁素体晶粒尺 寸的微裂纹扩展控制. 缺口试样IC型解理断裂遵循裂纹形核条 件$\varepsilon_{\rm p} \ge \varepsilon_{\rm pc}$和裂纹扩展条件$\sigma_{yy} \ge \sigma_{f}$, 而SIC型解理断裂条件则演化为$\varepsilon_{\rm p}+\varepsilon_{\rm ps} \ge \varepsilon_{\rm pc}$和$\sigma_{yy} +\sigma_{yy{\rm s}} \ge \sigma_{f}$.     

弹簧-金属丝网橡胶组合减振器迟滞力学模型及实验研究

金属丝网橡胶材料是一种完全由金属丝编织成的多孔复合材料,与传统螺旋卷制金属橡胶材料相比,其改进了成型工艺,剔除了制备过程中大量的手工工艺干扰,提高机械化程度,重合度更高,拥有更稳定的力学性能.由于金属丝网橡胶材料具有承载能力高、阻尼大、耐高温、耐低温、耐老化、抗油抗腐蚀等优良特性,在很多方面强于传统橡胶,多用于航空航天、船舶、军事武器等军工工业.弹簧$\!$-$\!$-$\!$金属丝网橡胶组合减振器具有可设计刚度和较高承载能力,但因其具有复杂的非线性迟滞特性,目前相关材料的本构模型还难以准确描述其力学特性.本文在弹簧$\!$-$\!$-$\!$金属丝网橡胶组合减振器静态迟滞力学性能实验的基础上,结合其干摩擦阻尼迟滞特性,提出了一种迟滞力学性能理论模型.根据减振器迟滞实验恢复力$\!$-$\!$-$\!$位移曲线特点,利用参数分离的方法将迟滞曲线分解为弹性恢复力和干摩擦阻尼力,分别建模求解等效刚度和干摩擦阻尼系数,以此建立了组合减振器理论模型,并与实验结果进行对比及进行误差分析,验证了理论模型的准确性.      

一类Markov过程的最大绝对值过程概率密度求解的新方法

随机过程或随机系统响应的最大绝对值概率分布往往是科学与工程中关心的重要挑战性问题.本文从理论与数值上进行了Markov过程的时变最大绝对值过程及其概率分布研究.文中,通过引入扩展状态向量,构造了最大绝对值$\!$-$\!$-$\!$状态量联合向量过程,由此将不具有Markov性的最大值过程转化为具有Markov性的向量随机过程.在此基础上,通过最大绝对值$\!$-$\!$-$\!$状态量之间的关系,建立了联合向量过程的转移概率密度函数.进而,结合Chapman-Kolmogorov方程和路径积分方法,提出了最大绝对值概率密度函数求解的数值方法.由此,可以得到Markov过程最大绝对值过程的时变概率密度函数,可进一步用于结构动力可靠度分析等.通过数值算例,验证了本文所提方法的有效性. 该方法有望推广到更一般随机系统的极值分布估计之中.      

中子辐照金属材料的脆化模型研究

金属材料的辐照脆化问题一直以来都是核能安全领域亟待解决的关键问题之一.为了更准确地预测金属材料的辐照脆化行为,基于Johnson-Cook本构模型,将未辐照金属材料的断裂真应力取作辐照材料的断裂真应力,建立了通过辐照退火态金属材料屈服强度就能够预测其整个真应力$\!$-$\!$-$\!$应变曲线,以及断裂真应变的辐照脆化模型.实验研究了不同中子剂量辐照退火态高纯铝的准静态拉伸真应力$\!$-$\!$-$\!$应变曲线、断裂真应力和断裂真应变随辐照剂量的变化规律.结果表明,辐照剂量越高,高纯铝的屈服强度越高,断裂真应变越低,但断裂真应力几乎不变.通过TEM显微分析获得了高纯铝内部辐照缺陷的尺寸和数密度随辐照剂量的变化规律,结果表明,辐照剂量越高,孔洞的尺寸和数密度越高,但位错环尺寸和数密度始终很小,难以准确统计.由辐照高纯铝实验数据拟合得到了辐照脆化模型所需参数,并检验了该模型的预测效果.结果表明,无论是通过实验还是显微分析得到辐照高纯铝的屈服强度,模型的预测结果均能够与实验结果较好地吻合,且模型对退火态高纯铝临界中子剂量的预测值也与文献结果一致.      

