The EHD-driven fluid flow and deformation of a liquid jet by a transverse electric field

The aim of this study is to investigate the effect of a uniform transverse electric field on the steady-state behavior of a liquid cylinder surrounded by another liquid of infinite extent. The governing electrohydrodynamic equations are solved for Newtonian and immiscible fluids in the framework of leaky-dielectric theory and in the limit of small electric field and fluid inertia. A detailed analysis of the electrical and hydrodynamic stresses acting on the interface separating the two fluids is presented, and an expression is found for the interface deformation for small distortions from a circular shape. The electrical stresses acting on the interface of two leaky-dielectric liquids are compared with those acting on an interface separating a perfect dielectric or infinitely conducting core fluid cylinder from a surrounding perfect dielectric fluid. A comparison is made between the results of this study and those of a similar study for fluids with permeable interfaces and the classical results for liquid drops.     

Propagation of long waves of finite amplitude at the interface of two viscous fluids

The propagation of long waves of finite amplitude at the interface of two viscous fluids has been studied theoretically. For plane Couette-Poiseuille flow of two superposed layers of fluids of different viscosity, an equation is derived to determine the development in time of the shape of these finite amplitude waves. The influence of the viscosity ratio, the density difference of the fluids and an imposed pressure gradient have been investigated.     

Bifurcation mechanism of interfacial electrohydrodynamic gravity-capillary waves near the minimum phase speed under a horizontal electric field

We investigate the evolution of interfacial gravity-capillary waves propagating along the interface between two dielectric fluids under the action of a horizontal electric field. There is a uniform background flow in each layer, and the relative motion tends to induce Kelvin–Helmholtz(KH) instability. The combined effects of gravity, surface tension and electrically induced forces are all taken into account. Under the short-wave assumption, the expansion and truncation method of Dirichlet-Neumann(DN) operators is applied to derive a reduced dynamical model. When KH instability is suppressed linearly by a considerably large electric field, our numerical results reveal that in certain regions of parameter space, nonlinear symmetric traveling wave solutions can be found near the minimum phase speed. Additionally, the detailed bifurcation structures are presented together with typical wave profiles.     

Finite-amplitude solitary waves in a two-layer fluid

Second-mode nonlinear internal waves at a thin interface between homogeneous layers of immiscible fluids of different densities have been studied theoretically and experimentally. A mathematical model is proposed to describe the generation, interaction, and decay of solitary internal waves which arise during intrusion of a fluid with intermediate density into the interlayer. An exact solution which specifies the shape of solitary waves symmetric about the unperturbed interface is constructed, and the limiting transition for finite-amplitude waves at the interlayer thickness vanishing is substantiated. The fine structure of the flow in the vicinity of a solitary wave and its effect on horizontal mass transfer during propagation of short intrusions have been studied experimentally. It is shown that, with friction at the interfaces taken into account, the mathematical model adequately describes the variation in the phase and amplitude characteristics of solitary waves during their propagation.     

Reflection and refraction of attenuated waves at boundary of elastic solid and porous solid saturated with two immiscible viscous fluids

The propagation of elastic waves is studied in a porous solid saturated with two immiscible viscous fluids.The propagation of three longitudinal waves is represented through three scalar potential functions.The lone transverse wave is presented by a vector potential function.The displacements of particles in different phases of the aggregate are defined in terms of these potential functions.It is shown that there exist three longitudinal waves and one transverse wave.The phenomena of reflection and refraction due to longitudinal and transverse waves at a plane interface between an elastic solid half-space and a porous solid half-space saturated with two immiscible viscous fluids are investigated.For the presence of viscosity in pore-fluids,the waves refracted to the porous medium attenuate in the direction normal to the interface.The ratios of the amplitudes of the reflected and refracted waves to that of the incident wave are calculated as a nonsingular system of linear algebraic equations.These amplitude ratios are used to further calculate the shares of different scattered waves in the energy of the incident wave.The modulus of the amplitude and the energy ratios with the angle of incidence are computed for a particular numerical model.The conservation of the energy across the interface is verified.The effects of variations in non-wet saturation of pores and frequencies on the energy partition are depicted graphically and discussed.     

