Convergence of the Cahn-Hilliard equation to the Hele-Shaw model

We prove that level surfaces of solutions to the Cahn-Hilliard equation tend to solutions of the Hele-Shaw problem under the assumption that classical solutions of the latter exist. The method is based on a new matched asymptotic expansion for solutions, a spectral analysis for linearizd operators, and an estimate for the difference between the true solutions and certain approximate ones.     

Quasiperiodic solutions of the Navier-Stokes equations induced by the quasiperiodic shape of the boundaries of a two-dimensional flow domain

Steady quasiperiodic solutions of the Navier-Stokes equations in an infinite two-dimensional layer with quasiperiodic boundaries are obtained on the Reynolds number range 0 < Re* < 200. The calculations are performed using a spectral-difference method based on the representation of the quasiperiodic solutions in the form of convergent double Fourier series. The properties of these solutions and the distinctive features of their spectra are studied and their fundamental differences from periodic solutions are shown. The possibility of applying the quasiperiodic solutions for modeling flows in fractal layers is discussed.     

An application of the operational calculus to the equations of the Rayleigh Problem in MHD

Summary The operational calculus in the form developed by Van der Pol and Bremmer is applied to obtain some solutions for the equations of the Rayleigh Problem in MHD. The diffusivities in these solutions can have arbitrary values. A limiting case of the solutions is studied, and it is shown that the solutions have some advantages as compared to the integrals of Chang and Yen, and of Hide and Roberts. Finally an example of Ludford is used to verify that the integrals obtained here will make their appearance in the solution of some boundary value problems.MHD stands for magnetohydrodynamics.     

The analyticity of solutions of the Stefan problem

The Stefan problem of a semi-infinite body with arbitrarily prescribed initial and boundary conditions is studied. One of the objectives of the paper is to investigate the analyticity of the solutions. For this purpose, the prescribed initial and boundary conditions are considered to be series of fractional powers of their arguments. It is found that the exact solutions of the problem for various forms of the initial and boundary conditions can be established in series of parabolic cylinder functions and time t. Existence and convergence of the series solutions are studied and proved. The present solutions include the known exact solutions as special cases. On the basis of the present solutions, the question of the analyticity of solutions of the Stefan problem, raised by Rubinstein in his book, can be answered. Conditions for analyticity of the solutions with various initial and boundary conditions are fully discussed.     

Unidirectional flows of binary mixtures within the framework of the Oberbeck–Boussinesq model

Existence conditions are investigated for the unidirectional flows of binary mixtures describable by the equations of motion in the Oberbeck–Boussinesq approximation with account for the thermodiffusion effect. Possible solutions are classified, known and novel classes of exact solutions being distinguished. For the solutions obtained different formulations of boundary-value problems are proposed. Flows between two rigid walls with given heat distribution laws are described.     

On the boundedness and periodicity of the solutions of a certain vector differential equation of third-order

IntroductionWeshallbeconcernedherewiththird-ordersystemofnon-lineardifferentialequationsoftheformX F(X,X)X Ba H(X)=P(t,X,X,X),(l)whereX6R",R"denotestherealn-dimensionalEuclldeanspaceRxRx'xR(nfactors),R=(--ac,ac),Fisannxn-matrixfunction,Bisarealnxn-constantsymmetricmatrix,H:R"~R"1andp:RxR"xR"xR"~R".ThedotsindicatCdifferentiationwithrespeCttot.ThefunctionsF,HandParecontinuousforallvaluesoftheirrespeCtivearguments.Moreover,theexistenceandtheuniquenessofthesolutionsof(l)willbeassu…     

Exact solution of the Navier-Stokes equations in a fluid layer between the moving parallel plates

This paper studies exact solutions of the Navier-Stokes equations for a layer between parallel plates the distance between which increases proportionally to the square root of time. A countable set of exact solutions and their derived countable set of continuous families of exact solutions are obtained. It is shown that certain intervals of the Reynolds parameter have two solutions and some of them one solution.     

A Collection of Analytical Solutions for the Flash Equilibrium Calculation Problem

Transport in Porous Media - We describe an interesting family of closed-form solutions for the flash equilibrium calculation problem. These solutions can be used as benchmark solutions for...     

