By adding minute concentrations of a high molecular weight polymer, liquid jets or bridges collapsing under the action of surface tension develop a characteristic shape of uniform threads connecting spherical fluid drops. In this paper, high-precision measurements of this beads-on-string structure are combined with a theoretical analysis of the limiting case of large polymer relaxation times, for which the evolution can be divided into two distinct regimes. This excludes the very late stages of the evolution, for which the polymers have become fully stretched. For times smaller than the polymer relaxation time, over which the beads-on-string structure develops, we give a simplified local description, which still contains the full complexity of the problem. At times much larger than the relaxation time, we show that the solution consists of exponentially thinning threads connecting almost spherical drops. Both experiment and theoretical analysis of a one-dimensional model equation reveal a self-similar structure of the corner where a thread is attached to the neighbouring drops.

We study the impact of a fluid drop onto a planar solid surface at high speed so that at impact, kinetic energy dominates over surface energy and inertia dominates over viscous effects. As the drop spreads, it deforms into a thin film, whose thickness is limited by the growth of a viscous boundary layer near the solid wall. Owing to surface tension, the edge of the film retracts relative to the flow in the film and fluid collects into a toroidal rim bounding the film. Using mass and momentum conservation, we construct a model for the radius of the deposit as a function of time. At each stage, we perform detailed comparisons between theory and numerical simulations of the Navier-Stokes equation.

Many key phenomena in physics and engineering are described as singularities in the solutions to the differential equations describing them. Examples covered thoroughly in this book include the formation of drops and bubbles, the propagation of a crack, and the formation of a shock in a gas. Aimed at a broad audience, this book provides the mathematical tools for understanding singularities and explains the many common features in their mathematical structure. Part I introduces the main concepts and techniques, using the most elementary mathematics possible so that it can be followed by readers with only a general background in differential equations. Parts II and III require more specialized methods of partial differential equations, complex analysis, and asymptotic techniques. The book may be used for advanced fluid mechanics courses and as a complement to a general course on applied partial differential equations.

We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist globally in time.

We survey rigorous, formal, and numerical results on the formation of point-like singularities (or blow-up) for a wide range of evolution equations. We use a similarity transformation of the original equation with respect to the blow-up point, such that self-similar behaviour is mapped to the fixed point of a dynamical system. We point out that analysing the dynamics close to the fixed point is a useful way of characterising the singularity, in that the dynamics frequently reduces to very few dimensions. As far as we are aware, examples from the literature either correspond to stable fixed points, low-dimensional centre-manifold dynamics, limit cycles, or travelling waves. For each "class" of singularity, we give detailed examples.

We investigate the collapse of an axisymmetric cavity or bubble inside a fluid of small viscosity, like water. Any effects of the gas inside the cavity as well as of the fluid viscosity are neglected. Using a slender-body description, we show that the minimum radius of the cavity scales like h0 ∝ t ′α , where t ′ is the time from collapse. The exponent α very slowly approaches a universal value according to α = 1/2 + 1/(4 − ln(t ′ )). Thus, as observed in a number of recent experiments, the scaling can easily be interpreted as evidence of a single non-trivial scaling exponent. Our predictions are confirmed by numerical simulations. PACS numbers: Valid PACS appear hereOver the last decade, there has been considerable progress in understanding the pinch-off of fluid drops, described by a set of universal scaling exponents, independent of the initial conditions [1,2]. The driving is provided for by surface tension, the value of the exponents depend on the forces opposing it: inertia, viscosity, or combinations thereof. Bubble collapse appears to be a special case of an inviscid fluid drop breaking up inside another inviscid fluid, which is a well studied problem [3,4,5]: the minimum drop radius scales like h 0 ∝ t ′2/3 , where t ′ = t 0 − t and t 0 is the pinch-off time. Thus, huge excitement was caused by the results of recent experiments on the pinch-off of an air bubble [6,7,8,9,10], or the collapse of a cavity [11] in water, which resulted in a radically different picture, in agreement with two earlier studies [12,13]. As demonstrated in detail in [10], the air-water system corresponds to an inner "fluid" of vanishing inertia, surrounded by an ideal fluid.Firstly, the scaling exponent α was found to be close to 1/2, (typical values reported in the literature are 0.56 [9] and 0.57 [10]), which means that breakup is much faster than in the fluid-fluid case, and surface tension must become irrelevant as a driving force. Secondly, the value of α appeared to depend subtly on the initial condition [11], and was typically found to be larger than 1/2. This raised the possibility of an "anomalous" exponent, selected by a mechanism as yet unknown. To illustrate the qualitative appearance of the pinch-off of a bubble, in Fig. 1 we show a temporal sequence of profiles, using a full numerical simulation of the inviscid flow equations [5]. We confine ourselves to axisymmetric flow, which experimentally is found to be preserved down to a scale of a micron [10], provided the experiment is aligned carefully [9].The only existing theoretical prediction [7,11,15] is based on treating the bubble as a (slightly perturbed) cylinder [12,13]. This leads to the exponent being 1/2 with logarithmic corrections, a result which harks back to the 1940's [16]. Our numerics, to be reported below, are inconsistent with this result. Moreover, a cylinder is not a particularly good description of the actual profiles (cf. Fig. 1), as has been remarked before [9]. In this Letter, we present a systematic expansion in the slenderness of the cavity, ...

