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J. Eggers

Publications and source records attributed to J. Eggers.

At least 19 recordsLinked to original sources
arXiv2024

When does the elastic regime begin in viscoelastic pinch-off?

In this experimental and numerical study, we revisit the question of the onset of the elastic regime in viscoelastic pinch-off. This is relevant for all modern filament thinning techniques which aim at measuring the extensional properties of low-viscosity polymer solutions such as the Slow Retraction Method (SRM) in Capillary Breakup Extensional Rheometry (CaBER) as well as the dripping method where a drop detaches from a nozzle. In these techniques, a stable liquid bridge is slowly brought to its stability threshold where capillary-driven thinning starts, slowing down dramatically at a critical radius $h_1$ marking the onset of the elastic regime where the bridge becomes a filament with elasto-capillary thinning dynamics. While a theoretical scaling for this transition radius exists for the classical step-strain CaBER protocol, where polymer chains stretch without relaxing during the fast plate separation, we show that it is not necessarily valid for a slow protocol such as in SRM since polymer chains only start stretching (beyond their equilibrium coiled configuration) when the bridge thinning rate becomes comparable to the inverse of their relaxation time. We derive a universal scaling for $h_1$ valid for both low and high-viscosity polymer solution which is validated by both CaBER (SRM) experiments with different polymer solutions, plate diameters and sample volumes and by numerical simulations using the FENE-P model.

cond-mat.soft↗
arXiv2022

Anomalous dimensions of the Smoluchowski coagulation equation

The coagulation (or aggregation) equation was introduced by Smoluchowski in 1916 to describe the clumping together of colloidal particles through diffusion, but has been used in many different contexts as diverse as physical chemistry, chemical engineering, atmospheric physics, planetary science, and economics. The effectiveness of clumping is described by a kernel $K(x,y)$, which depends on the sizes of the colliding particles $x,y$. We consider kernels $K = (xy)^γ$, but any homogeneous function can be treated using our methods. For sufficiently effective clumping $1 \ge γ> 1/2$, the coagulation equation produces an infinitely large cluster in finite time (a process known as the gel transition). Using a combination of analytical methods and numerics, we calculate the anomalous scaling dimensions of the main cluster growth, calling into question results much used in the literature. Apart from the solution branch which originates from the exactly solvable case $γ= 1$, we find a new branch of solutions near $γ= 1/2$, which violates scaling relations widely believed to hold universal.

cond-mat.soft↗
arXiv2021

Self-similarity in the breakup of very dilute viscoelastic solutions

When pushed out of a syringe, polymer solutions form droplets attached by long and slender cylindrical filaments whose diameter decreases exponentially with time before eventually breaking. In the last stages of this process, a striking feature is the self-similarity of the solution shape near the end of the filament. This means that shapes at different times, if properly rescaled, collapse onto one universal shape. A theoretical description inspired by this similarity observation and based on the Oldroyd-B model was recently shown to disagree with existing experimental results. By revisiting these measurements and analysing the interface profiles of very diluted polyethylene oxide solutions at high temporal and spatial resolution, we show that they are very well described by the model.

cond-mat.soft↗
arXiv2020

Influence of the surface viscous stress on the pinch-off of free surfaces loaded with nearly-inviscid surfactants

We analyze both theoretically and experimentally the breakup of a pendant water droplet loaded with Sodium Dodecyl Sulfate (SDS). The free surface minimum radius measured in the experiments is compared with that obtained from a numerical solution of the full Navier-Stokes equations for different values of the shear and dilatational surface viscosities. This comparison shows the small but measurable effect of the surface viscous stresses on the system dynamics for sufficiently small spatiotemporal distances from the breakup point, and allows to establish upper bounds for the values of the shear and dilatational viscosities. We study numerically the distribution of Marangoni and viscous stresses over the free surface as a function of the time to the pinching, and describe how surface viscous stresses grow in the pinching region as the free surface approaches its breakup. When Marangoni and surface viscosity stresses are taken into account, the surfactant is not swept away from the thread neck in the time interval analyzed. Surface viscous stresses eventually balance the driving capillary pressure in that region for small enough values of the time to pinching. Based on this result, we propose a scaling law to account for the effect of the surface viscosities on the last stage of the temporal evolution of the neck radius.

