On Singularity Formation of a Nonlinear Nonlocal System

We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei (Comm Pure Appl Math 62(4):501–564, 2009) for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the nonlocal system for a large class of smooth initial data with finite energy. We also prove global regularity for a class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model.     

Geometric Desingularization of a Cusp Singularity in Slow–Fast Systems with Applications to Zeeman’s Examples

The cusp singularity—a point at which two curves of fold points meet—is a prototypical example in Takens’ classification of singularities in constrained equations, which also includes folds, folded saddles, folded nodes, among others. In this article, we study cusp singularities in singularly perturbed systems for sufficiently small values of the perturbation parameter, in the regime in which these systems exhibit fast and slow dynamics. Our main result is an analysis of the cusp point using the method of geometric desingularization, also known as the blow-up method, from the field of geometric singular perturbation theory. Our analysis of the cusp singularity was inspired by the nerve impulse example of Zeeman, and we also apply our main theorem to it. Finally, a brief review of geometric singular perturbation theory for the two elementary singularities from the Takens’ classification occurring for the nerve impulse example—folds and folded saddles—is included to make this article self-contained.     

Method for determining power-type complex singularities in solutions of singular integral equations with generalized kernels and complex conjugate unknowns

We develop a method for determining power-type complex singularities of solutions for a class of one-dimensional singular integral equations with generalized kernels and complex conjugate unknown functions. By analyzing the characteristic part of a singular integral equation, we reduce the problem of determining the solution singularity exponents at the ends of the integration interval to two independent transcendental equations for these exponents. We show that the distribution of admissible singularity exponents is of continuous character. We present numerical results for a two-dimensional elasticity problem whose mathematical statement leads to a singular integral equation of the class under study. We also reveal the drawbacks of one classical approach to the determination of stress field singularities.     

Choice of a self-similar solution in boundary-layer theory

If the speed of the outer flow at the edge of the boundary layer does not depend on the time and is specified in the form of a power-law function of the longitudinal coordinate, then a self-similar solution of the boundary-layer equations can be found by integrating a third-order ordinary differential equation (see [1–3]). When the exponent of the power in the outerflow velocity distribution is negative, a self-similar solution satisfying the equations and the usually posed boundary conditions is not uniquely determinable [4], A similar result was obtained in [5] for flows of a conducting fluid in a magnetic field. In the present paper we study the behavior of non-self-similar perturbations of a self-similar solution, enabling us to provide a basis for the choice of a self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–46, July–August, 1974.     

On Barenblatt's Model of Spontaneous Countercurrent Imbibition

Water imbibition is a critical mechanism of secondary oil recovery from fractured reservoirs. Spontaneous imbibition also plays a significant role in storage of liquid waste by controlling the extent of rock invasion. In the present paper, we extend a model of countercurrent imbibition based on Barenblatt's theory of non-equilibrium two-phase flow by allowing the model's relaxation time to be a function of the wetting fluid saturation. We obtain two asymptotic self-similar solutions, valid at early and late times, respectively. At a very early stage, the time scale characterizing the cumulative volume of imbibed (and expelled) fluid is a power function with exponent between 1.5 and 1. At a later stage, the time scaling for this volume approaches asymptotically classical square root of time, whereas the saturation profile asymptotically converges to Ryzhik's self-similar solution. Our conclusions are verified against experiments. By fitting the laboratory data, we estimate the characteristic relaxation times for different pairs of liquids.     

On the pressure and stress singularities induced by steady flows of incompressible viscous fluids

Design for structural integrity requires an appreciation of where stress singularities can occur in structural configurations. While there is a rich literature devoted to the identification of such singular behavior in solid mechanics, to date there has been relatively little explicit identification of stress singularities caused by fluid flows. In this study, stress and pressure singularities induced by steady flows of viscous incompressible fluids are asymptotically identified. This is done by taking advantage of an earlier result that the Navier-Stokes equations are locally governed by Stokes flow in angular corners. Findings for power singularities are confirmed by developing and using an analogy with solid mechanics. This analogy also facilitates the identification of flow-induced log singularities. Both types of singularity are further confirmed for two global configurations by applying convergence-divergence checks to numerical results. Even though these flow-induced stress singularities are analogous to singularities in solid mechanics, they nonetheless render a number of structural configurations singular that were not previously appreciated as such from identifications within solid mechanics alone.     

Singularities in the boundary layer on a cone at incidence

The singularities in the three-dimensional laminar boundary layer on a cone at incidence are studied. It is shown that these singularities are formed in the outer part of the boundary layer and described by linear equations whose solutions are obtained in analytic form. The known results for the plane of symmetry are classified on this basis. Two solutions of the non-self-similar problem are found, one of which has a singularity at zero incidence and in the sink plane. The second branch goes over continuously into the solution for axisymmetric flow. However, as the angle of attack increases, in the sink plane a singularity is formed and all the self-similar solutions existing here lose their meaning. Starting from the critical angle of attack, the flow in the vicinity of the sink plane is no longer described by the boundary layer equations, so that the results can be used to construct an adequate physical model.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 25–33, November–December, 1993.     

