Smooth boundary integration method in a linear two-dimensional problem of incompressible fluid and solid

This paper presents a new application of a theoretical and computational method of smooth boundary integration which belongs to the methods of boundary integral equations. Smooth integration is not a method of approximation. In its final analytical form, a smooth-kernel integral equation is computerized easily and accurately.

Smooth integration is associated with a “pressure-vorticity” formulation which covers linear problems in elasticity and fluid mechanics. The solution presented herein is essentially the same as that reported in an earlier paper for regular elasticity. The constraint of incompressibility does not cause difficulties in the pressure-vorticity formulation.

The linear fluid mechanics problem formulated and solved in this paper covers Stokes' problem of a slow viscous flow, and has a wider interpretation. The translational inertia forces are incorporated in the linear problem, as in Euler's dynamic theory of inviscid flow. The centrifugal inertia forces are left for the non-linear problem. The linear problem is perceived as a step in solution of the non-linear problems.     

Free boundary problems with nonlinear source terms

Summary The method of lines is used to semi-discretize the non-linear Poisson equation over a domain with a free boundary. The resulting multipoint free boundary problem is solved with a line Gauss-Seidel method which is shown to converge monotonically. The method of lines solution is then shown to converge to the continuous solution of the variational inequality form of the obstacle problem. Some numerical results for the diffusion-reaction equation indicate that the method is applicable to more general free boundary problems for nonlinear elliptic equations.This research was supported by the U.S. Army Research Office under Contract DAAG-79-0145     

The regularized Benjamin-Ono and BBM equations: Well-posedness and nonlinear stability

Nonlinear stability of nonlinear periodic solutions of the regularized Benjamin-Ono equation and the Benjamin-Bona-Mahony equation with respect to perturbations of the same wavelength is analytically studied. These perturbations are shown to be stable. We also develop a global well-posedness theory for the regularized Benjamin-Ono equation in the periodic and in the line setting. In particular, we show that the Cauchy problem (in both periodic and nonperiodic case) cannot be solved by an iteration scheme based on the Duhamel formula for negative Sobolev indices.     

An asymptotic solution to non-linear transmission lines

This paper explores an asymptotic approach to the solution of a non-linear transmission line model. The model is based on a set of non-linear partial differential equations without analytical solution. The perturbations method is used to reduce the system of non-linear equations to a single non-linear partial differential equation, the modified Korteweg–de Vries equation (KdV). By using the Laplace transform, the solution is represented in integral form in terms of Green's functions. The solution for the non-linear case is obtained by means of asymptotic methods. Thus, an approximate explicit analytical solution to the problem is obtained where the errors can be controlled. This allows us to analyze the non-linear behavior of the solution. This kind of information is difficult to obtain by means of numerical methods due to the fact that for large periods of time greater computational resources are required and also accumulated errors increase. For this reason, asymptotic methods have a great importance like a natural complement to numerical methods. Computer simulations support the developments presented.     

Numerical Solution of Some Nonlocal, Nonlinear Dispersive Wave Equations

Summary. We use a spectral method to solve numerically two nonlocal, nonlinear, dispersive, integrable wave equations, the Benjamin-Ono and the Intermediate Long Wave equations. The proposed numerical method is able to capture well the dynamics of the solutions; we use it to investigate the behaviour of solitary wave solutions of the equations with special attention to those, among the properties usually connected with integrability, for which there is at present no analytic proof. Thus we study in particular the resolution property of arbitrary initial profiles into sequences of solitary waves for both equations and clean interaction of Benjamin-Ono solitary waves. We also verify numerically that the behaviour of the solution of the Intermediate Long Wave equation as the model parameter tends to the infinite depth limit is the one predicted by the theory. Received October 28, 1997; revised February 11, 1999; accepted April 7, 1999     

Reduction of quasilinear elliptic equations in cylindrical domains with applications

For certain types of semilinear partial differential equations in a cylindrical domain small bounded solutions are known to lie on a so-called centre manifold of finite dimension. In this paper we prove the result for quasilinear partial differential equations of elliptic and parabolic type. Possible applications include free surface waves and problems in non-linear elasticity. Here we investigate, pars pro toto, the stationary inviscid potential flow in a channel with a free surface and an obstacle at the bottom.     

