Permanent Wave Structures on an Open Two Layer Fluid

There is a significant nonlinear interaction between fast free surface waves and slow interface waves when the group velocity of the free surface waves is the same as the phase velocity of the interface waves. This interaction leads to permanent wave structures consisting of a wave group of permanent envelope on the free surface and a wave of permanent shape on the interface. A theory is developed for periodic permanent wave structures of this type, from which solutions are found numerically. The theory includes all significant quadratic nonlinear interactions between free surface harmonics and interface harmonics, as well as between the interface harmonics themselves. It is found that there is a sequence of forms of differing free surface group structure.     

Nonlinear dynamics of a simplified engine-propeller system

This paper presents a procedure for studying dynamical behaviors of a simplified engine-propeller dynamical system consisting of a number of bodies of plane motions. The equation of motion of the complex system is obtained using the Lagrange equation and solved numerically using the 4th order Runge–Kutta method. Various simulations were performed to investigate the transient and steady state behaviors of the multiple body system while taking into consideration the engine pressure pulsations, nonlinear inertia of moving bodies, and nonlinear aerodynamic load. Sub-harmonics and super harmonics in the steady state responses for different power and propeller pitch settings are obtained using the fast Fourier transform. Numerical simulations indicate that the 1.5 order is the dominant order of harmonics in the steady state oscillatory motion of the crankshaft. The findings and procedure presented in the paper are useful to the aerospace industry in certifying reciprocating engines and propellers. The crankshaft oscillatory velocities obtained from the simplified rigid body model are in good agreement with the experimental data for a SAITO-450 engine and a SOLO propeller at a 6″ pitch setting.     

各向异性压电介质摩擦接触界面波动特性分析

应用Fourier(傅立叶)分析及奇异积分方程技术研究了弹性波与一般压电介质单侧接触界面的相互作用问题.以界面只产生滑移不分离的情况为例,分析了问题的求解思路和过程,给出了弹性波引起界面滑移或分离的条件.以六角晶系极化碳酸钡陶瓷材料和三角晶系石英材料为例,分析了不同材料摩擦接触界面对弹性波入射的不同反应,给出了不同的入射角及不同的外加压力和电场对界面的影响特性.界面产生滑移和分离时,问题的边界条件具有非线性,导致高频谐波出现,以石英材料为例分析了高频成分随外加条件的变化特性.     

Novel methods of pulse wave diagnostics based on compression of a superficial artery

Pulse wave propagation and reflection in the branching systems of viscoelastic tubes and complex resistances as a model of arterial vasculature is considered. Axisymmetric wave motion of a viscous incompressible liquid in the system is investigated in application to the pressure and flow wave propagation in the system. Expression for the input admittance of the system taking into account wave reflection at the end of the tube is obtained. The dependence of the input admittance on the geometrical and mechanical parameters of the system is investigated. It is shown that the parameters of the pressure wave (the modules and phases of different harmonics) at the inlet of the system give information on the state (normal or pathological) of the inner organs. It is shown that after a proper compression (slight or deep) some harmonics can be amplified while the others are attenuated. Based on the theory of resonant harmonics of different inner organs the novel method of pulse wave palpation is proposed. The results are compared to some other methods as separation of the pressure wave into the forward and backward components, estimation of the parameters of the reflected wave and wave-intensity analysis. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An analytic–numerical method for the construction of the reference law of operation for a class of mechanical controlled systems

An analytic–numerical method for the construction of a reference law of operation for a class of dynamic systems describing vibrations in controlled mechanical systems is proposed. By the reference law of operation of a system, we mean a law of the system motion that satisfies all the requirements for the quality and design features of the system under permanent external disturbances. As disturbances, we consider polyharmonic functions with known amplitudes and frequencies of the harmonics but unknown initial phases. For constructing the reference law of motion, an auxiliary optimal control problem is solved in which the cost function depends on a weighting coefficient. The choice of the weighting coefficient ensures the design of the reference law. Theoretical foundations of the proposed method are given.     