A case of strong nonlinearity: Intermittency in highly turbulent flows

It has long been suspected that flows of incompressible fluids at large or infinite Reynolds number (namely at small or zero viscosity) may present finite time singularities. We review briefly the theoretical situation on this point. We discuss the effect of a small viscosity on the self-similar solution to the Euler equations for inviscid fluids. Then we show that single-point records of velocity fluctuations in the Modane wind tunnel display correlations between large velocities and large accelerations in full agreement with scaling laws derived from Leray's equations (1934) for self-similar singular solutions to the fluid equations. Conversely, those experimental velocity–acceleration correlations are contradictory to the Kolmogorov scaling laws.     

Singularities in Axisymmetric Free Boundaries for ElectroHydroDynamic Equations

We consider singularities in the ElectroHydroDynamic equations. In a regime where we are allowed to neglect surface tension, and assuming that the free surface is given by an injective curve and that either the fluid velocity or the electric field satisfies a certain non-degeneracy condition, we prove that either the fluid region or the gas region is asymptotically a cusp. Our proofs depend on a combination of monotonicity formulas and a non-vanishing result by Caffarelli and Friedman. As a by-product of our analysis we also obtain a special solution with convex conical air-phase which we believe to be new.     

Erosion Profile by a Global Model for Granular Flow

In this paper we study an integro-differential equation that models the erosion of a mountain profile caused by small avalanches. The equation is in conservative form, with a non-local flux involving an integral of the mountain slope. Under suitable assumptions on the erosion rate, the mountain profile develops several types of singularities, which we call kinks, shocks and hyper-kinks. We study the formation of these singularities and derive admissibility conditions. Furthermore, entropy weak solutions to the Cauchy problem are constructed globally in time, taking limits of piecewise affine approximate solutions. Entropy and entropy flux functions are introduced, and a Lax entropy condition is established for the weak solutions.     

Corner stress singularities in an FGM thin plate

Describing the behaviors of stress singularities correctly is essential for obtaining accurate numerical solutions of complicated problems with stress singularities. This analysis derives asymptotic solutions for functionally graded material (FGM) thin plates with geometrically induced stress singularities. The classical thin plate theory is used to establish the equilibrium equations for FGM thin plates. It is assumed that the Young’s modulus varies along the thickness and Poisson’s ratio is constant. The eigenfunction expansion method is employed to the equilibrium equations in terms of displacement components for an asymptotic analysis in the vicinity of a sharp corner. The characteristic equations for determining the stress singularity order at the corner vertex and the corresponding corner functions are explicitly given for different combinations of boundary conditions along the radial edges forming the sharp corner. The non-homogeneous elasticity properties are present only in the characteristic equations corresponding to boundary conditions involving simple support. Finally, the effects of material non-homogeneity following a power law on the stress singularity orders are thoroughly examined by showing the minimum real values of the roots of the characteristic equations varying with the material properties and vertex angle.     

Relationship Between Singularities in Classical Mechanics for Complex Initial Conditions and for Complex Time

A relationship is established between the functional forms of two kinds of singularities in dynamical variables that arise in complexified versions classical mechanics: singularities that are treated as a functions of complex initial conditions for real time and those that are treated as a functions of complex time for real initial conditions. The analysis is verified by numerical calculations. The results imply that Kowaleskaya–Painlevé condition for integrability can be phrased in terms of singularities with respect to initial conditions.     

Singularities of the boundary layer equations and the structure of the flow in the vicinity of the convergence plane on conical bodies

The singularities of the boundary layer equations and the laminar viscous gas flow structure in the vicinity of the convergence plane on sharp conical bodies at incidence are analyzed. In the outer part of the boundary layer the singularities are obtained in explicit form. It is shown that in the vicinity of a singularity a boundary domain, in which the flow is governed by the shortened Navier-Stokes equations, is formed; their regular solutions are obtained. The viscous-inviscid interaction effect predominates in a region whose extent is of the order of the square root of the boundary layer thickness, in which the flow is described by a two-layer model, namely, the Euler equations in the slender-body approximation for the outer region and the three-dimensional boundary layer equations; the pressure is determined from the interaction conditions. On the basis of an analysis of the solutions for the outer part of the boundary layer it is shown that interaction leads to attenuation of the singularities and the dependence of the nature of the flow on the longitudinal coordinate, but does not make it possible to eliminate the singularities completely.     

非结构混合网格高超声速绕流与磁场干扰数值模拟

对均匀磁场干扰下的二维钝头体无粘高超声速流场进行了基于非结构混合网格的数值模拟.受磁流体力学方程组高度非线性的影响及考虑到数值模拟格式的精度,目前在此类流场的数值模拟中大多使用结构网格及有限差分方法,因而在三维复杂外形及复杂流场方面的研究受到限制.本文主要探索使用非结构网格(含混合网格)技术时的数值模拟方法.控制方程为耦合了Maxwell方程及无粘流体力学方程的磁流体力学方程组,数值离散格式采用Jameson有限体积格心格式,5步Runge-Kutta显式时间推进.计算模型为二维钝头体,初始磁场均匀分布.对不同磁感应强度影响下的高超声速流场进行了数值模拟,并与有限的资料进行了对比,得到了较符合的结果.     

