Surface Waves In Fluids Research Articles

Nondiffracting Accelerating Wave Packets of Maxwell’s Equations

We present the nondiffracting spatially accelerating solutions of the Maxwell equations. Such beams accelerate in a circular trajectory, thus generalizing the concept of Airy beams to the full domain of the wave equation. For both TE and TM polarizations, the beams exhibit shape-preserving bending which can have subwavelength features, and the Poynting vector of the main lobe displays a turn of more than 90°. We show that these accelerating beams are self-healing, analyze their properties, and find the new class of accelerating breathers: self-bending beams of periodically oscillating shapes. Finally, we emphasize that in their scalar form, these beams are the exact solutions for nondispersive accelerating wave packets of the most common wave equation describing time-harmonic waves. As such, this work has profound implications to many linear wave systems in nature, ranging from acoustic and elastic waves to surface waves in fluids and membranes.

Horizon effects for surface waves in wave channels and circular jumps

Surface waves in classical fluids experience a rich array of black/white hole horizon effects. The dispersion relation depends on the characteristics of the fluid (in our case, water and silicon oil) as well as on the fluid depth and the wavelength regime. In some cases, it can be tuned to obtain a relativistic regime plus high-frequency dispersive effects. We discuss two types of ongoing analogue white-hole experiments: deep water waves propagating against a counter-current in a wave channel and shallow waves on a circular hydraulic jump.

Depth variation of energy transport in fluid gravity waves

We calculate the distribution of energy flux as a function of the distance below the surface for propagating polychromatic gravity fluid surface waves. Linear theory has been used to derive closed-form expressions for the energy flux as a function of depth. In this context we discuss the power distribution for real ocean waves measured off the west coast of Sweden and compare this to the energy flux distribution for waves with Pierson–Moskowitz and Bretschneider spectral distributions. This is done in order to get an improved understanding of how to improve the power absorption in wave energy converters, and this is also discussed in this paper.

Wave particles

We present a new method for the real-time simulation of fluid surface waves and their interactions with floating objects. The method is based on the new concept of wave particles, which offers a simple, fast, and unconditionally stable approach to wave simulation. We show how graphics hardware can be used to convert wave particles to a height field surface, which is warped horizontally to account for local wave-induced flow. The method is appropriate for most fluid simulation situations that do not involve significant global flow. It is demonstrated to work well in constrained areas, including wave reflections off of boundaries, and in unconstrained areas, such as an ocean surface. Interactions with floating objects are easily integrated by including wave forces on the objects and wave generation due to object motion. Theoretical foundations and implementation details are provided, and experiments demonstrate that we achieve plausible realism. Timing studies show that the method is scalable to allow simulation of wave interaction with several hundreds of objects at real-time rates.

Extremal periodic wave profiles

Abstract. As a contribution to deterministic investigations into extreme fluid surface waves, in this paper wave profiles of prescribed period that have maximal crest height will be investigated. As constraints the values of the momentum and energy integrals are used in a simplified description with the KdV model. The result is that at the boundary of the feasible region in the momentum-energy plane, the only possible profiles are the well known cnoidal wave profiles. Inside the feasible region the extremal profiles of maximal crest height are "cornered" cnoidal profiles: cnoidal profiles of larger period, cut-off and periodically continued with the prescribed period so that at the maximal crest height a corner results.

Modeling of Melt Flow and Surface Standing Waves in a Continuous Casting Mold

Majority of surface and subsurface defects in the continuously cast steel may be attributed to surface standing waves and turbulence. A lot of work has been done on surface defects and on understanding the behavior of thesurface waves but there is a dearth of literature connecting the two. Also 3-D modeling of surface waves with heat transfer and solidification has received little attention. In the present study, a 3-dimensional mathematical model incorporating turbulent fluid flow, heat transfer, and solidification using a commercial code, FLOW3D, was developed. Using the model, the applicability of Froude criterion for scaling down industrial flow systems was analyzed. Although the small scale models, using this criterion, correctly predict the general fluid flow but the prediction of surface waves is not very accurate. Also, the effect of temperature is difficult to incorporate in water models. The model was used to study the effect of temperature on surface waves and verify the water modeling results. The effects of casting and SEN parameters on the fluid flow and surface waves were studied.

Ferrofluid hydrodynamics: waves, jets and free drops

A novel approach to hydrodynamical studies uses the bulk force properties of magnetic liquids to probe the dynamics of (i) freely suspended drops; and (ii) unstable surface waves and jets. The fluid dynamics are imaged by a fast CCD camera, allowing thorough analysis of these time-dependent phenomena. Ferrofluid drops are freely suspended in air by using magnetic fields to create an attractive force opposing gravity. The suspended drop then undergoes forced oscillations by perturbing the supporting magnetic field, and exhibits high order nonlinear modes of oscillation which can be driven until the drop bifurcates. Fluid surface waves and jets are investigated in cylindrical geometry. Nonlinear waves are magnetically driven, resulting in dramatic jets when the critical amplitude is exceeded. Such jets are observed to have a maximum acceleration exceeding 70g.

