Rayleigh surface wave propagation in an orthotropic rotating magneto-thermoelastic medium subjected to gravity and initial stress

Abstract

The prime objective of the present article is to analyze the effects of rotation and initial stress on the propagation of Rayleigh surface waves in a homogeneous, orthotropic magneto-thermoelastic half space subjected to gravity field. The frequency equations in closed form are derived and the amplitude ratios of surface displacements, temperature change during the Rayleigh wave propagation on the surface of half space have been computed analytically. The highlights of this study are the effects of different parameters (rotation, magnetic field, initial stress, and gravity) on the velocity of Rayleigh waves. Variation in phase velocity of Rayleigh waves against a wave number is shown graphically. Some particular cases have been deduced. Also, the classical Rayleigh wave equation is obtained as a special case of the present study. Numerical example has been carried out and represented by the means of graphs. Impacts of various involved parameters appearing in the solutions are carefully analyzed. In fact, in the absence of various parameters, these equations are in agreement with the results for isotropic medium.     

A new features on S-waves propagation in a nonhomogeneous anisotropic incompressible medium under influence of gravity field and initial stress with and without electromagnetic field and rotation

The present article is concerned with the investigation of the propagation of shear waves in a nonhomogeneous anisotropic incompressible medium under the effect of the electromagnetic field, gravity field, rotation, and initial stress taking into account a comparison between presence and absence of magnetic field, initial stress, and rotation. Analytical analysis reveals that the velocity of propagation of the shear waves depends upon the direction of propagation, the anisotropy, magnetic field, rotation, gravity field, nonhomogeneity of the medium, and the initial stress. The frequency equation that determines the velocity of the shear waves has been obtained. Some special cases are also deduced from the present investigation. In fact, these equations are an agreement with the corresponding classical results when the medium is isotropic. Numerical results have been given and illustrated graphically in each case considered. The results indicate that the effects of gravity field, initial stress, magnetic field, electric field, anisotropy, and rotation are very pronounced. Also, the absence of initial stress, magnetic field, and rotation tends to increasing of the S-waves velocity compared with presence of them.     

Magneto-thermo-visco-elastic waves in an initially stressed conducting layer

The aim of this paper is to investigate magneto-thermo-viscoelastic surface waves in electrically and thermally conducting layers involving time rates of strain and stress of ordern, the media being under an initial stress in the nature of hydrostatic tension or compression. The theory of magneto-thermo-visco-elastic surface waves in the conducting medium involving strain rate and stress rate ofnth order is derived under initial stress. This theory is then employed to obtain wave velocity equations in specific cases. Results obtained in the above cases reduce to well-known classical results when additional fields are absent.     

Thermal effect on gravity waves in a compressible liquid layer over a solid half-space under initial hydrostatic stress

This paper deals with the effect of temperature on gravity waves in a compressible liquid layer over a solid half-space. It has been assumed that the liquid layer is under the action of gravity, while the solid half-space is under the influence of initial compressive hydrostatic stress. When the temperature of the half-space is altered, gravity waves propagate through the liquid layer along with sub-oceanic Rayleigh waves in the system. A new frequency equation has been derived here for gravity waves and sub-oceanic Rayleigh waves. It has been shown graphically that the phase velocity of gravity waves is influenced significantly by the initial compressive hydrostatic stress present in the solid half-space, for a particular value of the phase velocity of sub-oceanic Rayleigh waves and different coupling co-efficients of the temperature.     