Self-similar solutions of the problem of formation of a hydraulic fracture in a porous medium

The problem of hydraulic fracture formation in a porous medium is investigated in the approximation of small fracture opening and inertialess incompressible Newtonian fluid fracture flow when the seepage through the fracture walls into the surrounding reservoir is asymptotically small or large. It is shown that the system of equations describing the propagation of the fracture has self-similar solutions of power-law or exponential form only. A family of self-similar solutions is constructed in order to determine the evolution of the fracture width and length, the fluid velocity in the fracture, and the length of fluid penetration into the porous medium when either the fluid flow rate or the pressure as a power-law or exponential function of time is specified at the fracture entrance. In the case of finite fluid penetration into the soil the system of equations has only a power-law self-similar solution, for example, when the fluid flow rate is specified at the fracture entrance as a quadratic function of time. The solutions of the self-similar equations are found numerically for one of the seepage regimes.     

Similarity method for imploding strong shocks in a non-ideal relaxing gas

Self-similar solutions arise naturally as special solutions of system of partial differential equations (PDEs) from dimensional analysis and, more generally, from the invariance of system of PDEs under scaling of variables. Usually, such solutions do not globally satisfy imposed boundary conditions. However, through delicate analysis, one can often show that a self-similar solution holds asymptotically in certain identified domains. In the present paper, it is shown that self-similar phenomena can be studied through use of many ideas arising in the study of dynamical systems. In particular, there is a discussion of the role of symmetries in the context of self-similar dynamics. We use the method of Lie group invariance to determine the class of self-similar solutions to a problem involving plane and radially symmetric flows of a relaxing non-ideal gas involving strong shocks. The ambient gas ahead of the shock is considered to be homogeneous. The method yields a general form of the relaxation rate for which the self-similar solutions are admitted. The arbitrary constants, occurring in the expressions for the generators of the local Lie group of transformations, give rise to different cases of possible solutions with a power law, exponential or logarithmic shock paths. In contrast to situations without relaxation, the inclusion of relaxation effects imply constraint conditions. A particular case of the collapse of an imploding shock is worked out in detail for radially symmetric flows. Numerical calculations have been performed to determine the values of the self-similarity exponent and the profile of the flow variables behind the shock. All computations are performed using the computation package Mathematica.     

Restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities

The paper addresses the occurrence of possible restrictions on the flows defined by scalar retarded functional differential equations (FDEs), locally around certain simple singularities, compared with the possible flows of ordinary differential equations (ODEs) with the same singularities. It is found that for the Hopf and the Bogdanov-Takens singularities, there are no restrictions on the local flows defined by scalar FDEs, even when the nonlinearities depend on just one delayed value of the solutions. On the other hand, for the singularity associated with a zero and a conjugated pair of pure imaginary numbers as simple eigenvalues, it is shown that there occur restrictions on the flows defined by scalar FDEs with nonlinearities involving just one delay, as well as two delays satisfying a certain resonance condition. These restrictions are of geometric significance, since they amount to the impossibility of observing the homoclinic orbits that occur in arbitrarily small neighborhoods of the singularity for ODEs. Versal unfolfings for the considered singularities by FDEs and the possible restrictions on the associated flows are also studied.     

Heat Transfer in Incompressible Magnetic Fluid

In this paper we study the equations describing the dynamics of heat transfer in an incompressible magnetic fluid under the action of an applied magnetic field. The system is a combination of the Navier?CStokes equations, the magnetostatic equations and the temperature equation. We prove global-in-time existence of weak solutions with finite energy to the system posed in a bounded domain of ${\mathbb{R}^3}$ and equipped with initial and boundary conditions. The main difficulty comes from the singularity of the terms representing the Kelvin force due to the magnetization and the thermal power due to the magnetocaloric effect.     

Axisymmetric nuclei of strain and their applications for multilayered solids

In this paper, the effect of several axisymmetric elastic singularities (i.e., point forces, double forces, sum of two double forces and centers of dilatation) on the elastic response of a multilayered solid is investigated. The boundary conditions in an infinite solid at the plane passing through the singularity are derived first using Papkovich–Neuber harmonic functions. Then, a Green’s function solution for multilayered solids is obtained by solving a set of simultaneous linear algebraic equations using both the boundary conditions for the singularity and the layer interfaces. Finally, the elastic solutions in a single layer on an infinite substrate due to point defects and infinitesimal prismatic dislocation loops are presented to illustrate the application of these Green’s function solutions.     

层体边界元法在二维层体断裂分析中的应用

在刚度矩阵法的基础上建立了用于进行二维多层体结构断裂分析的边界单元法（ＢＥＭＬＭ）由于ＢＥＭＬＭ的基本方程中已经包含了层体表面和裂纹缝面的边界条件，因而不需要对这些边界进行单元离散，从而其断裂分析可望有较好的精度通过与柯西积分方程法进行结合，算例表明ＢＥ ＭＬＭ是可靠并有效的