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

Lu, Danni; Ng, Chiu‐On

doi:10.1007/s10665-007-9199-6

Cited by 10 publications

(6 citation statements)

References 24 publications

(31 reference statements)

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“…Parau et al [13] analyzed three-dimensional capillary-gravity interfacial flows due to an immersed disturbance that propagates to a constant velocity along the interface between two semi-infinite fluids. Lu and Ng [14] studied interfacial capillary-gravity waves due to a fundamental singularity in a system of two semi-infinite fluids in both two-and three-dimensional cases.…”

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confidence: 99%

On capillary gravity-wave motion in two-layer fluids

2011

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Generalized expansion formulae for the velocity potentials associated with plane gravity-wave problems in the presence of surface tension and interfacial tension are derived in both the cases of finite and infinite water depths in two-layer fluids. As a part of the expansion formulae, orthogonal mode-coupling relations, associated with the eigenfunctions of the velocity potential, are derived. The dispersion relations are analyzed to determine the characteristics of the two propagating modes in the presence of surface and interfacial tension in both the cases of deep-water and shallow-water waves. The expansion formulae are then generalized to deal with boundary-value problems satisfying higher-order boundary conditions at the free surface and interface. As applications of the expansion formulae, the solutions associated with the source potential, forced oscillation and reflection of capillary-gravity waves in the presence of interfacial tension are derived.

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confidence: 99%

On capillary gravity-wave motion in two-layer fluids

2011

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“…(22) shall be studied for large t with x/t held fixed. This approximation has been successfully used to study the generalized Cauchy-Poisson problems [29][30][31][32][33].…”

Section: Transient Waves Due To a 2d Instantaneous Stokesletmentioning

confidence: 99%

“…On the other hand, the transient responses to the oscillatory singularity [29,30] do not tend to zero as time goes to infinity, and thus the steady state cannot be attained. These are really singular behaviors from the physical point of view, which is caused by the potential theory in which the energy of wave motion is conservative [31]. One is therefore motivated to eliminate the singular behaviors by the inclusion of viscosity in the mathematical formulation such that the dissipative wave motion can be predicted.…”

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confidence: 99%

Generation of free-surface gravity waves by an unsteady Stokeslet

2008

Arch Appl Mech

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The interaction of unsteady Stokeslets with the free surface of an initially quiescent incompressible fluid of infinite depth is investigated analytically for two-and three-dimensional cases. The disturbed flows are generated by an unsteady singular force moving perpendicularly downwards away from the surface. The analysis is based on the assumption that the motion satisfies the linearized unsteady Navier-Stokes equations with linear kinematic and dynamic boundary conditions. Firstly, the asymptotic representation for the transient free-surface waves due to an instantaneous Stokeslet is derived for a large time with a fixed distance-to-time ratio. As is well known, the corresponding inviscid waves predicted by the potential theory do not decay to zero as the time goes to infinity. In the present study, the transient waves predicted by the viscous theory eventually vanish due to the presence of viscosity, which is consistent with reality from the physical point of view. Secondly, the asymptotic solutions are obtained for the unsteady free-surface waves due to a harmonically oscillating Stokeslet. It is found that the unsteady waves can be decomposed into steady-state and transient responses. The steady state can be attained as time approaches infinity. It is shown that the viscosity of the fluid plays an important role in the evolution of the singularity-induced waves.

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“…By a straightforward application of the Stokes method of stationary phase, the asymptotic representation of Eq. (21) can be given as…”

Section: Transient Waves Due To Instantaneous Line Disturbancesmentioning

confidence: 99%

“…6.4], [16,Sect. 3.7] and their generalized cases [1,4,5,[11][12][13][14][17][18][19][20][21][22][23]. It is found that the transient waves are dispersive.…”

Section: Introductionmentioning

confidence: 99%

Generation of unsteady waves by concentrated disturbances in an inviscid fluid with an inertial surface

Dai²

2008

Acta Mech Sin

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The surface waves generated by unsteady concentrated disturbances in an initially quiescent fluid of infinite depth with an inertial surface are analytically investigated for two-and three-dimensional cases. The fluid is assumed to be inviscid, incompressible and homogenous. The inertial surface represents the effect of a thin uniform distribution of non-interacting floating matter. Four types of unsteady concentrated disturbances and two kinds of initial values are considered, namely an instantaneous/oscillating mass source immersed in the fluid, an instantaneous/oscillating impulse on the surface, an initial impulse on the surface of the fluid, and an initial displacement of the surface. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The solutions in integral form for the surface elevation are obtained by means of a joint Laplace-Fourier transform. The asymptotic representations of the wave motion for large time with a fixed distanceto-time ratio are derived by using the method of stationary phase. The effect of the presence of an inertial surface on the wave motion is analyzed. It is found that the wavelengths of the transient dispersive waves increase while those of the steady-state progressive waves decrease. All the wave amplitudes decrease in comparison with those of conventional free-surface waves. The explicit expressions for the freesurface gravity waves can readily be recovered by the present results as the inertial surface disappears.

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

Cited by 10 publications

References 24 publications

On capillary gravity-wave motion in two-layer fluids

On capillary gravity-wave motion in two-layer fluids

Generation of free-surface gravity waves by an unsteady Stokeslet

Generation of unsteady waves by concentrated disturbances in an inviscid fluid with an inertial surface

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