考虑陀螺效应的铣削颤振分析1)

主要研究主轴旋转过程中产生的陀螺效应对切削颤振稳定性边界的影响。基于一种旋转下的主轴、刀杆和刀具模型,采用欧拉$\!$-$\!$-$\!$伯努利悬臂梁理论获得考虑陀螺效应的切削系统的传递函数。在此基础上,探究了考虑陀螺效应的高速切削系统的稳定性。结果表明,陀螺效应改变了高速铣削的临界切削深度。并且在考虑陀螺效应的情况下,刀具齿数越多临界切削深度越小,降低系统刚度将减小临界切削深度,增大系统阻尼比将增大临界切削深度。     

屈服准则及切线模量修正的弹塑性计算模型

为提高不同应力状态下金属弹塑性行为的模拟精度,采用含应力三轴度修正的von Mises屈服准则,材料弹塑性本构关系在等效应力$\!$-$\!$-$\!$等效应变曲线基础上,提出一个切线模量为主应力函数的理论修正项. 将这两个修正项结合,以子程序形式编程嵌入ABAQUS主程序,以此模拟几种不同形状试样的弹塑性行为,并将其他屈服准则在单一曲线假设下编程与之对比. 模拟结果与真实试验结果对比发现,对于屈服而言,含应力三轴度修正的vonMises屈服准则比其他屈服理论准确;对于弹塑性阶段计算而言,提出的切线模量为主应力函数这一假设比单一曲线假设更加接近真实试验.      

完整岩石拉压应力状态下的张裂破坏准则

一点的应力状态可由3个主应力$\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$来表示, 当规定主应力以压为正时, 沿最大主应力$\sigma_{1}$方向将产生收缩变形, 若中间主应力$\sigma_{2}$和最小主应力$\sigma_{3}$都远小于$\sigma_{1}$, 则沿$\sigma_{2}$和$\sigma_{3}$方向会产生横向扩张变形, 当横向扩张变形达到一定极限时, 将会在平行于$\sigma _{1}$的方向产生张裂破坏. 如何建立这种张裂破坏的强度准则目前尚缺乏研究, 最大拉应变理论(第二强度理论)有时被用来解释张裂破坏, 但最大拉应变理论难以应用于三向受力状态. 本文分别用$\varepsilon_{1}$, $\varepsilon_{2}$表示最大张应变和次大张应变, 则最大拉应变理论认为当$\varepsilon_{1}$达到单向拉伸屈服应变时, 材料将产生破坏. 而本文将根据$\varepsilon_{1}+\varepsilon_{2}$之和达到极限值$\varepsilon_u$来建立张裂破坏准则. 可以证明$\varepsilon_{1} +\varepsilon_{2}$所表示的是$\sigma_{1}$主平面的面积增长率. 当$\sigma_{3}<\sigma_{2} \ll \sigma_{1}$时, 大部分岩石都具有脆性破坏的特点, 所以可将破坏前的岩石视为满足广义胡克定律的线弹性材料, 这样用$\varepsilon_{1}$, $\varepsilon_{2}$表示的强度准则可通过$\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$来表示. 在这个过程中还可考虑岩石在拉伸和压缩时具有不同弹性参数和强度的特点, 并可通过单向拉伸和单向压缩的破坏状态来确定$\varepsilon_u$. 不管$\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$是压应力, 还是拉应力, 或者$\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$中有拉有压的情形, 基于$\varepsilon_{1} +\varepsilon_{2} =\varepsilon_u$都可建立相应的强度准则. 所建立的准则可以反映中间应力$\sigma_{2}$对强度的影响规律, 通过建立的强度准则还可以证明: 静水拉力能引起屈服, 而静水压力不能产生屈服; 压缩破坏能使塑性体积增大, 其结果比Mohr-Coulomb准则更能反映实际情形. 并通过拉压应力状态下的试验数据验证了所建立的强度准则, 所得理论计算结果和已有的试验数据吻合得很好. 通过提出的强度准则和圆盘劈裂的试验结果, 可获得更为可靠的岩石单轴抗拉强度.      

