Drop Behavior on Heterogeneous Ratchet-Structured Substrates Harmonically Vibrated in Lateral Direction.

We analyze numerically a new ratchet system: a liquid drop is sitting on a heterogeneous ratchet-structured solid plate. The coated plate is subject to a lateral harmonic oscillation. The systematic investigation performed in the frame of a phase field model shows the possibility of realizing a long-distance net-driven motion for isolated domains of the forcing parameters. The studied problem might be of considerable interest for controlled motion in micro- and nanofluidics.

Stochastic Compartment Model with Mortality and Its Application to Epidemic Spreading in Complex Networks.

We study epidemic spreading in complex networks by a multiple random walker approach. Each walker performs an independent simple Markovian random walk on a complex undirected (ergodic) random graph where we focus on the Barabási-Albert (BA), Erdös-Rényi (ER), and Watts-Strogatz (WS) types. Both walkers and nodes can be either susceptible (S) or infected and infectious (I), representing their state of health. Susceptible nodes may be infected by visits of infected walkers, and susceptible walkers may be infected by visiting infected nodes. No direct transmission of the disease among walkers (or among nodes) is possible. This model mimics a large class of diseases such as Dengue and Malaria with the transmission of the disease via vectors (mosquitoes). Infected walkers may die during the time span of their infection, introducing an additional compartment D of dead walkers. Contrary to the walkers, there is no mortality of infected nodes. Infected nodes always recover from their infection after a random finite time span. This assumption is based on the observation that infectious vectors (mosquitoes) are not ill and do not die from the infection. The infectious time spans of nodes and walkers, and the survival times of infected walkers, are represented by independent random variables. We derive stochastic evolution equations for the mean-field compartmental populations with the mortality of walkers and delayed transitions among the compartments. From linear stability analysis, we derive the basic reproduction numbers RM,R0 with and without mortality, respectively, and prove that RM1, the healthy state is unstable, whereas for zero mortality, a stable endemic equilibrium exists (independent of the initial conditions), which we obtained explicitly. We observed that the solutions of the random walk simulations in the considered networks agree well with the mean-field solutions for strongly connected graph topologies, whereas less well for weakly connected structures and for diseases with high mortality. Our model has applications beyond epidemic dynamics, for instance in the kinetics of chemical reactions, the propagation of contaminants, wood fires, and others.

Four-compartment epidemic model with retarded transition rates.

We study an epidemic model for a constant population by taking into account four compartments of the individuals characterizing their states of health. Each individual is in one of the following compartments: susceptible S; incubated, i.e., infected yet not infectious, C; infected and infectious I; and recovered, i.e., immune, R. An infection is visible only when an individual is in state I. Upon infection, an individual performs the transition pathway S→C→I→R→S, remaining in compartments C, I, and R for a certain random waiting time t_{C}, t_{I}, and t_{R}, respectively. The waiting times for each compartment are independent and drawn from specific probability density functions (PDFs) introducing memory into the model. The first part of the paper is devoted to the macroscopic S-C-I-R-S model. We derive memory evolution equations involving convolutions (time derivatives of general fractional type). We consider several cases. The memoryless case is represented by exponentially distributed waiting times. Cases of long waiting times with fat-tailed waiting-time distributions are considered as well where the S-C-I-R-S evolution equations take the form of time-fractional ordinary differential equations. We obtain formulas for the endemic equilibrium and a condition of its existence for cases when the waiting-time PDFs have existing means. We analyze the stability of healthy and endemic equilibria and derive conditions for which the endemic state becomes oscillatory (Hopf) unstable. In the second part, we implement a simple multiple-random-walker approach (microscopic model of Brownian motion of Z independent walkers) with random S-C-I-R-S waiting times in computer simulations. Infections occur with a certain probability by collisions of walkers in compartments I and S. We compare the endemic states predicted in the macroscopic model with the numerical results of the simulations and find accordance of high accuracy. We conclude that a simple random-walker approach offers an appropriate microscopic description for the macroscopic model. The S-C-I-R-S-type models open a wide field of applications allowing the identification of pertinent parameters governing the phenomenology of epidemic dynamics such as extinction, convergence to a stable endemic equilibrium, or persistent oscillatory behavior.

Simple model of epidemic dynamics with memory effects.