Water Redistribution in Irrigated Soil Profiles: Invariant Solutions of the Governing Equation

Exact solutions for a class of nonlinear partial differential equations modelling soil water infiltration and redistribution in irrigation systems are studied. These solutions are invariant under two-parameter symmetry groups obtained by the group classification of the governing equation. A general procedure for constructing invariant solutions is presented in a way convenient for investigating numerous new exact solutions.     

Qualitative analysis for the solution of Kuramoto-Sivashinsky equation

ＷｅｗｉｌｌｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇＫｕｒａｍｏｔｏ＿Ｓｉｖａｓｈｉｎｓｋｙｅｑｕａｔｉｏｎ［１,２］ｗｉｔｈｏｂｖｉｏｕｓｐｈｙｓｉｃａｌｍｅａｎｉｎｇｓａｎｄｍｅｃｈａｎｉｃａｌｍｏｄｅｌｓ：ｕｔ＋ε４ｕｘ４＋２ｕｘ２＋ｕｕｘ＝０,（１）ｗｉｔｈｔｈｅｉｎｉｔｉａｌｃｏｎｄｉｔｉｏｎ：ｕ（ｘ,０）＝φ（ｘ）,（２）ａｎｄｔｈｅｂｏｕｎｄａｒｙｃｏｎｄｉｔｉｏｎ：ｕ（－ｌ,ｔ）＝ｕ（ｌ,ｔ）＝ｕｘ２（－ｌ,ｔ）＝ｕｘ２（ｌ,ｔ）＝０,（３ａ）ｏｒｔｈｅｂｏｕｎｄ…     

Class of differentially invariant solutions of the submodel of axisymmetric flows of an ideal gas

A class of differentially invariant solutions of a problem with the pressure independent of the radial coordinate is considered for a submodel of steady axisymmetric flows of a polytropic gas. The overdetermined system turns out to be compatible and is integrated. All solutions defining transonic and supersonic flows with a limiting surface are found. These solutions are compared with invariant solutions obtained previously.     

Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

This paper presents two exact explicit solutions for the three dimensional dual-phase lag （DLP） heat conduction equation, during the derivation of which the method of trial and error and the authors＇ previous experiences are utilized. To the authors＇ knowledge, most solutions of 2D or 3D DPL models available in the literature are obtained by numerical methods, and there are few exact solutions up to now. The exact solutions in this paper can be used as benchmarks to validate numerical solutions and to develop numerical schemes, grid generation methods and so forth. In addition, they are of theoretical significance since they correspond to physically possible situations. The main goal of this paper is to obtain some possible exact explicit solutions of the dual-phase lag heat conduction equation as the benchmark solutions for computational heat transfer, rather than specific solutions for some given initial and boundary conditions. Therefore, the initial and boundary conditions are indeterminate before derivation and can be deduced from the solutions afterwards. Actually, all solutions given in this paper can be easily proven by substituting them into the governing equation.     

基于第二类四边形面积坐标的弹性力学纯弯解及MacNeal局限定理的破解

为了提高有限元的性能,弹性力学的解析解(齐次方程的通解)常常可用作有限元的试探函数。然而单元自由度数与完备的直角坐标解析解个数并不匹配,不完备的试函数会导致单元有方向依赖性。利用新型局部自然坐标——第二类四边形面积坐标QACM-II(S,T),给出了平面问题对应任意方向纯弯曲状态的应力函数解析解,即S3和T3的线性组合,并推导出了这两组应力函数对应的应力、应变和位移解析解。之后,利用QACM-II表示的解析解构造了非对称的平面4节点8自由度单元USQ4,该单元可以同时通过常应力/应变分片检验和纯弯测试,从而破解了MacNeal局限定理对平面低阶单元的限制。     

Exact solutions of the kinetic-moment equations of a mixture of monatomic gases

An extension is given of the class of exact solutions of the kinetic-moment equations for a monatomic gas in the absence of external forces [1] to the case of a mixture of monatomic Maxwellian gases with account for external forces. Very simple solutions of this class are obtained which are examples of the normal solutions of the Boltzmann equations in the Chapman-Enskog sense [2]. Conclusions are summarized concerning the applicability of the various methods of solving the Boltzmann equations and their properties, obtained on the basis of an analysis of the indicated exact solutions.The author wishes to thank M. N. Kogan and A. A. Nikol'skii for their interest in the study.     