Numerical simulation of plane and spatial nonlinear stationary waves in a two-layer fluid of arbitrary depth

The solution of a model differential equation for the three-dimensional perturbations of the interface between two immiscible fluids of different densities lying between a stationary nondeformable bottom and cover is presented. It is assumed that the waves have an arbitrary length and small, though finite, amplitude. The shapes of stationary traveling internal waves, both periodic in the two horizontal coordinates and soliton-like, are presented. These shapes depend on different parameters of the problem: the direction of the perturbation wave vector and the fluid layer depth and density ratios.     

Nonlinear analysis and solitary waves for two superposed streaming electrified fluids of uniform depths with rigid boundaries

The nonlinear modulation of the interfacial waves of two superposed dielectric fluids with uniform depths and rigid horizontal boundaries, under the influence of constant normal electric fields and uniform horizontal velocities, is investigated using the multiple-time scales method. It is found that the behavior of small perturbations superimposed on traveling wave trains can be described by a nonlinear Schrödinger equation in a frame of reference moving with the group velocity. Wave-like solutions to this equation are examined, and different types of localized excitations (envelope solitary waves) are shown to exist. It is shown that when these perturbations are neutrally stable and sufficiently long, solutions to the nonlinear Schrödinger equation may be approximated by the well-known Korteweg-de Vries equation. The speed of the solitary on the interface is seen to be reduced by the electric field. It is found that there are two critical values of the applied voltage that lead to (i) breaking up of the solitary waves, and (ii) bifurcation of solutions of the governing equations. On the other hand, the complex amplitude of standing wave trains near the marginal state is governed by a similar type of nonlinear Schrödinger equation in which the roles of time and space are interchanged. This equation, under a suitable transformation, is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equations admit a solitary wave type of solutions with variable speed. Using the tangent hyperbolic method, it is observed that the wave speed increases as well as decreases, with the increase of electric field values, according to the chosen wavenumbers range. Finally, the nonlinear stability analysis is discussed in view of the coefficients of nonlinear Schrödinger equation to show the effects of various physical parameters, and also to recover the some limiting cases studied earlier in the literature.     

结构中微裂纹与超声波的混频非线性作用数值仿真研究

针对结构中微裂纹检测难题,本文对结构中微裂纹与超声波的混频非线性作用进行了数值仿真研究。基于经典非线性理论,得到了两列超声纵波相互作用产生混频效应的理论条件。通过有限元仿真,研究了两列纵波与微裂纹相互作用产生混频的条件,并分析了界面处静应力、摩擦系数和裂纹方向对混频效应的影响。研究发现,超声波与微裂纹相互作用产生混频非线性效应的发生条件仍符合经典非线性理论下的混频产生条件。裂纹界面处施加的静应力对差频横波幅值有明显影响;当施加静应力与无裂纹模型得到的最大应力值接近时,混频非线性效应最强;裂纹界面的摩擦系数对超声波的混频非线性效应影响较小;透射差频横波传播方向与经典非线性理论预测的理论差频分量方向基本一致,且几乎不受裂纹方向变化的影响,而反射差频横波的传播方向随裂纹方向的改变而有所不同。本文研究工作为微裂纹检出及方向识别做了有益探索。     

Stability of the wave motion on the charged interface between two immiscible fluids in the presence of a tangential velocity field discontinuity

In the second-order approximation in the dimensionless wave amplitude, the problem of nonlinear periodic capillary-gravity wave motion of the uniformly charged interface between two immiscible ideal incompressible fluids, the lower of which is perfectly electroconductive and the upper, dielectric, moves translationally at a constant velocity parallel to the interface, is solved analytically. It is shown that on the uniformly charged surface of an electroconductive ideal incompressible fluid the positions of internal nonlinear degenerate resonances depend of the medium density ratio but are independent of the upper medium velocity and the surface charge density on the interface. All resonances are realized at densities of the upper medium smaller than the density of the lower medium. In the region of Rayleigh-Taylor instability with respect to density there is no resonant wave interaction.     