Experimental Investigation on Transport Properties of Cement-Based Materials Incorporating 2D Crack Networks

This paper investigates the correlation between the geometry of crack networks and the altered transport properties of cement-based porous materials. Cracks were artificially introduced into slice specimens to obtain bidimensional (2D) crack networks, and the network was characterized by the crack density, orientation, connectivity and crack opening aperture. For the permeability, the water vapor sorption isotherms were measured and an algorithm was established to solve the intrinsic permeability of cracked specimens with the help of moisture transport modeling and the data of drying tests. The electrical conductivity of cracked specimens was measured using an alternative current method. The study on the specimens with percolated cracks shows that: (1) the pertinent geometry parameters for altered transport properties include average-based crack density, crack opening and local crack connectivity; (2) the water permeability of cracked specimens is correlated to the combination \(b^{1.7}\rho f\) and electrical conductivity to \(b^{0.45}\rho f\); (3) the different exponents on the crack opening/length ratio reflect the resistance of tortuosity of crack paths to the water and current flow and this resistance is stronger for current flow.     

Lower Semicontinuity for Integral Functionals in the Space of Functions of Bounded Deformation Via Rigidity and Young Measures

We establish a general weak* lower semicontinuity result in the space BD(Ω) of functions of bounded deformation for functionals of the form

利用维氏和玻氏压头表征半导体材料断裂韧性

压痕法是测量材料断裂韧性 ($K_{\rm IC})$ 的常用方法之一, 如何根据不同的材料、不同的压头选择适合的公式, 是当前面临的一大问题. 因此,在不同载荷下对单晶硅 (111) 和碳化硅 (4H-SiC, 0001面) 这两种半导体材料进行了维氏微米硬度和玻氏纳米压痕实验, 对实验产生的裂纹长度$c$进行了统计分析, 并采用13个压痕公式计算材料的$K_{\rm IC}$, 开展了微米划痕实验, 验证压痕法评估半导体材料$K_{\rm IC}$的适用性. 研究结果表明: 为了消除维氏压痕实验产生的$c$的固有离散性, 需要多次测量取平均值; 裂纹长度与压痕尺寸的比值随压痕载荷的增大而增大; 材料的裂纹类型与载荷相关且低载荷下表现为巴氏裂纹, 高载荷下表现为中位裂纹; 与微米划痕实验得到的单晶硅和碳化硅材料的$K_{\rm IC}$平均值 (分别为0.96 MPa,$\cdot$,$\sqrt{\rm m}$和2.89 MPa,$\cdot$,$\sqrt{\rm m}$) 相比, 在同一压头下无法从13个公式中获得同时适用于单晶硅和碳化硅材料的压痕公式,但在同一材料下可以获得同时适用于维氏和玻氏压头的$K_{\rm IC}$计算公式; 基于中位裂纹系统发展而来的压痕公式更适合用于评估半导体材料的$K_{\rm IC}$, 且维氏压头下的$K_{\rm IC}$与玻氏压头下$K_{\rm IC}$的关系不是理论上的1.073倍, 应为1.13$\pm 压痕法是测量材料断裂韧性(K_(IC))的常用方法之一,如何根据不同的材料、不同的压头选择适合的公式,是当前面临的一大问题.因此,在不同载荷下对单晶硅(111)和碳化硅(4H-Si C, 0001面)这两种半导体材料进行了维氏微米硬度和玻氏纳米压痕实验,对实验产生的裂纹长度c进行了统计分析,并采用13个压痕公式计算材料的K_(IC),开展了微米划痕实验,验证压痕法评估半导体材料K_(IC)的适用性.研究结果表明:为了消除维氏压痕实验产生的c的固有离散性,需要多次测量取平均值;裂纹长度与压痕尺寸的比值随压痕载荷的增大而增大;材料的裂纹类型与载荷相关且低载荷下表现为巴氏裂纹,高载荷下表现为中位裂纹;与微米划痕实验得到的单晶硅和碳化硅材料的K_(IC)平均值(分别为0.96 MPa·m~(1/2)和2.89 MPa·m~(1/2))相比,在同一压头下无法从13个公式中获得同时适用于单晶硅和碳化硅材料的压痕公式,但在同一材料下可以获得同时适用于维氏和玻氏压头的K_(IC)计算公式;基于中位裂纹系统发展而来的压痕公式更适合用于评估半导体材料的K_(IC),且维氏压头下的K_(IC)与玻氏压头下K_(IC)的关系不是理论上的1.073倍,应为1.13±0.01.     

