Axisymmetric three-dimensional finger solutions in Mullins-Sekerka equation

In this paper, we study the existence problem of axisymmetricthree-dimensional finger solutions of Mullins–Sekerkaequation. The finger solutions are travelling wave solutionswhose finger-shaped interfaces are moving along a certain directionat a constant speed within a cylindrical domain. The existenceof at least one axisymmetric three-dimensional finger solutionis obtained through a fixed-point argument of the Hilbert transformation.     

Existence of solutions of laminar boundary layer equations with decelerating external flows

New results on the existence and nonexistence of solutions of laminar boundary layer equations with decelerating external flows are obtained. Some previous results treated these equations with accelerating or constant external flows. Our approach is to establish a system of two integral equations with singularities which will be proved to be equivalent to the laminar boundary layer equations and prove that the system has solutions by using the Leray–Schauder fixed point theorem and the Helly selection principle.     

The limiting properties of piecewise-potential subcritical and critical jet streams of an ideal gas

The limiting properties of subcritical and critical (with Mach numbers M ⩽ 1) plane-parallel jet streams are investigated in the approximation of an ideal (inviscid and non-heat-conducting) gas. Chaplygin's equation is used with the pressure and the angle of inclination of the velocity as the independent variables, which are measured from the limiting values corresponding to the cross-section of the equalizing of the jet with respect to these variables. The stream function in the neighbourhood of the “equalizing cross-section” is represented in the form of an expansion in powers of the “distance to the origin of the coordinates” (in the plane of the independent variables) with coefficients which depend on a previously unknown combination of the independent variables. The limiting property of the flow, i.e. the position of the equalizing cross-section at a finite of infinite distance, is defined by the leading terms of the expansion. A symmetric potential jet and symmetric piecewise-potential jets flowing out into a submerged space in the case of subcritical and critical pressure drops are considered as examples. The critical pressures of the potential parts of the composite jet can be different or identical (including when the thermodynamic properties of the gases are different).     

On the non-starlikeness of solutions to the starlike interior wake problem

We study examples of the starlike interior ``wake problem" for which no starlike solution exists relative to the natural star center of the problem. These examples show that the main result of D.E. Tepper in ``A mathematical model for a wake' (Michigan Math. J. 31 (1984), 161-165) is not correct.

     

Axisymmetric solutions of the two-dimensional Euler equations with a two-constant equation of state

A family of self-similar and global bounded weak solutions are constructed rigorously for all positive time to the two-dimensional isentropic Euler equations with the equation of state $p=A_1{\rho }^{{\gamma }_1}+A_2{\rho }^{{\gamma }_2}$ for initial pure radial velocity. The main difficulty is that the equations cannot be directly reduced to an autonomous system of ordinary differential equations. To deal with it, we first establish the global existence and the detailed structures of solutions to a new system under the axisymmetric and self-similar assumptions.     

Axisymmetric isospectral annular plates and membranes

A simpler ‘conditional’ form of the vibrating plateequation for which all the physical parameters may be spatiallyvarying is found when the relationship D(1 – v) = constantis satisfied between flexural rigidity and Poisson's ratio.A method of coordinate inversion is used to find a family ofinhomogeneous clamped annular plates (with axisymmetric spatiallyvarying density, thickness, Young's modulus, and Poisson's ratio)which have the same complete frequency spectrum as a standardhomogeneous clamped annular plate of the same inner and outerradii. In particular, these inhomogeneous plates have inversefourthpower radial dependence of density. General radial power-lawtransformations are also considered, leading to conclusionsfor the axisymmetric vibration spectra. Various results forcircular and annular membranes also emerge from the analysis.     

Existence of weak solutions to a system of nonlinear partial differential equations modelling ice streams

This paper deals with the mathematical analysis of a nonlinear system of three differential equations of mixed type. It describes the generation of fast ice streams in ice sheets flowing along soft and deformable beds. The system involves a nonlinear parabolic PDE with a multivalued term in order to deal properly with a free boundary which is naturally associated to the problem of determining the basal water flux in a drainage system. The other two equations in the system are an ODE with a nonlocal (integral) term for the ice thickness, which accounts for mass conservation and a first order PDE describing the ice velocity of the system. We first consider an iterative decoupling procedure to the system equations to obtain the existence and uniqueness of solutions for the uncoupled problems. Then we prove the convergence of the iterative decoupling scheme to a bounded weak solution for the original system.     

