Fully Localised Solitary-Wave Solutions of the Three-Dimensional Gravity–Capillary Water-Wave Problem

A model equation derived by Kadomtsev & Petviashvili (Sov Phys Dokl 15:539–541, 1970) suggests that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal spatial direction. This prediction is rigorously confirmed for the full water-wave problem in the present paper. The theory is variational in nature. A simple but mathematically unfavourable variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle with significantly better mathematical properties. The reduced functional is related to the functional associated with the Kadomtsev–Petviashvili equation, and a nontrivial critical point is detected using the direct methods of the calculus of variations.     

The displacement wave theory of blood vessel

On the basis of the medical and mechanical analysis and explanations in this paper the visco-elastic simply supported beam model is proposed to treat the displacement wave of the blood vessels. The relationships between the displacement wave and blood vessel elasticity as well as the viscous dissipation of the blood and blood vessel are obtained. The corresponding relations of such kinds of pulses in the traditional Chinese medicine as smooth pulse, surface pulse and deep pulse to the displacement waves of blood vessels are also found. The computational results are in good agreement with those acquired in the experiments with ultrasonic wave.The Projects Supported by National Science Fundation of China.     

Determination of hydrodynamic load on the wall of a wellbore formed by an electric discharge

Hydrodynamic processes in the discharge chamber of a device intended to increase the fluid conductivity of the porous medium in the well bottom zone are studied. Spatial and temporal characteristics of the pressure waves resulting from an electric discharge in a fluid are determined. The effect of the performance of the electricdischarge device and the wellbore fluid pressure on the dynamic load acting on the wellbore wall are determined.     

Non-equilibrium relativistic M.H.D. with finite electrical conductivity

A system of balance laws for relativistic m.h.d, with finite eIectrical conductivity, heat flux and viscosity is proposed, starting from the properties of the systems of conservation laws compatible with a supplementary balance law (entropy balance). Adopting a two-fluid scheme the plasma is treated as a mixture of a neutral fluid and a charged fluid. Following the approach ofextended thermodynamics heat flux, viscous stress and electric current density are considered as new field variables contributing to non equilibrium entropy density and flux.     

Internal waves generated by a moving sphere and its wake in a stratified fluid

The internal gravity waves and the turbulent wake of a sphere moving through stratified fluid were studied by the fluorescent dye technique. The Reynolds number Re=U·2a/v was kept nearly constant at about 3 · 10³ and the Froude number F;U/a N ranged from 0.5 to 12.5. It is observed that waves generated by the body are dominant only when F<4 and are replaced by waves generated by the large scale coherent structures of the wake when F>4.     

Boundary effects on the Bénard-Marangoni instability under an electric field

This article theoretically studies the Bénard-Marangoni instability problem for a liquid layer with a free upper surface, which is heated from below by a heating coil through a solid plate in ana.c. electric field. The boundary effects of the solid plate, which include its thermal conductivity, electric conductivity and thickness, have great influences on the onset of convective instability in the liquid layer. The stability analysis in this study is based on the linear stability theory. The eigenvalue equations obtained from the analysis are solved by using the fourth order Runge-Kutta-Gill's method with the shooting technique. The results indicate that the solid plate with a higher thermal or electric conductivity and a bigger thickness tends to stabilize the system. It is also found that the critical Rayleigh numberR _c, the critical Marangoni numberM _c, and the criticala.c. Rayleigh numberE _ac become smaller as the intensity of thea.c. electric field increases.     

The reflection phenomena of quasi-vertical transverse waves in piezoelectric medium under initial stresses

The propagation of plane vertical transverse waves at an interface of a semi-infinite piezoelectric elastic medium under the influence of the initial stresses is discussed. The free surface of the piezoelectric elastic medium is considered to be adjacent to vacuum. We assumed that the piezoelectric material is anisotropic of the type of a transversely isotropic crystals (hexagonal crystal structure, class 6 mm). For an incident of vertical transverse plane wave, four types (two for the displacement and two for the electric potential) of reflected plane waves, called quasi-longitudinal (qP) and quasi-shear vertical (qSV) waves are shown to be exist. The relations governing the reflection coefficients of these reflected waves for various boundary conditions (mixed-free-fixed) are derived. It has been shown analytically that reflected coefficients of (qP) and (qSV) waves depend upon the angle of incidence, the parameters of electric potential, the material constants of the medium as well as the initial stresses presented in the medium. The numerical computations of reflection coefficients for different values of initial stresses have been carried out by computer for aluminum nitride (AlN) as an example and the results are given in the form of graphs. Finally, particular cases are considered in the absence of the initial stresses and the electric potential. Some of earlier studies have been compared to the special cases and shown good agreement with them.     