具有初应力的多层空心圆柱体中的导波

基于“增量变形力学”理论,研究了轴向初应力和径向初应力对多层空心圆柱体中导波传播特性的影响,应用Legendre多项式方法求解了耦合波动方程,并通过数值算例讨论了该方法的收敛性,分析了初应力对纵向波和扭转波的影响,数值分析结果表明初应力对两者的影响是非常不同的。此外,轴向初应力对频散曲线的影响与径向初应力的影响也非常不同。     

Fully nonlinear gravity-capillary solitary waves in a two-fluid system of finite depth

Large-amplitude waves at the interface between two laminar immisible inviscid streams of different densities and velocites, bounded together in a straight infinite channel are studied, when surface tension and gravity are both present. A long-wave approximation is used to develop a theory for fully nonlinear interfacial waves allowing amplitudes as large as the channel thickness. The result is a set of evolution equations for the interfacial shape and the velocity jump across it. Traveling waves of permanent form are studied and it is shown that solitary waves are possible for a range of physical parameters. All solitary waves can be expressed implicitly in terms of incomplete elliptic integrals of the third kind. When the upper layer has zero density, two explicit solitary-wave solutions have been found whose amplitudes are equal to h/4 or h/9, where 2h is the channel thickness. In the absence of gravity solitary waves are not possible but periodic ones are. Numerically constructed solitary waves are given for representative physical parameters.     

Gravity Effect on the Axisymmetric Drop Spreading

The flattening (spreading) of the axisymmetrical drop on a plane horizontal surface under action of gravity force at zero tangential force (no shear at the gas–liquid interface) is investigated analytically and numerically. We determine the exact profile of compressed drop assuming the condition of drop volume conservation. 2D time dependant numerical model, based on a finite difference method, has been developed to describe the hydrodynamics inside the drop. The energy and Navier–Stokes equations are solved within the drop’s analytical profile. Effects of surface tension and thermocapillarity are taken into account. The effect of gravity has been studied to define main features of the drop dynamics. In calculations vector of gravitational acceleration is oriented perpendicularly to the surface, the Bond number is changed in the range from Bo = 0 to Bo = 151.6. Our results show that the gravity has a significant effect on the drop spreading.     

Love waves propagation in a piezoelectric layered structure with initial stresses

Summary. The propagation behavior of Love waves in a piezoelectric layered structure with inhomogeneous initial stress is studied. Solutions of the mechanical displacement and electrical potential function are obtained for the isotropic elastic layer and transversely isotropic piezoelectric substrate, respectively, by solving the coupled electromechanical field equations. Firstly, effects of the inhomogeneous initial stress on the dispersion relations and phase velocity of Love wave propagation are discussed. Then the influence of the initial stress gradient coefficient on the stress, mechanical displacement and electrical potential distribution in the layer and the substrate is investigated in detail. The results reported in this paper are not only meaningful for the design of surface acoustic wave (SAW) devices with high performance, but also effective for evaluating the residual stress distribution in the layered structures.     

Interfacial capillary–gravity waves due to a fundamental singularity in a system of two semi-infinite fluids

The interfacial capillary–gravity waves due to a transient fundamental singularity immersed in a system of two semi-infinite immiscible fluids of different densities are investigated analytically for two- and three- dimensional cases. The two-fluid system, which consists of an inviscid fluid overlying a viscous fluid, is assumed to be incompressible and initially quiescent. The two fluids are each homogeneous, and separated by a sharp and stable interface. The Laplace equation is taken as the governing equation for the inviscid flow, while the linearized unsteady Navier–Stokes equations are used for the viscous flow. With surface tension taken into consideration, the kinematic and dynamic conditions on the interface are linearized for small-amplitude waves. The singularity is modeled as a simple mass source when immersed in the inviscid fluid above the interface, or as a vertical point force when immersed in the viscous fluid beneath the interface. Based on the integral solutions for the interfacial waves, the asymptotic wave profiles are derived for large times with a fixed distance-to-time ratio by means of the generalized method of stationary phase. It is found that there exists a minimum group velocity, and the wave system observed will depend on the moving speed of the observer. Two schemes of expansion of the phase function are proposed for the two cases when the moving speed of an observer is larger than, or close to the minimum group velocity. Explicit analytical solutions are presented for the long gravity-dominant and the short capillary-dominant wave systems, incorporating the effects of density ratio, surface tension, viscosity and immersion depth of the singularity.     