Axisymmetric stress concentration problems in cosserat elasticity

Zusammenfassung Das stationäre thermoelastische Problem für den Halbraum, das sich für einen Wärmedipol auf der Begrenzungsebene ergibt, und das Boussinesquesche Problem für eine konzentrierte Last werden mit Hilfe der Cosseratschen Elastizitätstheorie behandelt. Dabei bleibt die Ordnung der Singularität in den Verzerrungen und Verschiebungen unverändert. Im Gegensatz zur klassischen Theorie erweist sich hier das thermoelastische Problem nicht als eben. Die Resultate der klassischen Aufgabe und jene des Problems mit Momentenspannungen stellen Grenzfälle dar.     

Axisymmetric nonconforming element method and its convergence analysis

By virtue of the weighted Sobolev space theory, three convergence tests of the axisyrnmetric nonconforming element method are established. They consist of the generalized patch test, the F-E-M test and a test which could be used conveniently, called the strong patch test (SPT). In the light of SPT, a class of axisymmetric nonconforming elements is established.     

Axisymmetric bodies of minimum drag in hypersonic flow

The problem of determining the longitudinal contour of a slender, axisymmetric body in hypersonic flow which has minimum drag is considered. The pressure distribution is assumed to be Newtonian, while the skin-friction distribution is for laminar flow and depends on body geometry. This investigation is conducted with the method of steepest descent, whose feasiblity is demonstrated by solving minimum drag problems having known analytical solutions.     

Axisymmetric stokes exterior problem and its numerical computation

We establish variational formulation and prove the existence and uniqueness of the three dimensional axisymmetric Stokes exterior problem in weighted spaces.Error estimates and convergence for P2-P0 elements with infinite element methods are also obtained.Numerical experiments are presented to verify the theoretical analysis.     

Ideal-Gas Axisymmetric Flows and Their Interaction with Obstacles

The interaction of ideal-gas flows with obstacles is analyzed numerically. The study examines the effect of rotation of the supersonic jet on its structure and on the parameters of the normal modes that arise in collision with obstacles. The origin of rotation when the gas is ejected from an orifice in a cavity is also investigated.     

海浪、地震对于轴对称海工建筑物的作用

当今海洋工程中广泛采用各种竖直轴对称结构,诸如贮油罐、采油平台等等。本文论述一种计算上述结构物与其周围的海洋相互作用的极其有效的计算方法,着重讨论如下几个问题的数值计算:1.作用在固定建筑物上的波浪荷载和爬高;2.作用在浮式建筑物上的附加质量、阻尼系数以及激荡力(扰动力);3.因地震荷载作用而产生的基面剪切力和转动力矩;4.浮式建筑物在波浪作用下的运动状态。本文所述方法是一种以轴对称格林函数为基础的边界单元法,并利用结构物本身的轴对称性提出了适用于小型计算机上进行快速有效计算的计算程序。最后,作者给出了不同荷载对于伸出水面的截圆锥式建筑物的作用的计算结果。     

刚性轴对称狭窄血管内压强及其梯度的研究

基于对Karman-Pohlhausen方法的改进,运用非线性多项式拟合和数值积分,导出了刚性轴对称狭窄管内压强及其梯度的轴向分布,讨论了该分布与雷诺数和狭窄管几何形状之间的关系．结果表明随狭窄度和雷诺数的增加,压强及其梯度在狭窄区域的振荡会迅速加剧,并逐渐导致舒张区出现负压强．尤其在狭窄的轴向区域变宽时,舒张区的血流状态会受到较大影响．在高雷诺数和重度狭窄时,理论计算与过去的实验结果基本一致．     

空间轴对称问题边界元法的程序处理

弹性力学的空间轴对称问题可以化为两个变量(r,z)的二维问题求解,但又比平面问题略复杂些.在轴对称问题边界元的程序处理上也会相应带来些麻烦.文章具体介绍了作者在调试轴对称问题边界元程序中遇到的一些问题及处理方法,并给出数值结果加以验证.     

横观各向同性轴对称问题的通解

本文按应力求解轴对称问题,以统一的格式导出了一系列有实用价值的通解,其中有的是已有的著名的通解,有的尚未见文献报导.同时证明了各种通解的完备性.     

A Posteriori Error Estimates for Axisymmetric and Nonlinear Problems

We propose and examine the primal and dual finite element method for solving an axially symmetric elliptic problem with mixed boundary conditions. We derive an a posteriori error estimate and generalize the method used for a nonlinear elliptic problem. Finally, an a posteriori error estimate for a nonlinear parabolic problem based on the concept of hierarchical finite element basis functions is introduced.     

Plane-Parallel and Axisymmetric Flows with a Straight Sonic Line

Computational Mathematics and Mathematical Physics - Features of plane-parallel and axisymmetric flows of an ideal (inviscid and non-heat-conducting) gas with a straight sonic line are studied. The...