The Dielectric Permittivity of Crystals in the Reduced Hartree–Fock Approximation

In a recent article (Cancès et al. in Commun Math Phys 281:129–177, 2008), we have rigorously derived, by means of bulk limit arguments, a new variational model to describe the electronic ground state of insulating or semiconducting crystals in the presence of local defects. In this so-called reduced Hartree–Fock model, the ground state electronic density matrix is decomposed as ${\gamma = \gamma^0_{\rm per} + Q_{\nu,\varepsilon_{\rm F}}}$ , where ${\gamma^0_{\rm per}}$ is the ground state density matrix of the host crystal and ${Q_{\nu,\varepsilon_{\rm F}}}$ the modification of the electronic density matrix generated by a modification ν of the nuclear charge of the host crystal, the Fermi level ε _F being kept fixed. The purpose of the present article is twofold. First, we study in more detail the mathematical properties of the density matrix ${Q_{\nu,\varepsilon_{\rm F}}}$ (which is known to be a self-adjoint Hilbert–Schmidt operator on ${L^2(\mathbb{R}^3)}$ ). We show in particular that if ${\int_{\mathbb{R}^3}\,\nu \neq 0, Q_{\nu,\varepsilon_{\rm F}}}$ is not trace-class. Moreover, the associated density of charge is not in ${L^1(\mathbb{R}^3)}$ if the crystal exhibits anisotropic dielectric properties. These results are obtained by analyzing, for a small defect ν, the linear and nonlinear terms of the resolvent expansion of ${Q_{\nu,\varepsilon_{\rm F}}}$ . Second, we show that, after an appropriate rescaling, the potential generated by the microscopic total charge (nuclear plus electronic contributions) of the crystal in the presence of the defect converges to a homogenized electrostatic potential solution to a Poisson equation involving the macroscopic dielectric permittivity of the crystal. This provides an alternative (and rigorous) derivation of the Adler–Wiser formula.     

A-Priori Direct Numerical Simulation Modelling of Co-variance Transport in Turbulent Stratified Flames

Three-dimensional Direct Numerical Simulations of statistically planar turbulent stratified flames at global equivalence ratios <???>?=?0.7 and <???>?=?1.0 have been carried out to analyse the statistical behaviour of the transport of co-variance of the fuel mass fraction Y _F and mixture fraction ξ (i.e. $\widetilde{Y_F^{\prime\prime} \xi ^{\prime\prime}}={\overline {\rho Y_F^{\prime\prime} \xi^{\prime\prime}} } \Big/ {\overline \rho })$ for Reynolds Averaged Navier Stokes simulations where $\overline q $ , $\tilde{q} ={\overline {\rho q} } \big/ {\overline \rho }$ and $q^{\prime\prime}= q-\tilde{q}$ are Reynolds averaged, Favre mean and Favre fluctuation of a general quantity q with ρ being the gas density and the overbar suggesting a Reynolds averaging operation. It has been found that existing algebraic expressions may not capture the statistical behaviour of $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ with sufficient accuracy in low Damköhler number combustion and therefore, a transport equation for $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ may need to be solved. The statistical behaviours of $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ and the unclosed terms of its transport equation (i.e. the terms originating from turbulent transport T ₁ , reaction rate T ₄ and molecular dissipation $\left( {-D_2 } \right))$ have been analysed in detail. The contribution of T ₁ remains important for all cases considered here. The term T ₄ acts as a major contributor in <???>?=?1.0 cases, but plays a relatively less important role in <???>?=?0.7 cases, whereas the term $\left( {-D_2 } \right)$ acts mostly as a leading order sink. Through an a-priori DNS analysis, the performances of the models for T ₁ , T ₄ and $\left( {-D_2 } \right)$ have been addressed in detail. A model has been identified for the turbulent transport term T ₁ which satisfactorily predicts the corresponding term obtained from DNS data. The models for T ₄ , which were originally proposed for high Damköhler number flames, have been modified for low Damköhler combustion. Predictions of the modified models are found to be in good agreement with T ₄ obtained from DNS data. It has been found that existing algebraic models for $D_2 =2\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} $ (where D is the mass diffusivity) are not sufficient for low Damköhler number combustion and therefore, a transport equation may need to be solved for the cross-scalar dissipation rate $\widetilde{\varepsilon }_{Y\xi } ={\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} } \big/ {\overline \rho }$ for the closure of the $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ transport equation.     

