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.     

Nonlinear aspects of high Reynolds number channel flows

This paper considers the flow in a two-dimensional channel at high Reynolds number, with wall deformations which can lead to flow separation. An asymptotic model is proposed by using the successive complementary expansion method with generalized asymptotic expansions. In particular, the model emphasizes the asymmetry of the channel geometry by introducing a change of variables. It is shown that the model is more general than the models developed with the method of matched asymptotic expansions. Comparisons with Navier–Stokes solutions show that the model is well founded and enables us to treat original problems.     

Approximate Solutions of the Cahn-Hilliard Equation via Corrections to the Mullins-Sekerka Motion

We develop an alternative method to matched asymptotic expansions for the construction of approximate solutions of the Cahn-Hilliard equation suitable for the study of its sharp interface limit. The method is based on the Hilbert expansion used in kinetic theory. Besides its relative simplicity, it leads to calculable higher order corrections to the interface motion. An erratum to this article is available at .     

Effect of the medium inhomogeneity on the evolution equation of plane shock waves

Using the method of matched asymptotic expansions, solutions of boundary-value problems of shock deformation of nonlinear elastic weakly inhomogeneous semispaces are obtained. It is shown that nonlinearity of the model and various properties of the inhomogeneity lead to new evolution equations in the frontal areas of longitudinal and transverse shock waves, the transition to which is due to transformations of all independent coordinates of the problem.     

On axisymmetric motion of an incompressible elastic medium under impact loading

The method of matched asymptotic expansions was employed to obtain approximate solutions to the one-dimensional boundary-value problems of nonlinear dynamic elasticity theory of impact loading on the surface of a cylindrical cavity of an incompressible medium that causes antiplane motion or torsion of the medium. The expansion of the solution in the near-front region is based on solutions of evolution equations different from the equations for quasi-simple waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 144–151, November–December, 2006.     

Vibrations of highly prestressed anisotropic plates via a numerical-perturbation technique

The method of matched asymptotic expansions is used to reduce the problem of the transverse vibrations of a highly prestressed anisotropic plate into the simpler problem of the vibration of an anisotropic membrane with modified boundary conditions that account for the bending effects. In the absence of an exact solution the membrane problem can be solved by any well-known numerical technique. The numerical-perturbation results for a clamped circular plate with rectangular orthotropy and a uniform tensile stress applied on its boundary show an excellent correlation with finite-element solutions for the original problem. Furthermore, the solutions obtained for annular plates form the basis for solutions to problems involving near-annular plates.     

Transient natural convection between horizontal concentric cylinders: Natural convection between concentric cylinders

The initial phase of transient natural convection between two horizontal concentric cylinders is investigated. Short-time solutions to the momentum and energy equations are obtained using the method of matched asymptotic expansions. The solutions for the velocity and temperature are expressed in terms of three expansions reflecting the existence of three distinct regions in the initial flow field, and four different patterns of motion may be distinguished, corresponding to four different types of thermal boundary conditions.     

Diffraction by a rigid barrier with a soft or perfectly absorbent end face

The reduction of noise levels in the shadow region of a rigid barrier of finite thickness is considered when the end face of the barrier is lined with either a soft or a perfectly absorbent material. Solutions, obtained by the method of matched asymptotic expansions, are given for both a semi-infinite barrier and for a finite length barrier placed on a rigid plane. Comparisons are made with existing solutions for barriers that have a rigid end face.     

Steady thermocapillary-buoyant convection in a shallow annular pool. Part 1: Single layer fluid

This paper examines the steady thermocapillarybuoyant convection in a shallow annular pool subjected to a radial temperature gradient. A matched asymptotic theory is used to obtain the asymptotic solutions of the flow and thermal fields in the case of small aspect ratios,which is defined as the ratio of the layer thickness to the gap width. The flow domain is divided into the core region away from the cylinder walls and two end regions near each cylinder wall. Asymptotic solutions are obtained in the core region by solving the core and end flows separately and then joining them through matched asymptotic expansions. For the system of silicon melt,the asymptotic solutions are compared with the results of numerical simulations. It is found that the two kinds of solutions have a good agreement in the core region for a small aspect ratio. With the increase of aspect ratio,the applicability of the present asymptotic solutions decreases gradually.     

An asymptotic expansion for the eigenvalues arising in perturbations about the blasius solution

Summary A further term is found in the asymptotic expansion suggested by Stewartson for the eigenvalues arising in perturbations about the Blasius solution. The method employed is that of matched asymptotic expansions, and good agreement is obtained with the numerical results of Libby.     

On Some Extension of Center Manifold Method to Functional Differential Equations with Oscillatory Decreasing Coefficients and Variable Delays

We propose an asymptotic integration method for certain class of functional differential systems. This class includes the delay differential equations with oscillatory decreasing coefficients and variable delays that are close to constants at infinity. Both the ideas of the centre manifold theory and the averaging method together with some classical asymptotic theorems are used to construct the asymptotics for solutions. We illustrate the asymptotic integration method by constructing the asymptotics for solutions of scalar differential equation with variable delay.     