SH wave propagation in a cylindrically layered piezoelectric structure with initial stress

Summary SH wave propagation in a cylindrically layered piezoelectric structure with initial stress is investigated analytically. By means of transformation, the governing equations of the coupled waves are reduced to Bessel and Laplace equations. The boundary conditions imply that the displacements, shear stresses, electric potential, and electric displacements are continuous across the interface between the layer and the substrate. The electrically open and short conditions at the cylindrical surface are applied to solve the problem. The phase velocity is numerically calculated for the electrically open and short cases, respectively, for different wavenumber and thickness of the layer. The effect of the initial stress on the phase velocity and the electromechanical coupling factor are discussed in detail for piezoelectric ceramics PZT-5H. We find that the initial stress has an important effect on the SH wave propagation in the cylindrically layered piezoelectric structures. The results also show that the ratio of the layer thickness to the wavelength has a remarkable effect on the SH wave phase velocity and electromechanical coupling factor.     

Numerical study of asteroid impact on the earth’s surface with allowance for gravitation and radiation energy transfer

The study presents a procedure for numerical modeling and results of gasdynamic calculations of asteroid impact on the surface in a two-dimensional axisymmetric formulation for impact velocities of ∼ 50 km/sec and asteroid dimensions of ∼ 1 km. The effect of gravity and radiation energy transfer are taken into account. Radiation transfer is calculated using the equations of radiation diffusion in the multigroup approximation with respect to the photon energy (10 spectral groups). The equations of radiation diffusion are solved by the method of alternating directions.     

Guided Waves in Functionally Graded Rods with Rectangular Cross-Section under Initial Stress

The characteristics of the guided waves propagation in functionally graded rods with rectangular cross-section (finite width and height) under initial stress are investigated in this paper based on Biot’s theory of incremental deformation. An extended orthogonal polynomial approach is present to solve the coupled wave equations with variable coefficients. By comparisons with the available results of a rectangular aluminum rod, the validity of the present approach is illustrated. The dispersion curves and displacement profiles of various rectangular functionally graded rods are calculated to reveal the wave characteristics, and the effects of different width to height ratios and initial stress and gradient functions on the guided waves are discussed in detail.     

A note on time integrators in water-wave simulations

The importance of a suitable temporal integrator for fully nonlinear simulations of surface gravity waves is emphasized. Via numerical examples, it is demonstrated that constant-step procedures are inefficient. This relates to the practice of energy-conserving symplectic integration, assuming constant time steps, and is compared to direct numerical simulations using Runge–Kutta integrators with variable time-step control. It is concluded that the latter with automatic variable time-step control is the more efficient, and should be applied. The practice and efficiency of a stabilization procedure for the time-step selection is described and illustrated. An important point of the method is that the linear part of the prognostic equations is integrated analytically, which means that this part is obtained to machine presicion for any (large) time step (time interval). The evolution and instabilities of highly nonlinear water waves in three dimensions are exemplified through an accurate and efficient time-integration procedure. We are grateful to Professor J. N. Newman for his longstanding and pioneering contributions to the research field of marine hydrodynamics. His analytical and numerical works on fundamental and industrial problems in relation to water waves and their interaction with floating bodies have inspired us and many fellow scientist world wide over long time.     

Active control of sloshing in containers with elastic baffle plates

We consider the finite element approximations of an optimal control problem consisting in the suppression of slosh arising in fluid–structure interaction problems with free surface. The vibration of a plate in contact with an incompressible fluid is considered as state equations in the optimization problem, and distributed controls on the plate are calculated to suppress the slosh. Locking‐free finite elements are used to discretize the plate, which is modeled by Reissner–Mindlin equations. The effect of the fluid is taken into account by means of an added mass formulation, discretized by standard piecewise linear tetrahedral finite elements, and the gravity waves on the free surface of the liquid are considered in the model. The control variable is the amplitude of a secondary force actuating on the structure. Implementation issues are discussed, and numerical experiments are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