In this work, we show evidence of the existence of singularities developing in finite time for a class of contour dynamics equations depending on a parameter 0 < ␣ < 1. The limiting case ␣ 3 0 corresponds to 2D Euler equations, and ␣ ‫؍‬ 1 corresponds to the surface quasi-geostrophic equation. The singularity is point-like, and it is approached in a self-similar manner.alpha-patches ͉ quasi-geostrophic equation ͉ blow-up ͉ Euler equations ͉ self-similar behavior O ne of the most important open problems in mathematical fluid dynamics is whether the solutions to the Euler and Navier-Stokes equations modeling the evolution of incompressible inviscid and viscous fluids, respectively, may develop singularities in finite time. Several possible scenarios for a singularity have been proposed in the past (see ref. 1 for an account of some of them), although none of them led to a rigorous proof of the formation of singularities. One of these scenarios corresponds to the so-called vortex patch problem that we briefly describe below.A vortex patch consist of a 2D region D(t) (simply connected and bounded) that evolves with a velocity given at each instant of time bywhere the stream function is such that ϭ Ϫ⌬ , and the vorticity has a constant value 0 over D(t). Vortex patches are, therefore, weak solutions of Euler equations. The appearance of finite time singularities at the contour of D(t) was the subject of strong debate based on numerical results showing that the curvature of the boundary might grow superexponentially in time (see refs. 2-4). Nevertheless, work of Chemin (5) and Bertozzi and Constantin (6) rigorously proved global existence of regular solutions for the dynamics of the vortex patch, and, at least in this case, singularities cannot appear.A very natural singularity scenario would correspond to a vortex patch of the so-called surface quasi-geostrophic equation (see ref. 7) for which the relation between the stream function and potential temperature (that play the role that vorticity plays in 2D Euler equations) is ϭ (Ϫ⌬) 1/2 . The interest of this equation lies in its strong analogies to the 3D Euler equation as it has been argued in ref. 8 and its physical relevance as a model for the formation of temperature fronts in some geophysical contexts (see refs. 9 and 10).Hence, a natural question to ask is whether models for which ϭ (Ϫ⌬) 1Ϫ(␣/2) representing an interpolation between 2D Euler and quasi-geostrophic patches develop singularities for 0 Ͻ ␣ Յ 1. In this work, we provide numerical evidence showing that this scenario is indeed the case and describe the self-similar structure of such singularities. This result represents a previously undescribed singularity scenario for incompressible flows. The Model Following Zabusky et al. (4), we can invert the relation betweenand to obtain the following formula for the velocity with 0where x ជ(␥, t) is the position of points over C(t), the boundary of D(t), parameterized with ␥, and 0 ϭ ⅐c ␣ . Here is the value of in the patch and the factor c ␣ ϭ ⌫(␣/2)͞2 1Ϫ␣ ⌫(2...

It is well known that a viscoelastic jet breaks up much more slowly than a Newtonian jet. Typically, it evolves into the so-called beads-on-string structure, where large drops are connected by thin threads. The slow breakup process provides the viscoelastic jet sufficient time to exhibit some new phenomena. The aim of this paper is to investigate the drop dynamics of the beads-on-string structure. This includes drop migration, drop oscillation, drop merging and drop draining. We will use a 1D Oldroyd-B model for the viscoelastic jet, and solve this model numerically by an explicit finite difference method. Close to exponential draining of the filament, we found that the variation of the axial elastic force in the filament is roughly four times larger than the variation of the capillary force with opposite sign. This fact implies that the elastic force is responsible for the drop migration and oscillation. Our study of the drop draining process shows that the elastic force also plays an important role here, allowing the liquid to flow from smaller drops into larger drops through the filament.