Fragmentation instabilities of a drop as it falls in a miscible fluid

We review here a series of experiments on the fragmentation instabilities that a liquid drop undergoes as it falls inside a fluid with which it is miscible, so called the solvent. Motivated by the original experiments initiated by Thomson and Newall in 1885, we started to investigate this subject more than one decade ago, encountering up to date a number of challenging problems in hydrodynamical instabilities, complicated by the presence of transient interfaces between the drop and the solvent. In particular, we have shown that when a drop of liquid is deposited over the surface of the same liquid, it falls down inside the solvent because the energy associated to its surface tension against air is instantaneously converted into kinetic energy. As a consequence, a very fast fluid injection takes place as the drop touches the free surface of the solvent and the drop enters inside the solvent. Due to the hydrodynamical instabilities related to the large velocity gradients, it develops into a ring that expands radially. The ring continues to go downwards inside the solvent until it stops at a certain height due to viscous dissipation. In the first stages of the ring expansion, a fluid membrane remains attached to the ring, so-called “turban" for its shape, which is concave or convex depending on the sign of the density difference between the drop and the solvent. When a small density difference is introduced between the drop and the solvent, the ring becomes unstable because of density gradients and it fragments into smaller droplets. On their turn, the secondary droplets may undergo the same instability and may fragment again, so that a cascade of fragmentation takes place. If the density difference is positive, that is, the drop is heavier than the solvent, the secondary droplets continue to go down deeper inside the solvent, until the whole process is washed out by the slow diffusion of the concentration gradients. When the density difference between the drop and the solvent is negative, then the secondary droplets rise up to the free surface of the solvent, where they are distorted by the equivalent of an impact with a rigid wall. Universal scaling laws for the cascade of fragmentation and for the dynamical behavior of the drop have been derived and accompany the experimental observations.     

Lateral vibration of a water drop and its motion on a vibrating surface

The resonant modes of sessile water drops on a hydrophobic substrate subjected to a small-amplitude lateral vibration are investigated using computational fluid dynamic (CFD) modeling. As the substrate is vibrated laterally, its momentum diffuses within the Stokes layer of the drop. Above the Stokes layer, the competition between the inertial and Laplace forces causes the formation of capillary waves on the surface of the drop. In the first part of this paper, the resonant states of water drops are illustrated by investigating the velocity profile and the hydrostatic force using a 3d simulation of the Navier-Stokes equation. The simulation also allows an estimation of the contact angle variation on both sides of the drop. In the second part of the paper, we investigate the effect of vibration on a water drop in contact with a vertical plate. Here, as the plate vibrates parallel to gravity, the contact line oscillates. Each oscillation is, however, rectified by hysteresis, thus inducing a ratcheting motion to the water droplet vertically downward. Maximum rectification occurs at the resonant states of the drop. A comparison between the frequency-dependent motion of these drops and the variation of contact angles on their both sides is made. The paper ends with a discussion on the movements of the drops on a horizontal hydrophobic surface subjected to an asymmetric vibration.     

Numerical investigation of the deformation mechanism of a bubble or a drop rising or falling in another fluid

A numerical method for simulating the motion and deformation of an axisymmetric bubble or drop rising or falling in another infinite and initially stationary fluid is developed based on the volume of fluid （VOF） method in the frame of two incompressible and immiscible viscous fluids under the action of gravity, taking into consideration of surface tension effects. A comparison of the numerical results by this method with those by other works indicates the validity of the method. In the frame of inviseid and incompressible fluids without taking into consideration of surface tension effects, the mechanisms of the generation of the liquid jet and the transition from spherical shape to toroidal shape during the bubble or drop deformation, the increase of the ring diameter of the toroidal bubble or drop and the decrease of its cross-section area during its motion, and the effects of the density ratio of the two fluids on the deformation of the bubble or drop are analysed both theoretically and numerically.     

