Solitary waves in initially stressed thin elastic tubes

In the present work, we study the propagation of non-linear waves in an initially stressed thin elastic tube filled with an inviscid fluid. Considering the physiological conditions of the arteries, in the analysis, the tube is assumed to be subjected to a uniform inner pressure P₀ and an axial stretch ratio λ_z. It is assumed that due to blood flow, a finite dynamical displacement field is superimposed on this static field and, then, the non-linear governing equations of the elastic tube are obtained. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the longwave approximation is investigated. It is shown that the governing equations reduce to the Korteweg-deVries equation which admits a solitary wave solution. It is observed that the present model equations give two solitary wave solutions. The results are also discussed for some elastic materials existing in the literature.     

Non-linear wave modulation in a prestressed viscoelastic thin tube filled with an inviscid fluid

In the present work, the propagation of weakly non-linear waves in a prestressed thin viscoelastic tube filled with an incompressible inviscid fluid is studied. Considering that the arteries are initially subjected to a large static transmural pressure P₀ and an axial stretch λ_z and, in the course of blood flow, a finite time-dependent displacement is added to this initial field, the governing non-linear equation of motion in the radial direction is obtained. Using the reductive perturbation technique, the propagation of weakly non-linear, dispersive and dissipative waves is examined and the evolution equations are obtained. Utilizing the same set of governing equations the amplitude modulation of weakly non-linear and dissipative but strongly dispersive waves is examined. The localized travelling wave solution to these field equations are also given.     

Non-linear combined axial and torsional shear wave propagation in an incompressible hyperelastic solid

Finite amplitude combined axial and torsional shear wave propagation in an incompressible isotropic hyperelastic solid is considered. When the strain energy function of the solid is a non-linear function offI₁,− 3) and (I₂− 3), where I₁, and I₂are the first and second basic invariants of the left Cauchy-Green tensor, the two second order partial differential equations governing the propagation of the axial and torsional waves are non-linear and coupled. These two coupled equations are equivalent to a hyperbolic system of first order partial differential equations and a modification of the MacCormack finite difference scheme is used to obtain numerical solutions of this system. Numerical results, which show the effect of the coupling, are presented for boundary-initial value problems of propagation into initially unstressed and initially stressed regions at rest.     

Validation of a 2-D semi-coupled numerical model for fluid–structure–seabed interaction

A 2-D semi-coupled model PORO-WSSI 2D (also be referred as FSSI-CAS 2D) for the Fluid-Structure-Seabed Interaction (FSSI) has been developed by employing RANS equations for wave motion in fluid domain, VARANS equations for porous flow in porous structures; and taking the dynamic Biot's equations (known as "u − p" approximation) for soil as the governing equations. The finite difference two-step projection method and the forward time difference method are adopted to solve the RANS, VARANS equations; and the finite element method is adopted to solve the "u − p" approximation. A data exchange port is developed to couple the RANS, VARANS equations and the dynamic Biot's equations together. The analytical solution proposed by Hsu and Jeng (1994) and some experiments conducted in wave flume or geotechnical centrifuge in which various waves involved are used to validate the developed semi-coupled numerical model. The sandy bed involved in these experiments is poro-elastic or poro-elastoplastic. The inclusion of the interaction between fluid, marine structures and poro-elastoplastic seabed foundation is a special point and highlight in this paper, which is essentially different with other previous coupled models The excellent agreement between the numerical results and the experiment data indicates that the developed coupled model is highly reliablefor the FSSI problem.     

Cubic non-linearity and longitudinal surface solitary waves

It is shown that for some seismic media both quadratic and cubic non-linearities should be taken into account in the governing equation for longitudinal waves. The new equation is obtained to account for non-linear surface waves in a medium surrounding a non-linearly elastic rod. Exact solutions of the equation allow us to describe simultaneous propagation of tensile and compressive localized strain waves. Various interactions between these waves give rise to both the multi-bump and “Mexican hat” localized wave structures closer to the surface waves recently observed in experiments.     

The Riemann problem for a hyperbolic model of two-phase flow in conservative form

This article is to continue the present author's work (International Journal of Computational Fluid Dynamics (2009) 23 (9), 623–641) on studying the structure of solutions of the Riemann problem for a system of three conservation laws governing two-phase flows. While existing solutions are limited and found quite recently for the Baer and Nunziato equations, this article presents the first instance of an exact solution of the Riemann problem for two-phase flow in gas–liquid mixture. To demonstrate the structure of the solution, we use a hyperbolic conservative model with mechanical equilibrium and without velocity equilibrium. The Riemann problem solution for the model equations comprises a set of elementary waves, rarefaction and discontinuous waves of various types. In particular, such a solution treats both the wave structure and the intermediate states of the two-phase gas–liquid mixture. The resulting exact Riemann solver is fully non-linear, direct and complete. On this basis then, we use locally the exact Riemann solver for the two-phase flow in gas–liquid mixture within the framework of finite volume upwind Godunov methods. In order to demonstrate the effectiveness and accuracy of the proposed solver, we consider a series of test problems selected from the open literature and compare the exact and numerical results by using upwind Godunov methods, showing excellent oscillation-free results in two-phase fluid flow problems.     

