Anomalous scaling for three‐dimensional Cahn‐Hilliard fronts

Korvola, Timo; Kupiainen, Antti; Taskinen, Jari

doi:10.1002/cpa.20055

Cited by 14 publications

(34 citation statements)

References 7 publications

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“…This solution exists and is stable in all dimensions although the decay rate of perturbations depends upon the dimensionality (Korvola, Kupiainen & Taskinen 2005). From (2.2) one immediately realizes that the sharp-interface limit is obtained for ǫ → 0: in this case tanh y/( √ 2ǫ) → sign(y).…”

Section: Equilibrium Statementioning

confidence: 82%

Phase-field model for the Rayleigh–Taylor instability of immiscible fluids

et al. 2009

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The Rayleigh-Taylor instability of two immiscible fluids in the limit of small Atwood numbers is studied by means of a phase-field description. In this method the sharp fluid interface is replaced by a thin, yet finite, transition layer where the interfacial forces vary smoothly. This is achieved by introducing an order parameter (the phase field) whose variation is continuous across the interfacial layers and is uniform in the bulk region. The phase field model obeys a Cahn-Hilliard equation and is two-way coupled to the standard Navier-Stokes equations. Starting from this system of equations we have first performed a linear analysis from which we have analytically rederived the known gravity-capillary dispersion relation in the limit of vanishing mixing energy density and capillary width. We have performed numerical simulations and identified a region of parameters in which the known properties of the linear phase (both stable and unstable) are reproduced in a very accurate way. This has been done both in the case of negligible viscosity and in the case of nonzero viscosity. In the latter situation only upper and lower bounds for the perturbation growth-rate are known. Finally, we have also investigated the weaklynonlinear stage of the perturbation evolution and identified a regime characterized by a constant terminal velocity of bubbles/spikes. The measured value of the terminal velocity is in perfect agreement with available theoretical prediction. The phase-field approach thus appears to be a valuable tecnhique for the dynamical description of the stages where hydrodynamic turbulence and wave-turbulence enter into play.

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Section: Equilibrium Statementioning

confidence: 82%

Phase-field model for the Rayleigh–Taylor instability of immiscible fluids

et al. 2009

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“…If we let u 1 and u 2 denote the minimizing values of such an F , then there exist precisely two monotonic planar wave connections between u 1 and u 2 ,ū(x 1 ) andū(−x 1 ). In the case of (1.6) (the case studied in the series of papers [9,36,37]) one readily verifies that u(x 1 ) = tanh(…”

Section: Introductionmentioning

confidence: 93%

“…(See the remarks following (1.2).) A different approach is taken in this setting in [36,37], quite similar to the method employed here, and the authors conclude stability in dimensions d ≥ 3 for the planar waveū (x 1 ) = tanh…”

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

“…The primary difference between the approach of [36,37] and the current analysis is the local tracking function δ(t,x),x = (x 2 , x 3 , ..., x d ). It is precisely this local tracking that allows us here to obtain stability for the case of dimension d = 2, which was left open in [36,37]. It is well known that for the case of one space dimension solutions u(t, x) of the Cahn-Hilliard equation initialized by u(0, x) near a standing wave solutionū(x) will not generally approachū(x) time-asymptotically, but rather will approach a translate ofū(x) determined by an integral of the initial perturbation.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic behavior near planar transition fronts for the Cahn–Hilliard equation

Howard

2007

Physica D: Nonlinear Phenomena

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We consider the asymptotic behavior of perturbations of planar wave solutions arising in the CahnHilliard equation in space dimensions d ≥ 2. Such equations are well known to arise in the study of spinodal decomposition, a phenomenon in which rapid cooling of a homogeneously mixed binary alloy causes separation to occur, resolving the mixture into regions in which one component or the other is dominant, with these regions separated by steep transition layers. A critical feature of the Cahn-Hilliard equation in one space dimension is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis, a feature that is known to complicate the step from spectral to nonlinear stability. The analysis of planar waves in multiple space dimensions is further complicated by the fact that the leading eigenvalue for this linearized operator (leading in the case of stability) moves into the negative-real half plane with cubic scaling λ ∼ |ξ| 3 , where ξ ∈ R d−1 denotes a Fourier variable associated with spatial components transverse to the planar wave. Under the assumption of spectral stability, described in terms of an appropriate Evans function, we develop detailed asymptotics for perturbations from planar wave solutions, establishing asymptotic stability for initial perturbations decaying with appropriate algebraic rate in an L 1 norm of the transverse variables.

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