Triple-deck regularization of weak laminar hydraulic jumps in a two layer shallow water flow

The inviscid, weakly nonlinear, shallow-water limit of the Navier–Stokes equations leads to a hyperbolic conservation law. In certain cases of a two-layer flow the flux-function is non-convex, thus leading to the possibility of shocks (in physical terms hydraulic jumps) violating the Oleinik-entropy criteria, so called non-classical shocks. To rule out the inadmissible shocks their internal structure is studied, based on an asymptotic approach consistent with the Navier–Stokes equations. This leads to a triple-deck problem with a novel, non-linear interaction equation in the form of a forced, extended KdV-equation. The limit of vanishing and weak influence of the displacement effect is studied analytically, and in addition representative numerical solutions of the full problem are presented. Of particular interest is a solution, which has a pronounced, almost vanishing minimum in he wall shear. Its local structure is studied in some detail. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two layer transient water waves over a viscoelastic ocean bed

A two-layer analysis of the transient development of water waves over a viscoelastic ocean bed is presented here. This is a two-dimensional initial value investigation of the transient development of surface and internal wave motions governed by harmonic pressure distribution acting on the free surface in an inviscid liquid over a viscous and elastic ocean bed. The equations of motion and the equation of continuity are described in terms of velocity potential and stream functions. The solution of this problem is obtained by using Laplace and Fourier transform methods. Limiting case of the layers to obtain free surface elevation is also presented.     

Resonant Interactions between Vortical Flows and Water Waves. Part II: Shallow Water

Any weak, steady vortical flow is a solution, to leading order, of the inviscid fluid equations with a free surface, so long as this flow has horizontal streamlines coinciding with the undisturbed free surface. This work considers the propagation of long irrotational surface gravity waves when such a vortical flow is present. In particular, when the vortical flow and the irrotational surface waves are both periodic and have comparable length scales, resonant interactions can occur between the various components of the flow. The interaction is described by two coupled Korteweg-de Vries equations and a two-dimensional streamfunction equation.     

A FLUID-PARTICLE MODEL WITH ELECTRIC FIELDS NEAR A LOCAL MAXWELLIAN WITH RAREFACTION WAVE

The paper is concerned with time-asymptotic behavior of solution near a local Maxwellian with rarefaction wave to a fluid-particle model described by the Vlasov-Fokker-Planck equation coupled with the compressible and inviscid fluid by Euler-Poisson equations through the relaxation drag frictions, Vlasov forces between the macroscopic and microscopic momentums and the electrostatic potential forces. Precisely, based on a new micro-macro decomposition around the local Maxwellian to the kinetic part of the fluid-particle coupled system, which was first developed in [16], we show the time-asymptotically nonlinear stability of rarefaction wave to the one-dimensional compressible inviscid Euler equations coupled with both the Vlasov-Fokker-Planck equation and Poisson equation.     

A simple and robust boundary treatment for the forced Korteweg–de Vries equation

In this paper, we propose a simple and robust numerical method for the forced Korteweg–de Vries (fKdV) equation which models free surface waves of an incompressible and inviscid fluid flow over a bump. The fKdV equation is defined in an infinite domain. However, to solve the equation numerically we must truncate the infinite domain to a bounded domain by introducing an artificial boundary and imposing boundary conditions there. Due to unsuitable artificial boundary conditions, most wave propagation problems have numerical difficulties (e.g., the truncated computational domain must be large enough or the numerical simulation must be terminated before the wave approaches the artificial boundary for the quality of the numerical solution). To solve this boundary problem, we develop an absorbing non-reflecting boundary treatment which uses outward wave velocity. The basic idea of the proposing algorithm is that we first calculate an outward wave velocity from the solutions at the previous and present time steps and then we obtain a solution at the next time step on the artificial boundary by moving the solution at the present time step with the velocity. And then we update solutions at the next time step inside the domain using the calculated solution on the artificial boundary. Numerical experiments with various initial conditions for the KdV and fKdV equations are presented to illustrate the accuracy and efficiency of our method.     

Exact Multisoliton Solution of the Reduced Maxwell-Bloch Equations of Non-linear Optics

An exact N soliton solution of the reduced Maxwell-Bloch equationsof non-linear optics is given and proved. These equations playan important dual role in non-linear optics and contain as aspecial case the sharp line self-induced transparency equationsand the sine-Gordon equation. Simple asymptotic forms of thesolutions are given.     