Nonlinear asymptotic short-wave models in fluid dynamics

Nonlinear monochromatic short surface waves in ideal fluids are studied and, by the general consideration of wave dynamics and perturbative methods a simple and effective multiscale approach is devised for nonlinear asymptotic short-wave dynamics in dispersive systems. In particular the evolution of a monochromatic surface wave in an ideal fluid is shown to lead to a modified Green-Naghdi system of equations and a Green-Naghdi system with surface tension. Short surface waves exist in these systems and the nonlinear asymptotic analysis produces the nonlinear model equations that govern their dynamics. Particular solutions are shown. Moreover the method allows for a general classification of classical model equations as Boussinesq, Benjamin-Bona-Mahony-Peregrine and Camassa-Holm equations. Their related nonlinear short-wave model limits are then derived. A relation between the short-wave limit of the integrable Camassa-Holm equation and the Harry-Dym hierarchy is finally unveiled.

Embedded solitons: solitary waves in resonance with the linear spectrum

It is commonly held that a necessary condition for the existence of solitons in nonlinear-wave systems is that the soliton’s frequency (spatial or temporal) must not fall into the continuous spectrum of radiation modes. However, this is not always true. We present a new class of codimension-one solitons (i.e., those existing at isolated frequency values) that are embedded into the continuous spectrum. This is possible if the spectrum of the linearized system has (at least) two branches, one corresponding to exponentially localized solutions, and the other to radiation modes. An embedded soliton (ES) is obtained when the latter component exactly vanishes in the solitary-wave’s tail. The paper contains both a survey of recent results obtained by the authors and some new results, the aim being to draw together several different mechanism underlying the existence of ESs. We also consider the distinctive properties of semi-stability of ESs, and moving ESs. Results are presented for four different physical models, including an extended fifth-order KdV equation describing surface waves in inviscid fluids, and three models from nonlinear optics. One of them pertains to a resonant Bragg grating in an optical fiber with a cubic nonlinearity, while two others describe second-harmonic generation (SHG) in the temporal or spatial domain (i.e., respectively, propagating pulses in nonlinear-optical fibers, or stationary patterns in nonlinear planar waveguides). Special attention is paid to the SHG model in the temporal domain for a case of competing quadratic and cubic nonlinearities. In particular, a new result is that when both harmonics have anomalous dispersion, an ES can exist which is, virtually, completely stable.

Large-amplitudeferrofluid surface waves and jets

A non-mechanical technique for investigating fluid surface waves and jets ispresented. The method uses a ferrofluid driven by electromagnets, eliminatingcomplicated mechanical wave generating systems. The waves are imaged by a fastdigital camera, and associated diagnostics. Experimental results, withaccompanying modelling, are presented for cylindrical standing waves, with themaximum amplitude wave being investigated. Exceeding the maximum amplituderesults in a dramatic jet, with a maximum acceleration exceeding 70g.

Viscosity effect on free surface waves in fluids

The frequencies and damping coefficients of gravitational-capillary waves are found for a wide range of controlling dimensionless parameters. The transition to the limiting cases of deep water and an ideal fluid is analyzed. In the parameter plane, the boundary between the regions of oscillatory and aperiodic perturbations is determined and the region of weak damping is indicated. The equilibrium state of thin liquid films with account for the Van der Waals forces is considered and the dispersion equation for the capillary-Van der Waals surface waves is obtained. For a suitably chosen frequency scale, this equation is the same as that for gravitational-capillary waves. The physical conditions making it possible to observe capillary and Van der Waals waves in thin fluid layers are estimated.

EIGENVALUE PROBLEMS FOR VIBRATING STRUCTURES COUPLED WITH QUIESCENT FLUIDS WITH FREE SURFACE

Vibrations of plates, shells and plate–shell systems coupled with sloshing, quiescent and inviscid fluid have been advantageously studied by inserting the sloshing condition into the eigenvalue problem. Here a formulation of this particular eigenvalue problem for symmetric matrices is obtained. In fact, in the previous studies, this technique has given eigenvalue problems for non-symmetric matrices for which the problem of the existence of complex eigenvalues arises. The present analysis deals with compressible and incompressible fluids and the discretization of the system is obtained by using the Rayleigh–Ritz method. The Rayleigh quotient of the system is manipulated to obtain expressions suitable for symmetric formulations of the eigenvalue problem. In particular, the Rayleigh quotient is transformed into a simpler expression where the potential energies of the compressible fluid and free surface waves do not appear. The method is applied to a vertical, simply supported, circular cylindrical shell partially filled by an incompressible sloshing liquid. A case with large interaction between sloshing and bulging modes is considered and interesting phenomena are observed.