Global solvability of the problem on the motion of two fluids without surface tension

Unsteady motion of viscous incompressible fluids is considered in a bounded domain. The liquids are separated by an unknown interface on which the surface tension is neglected. This motion is governed by an interface problem for the Navier-Stokes system. First, a local existence theorem is established for the problem in Hölder classes of functions. The proof is based on the solvability of a model problem for the Stokes system with a plane interface, which was obtained earlier. Next, for a small initial velocity vector field and small mass forces, we prove the existence of a unique smooth solution to the problem on an infinite time interval. Bibliography: 7 titles.     

Hamiltonian model for coupled surface and internal waves in the presence of currents

We examine a two dimensional fluid system consisting of a lower medium bounded underneath by a flatbed and an upper medium with a free surface. The two media are separated by a free common interface. The gravity driven surface and internal water waves (at the common interface between the media) in the presence of a depth-dependent current are studied under certain physical assumptions. Both media are considered incompressible and with prescribed vorticities. Using the Hamiltonian approach the Hamiltonian of the system is constructed in terms of ‘wave’ variables and the equations of motion are calculated. The resultant equations of motion are then analysed to show that wave–current interaction is influenced only by the current profile in the ‘strips’ adjacent to the surface and the interface. Small amplitude and long-wave approximations are also presented.     

Large time behavior of interface solutions to the heat equation with Fisher-Wright white noise

Summary The one-dimensional heat equation driven by Fisher-Wright white noise is studied. From initial conditions with compact support, solutions retain this compact support and die out in finite time. There exist interface solutions which change from the value 1 to the value 0 in a finite region. The motion of the interface location is shown to approach that of a Brownian motion under rescaling. Solutions with a finite number of interfaces are approximated by a system of annihilating Brownian motions.     

On the evolution of two-dimensional patterns in an inviscid liquid sheet

We perform an analysis of the pattern formation for a moving sheet of inviscid fluid. The sheet, which is assumed to have an infinite horizontal extent, moves at some prescribed velocity into a passive surrounding gas. The sheet’s thickness is assumed much smaller than the horizontal scale of the fluid motion. By considering a system that is symmetric with respect to the horizontal planes, long scale asymptotics are used to reduce the full governing equations in three dimensions to a set of three coupled nonlinear partial differential equations for the horizontal components of the velocity field and the height of the interface profile. The interfacial conditions consisting of the kinematic and normal stress balance are incorporated into these evolution equations. Investigations are carried out as function of the sole dimensionless parameter, namely the Weber number. A small amplitude stability analysis around the planar gas–liquid interface reveals that wave patterns in the form of traveling plane waves occur subcritically, and are therefore unstable. The reduced evolution equations are solved numerically for fixed values of the Weber number. Since the reduced system of equations is homogeneous, the wave motion is generated by initial conditions. Five initial conditions have been imposed: one-dimensional rolls, two-dimensional squares, two-dimensional hexagons, two-dimensional ridges, and smooth peaks. The ensuing evolution of the liquid sheet’s shape and corresponding flow fields are described by illustrations of the changes in the sheet’s morphology with time.     

Generation of interfaces for Lotka-Volterra competition-diffusion system with large interaction rates

We discuss the generation and motion of interfaces for Lotka-Volterra competition-diffusion system with large interaction. An asymptotic analysis of solutions shows that the two competing species are segregated and an interface appears on the common boundary of their habitats. The motion of the interface is governed by a free boundary problem. In this paper we establish a mathematical theory for the formation of interfaces (at the initial stage) by using an upper and lower solutions method. In addition, combining our results and a known result for the motion of interfaces (after the initial stage), we obtain some information on the generation and motion of interfaces for given almost any smooth initial data.     

A hydrodynamic model of human cochlea

A two-compartment model of the human cochlea is proposed. When stretched out, the bony spiral tube looks like two chambers separated by a membrane. Both chambers are filled with viscous fluid called perilymph; they communicate with one another via a canal. Sound vibrations enter the cochlea through the oval window and cause periodic change of pressure in the perilymph, which, in turn, causes the membrane to vibrate. The motion of the fluid is described by hydrodynamic equations, which are supplemented with the membrane vibration equation. The equations are linearized in the amplitude of the vibrations, and their solution is sought in the form of Fourier harmonics with a given frequency. To determine the harmonics, a system of linear boundary value problems for ordinary differential equations with variable coefficients is obtained. The numerical solution of this system using finite difference method fails because it involves a large parameter and the problem is close to a singular one. We propose a novel numerical method without saturation that enables us to obtain solutions in a wide range of frequencies up to an arbitrary and controllable accuracy. The computations confirm the Bekesy theory stating that high-frequency sounds cause the membrane to bend near the apex of the cochlea, and low-frequency sounds cause it to bend near the base of the cochlea.     