We introduce a compartment model with memory for the dynamics of epidemic spreading in a constant population of individuals. Each individual is in one of the states S=susceptible, I=infected, or R=recovered (SIR model). In state R an individual is assumed to stay immune within a finite-time interval. In the first part, we introduce a random lifetime or duration of immunity which is drawn from a certain probability density function. Once the time of immunity is elapsed an individual makes an instantaneous transition to the susceptible state. By introducing a random duration of immunity a memory effect is introduced into the process which crucially determines the epidemic dynamics. In the second part, we investigate the influence of the memory effect on the space-time dynamics of the epidemic spreading by implementing this approach into computer simulations and employ a multiple random walker's model. If a susceptible walker meets an infectious one on the same site, then the susceptible one gets infected with a certain probability. The computer experiments allow us to identify relevant parameters for spread or extinction of an epidemic. In both parts, the finite duration of immunity causes persistent oscillations in the number of infected individuals with ongoing epidemic activity preventing the system from relaxation to a steady state solution. Such oscillatory behavior is supported by real-life observations and not captured by the classical standard SIR model.

A Markovian random walk model of epidemic spreading.

We analyze the dynamics of a population of independent random walkers on a graph and develop a simple model of epidemic spreading. We assume that each walker visits independently the nodes of a finite ergodic graph in a discrete-time Markovian walk governed by his specific transition matrix. With this assumption, we first derive an upper bound for the reproduction numbers. Then, we assume that a walker is in one of the states: susceptible, infectious, or recovered. An infectious walker remains infectious during a certain characteristic time. If an infectious walker meets a susceptible one on the same node, there is a certain probability for the susceptible walker to get infected. By implementing this hypothesis in computer simulations, we study the space-time evolution of the emerging infection patterns. Generally, random walk approaches seem to have a large potential to study epidemic spreading and to identify the pertinent parameters in epidemic dynamics.

Rayleigh-Taylor and Kelvin-Helmholtz instability studied in the frame of a dimension-reduced model.

Introducing an extension of a recently derived dimension-reduced model for an infinitely deep inviscid and irrotational layer, a two-layer system is examined in the present paper. A second thin viscous layer is added on top of the original one-layer system. The set-up is a combination of a long-wave approximation (upper layer) and a deep-water approximation (lower layer). Linear stability analysis shows the emergency of Rayleigh-Taylor and Kelvin-Helmholtz instabilities. Finally, numerical solutions of the model reveal spatial and temporal pattern formation in the weakly nonlinear regime of both instabilities. This article is part of the theme issue 'Stokes at 200 (Part 1)'.

Drop Behavior Influenced by the Correlation Length on Noisy Surfaces.

We investigate numerically the role of the correlation length in drop behavior on noisy surfaces. To this aim, a phase field tool has been used. Theoretical results are confirmed by experiments of distilled water drops sitting on stainless steel and silicon surfaces textured by laser-induced periodic self-organized structures: an increase of the noise amplitude results in an amplification of the original behavior (i.e., hydrophobic is getting more hydrophobic, hydrophilic is getting more hydrophilic). Furthermore, computer simulations in two and three spatial dimensions allow for predictions of drop behavior on noisy sloped substrates under a gravitational force, a problem of large interest in controlled motion in micro- and nanofluidics.

Can vibrations control drop motion?

We discuss a mechanism for controlled motion of drops with applications for microfluidics and microgravity. The mechanism is the following: a solid plate supporting a liquid droplet is simultaneously subject to lateral and vertical harmonic oscillations. In this way the symmetry of the back-and-forth droplet movement along the substrate under inertial effects is broken and thus will induce a net driven motion of the drop. We study the dependency of the traveled distance on the oscillation parameters (forcing amplitude, frequency, and phase shift between the two perpendicular oscillations) via phase field simulations. The internal flow structure inside the droplet is also investigated. We make predictions on resonance frequencies for drops on a substrate with a varying wettability.

Nonlinear pattern formation in thin liquid films under external vibrations.

