A moving boundary problem motivated by electric breakdown, I: Spectrum of linear perturbations

Tanveer, S.; Schäfer, Lothar; Brau, Fabian; Ebert, Ute

doi:10.1016/j.physd.2009.02.012

Cited by 10 publications

(24 citation statements)

References 46 publications

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“…There are also linear stability results [7,34] and exact travelling waves for sufficiently large kinetic undercooling [7]. Closely related to the finger is the problem of a travelling finite bubble in a directed flow field, which is of recent interest in the literature on streamers [28,36,58]. Numerical solutions of the evolving shape of a bubble suggest that an initially smooth bubble boundary may not remain so for all time [36]; this property also arises in the present work.…”

Section: Introductionmentioning

confidence: 54%

Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation

Dallaston

McCue²

2014

Eur. J. Appl. Math

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We examine the effect of a kinetic undercooling condition on the evolution of a free boundary in Hele-Shaw flow, in both bubble and channel geometries. We present analytical and numerical evidence that the bubble boundary is unstable and may develop one or more corners in finite time, for both expansion and contraction cases. This loss of regularity is interesting because it occurs regardless of whether the less viscous fluid is displacing the more viscous fluid, or vice versa. We show that small contracting bubbles are described to leading order by a well-studied geometric flow rule. Exact solutions to this asymptotic problem continue past the corner formation until the bubble contracts to a point as a slit in the limit. Lastly, we consider the evolving boundary with kinetic undercooling in a Saffman-Taylor channel geometry. The boundary may either form corners in finite time, or evolve to a single long finger travelling at constant speed, depending on the strength of kinetic undercooling. We demonstrate these two different behaviours numerically. For the travelling finger, we present results of a numerical solution method similar to that used to demonstrate the selection of discrete fingers by surface tension. With kinetic undercooling, a continuum of corner-free travelling fingers exists for any finger width above a critical value, which goes to zero as the kinetic undercooling vanishes. We have not been able to compute the discrete family of analytic solutions, predicted by previous asymptotic analysis, because the numerical scheme cannot distinguish between solutions characterised by analytic fingers and those which are corner-free but non-analytic.

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Section: Introductionmentioning

confidence: 54%

Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation

Dallaston

McCue²

2014

Eur. J. Appl. Math

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“…Propagating fronts in Laplacian growth occur naturally in quite a number of physical problems including viscous fingering [1,2,3,4,5], electro-chemical growth, dendritic crystal growth for small undercooling [6,7,8], and void migration in a conductor [9,10,11]. More recently, it has been shown that this class of problems includes the 'streamer' stage of electric breakdown [12,13,14,15,16,17,18,19], which will be described below. A central issue in these problems is the stability of curved fronts.…”

Section: Introductionmentioning

confidence: 99%

“…In two dimensions, a simple solution to the free boundary problem posed by this model takes the form of a uniformly translating circle. Our previous work in [17,18,19] and the present paper are primarily concerned with the linear and nonlinear stability of this solution to small perturbations. It is to be noted that the circular shape differs from an actual streamer shape.…”

Section: Introductionmentioning

confidence: 99%

A moving boundary model motivated by electric breakdown: II. Initial value problem

Kao

Brau

Ebert

et al. 2010

Physica D: Nonlinear Phenomena

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An interfacial approximation of the streamer stage in the evolution of sparks and lightning can be formulated as a Laplacian growth model regularized by a 'kinetic undercooling' boundary condition. Using this model we study both the linearized and the full nonlinear evolution of small perturbations of a uniformly translating circle. Within the linear approximation analytical and numerical results show that perturbations are advected to the back of the circle, where they decay. An initially analytic interface stays analytic for all finite times, but singularities from outside the physical region approach the interface for t → ∞, which results in some anomalous relaxation at the back of the circle. For the nonlinear evolution numerical results indicate that the circle is the asymptotic attractor for small perturbations, but larger perturbations may lead to branching. We also present results for more general initial shapes, which demonstrate that regularization by kinetic undercooling cannot guarantee smooth interfaces globally in time.

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“…Model reduction and analytical theory explained branching as a Laplacian instability that develops when the space charge layer around the streamer head is much thinner than the streamer radius [14]. Nevertheless, the analysis of a reduced movingboundary streamer model suggests that streamer heads are linearly stable [15,16] even if the stabilizing effects of electron diffusion and photo-ionization are neglected. Streamer fingers are therefore similar to laminar pipe flow: a finite perturbation is required to let streamers branch or to make the pipe flow turbulent.…”

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confidence: 99%

Electron density fluctuations accelerate the branching of positive streamer discharges in air

Luque

Ebert

2011

Phys. Rev. E

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Branching is an essential element of streamer discharge dynamics. We review the current state of theoretical understanding and recall that branching requires a finite perturbation. We argue that, in current laboratory experiments in ambient or artificial air, these perturbations can only be inherited from the initial state, or they can be due to intrinsic electron-density fluctuations owing to the discreteness of electrons. We incorporate these electron-density fluctuations into fully three-dimensional simulations of a positive streamer in air at standard temperature and pressure. We derive a quantitative estimate for the ratio of branching length to streamer diameter that agrees within a factor of 2 with experimental measurements. As branching without this noise would occur considerably later, if at all, we conclude that the intrinsic stochastic particle noise triggers branching of positive streamers in air at atmospheric pressure.

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A moving boundary problem motivated by electric breakdown, I: Spectrum of linear perturbations

Cited by 10 publications

References 46 publications

Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation

Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation

A moving boundary model motivated by electric breakdown: II. Initial value problem

Electron density fluctuations accelerate the branching of positive streamer discharges in air

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