Visualization in mechanics: the dynamics of an unbalanced roller

It is well known that mechanical engineering students often find mechanics a difficult area to grasp. This article describes a system of equations describing the motion of a balanced and an unbalanced roller constrained by a pivot arm. A wide range of dynamics can be simulated with the model. The equations of motion are embedded in a graphical user interface for its numerical solution in MATLAB. This allows a student's focus to be on the influence of different parameters on the system dynamics. The simulation tool can be used as a dynamics demonstrator in a lecture or as an educational tool driven by the imagination of the student. By way of demonstration the simulation tool has been applied to a range of roller–pivot arm configurations. In addition, approximations to the equations of motion are explored and a second-order model is shown to be accurate for a limited range of parameters.     

Growth rates for the linearized motion of fluid interfaces away from equilibrium

We consider the motion of a two-dimensional interface separating an inviscid, incompressible, irrotational fluid, influenced by gravity, from a region of zero density. We show that under certain conditions the equations of motion, linearized about a presumed time-dependent solution, are wellposed; that is, linear disturbances have a bounded rate of growth. If surface tension is neglected, the linear equations are well-posed provided the underlying exact motion satisfies a condition on the acceleration of the interface relative to gravity, similar to the criterion formulated by G. I. Taylor. If surface tension is included, the linear equations are well-posed without qualifications, whether the fluid is above or below the interface. An interesting qualitative structure is found for the linear equations. A Lagrangian approach is used, like that of numerical work such as [3], except that the interface is assumed horizontal at infinity. Certain integral equations which occur, involving double layer potentials, are shown to be solvable in the present case. © 1993 John Wiley & Sons, Inc.     

A numerical model for the trans-membrane voltage of vesicles

The Immersed Interface Method is employed to solve the time-varying electric field equations around a three-dimensional vesicle. To achieve second-order accuracy the implicit jump conditions for the electric potential, up to the second normal derivative, are derived. The trans-membrane potential is determined implicitly as part of the algorithm. The method is compared to an analytic solution based on spherical harmonics and verifies the second-order accuracy of the underlying discretization even in the presence of solution discontinuities. A sample result for an elliptic interface is also presented.     

Resonance interactions of waves in an ice channel

The properties of low-amplitude surface waves propagating in an ice channel are investigated in the shallow-water approximation. The ice cover is modelled either by a rigid cap or by a thin elastic plate floating on a liquid surface. It is shown that an ice channel is a waveguide for surface waves. The dispersive properties of the natural oscillations of the liquid in the channel are investigated. The resonance velocities of the motion of the load on the channel surface, at which the amplitude of the forced oscillations of the liquid increases without limit in time, are determined. The decay instability of the natural oscillations of high harmonics with respect to waves of the first mode is demonstrated. The process is described by the standard equations for non-linear three-wave interaction. The investigations lead to the conclusion that critical modes of motion of a boat are realizable in an ice channel.     

A New approach to finding the control that transports a system from one phase state to another

