This paper is an introductory review of the problem of front propagation into unstable states. Our presentation is centered around the concept of the asymptotic linear spreading velocity v*, the asymptotic rate with which initially localized perturbations spread into an unstable state according to the linear dynamical equations obtained by linearizing the fully nonlinear equations about the unstable state. This allows us to give a precise definition of pulled fronts, nonlinear fronts whose asymptotic propagation speed equals v*, and pushed fronts, nonlinear fronts whose asymptotic speed v^dagger is larger than v*. In addition, this approach allows us to clarify many aspects of the front selection problem, the question whether for a given dynamical equation the front is pulled or pushed. It also is the basis for the universal expressions for the power law rate of approach of the transient velocity v(t) of a pulled front as it converges toward its asymptotic value v*. Almost half of the paper is devoted to reviewing many experimental and theoretical examples of front propagation into unstable states from this unified perspective. The paper also includes short sections on the derivation of the universal power law relaxation behavior of v(t), on the absence of a moving boundary approximation for pulled fronts, on the relation between so-called global modes and front propagation, and on stochastic fronts.Comment: final version with some added references; a single pdf file of the published version is available at http://www.lorentz.leidenuniv.nl/~saarloo

Fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes: pulled fronts and pushed fronts. The term "pulled front" expresses that these fronts are "pulled along" by the spreading of linear perturbations about the unstable state. Accordingly, their asymptotic speed v * equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state. The central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towards v * . We show that when such fronts evolve from "sufficiently steep" initial conditions, which initially decay faster than e −λ * x for x → ∞, they have a universal relaxation behavior as time t → ∞: the velocity of a pulled front always relaxes algebraically likeThe parameters v * , λ * , and D are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state. This front velocity is independent of the precise value of the front amplitude, which one tracks to measure the front position. The interior of the front is essentially slaved to the leading edge, and develops universally asis a uniformly translating front solution with velocity v < v * .Our result, which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well-known nonlinear diffusion equation of type ∂ t φ = ∂ 2 x φ + φ − φ 3 , where the invaded unstable state is φ = 0. Even for this simple case, the subdominant t −3/2 term extends an earlier result of Bramson. Our analysis is then generalized to more general (sets of) partial differential equations with higher spatial or temporal derivatives, to PDEs with memory kernels, and also to difference equations such as those that occur in numerical finite difference codes. Our universal result for pulled fronts thus implies independence (i) of the level curve which is used to track the front position, (ii) of the precise nonlinearities, (iii) of the precise form of the linear operators in the dynamical equation, and (iv) of the precise initial conditions, as long as they are sufficiently steep. The only remainders of the explicit form of the dynamical equation are the nonlinear solutions v and the three saddle point parameters v * , λ * , and D. As our simulations confirm all our analytical predictions in every detail, it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts, if they are uniformly translating for t → ∞. An immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions. In addition to our main result for pulled fronts, we also discuss the propagation and convergence of fronts emerging from * Corresponding author. Present address: CWI, Postbus 94079, 1090 GB Amsterdam, Netherlands 0167-2789/00/$ -see front matter © 2000 Else...

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We study the origin of the scaling behavior in frictionless granular media above the jamming transition by analyzing their linear response. The response to local forcing is non-self-averaging and fluctuates over a length scale that diverges at the jamming transition. The response to global forcing becomes increasingly nonaffine near the jamming transition. This is due to the proximity of floppy modes, the influence of which we characterize by the local linear response. We show that the local response also governs the anomalous scaling of elastic constants and contact number.