physics.flu-dyn↗
arXiv2019

Self-similar breakup of polymeric threads as described by the Oldroyd-B model

When a drop of fluid containing long, flexible polymers breaks up, it forms threads of almost constant thickness, whose size decreases exponentially in time. Using an Oldroyd-B fluid as a model, we show that the thread profile, rescaled by the thread thickness, converges to a similarity solution. Using the correspondence between viscoelastic fluids and non-linear elasticity, we derive similarity equations for the full three-dimensional axisymmetric flow field in the limit that the viscosity of the solvent fluid can be neglected. A conservation law balancing pressure and elastic energy permits to calculate the thread thickness exactly. The explicit form of the velocity and stress fields can be deduced from a solution of the similarity equations. Results are validated by detailed comparison with numerical simulations.

physics.flu-dyn↗
arXiv2019

The relationship between viscoelasticity and elasticity

We consider models for elastic liquids, such as solutions of flexible polymers. They introduce a relaxation time $λ$ into the system, over which stresses relax. We study the kinematics of the problem, and clarify the relationship between Lagrangian and Eulerian descriptions, thereby showing which polymer models correspond to a nonlinear elastic deformation in the limit $λ\rightarrow\infty$. This allows us to split the change in elastic energy into reversible and dissipative parts, and thus to write an equation for the total energy, the sum of kinetic and elastic energies. As an illustration, we show how the presence or absence of an elastic limit determines the fate of an elastic thread during capillary instability, using novel numerical schemes based on our insights into the flow kinematics.

physics.flu-dyn↗
arXiv2017

Cusp-shaped Elastic Creases and Furrows

The surfaces of growing biological tissues, swelling gels, and compressed rubbers do not remain smooth, but frequently exhibit highly localized inward folds. We reveal the morphology of this surface folding in a novel experimental setup, which permits to deform the surface of a soft gel in a controlled fashion. The interface first forms a sharp furrow, whose tip size decreases rapidly with deformation. Above a critical deformation, the furrow bifurcates to an inward folded crease of vanishing tip size. We show experimentally and numerically that both creases and furrows exhibit a universal cusp-shape, whose width scales like $y^{3/2}$ at a distance $y$ from the tip. We provide a similarity theory that captures the singular profiles before and after the self-folding bifurcation, and derive the length of the fold from large deformation elasticity.

cond-mat.soft↗
arXiv2016

Spatial structure of shock formation

The formation of a singularity in a compressible gas, as described by the Euler equation, is characterized by the steepening, and eventual overturning of a wave. Using a self-similar description in two space dimensions, we show that the spatial structure of this process, which starts at a point, is equivalent to the formation of a caustic, i.e. to a cusp catastrophe. The lines along which the profile has infinite slope correspond to the caustic lines, from which we construct the position of the shock. By solving the similarity equation, we obtain a complete local description of wave steepening and of the spreading of the shock from a point.

physics.flu-dyn↗
arXiv2016

Shock formation in the dispersionless Kadomtsev-Petviashvili equation

The dispersionless Kadomtsev-Petviashvili (dKP) equation $(u_t+uu_x)_x=u_{yy}$ is one of the simplest nonlinear wave equations describing two-dimensional shocks. To solve the dKP equation we use a coordinate transformation inspired by the method of characteristics for the one-dimensional Hopf equation $u_t+uu_x=0$. We show numerically that the solutions to the transformed equation do not develop shocks. This permits us to extend the dKP solution as the graph of a multivalued function beyond the critical time when the gradients blow up. This overturned solution is multivalued in a lip shape region in the $(x,y)$ plane, where the solution of the dKP equation exists in a weak sense only, and a shock front develops. A local expansion reveals the universal scaling structure of the shock, which after a suitable change of coordinates corresponds to a generic cusp catastrophe. We provide a heuristic derivation of the shock front position near the critical point for the solution of the dKP equation, and study the solution of the dKP equation when a small amount of dissipation is added. Using multiple-scale analysis, we show that in the limit of small dissipation and near the critical point of the dKP solution, the solution of the dissipative dKP equation converges to a Pearcey integral. We test and illustrate our results by detailed comparisons with numerical simulations of both the regularized equation, the dKP equation, and the asymptotic description given in terms of the Pearcey integral.