Certain reducible helical hydromagnetic flows

Summary Intrinsic formulations are obtained of the equations governing three-dimensional steady flows of an inviscid magnetic fluid with infinite electrical conductivity both when the streamlines of the magnetic field H and the fluid velocity v everywhere have the same unit tangent, and when H lies in the normal plane to the fluid streamlines at every point, so that v×H=0 and v·H=0 respectively. Using these intrinsic equations certain results concerning hydromagnetic helical flows are derived, and particular classes of such flows are reduced (in the sense of Power and Walker¹)) to associated non-magnetic two-dimensional flows. Finally, reciprocal relations are used to link classes of hydromagnetic helical flows to four-parameter classes of ordinary fluid flows.     

Quasi-periodic waves and quasi-solitary waves in stratified fluid of slowly varying depth

The nonlinear waves in a stratified fluid of slowly varying depth are investigated in this paper. The model considered here consists of a two-layer incompressible constantdensity inviscid fluid confined by a slightly uneven bottom and a horizontal rigid wall. The Korteweg-de Vries (KdV) equation with varying coefficients is derived with the aid of the reductive perturbation method. By using the method of multiple scales, the approximate solutions of this equation are obtained. It is found that the unevenness of bottom may lead to the generation of so-called quasi-periodic waves and quasisolitary waves, whose periods, propagation velocities and wave profiles vary slowly. The relations of the period of quasiperiodic waves and of the amplitude, propagation velocity of quasi-solitary waves varying with the depth of fluid are also presented. The models with two horizontal rigid walls or single-layer fluid can be regarded as particular cases of those in this paper.Project Supported by National Natural Science Foundation of China.     

Localised knife waves in a structured interface

We consider a Mode III lattice with an interface layer where the dynamic crack growth is caused by a localised sinusoidal wave. In the wave–fracture scenario, the ‘feeding wave’ (here also called the knife wave) delivers energy to the moving crack front, while the dissipative waves carry a part of this energy away from the front. The questions addressed here are:

• What are the conditions of existence of the localised knife wave?

• What is the lower bound of the amplitude of the feeding wave, which supports the crack propagation, for a given deformational fracture criterion?

• How does the crack speed depend on the amplitude of the feeding wave?

• What are the dissipative waves? How much energy is irradiated by these waves and what is the total dissipation?

• What are the conditions of existence of the steady-state regime for the propagating crack?

We consider analytically two established regimes: the steady-state regime, where the motion of neighbouring masses (along the interface) differs only by a constant shift in time, and an alternating-strain regime, where the corresponding amplitudes differ by sign. We also present the numerical simulation results for a model of a high-contrast interface structure. Along with the energy of the feeding and dissipative waves, an energy radiated to the bulk of the lattice is identified.

Extended thermodynamics of a relativistic plasma with finite electric conductivity

A system of equations for relativistic m.h.d, with finite electric conductivity and no heat flux is proposed, starting from the properties of the systems of conservation laws compatible with a supplementary balance law (entropy balance) with convex density (symmetric-hyperbolic systems). The electric current density is treated as a new field variable which contributes to non equilibrium entropy density (extended thermodynamics). The result is a theory in which only one new constitutive function, representing entropy increment respect to equilibrium, is necessary to characterize the properties of the medium related to electric conductivity.G.N.F.M. of the C.N.R.     

Rheological characterization of complex fluids in electro-magnetic fields

The paper is focused on the experimental investigations and rheological characterisations in magnetic and electric fields of liquids based on water in crude oils emulsions, added with ferrofluids (two types of crude oils are used in experiments: asphaltic and paraffinic, respectively). The final samples disclose weakly effects in the presence of magnetic field (saturated magnetization: M_n < 300 [G]) and behave almost as isolators in electric field (conductivity: σ < 10⁻⁵ [S/m]). The main goal of the study is to explore to what extent rheometry of complex fluids in electric and magnetic fields is able to offer value information about the internal structure of the samples. The experimental results prove that anomalous rheological behaviour (thixotropy, non-monotonic flow curve or viscosity function) of a complex fluid (in our case, emulsions based on paraffinic oil) generate also thixotropic properties and non-monotonic answers in the presence ferrofluids, under low magnetic and/or electric fields intensity. Our prospective study suggests that novel experimental procedures based on interaction: electro-magnetic field–complex fluids can be developed, in order to determine indirectly some relevant rheological properties of the complex fluids with internal network structure.     