Parametrically Excited Surface Waves in Magnetic Fluids: Observation of Domain Structures

Observations of parametrically excited surface waves in a magnetic fluid are presented. Under the influence of a magnetic field these waves have a non--monotonic dispersion relation, which leads to a richer behavior than in ordinary liquids. We report observation of three novel effects, namely: i) domain structures, ii) oscillating defects and iii) relaxational phase oscillations.

Numerical study of pattern formation in weakly damped parametric surface waves

We present a numerical study of parametrically driven surface waves in fluids of low viscosity. The equations governing fluid motion in the bulk and the appropriate boundary conditions at the free surface are approximated by a nonlocal set of equations involving surface coordinates alone, thus allowing the numerical analysis of the large aspect ratio limit (the aspect ratio being the ratio between the size of the system and the wavelength of the waves). The numerical calculation supports the predictions of a previously given multi-scale asymptotic analysis, yielding stable patterns of square symmetry above onset in the capillary dominated regime. In the case of mixed capillary-gravity waves, patterns of eight-fold symmetry are obtained, again in agreement with the predictions of the asymptotic expansion. These calculations provide indirect evidence about the stability of such patterns, analysis that is too involved to be carried out analytically.

Non-monotonic dispersion of surface waves in magnetic fluids

The dispersion relation of surface waves of a magnetic fluid in a magnetic field is studied experimentally. We verify the theoretically predicted existence of a non-monotomic dispersion relation. In particular, we demonstrate the existence of two different wave numbers occuring at the same frequency in an annular geometry.

An experimental study of the onset of parametrically pumped surface waves in viscous fluids

We measure the threshold accelerations necessary to excite surface waves in a vertically vibrated fluid container (the Faraday instability). Under the proper conditions, the thresholds and onset wavelengths agree with recent theoretical predictions for a laterally infinite, finite-depth container filled with a viscous fluid. Experimentally, we show that by using a viscous, non-polar fluid, the finite-size effects of sidewalls and the effects of surface contamination can be made negligible. We also show that finite-size corrections are of order h/L, where h is the fluid depth and L the container size. Based on these measurements, one can more easily interpret certain unexpected observations from previous experimental studies of the Faraday instability.

Large-time existence of surface waves in incompressible viscous fluids with or without surface tension

Large-time existence of surface waves in incompressible viscous fluids with or without surface tension

Nonlinear stability of surface waves in magnetic fluids: effect of a periodic tangential magnetic field

Nonlinear wave propagation on the surface between two superposed magnetic fluids stressed by a tangential periodic magnetic field is investigated using the method of multiple scales. A stability analysis reveals the existence of both nonresonant and resonant cases. From the solvability conditions, three types of nonlinear Schrodinger equation are obtained. The necessary and sufficient conditions for stability are obtained in each case. Formulae for the surface elevation are also obtained in both the non-resonant and the resonant cases. It is found from the numerical calculation that the tangential periodic magnetic field plays a dual role in the stability criterion, while the field frequency has a destabilizing influence.

Shear instability in stratified viscoelastic and gas-liquid media

A number of problems on the development of surface and internal waves in fluids with complex rheophysical properties is solved in a linear formulation. The influence of the viscoelastic properties of non-Newtonian fluids and the dissipation properties and local deformation inertia of fluids with gas bubbles on the stability of such waves is investigated. The question of the stability of internal waves in a two-layered incompressible viscous fluid with a tangential velocity discontinuity has been studied in detail /1, 2/. It is known that a Kelvin-Helmholtz instability occurs in the short-wave domain while there is no instability in the long-wave domain because of the stabilizing action of stratification. However, waves with negative energies exist in this domain that become unstable in the presence of viscous dissipation. In this connection, investigation of the stability of waves with negative energies in two-layered non-Newtonian fluids characterized by more complex rheological behaviour is of interest.

Surface Waves in Solids and Fluids

A formulation of the theory of surface waves is presented in a way which allows for a unified treatment of the various interfaces which can occur between solids, simple viscous fluids and vacuum. By treating the fluids as compressible these can be studied on the same footing as elastic solids. The method is applied to different problems of physical interest, namely, load effects of fluids on solid surfaces (a situation arising in the ultrasonic detection of defects), Brillouin scattering from fluid surfaces at lower (kHz) and higher (MHz) frequencies (which exhibit, respectively, capillary wave and bulk theshold effects) and the fluid-fluid interface, a problem of practical interest in physical chemistry. The possibility of other applications is indicated.