math-ph↗
arXiv2008

Vibration-induced climbing of drops

We report an experimental study of liquid drops moving against gravity, when placed on a vertically vibrating inclined plate, which is partially wetted by the drop. The frequency of vibrations ranges from 30 to 200 Hz, and, above a threshold in vibration acceleration, drops experience an upward motion. We attribute this surprising motion to the deformations of the drop, as a consequence of an up or down symmetry breaking induced by the presence of the substrate. We relate the direction of motion to contact angle measurements. This phenomenon can be used to move a drop along an arbitrary path in a plane, without special surface treatments or localized forcing.

physics.flu-dyn↗
arXiv2008

Thick films coating a plate withdrawn from a bath

We consider the deposition of a film of viscous liquid on a flat plate being withdrawn from a bath, experimentally and theoretically. For any plate speed $U$, there is a range of ``thick'' film solutions whose thickness scales like $U^{1/2}$ for small $U$. These solutions are realized for a partially wetting liquid, while for a perfectly wetting liquid the classical Landau-Levich-Derjaguin (LLD) film is observed, whose thickness scales like $U^{2/3}$. The thick film is distinguished from the LLD film by a dip in its spatial profile at the transition to the bath. We calculate the phase diagram for the existence of stationary film solutions as well as the film profiles, and find excellent agreement with experiment.

physics.flu-dyn↗
arXiv2007

Blistering of viscoelastic filaments

When a dilute polymer solution experiences capillary thinning, it forms an almost uniformly cylindrical thread, which we study experimentally. In the last stages of thinning, when polymers have become fully stretched, the filament becomes prone to instabilities, of which we describe two: A novel "breathing" instability, originating from the edge of the filament, and a sinusoidal instability in the interior, which ultimately gives rise to a "blistering" pattern of beads on the filament. We describe the linear instability with a spatial resolution of 80 nm in the disturbance amplitude. For sufficiently high polymer concentrations, the filament eventually separates out into a "solid" phase of entangled polymers, connected by fluid beads. A solid polymer fiber of about 100 nanometer thickness remains, which is essentially permanent.

physics.flu-dyn↗
arXiv2007

Film transitions of receding contact lines

When a solid plate is withdrawn from a liquid bath, a receding contact line is formed where solid, liquid, and gas meet. Above a critical speed $U_{cr}$, a stationary contact line can no longer exist and the solid will eventually be covered completely by a liquid film. Here we show that the bifurcation diagram of this coating transition changes qualitatively, from discontinuous to continuous, when decreasing the inclination angle of the plate. We show that this effect is governed by the presence of capillary waves, illustrating that the large scale flow strongly effects the maximum speed of dewetting.

physics.flu-dyn↗
arXiv2006

Theory of the collapsing axisymmetric cavity

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 $h_0 \propto t'^α$, where $t'$ is the time from collapse. The exponent $α$ very slowly approaches a universal value according to $α=1/2 + 1/(4\sqrt{-\ln(t')})$. 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.

physics.flu-dyn↗
arXiv2006

To knot or not to knot

We study the formation of knots on a macroscopic ball-chain, which is shaken on a horizontal plate at 12 times the acceleration of gravity. We find that above a certain critical length, the knotting probability is independent of chain length, while the time to shake out a knot increases rapidly with chain length. The probability if finding a knot after a certain time is the result of the balance of these two processes. In particular, the knotting probability tends to a constant for long chains.

cond-mat.stat-mech↗
arXiv2005

Sink flow deforms the interface between a viscous liquid and air into a tip singularity

In our experiment, an interface between a viscous liquid and air is deformed by a sink flow of constant flow rate to form a sharp tip. Using a microscope, the interface shape is recorded down to a tip size of 1 $\rm{μm}$. The curvature at the tip is controlled by the distance $h$ between the tip and the sink. As a critical distance $h^{\star}$ is approached, the curvature diverges like $1/(h-h^{\star})^3$ and the tip becomes cone-shaped. As the distance to the sink is decreased further, the opening angle of the cone vanishes like $h^2$. No evidence for air entrainment was found, except when the tip was inside the orifice.

physics.flu-dyn↗
arXiv2004

Very viscous drops cannot break up

We consider an axisymmetric, freely suspended fluid drop with surface tension, whose viscosity is so large that both inertia and forcing by an external fluid can be ignored. We show that whatever be the initial condition, pinchoff can never occur.

physics.flu-dyn↗