Molecular Dynamics simulation of a polymer chain translocating through a nanoscopic pore

The detection of linear polymers translocating through a nanoscopic pore is a promising idea for the development of new DNA analysis techniques. However, the physics of constrained macromolecules and the fluid that surrounds them at the nanoscopic scale is still not well understood. In fact, many theoretical models of polymer translocation neglect both excluded-volume and hydrodynamic effects. We use Molecular Dynamics simulations with explicit solvent to study the impact of hydrodynamic interactions on the translocation time of a polymer. The translocation time τ that we examine is the unbiased (no charge on the chain and no driving force) escape time of a polymer that is initially placed halfway through a pore perforated in a monolayer wall. In particular, we look at the effect of increasing the pore radius when only a small number of fluid particles can be located in the pore as the polymer undergoes translocation, and we compare our results to the theoretical predictions of Chuang et al. (Phys. Rev. E 65, 011802 (2001)). We observe that the scaling of the translocation time varies from τ ∼ N ^11/5 to τ ∼ N ^9/5 as the pore size increases (N is the number of monomers that goes up to 31 monomers). However, the scaling of the polymer relaxation time remains consistent with the 9/5 power law for all pore radii.     

Structure of a tethered polymer under flow using molecular dynamics and hybrid molecular-continuum simulations

We analyse the structure of a single polymer tethered to a solid surface undergoing a Couette flow. We study the problem using molecular dynamics (MD) and hybrid MD-continuum simulations, wherein the polymer and the surrounding solvent are treated via standard MD, and the solvent flow farther away from the polymer is solved by continuum fluid dynamics (CFD). The polymer represents a freely jointed chain (FJC) and is modelled by Lennard-Jones (LJ) beads interacting through the FENE potential. The solvent (modelled as a LJ fluid) and a weakly attractive wall are treated at the molecular level. At large shear rates the polymer becomes more elongated than predicted by existing theoretical scaling laws. Also, along the normal-to-wall direction the structure observed for the FJC is, surprisingly, very similar to that predicted for a semiflexible chain. Comparison with previous Brownian dynamics simulations (which exclude both solvent and wall potential) indicates that these effects are due to the polymer–solvent and polymer–wall interactions. The hybrid simulations are in perfect agreement with the MD simulations, showing no trace of finite size effects. Importantly, the extra cost required to couple the MD and CFD domains is negligible.     

Instability of a charged spherical drop moving relative to a medium

It is shown that the critical self-charge for the onset of instability of a charged drop in a flow of an ideal fluid decreases as the flow velocity of the fluid past the drop increases, i.e., a complex instability arises which is a superposition of the instabilities of the free surface of the drop with respect to the tangential discontinuity of the velocity field at the free surface of the drop and with respect to the self-charge. Zh. Tekh. Fiz. 69, 7–14 (May 1999)     

Roughness scaling and sensitivity to initial conditions in a symmetric restricted ballistic deposition model

In this work, we introduce a restricted ballistic deposition model with symmetric growth rules that favors the formation of local finite slopes. It is the simplest model which, even without including a diffusive relaxation mode of the interface, leads to a macroscopic groove instability. By employing a finite-size scaling of numerical simulation data, we determine the scaling behavior of the surface structure grown over a one-dimensional substrate of linear size L. We found that the surface profile develops a macroscopic groove with the asymptotic surface width scaling as , with . The early-time dynamics is governed by the scaling law , with . We further investigate the sensitivity to initial conditions of the present model by applying damage spreading techniques. We find that the early-time distance between two initially close surface configurations grows in a ballistic fashion as , but a slower Brownian-like scaling () sets up for evolution times much larger than a characteristic time scale . Received 26 May 2000     

From bouncing to floating: noncoalescence of drops on a fluid bath

When a drop of a viscous fluid is deposited on a bath of the same fluid, it is shown that its coalescence with this substrate is inhibited if the system oscillates vertically. Small drops lift off when the peak acceleration of the surface is larger than g. This leads to a steady regime where a drop can be kept bouncing for any length of time. It is possible to inject more fluid into the drop to increase its diameter up to several centimeters. Such a drop remains at the surface, forming a large sunk hemisphere. When the oscillation is stopped, the two fluids remain separated by a very thin air film, which drains very slowly (approximately 30 min). An analysis using lubrication theory accounts for most of the observations.     

