Pearling and Pinching: Propagation of Rayleigh Instabilities

Powers, Thomas; Goldstein, Raymond E.

doi:10.1103/physrevlett.78.2555

Cited by 69 publications

(65 citation statements)

References 25 publications

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“…First of all, taking the spreading velocity v * of a linear perturbation of the unstable state (Eqs. (1.4) and (1.5)) as the generalization of v * = 2f (0) 1/2 for (1.1), we observe that there are numerous examples [9,12,17,22,61,[64][65][66]116] of fronts whose asymptotic velocity approaches the pulled value v * given by (1.5). So there is no doubt that the mechanism of fronts "being pulled along" by the leading edge generalizes to a large class of equations.…”

Section: Introductionmentioning

confidence: 94%

“…In [65], it was argued that the factor 3 2 in this result applies more generally to higher order equations as well, but a systematic analysis or an argument for why the convergence is uniform, was missing. Apart from this and a recent rederivation [70] of Bramson's result along lines similar in spirit to ours 8 and a few papers similar in spirit to that of Bramson [53,76,77] 9 , we are not aware of systematic calculations of the velocity and profile relaxation. Even for the convergence of the velocity in the celebrated nonlinear diffusion equation, our 1/t 3/2 term appears to be new.…”

Section: Outline Of the Problemmentioning

confidence: 99%

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Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts

Ebert

Saarloos

2000

Physica D: Nonlinear Phenomena

359

596

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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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Section: Introductionmentioning

confidence: 94%

Section: Outline Of the Problemmentioning

confidence: 99%

Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts

Ebert

Saarloos

2000

Physica D: Nonlinear Phenomena

359

596

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“…Leonhard Euler derived the celebrated Euler buckling formula for slender columns in the mid-18th century and the buckling of plates and shells was analyzed in the early 20th century (20). In the last few decades, buckling instabilities have been used to model and understand the force-deformation response of a wide variety of biological structures including DNA (32), cytoskeletal filaments (33)(34)(35)(36), membranes (37)(38)(39)(40)(41), white blood cells (42), viruses (43), and tissues (44)(45)(46)(47). Here we present an example in which the buckling of curved membranes can explain the ultradonut topology of the nuclear envelope.…”

Section: Discussionmentioning

confidence: 99%

Ultradonut topology of the nuclear envelope

Torbati

Lele

Agrawal

2016

Proc. Natl. Acad. Sci. U.S.A.

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The nuclear envelope is a unique topological structure formed by lipid membranes in eukaryotic cells. Unlike other membrane structures, the nuclear envelope comprises two concentric membrane shells fused at numerous sites with toroid-shaped pores that impart a "geometric" genus on the order of thousands. Despite the intriguing architecture and vital biological functions of the nuclear membranes, how they achieve and maintain such a unique arrangement remains unknown. Here, we used the theory of elasticity and differential geometry to analyze the equilibrium shape and stability of this structure. Our results show that modest inand out-of-plane stresses present in the membranes not only can define the pore geometry, but also provide a mechanism for destabilizing membranes beyond a critical size and set the stage for the formation of new pores. Our results suggest a mechanism wherein nanoscale buckling instabilities can define the global topology of a nuclear envelope-like structure.nuclear envelope | lipid membranes | topology | buckling instability T he cell nucleus is bounded by two lipid bilayers arranged in a unique geometry called the nuclear envelope. These two bilayers are shaped into concentric spheres that are maintained at a remarkably uniform spacing of ∼ 30−50 nm (1), and yet are fused together at thousands of pores (holes) at an average spacing of ∼ 250−500 nm from each other [based on areal density measurements (2-4)]. At these pores, the membranes locally take on a toroid shape with a radius of curvature on the order of ∼ 20 nm ( Fig. 1). Lipid structures such as vesicles, spherocylinders (bacterial membranes), and biconcave discoids (red blood cell membranes) are all closed structures with no holes, and hence possess a zero genus (5) (Fig. 1). A donut, on the other hand, is also a closed structure but has one hole, and as a result has a genus of 1 (Fig. 1). If we fuse two donuts, we get a shape with a genus of 2, and if we fuse thousands of donuts and bend them to form a sphere, we obtain a nucleus-membrane-like structure ( Fig. 1). This structure, therefore, can be thought of as an "ultradonut" with a genus on the order of thousands. How the two membranes assemble in a unique arrangement with such a large number of local fusions is a fundamental question in both physics and biology that is still unresolved (6-10). To seek an answer to this puzzle, we ask three natural questions based on the common notions in the field of membrane physics.First, can membrane curvature-mediated interactions determine optimal pore number and interpore separation? This principle has been successfully used to predict the interactions of membraneembedded proteins and nanoparticles (11-13). However, in the case of nuclear envelope, Fig. 1C (14) shows that the curved shapes of the membranes at the pores do not persist over interpore length scales; The "curvature memory" of the membranes is lost beyond ∼ 100 nm and they remain essentially flat in between the pores. As a result, a pore does not sense the presence of other p...

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“…Thus, with the purpose of calculating the boundary separating the two different drop formation processes depicted in figures 1, 2 and in the movies provided as supplementary material, we have determined the values of the capillary number for which the temporal growth rate of the perturbations with zero group velocity is also equal to zero. The result obtained using Tomotika's dispersion relation (Tomotika 1935;Powers & Goldstein 1997;Powers et al 1998;Gañán Calvo 2008), which describes the growth and propagation of perturbations in a cylindrical jet immersed into another immiscible liquid, is represented together with the experimental data in figure 3. Although the conditions of validity of Tomotika's analysis are fulfilled at large distances from the injector, where the diameter of the liquid jet is nearly constant, figure 3 reveals that the parallel flow stability analysis predicts larger values for the critical capillary number than those measured experimentally.…”

Section: Introductionmentioning

confidence: 99%

Global stability of stretched jets: conditions for the generation of monodisperse micro-emulsions using coflows

2013

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In this paper we reveal the physics underlaying the conditions needed for the generation of emulsions composed of uniformly sized drops of micrometric or submicrometric diameters when two immiscible streams flow in parallel under the so-called tip streaming regime after Suryo & Basaran (2006). Indeed, when inertial effects in both liquid streams are negligible, the inner to outer flow-rate and viscosity ratios are small enough and the capillary number is above an experimentally determined threshold which is predicted by our theoretical results with small relative errors, a steady micron-sized jet is issued from the apex of a conical drop. Under these conditions, the jet disintegrates into drops with a very well defined mean diameter, giving rise to a monodisperse microemulsion. Here, we demonstrate that the regime in which uniformly-sized drops are produced corresponds to values of the capillary number for which the cone-jet system is globally stable. Interestingly enough, our general stability theory reveals that liquid jets with a cone-jet structure are much more stable than their cylindrical counterparts thanks, mostly, to a capillary stabilization mechanism described here for the first time. Our findings also limit the validity of the type of stability analysis based on the common parallel flow assumption to only those situations in which the liquid jet diameter is almost constant.

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Pearling and Pinching: Propagation of Rayleigh Instabilities

Cited by 69 publications

References 25 publications

Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts

Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts

Ultradonut topology of the nuclear envelope

Global stability of stretched jets: conditions for the generation of monodisperse micro-emulsions using coflows

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