旋转圆盘受均布压力作用时的摄动分析

本文对中心带轴匀速旋转圆盘在均布压力作用下的受力及变形问题进行了摄动分析，得到了受载条件为Ｊ＾２／ＪＨ＜＜１时，圆盘应力及形变的渐近解。其形变渐近解为Ｗｈｉｔａｋｅｒ函数形式比较表明，渐近解与Ｃｈｅｂｙｓｈｅｖ数值解有较好的一致性。     

On the boundary layer methods

In this paper, the defect of the traditionary boundary layer methods (including the method of matched asymptotic expansions and the method of Visik-Lyusternik) is noted, from those methods we can not construct the asymptotic expansion of boundary layer term substantially. So the method of multiple scales is proposed for constructing the asymptotic expansion of boundary layer term, the reasonable result is obtained. Furthermore, we compare this method with the method used by Levinson, and find that both methods give the same asymptotic expansion of boundary layer term, but our method is simpler.Again, we apply this method to study some known works on singular perturbations. The limitations of those works have been noted, and the asymptotic expansion of solution is constructed in general condition.     

Laws of Incompressible-Fluid Turbulent Near-Wall Flows Following from the Limiting Asymptotic Properties

The averaged layered turbulent wall flows of an incompressible fluid are considered for an arbitrary wall roughness and a friction coefficient tending to zero. It is assumed that in the asymptotic limit the flow may be divided into two regions: an outer, only slightly disturbed region, and a thin wall region. A similarity law for the velocity disturbances in the outer region is established for cylindrical tubes with an arbitrary cross section. Using the asymptotically matched solutions in the inner and outer regions, the existence of a logarithmic velocity profile in the matching zone is proved. On the basis of a unique asymptotic approach, the existing semi-empirical solutions for tube flows are brought into consistency. A number of semi-empirical formulas for the velocity profiles, which refine and expand the range of applicability of existing relations, are proposed.     

Oscillatory Convection Onset in a Porous Rectangle with Non-analytical Corners

The analytical theory on Darcy–Bénard convection is dominated by normal-mode approaches, which essentially reduce the spatial order from four to two. This paper goes beyond the normal-mode paradigm of convection onset in a porous rectangle. A handpicked case where all four corners of the rectangle are non-analytical is therefore investigated. The marginal state is oscillatory with one-way horizontal wave propagation. The time-periodic convection pattern has no spatial periodicity and requires heavy numerical computation by the finite element method. The critical Rayleigh number at convection onset is computed, with its associated frequency of oscillation. Snapshots of the 2D eigenfunctions for the flow field and temperature field are plotted. Detailed local gradient analyses near two corners indicate that they hide logarithmic singularities, where the displayed eigenfunctions may represent outer solutions in matched asymptotic expansions. The results are validated with respect to the asymptotic limit of Nield (Water Resour Res 11:553–560, 1968).

     

Response of a non-linear system with strong damping to multifrequency excitations

Summary In this paper, a new asymptotic calculation method is presented for a class of nonlinear nonautonomous vibration systems with multiple external periodic interferences. Simple calculation formulae of an asymptotic solution for resonance and off-resonance vibration are derived. The paper is concerned with a vibration system representing a class of nonlinear oscillators. Consequently, the calculation of a class of nonlinear oscillators is routinized. Two different Duffing equations are verified which shows that the results are completely in accordance with the solutions of references [3, 4, 6]. The derivation of solutions of Duffing equations becomes easier and simpler. In addition, some errors in reference [6] are pointed out.     

Non-Isothermal Lava Flows over a Conical Surface

Using the equations of a non-isothermal thin layer of viscous fluid with an exponential dependence of the viscosity on temperature, a family of hydrodynamic models of a cooling lava flow over a conical surface in the presence of mass supply is constructed. These models correspond to asymptotically different rates of heat exchange with the ambient medium. The evolution of the free-surface shape and the temperature fields is investigated numerically for a stationary mass supply. Using the matched asymptotic expansions method, solutions valid both near and very far from the mass supply region are constructed. The solutions obtained are compared with known analytical solutions for isothermal flow.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 62–75.Original Russian Text Copyright © 2005 by Osiptsov.     

Structure verticale des ondes longues océaniques en présence de points tournants

An asymptotic analysis is presented to investigate the vertical structure of long ocean waves in presence of two turning points and a sharp density interface. After implementing the appropriate matching procedure on either side of these points we derive generally matched solutions, even for a very thin density interface.     

Asymptotic analysis of a 3D elasticity problem for a radially inhomogeneous transversally isotropic hollow cylinder

The method of asymptotic integration of equations of elasticity [1] is used to study the behavior of the solution of a 3D elasticity problem for a radially inhomogeneous transversally isotropic hollow cylinder of small thickness. Under the assumption that the load is sufficiently smooth, the asymptotic method [1] is used to construct inhomogeneous solutions. An algorithm for constructing exact particular solutions of the equilibrium equations is given for loads of specific types in the case where the cylinder lateral surface is loaded by forces polynomially depending on the axial coordinate. Then the homogeneous solutions are constructed. The asymptotic expansions of homogeneous solutions are obtained, and the above analysis is used to explain the character of the stress-strain state.