Initial value problem for Rayleigh–Taylor instability of viscous fluids

Menikoff, Ralph; Mjolsness, R. C.; Sharp, David H.; Zemach, Charles; Doyle, Brendon J.

doi:10.1063/1.862107

Cited by 42 publications

(23 citation statements)

References 17 publications

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“…X), and recent surveys of previous work are given by Sharp (1984) and Kull (1991). Many of the analyses (e.g., Bellman andPennington, 1954, andMenikoff et al, 1978) have focused on the initial small-amplitude growth, where linearized equations are appropriate. One approach for examining the nonlinear effects of large amplitude is direct numerical simulation, e.g., Harlow and Welch (1966) and Tryggvason (1988), or through boundaryintegral computations, e.g., Baker and Meiron (1984), and Newhouse and Pozrikidis (1990).…”

Section: Introductionmentioning

confidence: 99%

Buoyant instability of a viscous film over a passive fluid

Canright

Morris

1993

J. Fluid Mech.

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In certain geophysical contexts such as lava lakes and mantle convection, a cold, viscous boundary layer forms over a deep pool. The following model problem investigates the buoyant instability of the layer. Beneath a shear-free horizontal boundary, a thin layer (thickness d 1 ) of very viscous fluid overlies a deep layer of less dense, much less viscous fluid; inertia and surface tension are negligible. After the initial unstable equilibrium is perturbed, a long-wave analysis describes the growth of the disturbance, including the nonlinear effects of large amplitude. The results show that nonlinear effects greatly enhance growth, so that initial local maxima in the thickness of the viscous film grow to infinite thickness in finite time, with a timescale 8µ/∆ρ gd 1 . In the final catastrophic growth the peak thickness is inversely proportional to the remaining time. (A parallel analysis for fluids with power-law rheology shows similar catastrophic growth.) While the small-slope approximation must fail before this singular time, the failure is only local, and a similarity solution describes how the peaks become downwelling plumes as the viscous film drains away. BUOYANT INSTABILITY OF A VISCOUS FILM OVER A PASSIVE FLUID INTRODUCTIONThis work examines the strongly nonlinear effects of finite amplitude in the RayleighTaylor instability of a horizontal viscous film under a shear-free boundary and over a much less viscous fluid. Inertia and surface tension are neglected, and, in the parameter range considered, the motion is limited by normal stresses in the more viscous fluid. The analysis exploits the fact that the most unstable wavelengths are long compared to the thickness of the film. The results show how the growth of disturbances to the interface becomes greatly enhanced when the disturbance amplitude becomes large, leading to the formation of downwelling sheets or plumes in a finite time.The motivation for this problem comes from certain geophysical situations, particularly the stability of the Earth's lithospheric plates. In simplified terms, the oceanic lithosphere (tectonic plates) can be considered a cold, stiff thermal boundary layer above the convecting mantle. Where two plates come together, one subducts under the other and flows downward due to its negative buoyancy. The question of how a new subduction zone is formed, how one large plate may break into two and thus allow some of the dense material to flow back down into the mantle, is not yet resolved. Other closely related geophysical situations include the surfaces of lava lakes, thermal convection in the mantles of other planets, and possibly convection in the Earth's solid core.This work examines a simple model of one possible mechanism for the initiation of subduction: the Rayleigh-Taylor instability. In this model, the lithosphere and the mantle are treated as distinct, highly viscous fluids, the lithosphere being denser (and much more viscous) than the mantle. In this unstable configuration, any variations in the lithosphere thickness t...

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Section: Introductionmentioning

confidence: 99%

Buoyant instability of a viscous film over a passive fluid

Canright

Morris

1993

J. Fluid Mech.

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“…Thus, as had been found in the study of related problems [3][4][5][6][7][8], the equation of motion of the free surface exhibits a " memory " effect of the free surface velocity. …”

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“…While this representation has been used more or less explicitly in several previous studies [2,[5][6][7], it has never been discussed in the detail desirable. Our reasons for discussing this matter are threefold.…”

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Small-amplitude waves on the surface of a layer of a viscous liquid

Cortelezzi¹,

Prosperetti²

1981

Quart. Appl. Math.

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“…This survey showed that the Rayleigh-Taylor instability exhibits essentially different behavior above and below a certain critical wavenumber k c . Menikoff et al 19 studied the IVP problem associated with the development of small amplitude disturbances in Rayleigh-Taylor unstable, viscous, incompressible fluids. In addition to normal modes, a set of continuum modes, not treated explicitly in the literature, makes an important contribution to the development of the fluid motion.…”

Section: Introductionmentioning

confidence: 99%

Effects of a semi-infinite stratification on the Rayleigh-Taylor instability in an interface with surface tension

González

2017

AIP Advances

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The Rayleigh-Taylor instability (RTI) in an infinite slab where a constant density lower fluid is initially separated from an upper stratified fluid is discussed in linear regime. The upper fluid is of increasing exponential density and surface tension is considered between both of them. It was found useful to study stability by using the initial value problem approach (IVP), so that we ensure the inclusion of certain continuum modes, otherwise neglected. This methodology includes the branch cut in the complex plane, consequently, in addition to discrete modes (surface RTI modes), a set of continuum modes (internal RTI modes) also appears. As a result, the usual information given by the normal mode method is now complete. Furthermore, a new role is found for surface tension: to transform surface RTI modes (discrete spectrum) into internal RTI modes belonging to a continuous spectrum at a critical wavenumber. As a consequence, the cut-off wavenumber disappears: i.e. the growth rate of the RTI surface mode does not decay to zero at the cut-off wavenumber, as previous researchers used to believe. Finally, we found that, due to the continuum, the asymptotic behavior of the perturbation with respect to time is slower than the exponential when only the continuous spectrum exists.

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Initial value problem for Rayleigh–Taylor instability of viscous fluids

Cited by 42 publications

References 17 publications

Buoyant instability of a viscous film over a passive fluid

Buoyant instability of a viscous film over a passive fluid

Small-amplitude waves on the surface of a layer of a viscous liquid

Effects of a semi-infinite stratification on the Rayleigh-Taylor instability in an interface with surface tension

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