We study a thin liquid film with a free surface on a planar horizontal substrate. The substrate is subjected to oscillatory accelerations in the normal and/or in the horizontal direction(s). The description is based on the longwave approximation including inertia effects, which are important due to the large velocities imparted by external vibrations. The linearized system is examined using the Floquet analysis. Pattern formation in the nonlinear regime is computed numerically from the longwave equations for the thickness and the flow rate of the fluid in two and in three spatial dimensions. For certain amplitude and frequency ranges, combined lateral and normal oscillations can give rise to one or more traveling drops, similar to recent experimental findings by Brunet et al. [Phys. Rev. Lett. 99, 144501 (2007)].

Partial coalescence of sessile drops with different miscible liquids.

Using computer simulations in three spatial dimensions, we examine the interaction between two deformable drops consisting of two perfectly miscible liquids sitting on a solid substrate under a given contact angle. Driven by capillarity and assisted by Marangoni effects at the droplet interfaces, several distinct coalescence regimes are achieved after the droplets' collision.

Nonlinear dynamics of thin liquid films consisting of two miscible components.

Recently, we systematically derived a system of two coupled conservation equations governing a thin liquid layer with a deformable surface composed of two completely miscible components [Phys. Fluids 22, 104102 (2010)]. One equation describes the location of the free surface and the second one the evolution of the mean concentration. This lubrication model was investigated previously in linearized form. The study is now extended to the fully nonlinear case of thin liquid films of a binary mixture (in one and two horizontal spatial dimensions) with and without heat transport. For an initially flat and motionless film heated from below, we analyze the component separation induced by the Soret effect. Nonlinear simulations show that the Soret effect can cause a multitude of interesting behaviors, such as oscillatory patterns and solitonlike structures (localized traveling drops or holes). A stronger component separation induced by stronger Soret effects favors faster-moving localized structures. For isothermal systems, we study the fusion and the mixing of two thin liquid films of different but perfectly miscible liquids. Marangoni-driven forces can cause delayed coalescence, ripple formation, and fingering patterns at the borderline between the two liquid layers. A systematic analysis for ripple pattern formation and finger instabilities at different diffusion constants shows that these phenomena appear more pronounced for lower diffusion in the system.

Modeling of the laser polarization as control parameter in self-organized surface pattern.

To shed light on nanopattern formation upon femtosecond laser ablation, an adopted surface erosion model is developed, based on the description for ion beam sputtering. In particular, the dependence of generated patterns on the laser polarization is taken into account. We find that an asymmetry in deposition and dissipation of incident laser energy results in a respective dependence of coefficients in a nonlinear equation of the Kuramoto-Sivashinsky type. Surface morphologies obtained by this model for different polarization of the laser beam are presented and the time evolution of the nanopattern is discussed. A comparison of these numerical results with experimental data shows an excellent agreement. Dependence of femtosecond laser induced formation on the polarization of the incident beam within an adopted surface erosion model is considered. A continuum theory of erosion by polarized laser radiation is developed. We exploit the similarity to ion-beam sputtering and extend a corresponding model for laser ablation by including laser polarization. This yields a respective dependence of coefficients in a nonlinear equation of the Kuramoto-Sivashinsky type. We present the surface morphologies obtained by this model for different polarization of the laser beam and discuss a time evolution of the nanopattern. These numerical results are in a good agreement with numerous experimental data. We show that the correlation of ripples orientation with laser polarization can be described within a model where the polarization causes the breaking of symmetry at the surface. Our results support the non-linear self-organization mechanism of pattern formation on the surface of solids.

Different behaviors of delayed fusion between drops with miscible liquids.

Fusion of sessile droplets with body of different liquids is delayed when the approaching drops are sitting on a highly wettable solid substrate. Owing to the surface tension gradients between the mixing drops, a Marangoni driven flow through the connecting channel appears. Experiments of delayed coalescence were recently reported in [Langmuir 24, 6395 (2008)10.1021/la800630w] for millimeters sized drops. For droplets of submillimeter dimensions, capillary forces dominate. The control of interfacial energies becomes an important strategy for manipulating tiny droplets along the solid surfaces. In this paper we present phase field simulations in two spatial dimensions for microdroplets of perfectly miscible liquids. For drops with a given geometry, systematic investigations were performed for different fluid viscosities. Different behaviors are observed from chasing droplets to droplets repelling.

Controlled pattern formation in thin liquid layers.