Inhomogeneous electromagnetic plane waves in crystals

Summary The propagation of inhomogeneous, time harmonic, elliptically polarised, electromagnetic plane waves in non-absorbing, magnetically isotropic, but electrically anisotropic, crystals is considered. The electric displacement and the magnetic induction are assumed to have the forms D exp l(S · x–t) and B exp l(S · x–t), respectively, at the place x and time t, where D, S, B are Gibbs bivectors (complex vectors) and is real. The implications of Maxwell's equations for the various field quantities are interpreted simply and directly through the use of bivectors and their associated ellipses.The propagation of circularly polarised waves is considered in detail. For such waves the electric displacement bivector is isotropic: D · D = 0. In order that such waves may propagate it is found that either (i) D is parallel to the slowness bivector S, so that both D and S are isotropic and coplanar, or (ii) D is parallel to the magnetic induction bivector B, so that both D and B are isotropic and coplanar. It is shown that for type (ii) the secular equation must have a double root for the slowness and conversely if the secular equation has a double root then there exists an isotropic electric displacement right eigenbivector of the optical tensor.Both types of waves are possible in a biaxial crystal. They complement each other in the following way. For type (i) all but two great circles on the unit sphere are possible circles of polarisation for circularly polarised waves with D and S parallel. Each of the exceptional great circles is such that an optic axis is normal to the plane of the circle. These two exceptional circles are the only possible circles of polarisation for circularly polarised waves of type (ii) when D and B are parallel.The situation for uniaxial crystals is similar—the only essential difference being that for uniaxial crystals there is only one exceptional circle since there is only one optic axis.For isotropic crystals the situation is quite different. Circularly polarised waves of type (i) are not possible. All great circles on the unit sphere are possible circles of polarisation for circularly polarised waves of type (ii) with D and B parallel.     

Reflection of magnetoacoustic waves

The problem of the reflection of magnetoacoustic waves at the boundary dividing an elastic medium from a fluid medium with infinite conductivity in the presence of an arbitrary constant magnetic field was treated in [1]. In writing down the boundary conditions the continuity of the tangential component of the magnetic field was used. This condition is valid when the conductivity of the medium is finite but not when the conductivity is infinite. In this connection a problem similar to that in [1] is solved, without employing this particular boundary condition. The amplitude conversion coefficients found for the limiting cases of weak and strong magnetic fields coincide with the respective coefficients given in [2,3] for media with a finite conductivity, if we allow the conductivity in the latter expressions to become infinite.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 1, January–February, pp. 56–61, 1970.     

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.     

Formation of Regions with High Energy and Pressure Gradients at the Free Surface of Liquid Dielectric in a Tangential Electric Field

The nonlinear dynamics of the free surface of an ideal incompressible non-conducting fluid with a high dielectric constant subjected to a strong horizontal electric field is simulated using the method of conformal transformations. It is shown that in the initial stage of interaction of counter-propagating periodic waves of significant amplitude, there is a direct energy cascade leading to energy transfer to small scales. This results in the formation of regions with a steep wave front at the fluid surface, in which the dynamic pressure and the pressure exerted by the electric field undergo a discontinuity. It has been demonstrated that the formation of regions with high gradients of the electric field and fluid velocity is accompanied by breaking of surface waves; the boundary inclination angle tends to 90?, and the surface curvature increases without bound.     