On 1:2 resonant waves which arise between two stratified fluids of finite vertical extent in the presence of mean flow effects

A study is made of the profiles which arise on the interface of two fluids of different densities and which arise from the interaction of the first and second harmonics of the motion. The fluids are constrained above and below by impermeable plates and are subjected to the twin restoring forces of gravity and surface tension. The method of multiple scales in both space and time is employed to derive a system of coupled nonlinear partial differential equations which models the evolution of the interface. In addition, an examination of the stability of the interface profiles was undertaken. A fairly simple stability criterion was found and the waves were shown to be highly stable. These results therefore provide encouragement to experimentalists who might be able to generate these waves in the laboratory.     

A new system of equations which predicts the evolution of a wave packet due to a fluid-fluid interaction under a narrow bandwidth assumption

The problem of the capillary-gravity waves which may arise at an interface between two stratified fluids of different densities is investigated. Particular attention is paid to the case when two different wave modes move at the same speed and to the wave train produced by the ensuing interaction. In contrast to most previous studies, the wave steepness and the wave bandwidth are not taken to be of the same order of magnitude, but the latter is of one order smaller. This leads to a system of nonlinear evolution equations which can be used to predict the subsequent progression of the wave field. These equations may be compared with the more usual nonlinear Schrödinger set which are valid under the equal bandwidth assumption and also a recently derived set which describe broader bandwidth waves. A large class of solutions to the equations is found and the corresponding wave profiles are presented.     

The Cauchy-Poisson problem in electrohydrodynamics

The Cauchy-Poisson problem of wave propagation along the interface between two fluids under the action of an electric field is solved for the case in which the field strength is below a certain critical value at which a horizontal interface loses stability. The upper fluid layer is an ideal dielectric, while the lower one is an ideal conductor. For the shape of the wave crest a solution in integral form is obtained. Numerical results concerning the spatial-temporal wavefield pattern are also presented.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 133–138, May–June, 1995.The work was financially supported by the Russian Foundation for Fundamental Research (project No. 93-013-17594).     

Electromagnetic diffraction by a thin conducting circular disk placed at the plane interface of two different media

In this paper, the problem of diffraction of time harmonic, electromagnetic waves by a thin ideally conducting disk lying at the plane interface of two different media is considered. In this analysis, the incident wave is a plane wave travelling in a direction perpendicular to the plane interface of the two media. A Hertz vector formulation is applied to reduce our electromagnetic diffraction problem to a system of two scalar problems which are solved by the help of two pairs of Fredholm integral equations of the second kind. Low frequency approximations to the tangential components of the magnetic intensity associated with the diffracted field at the surfaces of the disk, the induced surface current density on the disk and the scattering cross section are obtained.     

具有小密度差的两层流体中运动点源的二阶内波解

在具有自由面的两层流体中，运动点源生成的Kelvin船波存在两种模式，即表面波模式和内波模式。当上、下层流体密度比趋于1时，由内波模式计算的界面波幅趋于无穷大，这与实验事实相违背。为克服此困难，在自由面和界面作小波幅运动的假设，引入一个小密度差参数。研究了运动点源在无粘、不可压且具有小密度差的两层有限深流体中生成的高阶波动。首先利用摄动方法推导了各阶小参数满足的边值问题；其次，给出了小密度差情形下的可解性条件。证明了在密度比趋于1的极限情形，不存在导致界面波幅无穷大的内波模式；最后，利用Phillips的非线性共振相互作用理论，构造了具有自由面的两层有限深流体中Kelvin船波系的二阶一致有效波动解，并证明了该解在深水情形下退化为Newman关于均匀流体中自由面的二阶波动解。     