We examine the fully nonlinear behavior of a thin liquid film on a hydrophobic/hydrophilic solid support in three dimensions using a phase field model. For flat homogeneous substrates, the stability of thin liquid layers is investigated under the action of gravity. The coarsening process at the solid boundary can be controlled on inhomogeneous substrates. On substrates chemically patterned in an adequate way with hydrophobic and hydrophilic spots, one can obtain stable, regular liquid droplets and even design liquid structures (PACS numbers: 47.54.-r, 68.18.Jk, and 05.70.Np).

Drops on an arbitrarily wetting substrate: a phase field description.

We propose a scheme for studying thin liquid films on a solid substrate using a phase field model. For a van der Waals fluid-far from criticality-the most natural phase field function is the fluid density. The theoretical description is based on the Navier-Stokes equation with extra phase field terms and the continuity equation. In this model free of interface conditions, the contact angle can be controlled through the boundary conditions for the density field at the solid walls [L. M. Pismen and Y. Pomeav, Phys. Rev. E 62, 2480 (2000)]. We investigate the stability of a thin liquid film on a flat homogeneous solid support with variable wettability. For almost hydrophobic surfaces, the liquid film breaks up and transitions from a flat film to drops occur. Finally, we report on two-dimensional numerical simulations for static liquid drops resting on a flat horizontal solid support and for drops sliding down on inclined substrates under gravity effects.

Phase-field simulations for drops and bubbles.

Recently we proposed a phase field model to describe Marangoni convection in a compressible fluid of van der Waals type far from criticality [Eur. Phys. J. B 44, 101 (2005)]. The model previously developed for a two-layer geometry is now extended to drops and bubbles. A randomly distributed initial density evolves towards phase separation and single droplet formation. For a two-component liquid-liquid system we report on numerical simulations for drop Marangoni migration in a vertical thermal gradient.

Regular surface patterns on Rayleigh-Taylor unstable evaporating films heated from below.

We study a thin liquid film with a free surface on the underside of a cooled horizontal substrate. We show that if the fluid is initially in equilibrium with its own vapor in the gas phase below, regular surface patterns in the form of long-wave hexagons having a well-defined lateral length scale are observed. This is in sharp contrast to the case without evaporation where rupture or coarsening to larger and larger patterns is seen in the long time limit. In this way, evaporation could be used for regular structuring of the film surface. Finally, we estimate the finite wave length for the simplified case of an extended Cahn-Hilliard equation.

Morphology changes in the evolution of liquid two-layer films.

We consider a thin film consisting of two layers of immiscible liquids on a solid horizontal (heated) substrate. Both the free liquid-liquid and the liquid-gas interface of such a bilayer liquid film may be unstable due to effective molecular interactions relevant for ultrathin layers below 100-nm thickness, or due to temperature-gradient-caused Marangoni flows in the heated case. Using a long-wave approximation, we derive coupled evolution equations for the interface profiles for the general nonisothermal situation allowing for slip at the substrate. Linear and nonlinear analyses of the short- and long-time film evolution are performed for isothermal ultrathin layers, taking into account destabilizing long-range and stabilizing short-range molecular interactions. It is shown that the initial instability can be of a varicose, zigzag, or mixed type. However, in the nonlinear stage of the evolution the mode type, and therefore the pattern morphology, can change via switching between two different branches of stationary solutions or via coarsening along a single branch.

Alternative pathways of dewetting for a thin liquid two-layer film.

We consider two stacked ultrathin layers of different liquids on a solid substrate. Using long-wave theory, we derive coupled evolution equations for the free liquid-liquid and liquid-gas interfaces. Depending on the long-range van der Waals forces and the ratio of the layer thicknesses, the system follows different pathways of dewetting. The instability may be driven by varicose or zigzag modes and leads to film rupture either at the liquid-gas interface or at the substrate. We predict that the faster layer drives the evolution and may accelerate the rupture of the slower layer by orders of magnitude, thereby promoting the rupture of rather thick films.

Activity dynamics in nonlocal interacting neural fields.

We study the activity of a synaptically coupled neuronal network consisting of an excitatory and an inhibitory layer with isotropic connections and nonlinear interactions. Using the mathematical model of Wilson and Cowan in two spatial dimensions, we first discuss a spatial hysteresis phenomenon. Then we analyze special traveling wave solutions with stationary shape. We establish existence conditions, derive analytic expressions of the particular solutions and their velocity, and finally present numerical simulations.