Fully Developed Forced Convection in a Parallel Plate Channel with a Centered Porous Layer

In this study, fully developed heat and fluid flow in a parallel plate channel partially filled with porous layer is analyzed both analytically and numerically. The porous layer is located at the center of the channel and uniform heat flux is applied at the walls. The heat and fluid flow equations for clear fluid and porous regions are separately solved. Continues shear stress and heat flux conditions at the interface are used to determine the interface velocity and temperature. The velocity and temperature profiles in the channel for different values of Darcy number, thermal conductivity ratio, and porous layer thickness are plotted and discussed. The values of Nusselt number and friction factor of a fully clear fluid channel (Nu _cl = 4.12 and fRe _cl = 24) are used to define heat transfer increment ratio (e_th = Nu_p/Nu_cl)({\varepsilon _{\rm th} =Nu_{\rm p}/Nu_{\rm cl})} and pressure drop increment ratio (e_p = fRe_p/fRe_cl )({\varepsilon_{\rm p} =fRe_{\rm p}/fRe_{\rm cl} )} and observe the effects of an inserted porous layer on the increase of heat transfer and pressure drop. The heat transfer and pressure drop increment ratios are used to define an overall performance (e = e_th/e_p)({\varepsilon = \varepsilon_{\rm th}/\varepsilon_{\rm p})} to evaluate overall benefits of an inserted porous layer in a parallel plate channel. The obtained results showed that for a partially porous filled channel, the value of e{\varepsilon} is highly influenced from Darcy number, but it is not affected from thermal conductivity ratio (k _r) when k _r > 2. For a fully porous material filled channel, the value of e{\varepsilon} is considerably affected from thermal conductivity ratio as the porous medium is in contact with the channel walls.     

Seismoelectric Fluid/Porous-Medium Interface Response Model and Measurements

Coupled seismic and electromagnetic (EM) wave effects in fluid-saturated porous media are measured since decades. However, direct comparisons between theoretical seismoelectric wavefields and measurements are scarce. A seismoelectric full-waveform numerical model is developed, which predicts both the fluid pressure and the electric wavefields in a fluid in which a porous disc is embedded. An experimental setup, in which pressure and electric signals in the fluid are simultaneously measured, is presented. The setup allows the detection of the EM field that is generated when an acoustic wave crosses the interface between the fluid and the thin porous disc, without interference of electrical fields that are present within seismic body waves. The predicted pressure wavefield agrees well with the measurements in terms of acoustic wave travel times, waveforms, and amplitudes. The electric wavefield predictions agree with the recordings in terms of travel times, waveforms, and spatial amplitude decay. A discrepancy in amplitude of the converted EM signal is observed. Theoretical amplitudes that are smaller than the measurements were also reported in previous literature. These results seem to validate seismoelectric theory.     

Self-similar magnetogasdynamic flows accompanied by detonation and combustion waves

Magnetogasdynamic (MGD) flows with detonation waves and combustion fronts have attracted more and more attention in recent years. Intensive heat supply assures such a significant increase in the temperature and pressure behind the heat liberation fronts that the gaseous combustion products become conductive so that the flow map in the electric and magnetic fields can vary substantially as compared with ordinary gasdynamics. In the case of finite gas conductivity, when the magnetic Reynolds numbers R_m are low, the asymptotic laws of detonation wave propagation which either go over into the Chapman-Jouguet (CJ) mode (in a number of cases at a finite distance from the initiation source) or remain overcompressed, have been studied [1]. Stationary flow modes behind detonation waves have been investigated in [2] and the problem of the detonation wave originating at the closed end of the tube emerging in the stationary mode in crossed homogeneous magnetic and electric fields has been examined. Results are presented in this paper of an investigation of one-dimensional self-similar flows caused by piston motion in a hot gas mixture in which a detonation wave or combustion front is propagated. The motion is realized in external electric and magnetic fields which exert a substantial effect on the flow of the conductive combustion products. Domains of application of the governing parameters in which the various flow modes are realized are found by using a qualitative and numerical analysis. The results obtained are used to solve problems about the hypersonic gas flow around a thin wedge in an axial magnetic field.     

The effects of temperature-dependent fluid properties on free convective flow along a semi-infinite vertical plate by the presence of radiation

The effects of temperature-dependent density, viscosity and thermal conductivity on the free convective steady laminar boundary layer flow by the presence of radiation for large temperature differences, are studied. The fluid density and the thermal conductivity are assumed to vary linearly with temperature. The fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature. The usual Boussinesq approximation is neglected due to the large temperature difference between the plate and the fluid. The nonlinear boundary layer equations, governing the problem under consideration, are solved numerically by applying an efficient numerical technique based on the shooting method. The effects of the density/temperature parameter n, the thermal conductivity parameter , the viscosity/temperature parameter _r and the radiation parameter F are examined on the velocity and temperature fields as well as the coefficient of heat flux and the shearing stress at the plate.