On the self-induced motion of vortex sheets

An accurate numerical scheme has been devised to study the self-induced motion of an infinitely thin, free vortex sheet of finite span in an unbounded, inviscid, incompressible fluid. The new numerical scheme has been tested against two vortex sheet problems for which exact solutions have also been obtained. The agreement between the numerical and exact solutions is excellent. The scheme has been further tested against two more examples for which analytical solutions for small times were available. Here too the agreement is excellent.     

Resolving Controversies in the Application of the Method of Multiple Scales and the Generalized Method of Averaging

I compare application of the method of multiple scales with reconstitution and the generalized method of averaging for determining higher-order approximations of three single-degree-of-freedom systems and a two-degree-of-freedom system. Three implementations of the method of multiple scales are considered, namely, application of the method to the system equations expressed as second-order equations, as first-order equations, and in complex-variable form. I show that all of these methods produce the same modulation equations.I address the problem of determining higher-order approximate solutions of the Duffing equation in the case of primary resonance. I show that the conclusions of Rahman and Burton that the method of multiple scales, the generalized method of averaging, and Lie series and transforms might lead to incorrect results, in that spurious solutions occur and the obtained frequency–response curves bear little resemblance to the actual response, is the result of their using parameter values for which the neglected terms are the same order as the retained terms. I show also that spurious solutions cannot be avoided, in general, in any consistent expansion and their presence does not constitute a limitation of the methods. In particular, I show that, for the Duffing equation, the second-order frequency–response equation does not possess spurious solutions for the case of hardening nonlinearity, but possesses spurious solutions for the case of softening nonlinearity. For sufficiently small nonlinearity, the spurious solutions are far removed from the actual response. But as the strength of the nonlinearity increases, these solutions move closer to the backbone and eventually distort it. This is not a drawback of the perturbation methods but an indication of an application of the analysis for parameter values outside the range of validity of the expansion.Also, I address the problem of obtaining non-Hamiltonian modulation equations in the application of the method of multiple scales to multi-degree-of-freedom Hamiltonian systems written as second-order equations in time and how this problem can be overcome by attacking the state-space form of the governing equations. Moreover, I show that application of a variation of the method of Rahman and Burton to multi-degree-of-freedom systems leads to results that do not agree with those obtained with the generalized method of averaging.Contributed by Prof. R.A. Ibrahim.     

A new method for obtaining the exact solutions of the Navier-Stokes (NS) equations and its applications

In this paper we propose a new method for obtaining the exact solutions of the Mavier-Stokes (NS) equations for incompressible viscous fluid in the light of the theory of simplified Navier-Stokes (SNS) equations developed by the first author^[1,2], Using the present method we can find some new exact solutions as well as the well-known exact solutions of the NS equations. In illustration of its applications, we give a variety of exact solutions of incompressible viscous fluid flows for which NS equations of fluid motion are written in Cartesian coordinates, or in cylindrical polar coordinates, or in spherical coordinates. The project supported by National Natural Science Foundation of China.     

Axisymmetric Solutions of the Euler Equations for Polytropic Gases

We construct rigorously a three‐parameter family of self‐similar, globally bounded, and continuous weak solutions in two space dimensions for all positive time to the Euler equations with axisymmetry for polytropic gases with a quadratic pressure‐density law. We use the axisymmetry and self‐similarity assumptions to reduce the equations to a system of three ordinary differential equations, from which we obtain detailed structures of solutions besides their existence. These solutions exhibit familiar structures seen in hurricanes and tornadoes. They all have finite local energy and vorticity with well‐defined initial and boundary values. These solutions include the one‐parameter family of explicit solutions reported in a recent article of ours. (Accepted October 29, 1996)     

Unified complete solutions for the theories of thermoelasticity and poroelasticity

Complete solutions for the basic system of equations governing linear elastic materials with pores are extended to complete solutions for the basic system of equations for thermoelasticity and poroelasticity.     

On the high Weissenberg number limit of the upper convected Maxwell fluid

We consider the equations for time-dependent creeping flow of an upper convected Maxwell fluid in the limit of infinite Weissenberg number. We identify a particular class of solutions which is analogous to potential flow and discuss several examples. We also discuss more general solutions for two-dimensional flow.