On the admissibility of hydraulic jumps in two-layer fluids

Summary The present paper deals with the propagation of nearly resonant gravity waves in two-layer fluids assuming that both fluids are incompressible and inviscid. Also, in accordance with hydraulic theory, the pressure distribution inside both layers is taken to be hydrostatic. As a consequence, equations which govern the evolution of weakly nonlinear surface layer and internal layer waves are found to agree in form with transport equations known from investigations of acoustic waves in dense gases. In general, these solutions of steady as well as unsteady flows contain regions of multivaluedness, which have to be eliminated by the insertion of discontinuities in the flow quantities, but now representing hydraulic jumps or bores rather than shocks. These discontinuities are not uniquely defined by the initial data and boundary conditions for the problem under consideration and, therefore, a central issue is the identification of physically acceptable, i.e. admissible weak solutions. In contrast to gas dynamics, however, the balance of mass and total momentum for a two-layer flow are not sufficient to derive the jump relationships. To resolve this difficulty, a number of additional model equations have been proposed in the literature. Here we investigate their consequences and, in particular, their compatibility with the second law in the small amplitude limit. Special emphasis is placed on the existence of inviscid bores, which have no counterpart in gas dynamics. Dedicated to Professor Franz Ziegler on the occasion of his 70th birthday     

Corner flow in free liquid films

A lubrication-flow model for a free film in a corner is presented. The model, written in the hyperbolic coordinate system ξ = x ² −y ², η = 2xy, applies to films that are thin in the η-direction. The lubrication approximation yields two coupled evolution equations for the film thickness and the velocity field which, to lowest order, describes plug flow in the hyperbolic coordinates. A free film in a corner evolving under surface tension and gravity is investigated. The rate of thinning of a free film is compared to that of a film evolving over a solid substrate. Viscous shear and normal stresses are both captured in the model and are computed for the entire flow domain. It is shown that normal stress dominates over shear stress in the far field, while shear stress dominates close to the corner. This revised version was published online in October 2005 with corrected page numbers.     

Magneto-visco-elastic surface waves in stressed conducting media

The present paper is concerned with magneto-visco-elastic surface waves in conducting media involving time rate of strain and stress of first order, the media being under an initial stress of hydrostatic tension or compression. The theory of magneto-visco-elastic surface waves in a conducting medium involving time rate of strain and stress of first order is derived under an initial stress. The above general theory is then employed to characterise Rayleigh, Love and Stoneley waves. Results obtained in the above cases reduce to well-known classical results when viscosity and magnetic field are absent.     

A two-layer model for a non-Newtonian gravity current subjected to wind shear

Summary.  A theoretical model is developed for the gravity current resulting from the sudden release of a fixed volume of fluid of non-Newtonian power law rheology on top of a slightly denser Newtonian fluid layer in the presence of wind stress. The model incorporates the flow of both layers and accounts for the effects of inertial and viscous forces, and is suited for moderate Reynolds number flows. The governing equations are obtained by depth-averaging the unsteady equations of motion in accordance with the von Kármán's momentum integral method, and constitute a hyperbolic system of four equations for the flow rates and thicknesses of the fluid layers. Results are obtained by a well established numerical scheme for systems of nonlinear hyperbolic equations. For a particular case analytical results are obtained by employing an asymptotic matching approach. Good agreement is obtained between the numerical and analytical results. The effects of the thickness of the ambient layer, wind stress, Reynolds numbers, and rheology on the gravity current are discussed. Received July 22, 2002; revised November 27, 2002 Published online: May 8, 2003     

Effect of magnetic field and initial stress on the propagation of interface waves in transversely isotropic perfectly conducting media

Elasto-dynamical equations for transversely isotropic solids have been employed to investigate the general theory of transversely isotropic magneto-elastic interface waves in conducting media under initial hydrostatic tension or compression. Particular cases of interface waves such as Rayleigh, Love and Stoneley waves have been investigated in details. In all cases, the wave velocity equations have been deduced which are in complete agreement with the corresponding results of classical surface waves of the same types where magnetic fields and initial stresses are absent. Results obtained in this paper may be considered as more general and important in the sense that the corresponding results of classical surface waves due to Rayleigh, Love and Stoneley can readily be deduced from our results as special cases. Numerical calculations and graphs have been presented in the case of Love waves and conclusions are drawn.