孤立波与多孔介质结构物的非线性相互作用

基于精确至Ｏ（εμ＾２,μ＾４）的多孔介质无压渗流模型方程和均匀流体质波动的Ｂｏｕｓｓｉｎｅｓｑ方程,本文对孤立波与多孔介质结构物的相互作用了较系统的数值实验。控制方程采用基于有限差分方程离散,在时域上采用了预估－校正方法进行了时间积分。在求解演化方程的过程中,引入“内迭代”过程实现流体域和多孔介质交界面的连接条件。结果表明孤立波在多孔介质上的反射与在不可渗透的界面上的反射类似,形成反向的孤立波但     

The generalized Boussinesq equations and obliquely interacting solitary waves in a stratified fluid

In this paper, on the basis of Boussinesq’s shallow water theory, we establish the basic equations governing the motion of a stratified fluid, a kind of the generalized Boussinesq equations. And then by way of them, we study the weak interaction of two pairs of obliquely colliding solitary waves, give the second-order approximate solutions for wave profiles and maximum amplitudes, as well as conclude that when the included angle between the directions of propagation of impinging solitary waves is less than 120°, the effect of oblique interaction is stronger than that of the head-on one, but when the angle concerned is greater than 120°, the former is slightly weaker than the latter.     

Solution of Riemann problem for dusty gas flow

A direct approach is used to solve the Riemann problem for a quasilinear hyperbolic system of equations governing the one dimensional unsteady planar flow of an isentropic, inviscid compressible fluid in the presence of dust particles. The elementary wave solutions of the Riemann problem, that is, shock waves, rarefaction waves and contact discontinuities are derived and their properties are discussed for a dusty gas. The generalised Riemann invariants are used to find the solution between rarefaction wave and the contact discontinuity and also inside rarefaction fan. Unlike the ordinary gasdynamic case, the solution inside the rarefaction waves in dusty gas cannot be obtained directly and explicitly; indeed, it requires an extra iteration procedure. Although the case of dusty gas is more complex than the ordinary gas dynamics case, all the parallel results for compressive waves remain identical. We also compare/contrast the nature of the solution in an ordinary gasdynamics and the dusty gas flow case.     

Nonlinear travelling waves in a hyperelastic rod composed of a compressible Mooney-Rivlin material

In this paper, we study strongly nonlinear axisymmetric waves in a circular cylindrical rod composed of a compressible Mooney-Rivlin material. To consider the travelling wave solutions for the governing partial differential system, we first reduce it to a nonlinear ordinary differential equation. By using the bifurcation theory of planar dynamical systems, we show that the reduced system has seven periodic annuluses with different boundaries which depend on four parameters. We further consider the bifurcation behavior of the phase portraits for the reduced one-parameter vector fields when other three parameters are fixed. Corresponding to seven different periodic annuluses, we obtain seven types of travelling wave solutions, including solitary waves of radial contraction, solitary waves of radial expansion, solitary shock waves of radial contraction, solitary shock waves of radial expansion, periodic waves and two types of periodic shock waves. These are physically acceptable solutions by the governing partial differential system. The rigorous parameter conditions for the existence of these waves are given.     

Non-linear temporal dynamics of two-mode interactions of magnetized flow

This paper considers, in the frame work of the model of two superposed layers of viscous-potential incompressible magnetic fluids, the problem on formation of resonant waves of two modes on the interface between fluids that arisen as a result of second-harmonic resonance. The fluids moving with uniform velocities parallel to their interface are stressed by a tangential magnetic field. The analysis includes the linear, as well as the non-linear effects where the analytical solutions are constructed using the method of multiple scales, in both space and time, and hence the solvability conditions correspond to the uniform (convergent) solutions are obtained. The solvability conditions are then exploited to derive a more general system of non-linear partial differential equations with complex coefficients governing the amplitudes of the resonant waves. These equations are examined for solutions corresponding to sinusoidal wavetrains consequently different kinds of instabilities are demonstrated. The stability criterion in each case is derived and discussed both analytically and graphically.     

Cubically non-linear effects of plane waves in isotropic soft solid materials

In this paper we analyze mathematical properties of an isotropic soft solid model which is characterized by three elastic constants. The model was proposed to interpret measurements of weakly non-linear shear waves in gel-like and tissue-like media. In our analysis we are particularly interested in third order non-linear terms. We present for the first time the full equations of elastodynamics, as well as the equations for plane waves for this model, with cubically non-linear terms. Next, the interaction coefficients for non-linear interactions of three plane waves to produce the fourth wave are explicitly calculated. These coefficients show which of the three waves interact with each other and determine how strong the effect of interaction is on the produced fourth wave. It turns out that these coefficients are expressed in terms of some combinations of three elastic constants. The obtained results can be helpful in experimental determination of elastic constants which describe the model.     