基于势流理论的内孤立波与立管作用数值研究

内孤立波是一种发生在水面以下的在世界各个海域广泛存在的大幅波浪, 其剧烈的波面起伏所携带的巨大能量对以海洋立管为代表的海洋结构物产生严重威胁, 分析其传播演化过程的流场特征及立管在内孤立波作用下的动力响应规律对于海洋立管的设计具有重要意义. 本文基于分层流体的非线性势流理论, 采用高效率的多域边界单元法, 建立了内孤立波流场分析计算的数值模型, 可以实时获得内孤立波的流场特征. 根据获得的流场信息, 采用莫里森方程计算内孤立波对海洋立管作用的载荷分布. 将内孤立波流场非线性势流计算模型与动力学有限元模型结合来求解内孤立波作用下海洋立管的动力响应特征, 讨论了内孤立波参数、顶张力大小以及内部流体密度对立管动力响应的影响. 发现随着内孤立波波幅的增大, 海洋立管的流向位移和应力明显增大. 由于上层流体速度明显大于下层, 且在所研究问题中拖曳力远大于惯性力, 因此管道顺流向的最大位移发生在上层区域. 顶张力通过改变几何刚度阵的值进而对立管的响应产生明显影响. 对于弱约束立管, 内部流体的密度对管道的流向位移影响较小.      

Head-on collision between two mKdV solitary waves in a two-layer fluid system

In this paper, based on the equations presented in [2], the head-on collision between two solitary waves described by the modified KdV equation (the mKdV equation, for short) is investigated by using the reductive perturbation method combined with the PLK method. These waves propagate at the interface of a two-fluid system, in which the density ratio of the two fluids equals the square of the depth ratio of the fluids. The second order perturbation solution is obtained. It is found that in the case of disregarding the nonuniform phase shift, the solitary waves preserve their original profiles after collision, which agrees with Fornberg and Whitham's numerical result of overtaking collision161 whereas after considering the nonuniform phase shift, the wave profiles may deform after collision.     

Theory of scattering from multilayered bodies of arbitrary shape

Green 's functions and boundary integral equation methods are used to derive a matrix set of equations for scattering from a multilayered homogeneous elastic body embedded in an infinite elastic material. The surfaces separating the layers have arbitrary shape. The formalism for the three-layer material is derived in detail and generalized to N-layers. A matrix factorization method (MFM) is shown to considerably simplify the computational problem. The relation to the problems of acoustic waves in fluids and electromagnetic waves in a dielectric material is briefly indicated.     

Wilton ripples between two uniform streaming magnetic fluids

An analysis is made of the small-amplitude capillary-gravity waves which occur on the interface of two incompressible inviscid magnetic fluids of different densities. The waves arise as a result of second harmonic resonance. The fluids moving with uniform velocities parallel to their interface are stressed by an oblique magnetic field. The linear relations between the oblique magnetic field and the instability criteria of the linear waves are analyzed. At the stability region (away from the neutral curve) of the linear theory, a pair of coupled non-linear partial differential equations are presented. On the neutral curve, a pair of coupled non-linear partial differential equations are introduced. The last pair of equations may be regarded as the counterparts of the single Klein-Gordon equation which occurs in the non-resonant case. In all cases, the wave profile and its stability conditions are obtained. These conditions are discussed analytically and graphically.     

Nonlinear analysis of governing laws of realization of the wave motion on the surface of a charged jet moving with respect to a material medium

On the base of analytic asymptotic calculations which are quadratic with respect to the ratio of the wave amplitude and the jet radius it is shown that the presence of a tangential jump in the velocity field on the jet surface leads to generation of a periodic wave motion on the interface between the media and has the destabilizing effect for both axisymmetric and bending and bending-deformation waves. It is found that there is a degenerate internal nonlinear resonance interaction between waves on the jet surface. This interaction may be of six different types in which the energy can be transferred between the interacting waves including waves of different symmetry. In the last case the energy is transferred from waves determining the initial deformation to axisymmetric waves.     

Evolution of long nonlinear waves on the interface of a stratified viscous fluid flow in a channel

The dynamics of disturbances of the interface between two layers of incompressible immiscible fluids of different densities in the presence of a steady flow between the horizontal bottom and lid is studied analytically and numerically. A model integrodifferential equation is derived, which takes into account long-wave contributions of inertial layers and surface tension of the fluids, small but finite amplitude of disturbances, and unsteady shear stresses on all boundaries. Numerical solutions of this equation are given for the most typical nonlinear problems of transformation of both plane waves of different lengths and solitary waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 49–61, July–August, 2007.