Transition from symmetric to asymmetric scaling function before drop pinch-off

The drop pinch-off at a nozzle is studied experimentally for a glycerin-water mixture in surrounding air. The neck diameter of the fluid shrinks with constant velocity. After a distinct transition point, the shrink velocity switches to a smaller value. Before that transition point, the shape of the neck can well be described by a symmetric scaling function, as obtained from Stokes-flow theory of drop formation. This function gives way to an asymmetric scaling function in the final stage before pinch-off.     

Surface diffusion dynamics of a single polymer chain in dilute solution

Comprehensive three-dimensional dissipative particle dynamics simulations are carried out to elucidate the diffusion mechanism of a strongly adsorbed polymer chain on a solid surface in dilute solutions. We find Rouse and reptation dynamics for polymer chain diffusing on smooth and rough surfaces (with obstacles or sticking points), respectively. Combining with scaling analysis, we find that the interactions between the surface and the fluid screen the hydrodynamic interaction. The different scaling as found for a polymer chain diffusing on a fluid membrane [Phys. Rev. Lett. 82, 1911 (1999)] and on a solid surface [Nature (London) 406, 146 (2000)] may be explained by the solid surface inhomogeneity that induces reptation.     

Holographic flow visualization as a tool for studying three-dimensional coherent structures and instabilities

Holography is capable of three-dimensional (3D) representation of spatial objects such as fluid interfaces and particle ensembles. Based on this, we adapt it into a 3D flow visualization tool called Holographic Flow Visualization (HFV). This technique provides a novel means of studying spatially and temporally evolving complex fluid flow structures marked by a disperse phase or interfaces of different fluids. This paper demonstrates that HFV is a straightforward technique, especially when the In-line Recording Off-axis Viewing (IROV) configuration is used. The technique can be applied either as a stand-alone experimental tool for studying scalar-based coherent structures, flow instabilities, interactions of different fluids driven by fluid dynamics, interfacial phenomena, or as a precursor to volumetric 3D velocity vector field measurement of complex transient flow dynamics. Experimental results in several complex fluid flows and flames demonstrate the effectiveness of HFV. Different methods are used to mark flow structures undergoing different instabilities: 1) a vortex ring grown out of a drop of polymer suspension falling in water, 2) cascade of a bag-shaped drop of milk in water, and 3) internal flow structures of a jet diffusion flame.     

Scaling and self-similarity in an unforced flow of inviscid fluid trapped inside a viscous fluid in a Hele-Shaw cell

We investigate quasi-two-dimensional relaxation, by surface tension, of a long straight stripe of inviscid fluid trapped inside a viscous fluid in a Hele-Shaw cell. Combining analytical and numerical solutions, we describe the emergence of a self-similar dumbbell shape and find nontrivial dynamic exponents that characterize scaling behavior of the dumbbell dimensions.     

Adsorption of hydrophobic polyelectrolytes onto oppositely charged surfaces

We present a scaling theory for the adsorption of a weakly charged polyelectrolyte chain in a poor solvent onto an oppositely charged surface. Depending on the fraction of charged monomers and on the solvent quality for uncharged monomers, the globule in the bulk of the solution has either a spherical conformation or a necklace structure. At sufficiently high surface charge density, a chain in the globular conformation adsorbs in a flat pancake conformation due to the Coulombic attraction to the oppositely charged surface. Different adsorption regimes are predicted depending on two screening lengths (the Debye screening length monitored by the salt concentration and the Gouy-Chapman length monitored by the surface charge density), on the degree of ionization of the polymer and on the solvent strength. At low bulk ionic strength, an increase in the surface charge density may induce a transition from an adsorbed necklace structure to a uniform pancake due to the enhanced screening of the intra-chain Coulombic repulsion by the counterions localized near the surface. Received 12 April 2001     

Fluid adsorption at the base of a cylinder

We consider the adsorption of fluid at a cylinder protruding from a flat substrate. For small contact angles θ, a liquid drop condenses at the base, the size of which is determined by macroscopic arguments. The adsorption exhibits scaling behavior related to a number of phase transitions and, for systems with short-ranged forces, shows a remarkable property: for small θ, the height and width of the drop are near identical to expressions for the thickness and parallel correlation length for microscopic wetting films. The only difference is that the bulk correlation length is replaced by the cylinder radius. This geometrical amplification of the microscopic lengths occurs for second-order, first-order, and complete wetting transitions, and is specific to three dimensions. Similar phenomena occurs for long-ranged forces, and shows crossover scaling behavior.     