Discontinuous solutions of the shallow water equations on a rotatable attracting sphere

The shallow water equations on a rotatable attracting sphere represent a system of hyperbolic equations on a compact manifold. These equations are derived in a spherical coordinate system from the integral laws of mass and total momentum conservation with account for the Coriolis and centrifugal forces. An analysis of the stability of discontinuous solutions with discontinuous waves and contact discontinuities is made using the closing law of total energy conservation, which represents a convex extension of the basic conservation-law system. The classes of stationary, one-dimensional (latitude-dependent only) exact solutions with contact discontinuities and discontinuous waves are constructed. Within the framework of the one-dimensional equations the test problem of wave flows resulting from the simultaneous break of two dams confining a fluid at rest in the vicinities of the poles is numerically modeled.     

浅水孤立波在三维浮体上的绕射

浅水域中非线性水波运动的控制方程通常是经过深度平均的Boussinesq方程。然而,这一方程在浮体近旁或水下障碍物附近不再适用,在这些区域,流动在水深方向的变化不容忽略,本文应用匹配渐近展开法和边缘层(edge layer)思想,建立了浅水弱非线性波与三维浮体相互作用的数学模型,作为算例,求解了浅水孤立波在垂直圆柱形浮体上的绕射.本方法可以推广到波在一般浮体上绕射的情况。     

Theory of non-linear wave propagation with application to the interaction and inverse problems

The propagation of non-linear deformation waves in a dissipativc medium is described by a unified asymptotic theory, making use of wave front kinematics and the concepts of progressive waves. The mathematical models are derived from the theories of thermoclasticity or viscoclasticity taking into account the geometric and physical non-linearities and dispersion. On the basis of eikonal equations for the associated linear problem the transport equations of the nth order are obtained. In the multidimensional case the method of matched separation of initial equations is proposed. The interaction problems which occur in head-on collisions and in reflection from boundaries or interfaces are analyzed. Conditions are also studied when the interaction of non-linear waves does not take place. The inverse problem of determining materials properties according to pulse shape changes is discussed.     

Classification of Traveling Waves for a Class of Nonlinear Wave Equations

We classify the weak traveling wave solutions for a class of one-dimensional non-linear shallow water wave models. The equations are shown to admit smooth, peaked, and cusped solutions, as well as more exotic waves such as stumpons and composite waves. We also explain how some previously studied traveling wave solutions of the models fit into this classification.     

On solitary waves in plasma

From the equations of hydrodynamics and electrodynamics, a system of coupled nonlinear equations governing the propagation of plane electromagnetic waves in a collisionless electron plasma is obtained. It is shown that solitary wave solutions exists for both the longitudinal and transverse components of the electromagnetic field. It is found that the velocity of the electromagnetic vector solitary wave depends on the amplitudes of all components of the field linearly. The relations among the longitudinal and transverse components that support the solitary waves are determined for different values of plasma temperature. It is shown that while transverse solitary waves cannot exist, except when they are supported by longitudinal waves, the latter can exist by themselves. The interaction of the longitudinal solitary waves with each other is studied and an upper bound on the amplitudes of these waves is obtained. A Lagrangian density function and two conservation laws for the longitudinal wave equation are found. Frequency spectra of the solitary waves are calculated and their low frequency content is emphasized.     

Nonlinear Stability for Multidimensional Fourth-Order Shock Fronts

We consider the question of stability for planar wave solutions that arise in multidimensional conservation laws with only fourth-order regularization. Such equations arise, for example, in the study of thin films, for which planar waves correspond to fluid coating a pre-wetted surface. An interesting feature of these equations is that both compressive, and undercompressive, planar waves arise as solutions (compressive or undercompressive with respect to asymptotic behavior relative to the un-regularized hyperbolic system), and numerical investigation by Bertozzi, Münch, and Shearer indicates that undercompressive waves can be nonlinearly stable. Proceeding with pointwise estimates on the Green's function for the linear fourth-order convection–regularization equation that arises upon linearization of the conservation law about the planar wave solution, we establish that under general spectral conditions, such as appear to hold for shock fronts arising in our motivating thin films equations, compressive waves are stable for all dimensions d≧2 and undercompressive waves are stable for dimensions d≧3. (In the special case d=1, compressive waves are stable under a very general spectral condition.) We also consider an alternative spectral criterion (valid, for example, in the case of constant-coefficient regularization), for which we can establish nonlinear stability for compressive waves in dimensions d≧3 and undercompressive waves in dimensions d≧5. The case of stability for undercompressive waves in the thin films equations for the critical dimensions d=1 and d=2 remains an interesting open problem.