A bidimensional phase-field model with convection for change phase of an alloy

The article analyzes a two-dimensional phase-field model for a non-stationary process of solidification of a binary alloy with thermal properties. The model allows the occurrence of fluid flow in non-solid regions, which are a priori unknown, and is thus associated to a free boundary value problem for a highly non-linear system of partial differential equations. These equations are the phase-field equation, the heat equation, the concentration equation and a modified Navier-Stokes equations obtained by the addition of a penalization term of Carman-Kozeny type which accounts for the mushy effects. A proof of existence of weak solutions for such system is given. The problem is firstly approximated and a sequence of approximate solutions is obtained by Leray-Schauder fixed point theorem. A solution is then found by using compactness argument.     

关于广义Benjamin－Ono方程的整体适定性

对Ｋ＞α＋３／２的广义Ｂｅｎｊａｍｉｎ－Ｏｎｏ方程，证明了当初值为小初值时相应的非线性初值问题的整体解的存在性以及唯一性．还得到了该整体在Ｌ２的意义下的渐近估计．     

On Schwarz Alternating Methods for the Subsonic Full Potential Equation

The Schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. The full potential equation is derived from the Navier–Stokes equations assuming the fluid is compressible, inviscid, irrotational and isentropic. It is being used by the aircraft industry to model flow over an airfoil or even an entire aircraft. This paper shows that the additive and multiplicative versions of the Schwarz alternating method, when applied to the full potential equation in three dimensions, converge to the true solution geometrically. The assumptions are that the initial guess and the true solution are everywhere subsonic. We use the convergence proof by Tai and Xu and modify it for certain closed convex subsets.     

Exact self-similar solution of a certain equation of non-linear diffusion with dissipation

Using the renormalization-group method, an exact solution of the equation of non-linear diffusion with dissipation of the Cahn–Hilliard type is obtained. When finding an admissible form of the self-similar solution, only dimensionality arguments and the property of renormalization invariance are used. The form of the initial and boundary conditions, consistent with the non-linear equation cannot be given in advance, but is found from the solution of the renormalization-group equations.     

The contact angle in inviscid fluid mechanics

We show that in general, the specification of a contact angle condition at the contact line in inviscid fluid motions is incompatible with the classical field equations and boundary conditions generally applicable to them. The limited conditions under which such a specification is permissible are derived; however, these include cases where the static meniscus is not flat. In view of this situation, the status of the many ‘solutions’ in the literature which prescribe a contact angle in potential flows comes into question. We suggest that these solutions which attempt to incorporate a phenomenological, but incompatible, condition are in some, imprecise sense ‘weak-type solutions’; they satisfy or are likely to satisfy, at least in the limit, the governing equations and boundary conditions everywhere except in the neighbourhood of the contact line. We discuss the implications of the result for the analysis of inviscid flows with free surfaces.     

The inviscid limit for an inflow problem of compressible viscous gas in presence of both shocks and boundary layers

In this paper, we study the inviscid limit problem for the Navier-Stokes equations of one-dimensional compressible viscous gas on half plane. We prove that if the solution of the inviscid Euler system on half plane is piecewise smooth with a single shock satisfying the entropy condition, then there exist solutions to Navier-Stokes equations which converge to the inviscid solution away from the shock discontinuity and the boundary at an optimal rate of ε¹ as the viscosity ε tends to zero.     

Global Solutions and Their Large-time Behavior of Cauchy Problem for Equations of Deep Water Type

We consider the large time behavior of the global solution of the Cauchy problem for the equation of Benjamin-Ono type. A series of large-time global estimates for Cauchy problem arc constructed. By means of the obtained global estimates uniformly for 0 ≤ t < ∞, the attractors of the Cauchy problem for the mentioned nonlinear equations are considered. And also the dimensions of the global attractor are estimated.     

Five-wave classical scattering matrix and integrable equations

We study the five-wave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type u?u/?x. Our aim is to find the most general nontrivial form of the dispersion relation ω(k) for which the five-wave interaction scattering matrix is identically zero on the resonance manifold. As could be expected, the matrix in one dimension is zero for the Korteweg-de Vries equation, the Benjamin-Ono equation, and the intermediate long-wave equation. In two dimensions, we find a new equation that satisfies our requirement.