Effective scaling behavior of magnetization and Hamming distance in Ising thin films under a surface field

The recent improvements on the technology for developing high-quality thin magnetic films has renewed the interest in the study of surface effects in both static and dynamic magnetic responses. In this work, we use a Monte-Carlo algorithm with Metropolis dynamics together with a spreading of damage technique to study the interplay between the effects of finite thickness and surface ordering field in thin ferromagnetic Ising (S=1/2) films. We calculate, near the bulk critical temperature and several values of the surface field, the dependence on the film thickness of the average magnetization M and Hamming distance D. We employ a finite size scaling analysis to show that both obey an effective one-parameter scaling but exhibit distinct characteristic surface fields. At their corresponding characteristic surface fields both M and D become roughly thickness independent and we estimate the critical exponent characterizing the behavior of the typical scaling lengths. Received 29 March 1999 and Received in final form 21 April 1999     

Self-assembly of gradient concentric rings via solvent evaporation from a capillary bridge

A drop of solution containing nonvolatile solute is allowed to evaporate from a sphere-on-flat geometry. Left behind is a striking pattern of gradient concentric rings with unprecedented regularity. The center-to-center distance between adjacent rings, lambda(C-C), and the height of the ring, h(d), are strongly affected by the concentration of the solution and the properties of the solvent. The nature of the formation of regular gradient ring patterns during the course of irreversible solvent evaporation is revealed through theoretical calculations based on the mass conservation of the solution.     

Phase separation and pattern formation in a binary Bose-Einstein condensate

The miscibility-immiscibility phase transition in binary Bose-Einstein condensates (BECs) can be controlled by a coupling between the two components. Here we propose a new scheme that uses coupling-induced pattern formation to test the Kibble-Zurek mechanism (KZM) of topological-defect formation in a quantum phase transition. For a binary BEC in a ring trap we find that the number of domains forming the pattern scales as a function of the coupling quench rate with an exponent as predicted by the KZM. For a binary BEC in an elongated harmonic trap we find a different scaling law due to the transition being spatially inhomogeneous. We perform a "quantum simulation" of the harmonically trapped system in a ring trap to verify the scaling exponent.     

Influence of hydrodynamics on the dynamics of a homopolymer

We present an analytical approach of the dynamics of a polymer when it is quenched from a solvent into a good or bad solvent. The dynamics is studied by means of a Langevin equation, first in the absence of hydrodynamic effect, then taking into account the hydrodynamic interactions with the solvent. The variation of the radius of gyration is studied as a function of time. In both cases, for the first stage of collapse or swelling, the evolution is described by a power law with a characteristic time proportional to N ^4/3 (N), where N is the number of monomers, without (with) hydrodynamic interactions. At larger times, scaling laws are derived for the diffusive relaxation time. Received: 10 March 1998 / Received in final form: 15 September 1998 / Accepted: 25 September 1998     

Short-time dynamics of partial wetting

When a liquid drop contacts a wettable surface, the liquid spreads over the solid to minimize the total surface energy. The first moments of spreading tend to be rapid. For example, a millimeter-sized water droplet will wet an area having the same diameter as the drop within a millisecond. For perfectly wetting systems, this spreading is inertially dominated. Here we identify that even in the presence of a contact line, the initial wetting is dominated by inertia rather than viscosity. We find that the spreading radius follows a power-law scaling in time where the exponent depends on the equilibrium contact angle. We propose a model, consistent with the experimental results, in which the surface spreading is regulated by the generation of capillary waves.