Stochastic Variational Inequalities and Applications to the Total Variation Flow Perturbed by Linear Multiplicative Noise

In this work, we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation ${{\rm d}X(t) = {\rm div} \left[\frac{\nabla X(t)}{|\nabla X(t)|}\right]{\rm d}t + X(t){\rm d}W(t) {\rm in} (0, \infty) \times \mathcal{O},}$ where ${\mathcal{O}}$ is a bounded and open domain in ${\mathbb{R}^N, N \geqq 1}$ and W(t) is a Wiener process of the form ${W(t) = \sum^{\infty}_{k = 1}\mu_{k}e_{k}\beta_{k}(t), e_{k} \in C^{2}(\overline{\mathcal{O}}) \cap H^{1}_{0}(\mathcal{O}),}$ and ${\beta_{k}, k \in \mathbb{N}}$ are independent Brownian motions. This is a stochastic diffusion equation with a highly singular diffusivity term. One main result established here is that for all initial conditions in ${L^2(\mathcal{O})}$ , it is well posed in a class of continuous solutions to the corresponding stochastic variational inequality. Thus, one obtains a stochastic version of the (minimal) total variation flow. The new approach developed here also allows us to prove the finite time extinction of solutions in dimensions ${1\leqq N \leqq3}$ , which is another main result of this work.     

Regularization of Point Vortices Pairs for the Euler Equation in Dimension Two

In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem $$\left\{\begin{array}{l@{\quad}l} -\varepsilon^2 \Delta u = \sum\limits_{i=1}^m \chi_{\Omega_i^{+}} \left(u - q - \frac{\kappa_i^{+}}{2\pi} {\rm ln} \frac{1}{\varepsilon}\right)_+^p\\ \quad - \sum_{j=1}^n \chi_{\Omega_j^{-}} \left(q - \frac{\kappa_j^{-}}{2\pi} {\rm \ln} \frac{1}{\varepsilon} - u\right)_+^p , \quad \quad x \in \Omega,\\ u = 0, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad x \in \partial \Omega,\end{array}\right.$$ where p > 1, ${\Omega \subset \mathbb{R}^2}$ is a bounded domain, ${\Omega_i^{+}}$ and ${\Omega_j^{-}}$ are mutually disjoint subdomains of Ω and ${\chi_{\Omega_i^{+}} ({\rm resp}.\; \chi_{\Omega_j^{-}})}$ are characteristic functions of ${\Omega_i^{+}({\rm resp}. \;\Omega_j^{-}})$ , q is a harmonic function. We show that if Ω is a simply-connected smooth domain, then for any given C ¹-stable critical point of Kirchhoff–Routh function ${\mathcal{W}\;(x_1^{+},\ldots, x_m^{+}, x_1^{-}, \ldots, x_n^{-})}$ with ${\kappa^{+}_i > 0\,(i = 1,\ldots, m)}$ and ${\kappa^{-}_j > 0\,(j = 1,\ldots,n)}$ , there is a stationary classical solution approximating stationary m + n points vortex solution of incompressible Euler equations with total vorticity ${\sum_{i=1}^m \kappa^{+}_i -\sum_{j=1}^n \kappa_j^{-}}$ . The case that n = 0 can be dealt with in the same way as well by taking each ${\Omega_j^{-}}$ as an empty set and set ${\chi_{\Omega_j^{-}} \equiv 0,\,\kappa^{-}_j=0}$ .     

Incompressible 3D Navier–Stokes Equations as a Limit of a Nonlinear Parabolic System

In this paper, we consider the Cauchy problem for a nonlinear parabolic system ${u^\epsilon_t - \Delta u^\epsilon + u^\epsilon \cdot \nabla u^\epsilon + \frac{1}{2}u^\epsilon\, {\rm div}\, u^\epsilon - \frac{1}{\epsilon}\nabla\, {\rm div}\, u^\epsilon = 0}$ in ${\mathbb {R}^3 \times (0,\infty)}$ with initial data in Lebesgue spaces ${L^2(\mathbb {R}^3)}$ or ${L^3(\mathbb {R}^3)}$ . We analyze the convergence of its solutions to a solution of the incompressible Navier?CStokes system as ${\epsilon \to 0}$ .