In their previous papers, the authors have considered the possibility of applying the theory of motion for nonholonomic systems with high-order constraints to solving one of the main problems of the control theory. This is a problem of transporting a mechanical system with a finite number of degrees of freedom from a given phase state to another given phase state during a fixed time. It was shown that, when solving such a problem using the Pontryagin maximum principle with minimization of the integral of the control force squared, a nonholonomic high-order constraint is realized continuously during the motion of the system. However, in this case, one can also apply a generalized Gauss principle, which is commonly used in the motion of nonholonomic systems with high-order constraints. It is essential that the latter principle makes it possible to find the control as a polynomial, while the use of the Pontryagin maximum principle yields the control containing harmonics with natural frequencies of the system. The latter fact determines increasing the amplitude of oscillation of the system if the time of motion is long. Besides this, a generalized Gauss principle allows us to formulate and solve extended boundary problems in which along with the conditions for generalized coordinates and velocities at the beginning and at the end of motion, the values of any-order derivatives of the coordinates are introduced at the same time instants. This makes it possible to find the control without jumps at the beginning and at the end of motion. The theory presented has been demonstrated when solving the problem of the control of horizontal motion of a trolley with pendulums. A similar problem can be considered as a model, since when the parameters are chosen correspondingly it becomes equivalent to the problem of suppression of oscillations of a given elastic body some cross-section of which should move by a given distance in a fixed time. The equivalence of these problems significantly widens the range of possible applications of the problem of a trolley with pendulums. The previous solution of the problem has been reduced to the selection of a horizontal force that is a solution to the formulated problem. In the present paper, it is offered to seek an acceleration of a trolley with which it moves by a given distance in a fixed time, as a time function but not a force applied to the trolley, while the velocities and accelerations are equal to zero at the beginning and end of motion. In this new problem, the rotation angles of pendulums are the principal coordinates. This makes it possible to find a sought acceleration of a trolley on the basis of a generalized Gauss principle according to the technique developed before. Knowing the motion of a trolley and pendulums it is easy to determine the required control force. The results of numerical calculations are presented.     

黏性流体中运动界面上的速度分解

本文研究黏性流体与零厚度弹性材料界面上的耦合运动问题.利用沉浸边界法与时间步长法,半Lagrange离散法,对速度分解后的Stokes部分与正则部分进行求解,获得了关于耦合运动中二阶PDE求解的一般方法,推广了文献[14,15,18]的结果.     

Observing nonlinear stochastic resonance with piecewise constant driving forces by the method of moments

The original method of moments confined within linear response theory is improved to calculate the nonlinear dynamic response of the standard noisy bistable stochastic systems in the general response sense by proposing a different operating technique. Especially, the proposed technique is simple and efficient to be used to the cases where the driving forces are not harmonics. Using the piecewise constant driving forces for demonstration, our comparative analysis shows that the long time ensemble average and the first three harmonic susceptibilities calculated by the proposed technique are of high accuracy. The dependence of the spectral amplification parameters at the first three harmonics on the noise intensity is also investigated, and the analysis to the resonant curves suggests a possible way to induce the even-order harmonic stochastic resonance.     

Stability of reflection and refraction of shocks on interface

In this paper we study the stability of the nonlinear wave structure caused by the attack of an incident shock on an interface of two different kinds of media. The attack will produce a reflected wave and a refracted wave, and also let the interface deflected. In this paper we will mainly study the case, when the reflected wave is a shock, and the flow between the reflected wave and the refracted shock is relatively subsonic. Our result indicates that the wave structure and the flow field for the reflection-refraction problem in this case is conditionally stable.To describe the motion of the fluid we use the inviscid Euler system as the mathematical model. The reflection-refraction problem can be reduced to a free boundary value problem, where the unknown reflected shock and refracted shock are free boundaries, and the deflected interface is also to be determined. In the proof of the existence and the stability of the corresponding wave structure we apply the Lagrange transformation to fix the interface and the decoupling technique to decouple the elliptic-hyperbolic composite system in its principal part. Meanwhile, some efficient weighted Sobolev estimates are established to derive the existence for corresponding nonlinear problems.     

Wavelet Decomposition of Spherical Vector Fields with Respect to Sources

This article is concerned with an approach of modelling the Earth’s magnetic field as measured by satellites in terms of a special system of vector spherical harmonics and in terms of vector kernel functions, called vector scaling functions and wavelets. The main ingredient is the presentation of a system of vector spherical harmonics which separates a given spherical vector field with respect to its sources, i.e., the spherical vector field is separated into a part which is induced by sources inside the sphere, a part which is induced by sources outside the sphere and a part which is induced by sources on the sphere, which are, for example, currents crossing the sphere. Using this special system of vector spherical harmonics vector scaling functions and wavelets are constructed which keep the advantageous property of separating with respect to sources but which also allow a locally reflected modelling of the respective vector field. At the end of the article, the method is tested on real magnetic field data measured by the German geoscientific research satellite CHAMP.