We probe flows of soft, viscous spheres near the jamming point, which acts as a critical point for static soft spheres. Starting from energy considerations, we find nontrivial scaling of velocity fluctuations with strain rate. Combining this scaling with insights from jamming, we arrive at an analytical model that predicts four distinct regimes of flow, each characterized by rational-valued scaling exponents. Both the number of regimes and values of the exponents depart from prior results. We validate predictions of the model with simulations.PACS numbers: 47.57. Bc, 83.50.Rp, 83.80.Iz The past few years have seen enormous progress towards understanding the static, "jammed" state that occurs when soft athermal particles are packed sufficiently densely that they attain a finite rigidity [1][2][3]. Such systems may flow when shear stresses are applied, and in seminal work, Olsson and Teitel addressed the relation between strain rate, shear stress and packing fraction in a simplified numerical model for the flow of soft viscous spheres [4]. When rescaled appropriately, the data for strain rateγ, shear stress σ and packing fraction φ were found to collapse to two curves, reminiscent of second order-like scaling functions, and a large length scale was found to emerge near jamming. Since then, qualitatively similar results have been obtained in simulations of a number of flowing systems [5][6][7][8][9], but there is little agreement on the actual value of scaling exponents, nor on the relation to jamming in static systems.Here we describe an analytical model that connects the scaling of static systems to the scaling of both the velocity fluctuations and the shear stress of flowing systems near jamming. The model is built around a "viscoplastic" effective strain γ eff = γ y + γ dyn , where γ dyn is a dynamic contribution set by the strain rate, and γ y stems from the (dynamical) yield stress and is controlled by the distance to jamming. We show that steady state power balance dictates nontrivial scaling of γ dyn with strain rate, and propose a nonlinear stress-strain relation that leads to a closed set of equations predicting a rich scaling scenario for flows near jamming. We verify central ingredients of the model and our predictions for the rheology numerically in Durian's bubble model for foams [11]. Our simple model captures and predicts the rheology and fluctuations starting from the microscopic interactions; it also indicates the need for, and provides, new ways to present and analyze rheological data near jamming.Numerical Model -The two-dimensional Durian bubble model stipulates overdamped dynamics in which the sums of elastic and dissipative forces on each bubble, represented by a disk, balance at all times [11]. Forces are pairwise and occur only between contacting bubbles. Elastic interaction forces are proportional to the disk overlap, f el ij = k(R i + R j − r ij ) α el , where r ij := r j − r i points from one bubble center to another and R i labels the radius of disk i. In the full model that we focus...

Investigations of counter-rotating Taylor-Couette flow (TCF) in the narrow gap limit are conducted in a very large aspect ratio apparatus. The phase diagram is presented and compared to that obtained by Andereck et al. [1]. The spiral turbulence regime is studied by varying both internal and external Reynolds numbers. Spiral turbulence is shown to emerge from the fully turbulent regime via a continuous transition appearing first as a modulated turbulent state, which eventually relaxes locally to the laminar flow. The connection with the intermittent regimes of the plane Couette flow (pCf) is discussed. 47.20.Ft 47.20.Ky 47.54.+r, 47.27.Cn The "barber pole structure of turbulence" [2] between two counter-rotating cylinders, also called spiral turbulence, is commonly described as alternating helical stripes of laminar and turbulent flow. There are few quantitative studies of this puzzling regime, where long range order coexists with small scale turbulence. In early studies Coles and Van Atta [3][4][5] [6,7] described spiral turbulence within the framework of phase dynamics. All these studies were limited by their relatively small size. Only one helical turbulent stripe, winding no more than twice along the cylinder axis, could be observed. Altogether, the origin of this flow pattern remains unknown.Performing measurements in large aspect ratio TaylorCouette flow, we show that the spiral turbulence bifurcates continuously from the turbulent flow, appearing as a modulated turbulent state. After a rapid description of the experimental set up, we present the phase diagram and compare it to the one obtained by Andereck et al. [1] with a different cylinder radius ratio. Then we describe the successive steps leading to the fully turbulent flow before discussing the origin of spiral turbulence. Finally, we examine its breakdown into a spatio-temporal disordered regime similar to the laminar-turbulent coexistence dynamics observed in plane Couette flow [8,9]. We visualize the flow by a "fluorescent lighting "technique [10] developed for this study. The water flow is seeded with Kalliroscope AQ 1000 (6 × 30 × 0.07µm platelets). The inner cylinder is covered by a fluorescent film and the entire apparatus is UV-lighted. The fluorescent film re-emits a uniform visible lighting, transmitted through the fluid layer: the more turbulent the flow, the brighter it appears. As the gap is very thin, the Kalliroscope concentration is increased up to 25% by volume to enhance the contrast. A rheological study has shown that the fluid remains Newtonian, so that the only impact is a viscosity increase up to ν = 1.13 10 −6 m 2 /s at 20 • C. The flow is thermalized by water circulation inside the inner cylinder. At thermal equilibrium the temperature is uniform in space up to 0.1 • C and does not vary more than 0.1 • C/hour. Images and spatio-temporal diagrams (temporal recording of one line along the cylinder axis) are recorded by a CCD camera. Two plane mirrors reflect the two thirds of the flow hidden to the camera so that the whole cy...