Exact solutions of the equations of motion of liquid helium with a charged free surface

The dynamics of the development of instability of the free surface of liquid helium, which is charged by electrons localized above it, is studied. It is shown that, if the charge completely screens the electric field above the surface and its magnitude is much larger than the instability threshold, the asymptotic behavior of the system can be described by the well-known 3D Laplacian growth equations. The integrability of these equations in 2D geometry makes it possible to describe the evolution of the surface up to the formation of singularities, viz., cuspidal point at which the electric field strength, the velocity of the liquid, and the curvature of its surface assume infinitely large values. The exact solutions obtained for the problem of the electrocapillary wave profile at the boundary of liquid helium indicate the tendency to a change in the surface topology as a result of formation of charged bubbles.     

Instability of a flat horizontal interface between a thin layer of a ferrofluid and a thin layer of a nonmagnetic liquid in the presence of a vertical magnetic field

An asymptotic analysis of the equations and boundary conditions of fluid dynamics is performed, and a nonlinear model is constructed for the onset of the development of Rosensweig instability in a thin horizontal ferrofluid layer at rest covered with a thin layer of a lighter nonmagnetic liquid. The surface of a nonmagnetized slab is the lower boundary of the ferrofluid, and the interface with a gas is the upper boundary of the nonmagnetic liquid. The pressure in the gas is constant. The instability being considered arises upon the application of a rather strong uniform vertical magnetic field. The proposed model involves five dimensionless parameters. The critical magnetization of the initial ferrofluid layer with a flat upper boundary and the threshold wave number are found. The effect of the governing parameters on the instability region and on the wavelength of the fastest growing mode is studied in the linear formulation of the problem.     

Exact partial solutions for the surface dynamics of a dielectric liquid with a charged surface in the gravitational field

We study the evolution of instability in the boundary of a perfect dielectric liquid with a free surface charge in an external electric field. Conformal variables are used to find exact partial solutions to the equations of motion for the case when the charge completely shields the field above the liquid, the electrostatic and gravitational forces being taken into account. The solutions describe the development of instability of the initially planar boundary until sharp dimples are formed on it.     

Rosenzweig instability in a thin layer of a magnetic fluid

A simple mathematical model of the initial stage of nonlinear evolution of the Rosenzweig instability in a thin layer of a nonlinearly magnetized viscous ferrofluid coating a horizontal nonmagnetizable plate is constructed on the basis of the system of equations and boundary conditions of ferrofluid dynamics. A dispersion relation is derived and analyzed using the linearized equations of this model. The critical magnetization of the initial layer with a flat free surface, the threshold wavenumber, and the characteristic time of evolution of the most rapidly growing mode are determined. The equation for the neutral stability curve, which is applicable for any physically admissible law of magnetization of a ferrofluid, is derived analytically.     

Effect of tangential electric field on the evolution of the Rayleigh-Taylor instability of a dielectric liquid film

The effect of surface ponderomotive forces produced by a uniform tangential electric field on the evolution of the Rayleigh-Taylor instability of an isothermal film of a homogeneous incompressible dielectric liquid coating the lower surface of a horizontal insulating plate is studied in the long-wavelength approximation for the hydrodynamic equations and a simplified system of electrostatics equations. The lower boundary of the liquid is the interface with a stationary gas. It is shown in the framework of the linear theory that during the disruption of the continuous film, ponderomotive forces induce the formation of liquid billows extended along the applied field lines.     

On diffusion-elastic instabilities in a solid half-space

A model of the diffusion-elastic instability that appears in an ensemble of non-equilibrium atomic defects in unbounded condensed media as well as on the free surface of a half-space is introduced and studied. The dynamical model developed here is based on coupled evolution equations for the elastic displacement of the medium and atomic defect density fields. The idea of an instability model is related to a drift of atomic defects under the influence of elastic fields. It is shown that the development of this instability creates ordered structures of coupled strain and defect-concentration fields. Dispersion relationships for the growth increment of these structures are derived and their characteristic scales are obtained.     

Film flows down a fiber: Modeling and influence of streamwise viscous diffusion

A two-equation model is formulated in terms of two coupled evolution equations for the film thickness h and the local flow rate q within the framework of lubrication theory. Consistency is achieved up to first order in the film parameter epsilon and streamwise diffusion effects are accounted for. The evolution equation obtained by Craster and Matar [1] is recovered in the appropriate limit. Comparisons to the experimental results by [2] and [3] show good agreement in the linear and nonlinear regimes. Second-order viscous diffusion terms are found to potentially enhance the speed and amplitude of nonlinear waves triggered by the Rayleigh-Plateau instability mechanism. Time-dependent computations of the spatial evolution of the film reveal a strong influence of streamwise diffusion on the dynamics of the flow and the wave selection process.     

Dynamics of reorientation of a constrained nematic elastomer

A model is proposed for the reorientation dynamics of a confined nematic liquid crystal elastomer, where the effect of crosslinks is to couple the director to deformations of the elastic matrix. The model combines the (equilibrium) `neo-classical' theory of liquid crystal rubber elasticity with the simplest time evolution equations for a system described by two coupled, non-conserved order parameters. Relaxation from an orientation imposed by an electric field is studied as a function of elastic softness, starting angle, surface pretilt, and the relative mobilities of director and strain. Most importantly, the absence of a `semi-soft' elastic threshold changes the long-time behaviour of the effective refractive index of the medium from exponential to inverse power law decay. Predictions are compatible with recent experimental results by Chang, Chien and Meyer [Phys. Rev. E 56, 595 (1997)]. Received 22 June 1998     

On the time evolution of the liquid meniscus at the end of a capillary placed in an external electric field

Using a linearized set of equations of electrodynamics, the stability of the uniformly charged meniscus of a viscous conducting incompressible liquid at the end of a capillary is investigated and analytical expressions are derived for the electric field outside the meniscus, velocity fields in the liquid flow and meniscus, and generatrix of the meniscus shape. It is found that, if an external electric field near the meniscus exceeds that at which the free liquid surface becomes unstable against the surface charge, a finite number of longest waves become unstable with their instability growth rates nonmonotonically depending on the wavenumber. Analysis of the time evolution of the meniscus shape under various initial conditions shows that cylindrical waves with the highest instability growth rates play a decisive role in this process, while the influence of the initial deformation amplitude is insignificant.     

Orientational instability in a nematic liquid crystal in a decaying Poiseuille flow

The results of studies of orientational dynamics and instability in an MBBA nematic liquid crystal in a decaying Poiseuille flow are considered. The experiments were made on a wedge cell with a gap width varying in a direction perpendicular to the flow. Confining surfaces ensured homeotropic adhesion of the nematic to the surface. Above a certain critical value of the initial pressure drop, a uniform orientational instability is observed, which corresponds to the emergence of the director from the plane of the flow. The dependence of the critical pressure drop on the local thickness of the liquid crystal layer and on the external destabilizing electric field is determined. Simulation of nematodynamics equations is carried out. The results of theoretical calculations are in qualitative and quantitative agreement with the experimental data.     

磁场中液氦薄膜表面电子涟波子系统的性质

研究了磁场中液氦薄膜表面电子与涟波子强耦合和弱耦合的性质。采用线性组合算符方法导出磁场中液氦薄膜表面电子 涟波子系统的振动频率和基态能量。讨论磁场对表面电子 涟波子系统的振动频率和基态能量的影响。     

Convective instabilities in films of binary mixtures

We present a model for the evolution of films of isothermal binary liquid mixtures with a free evolving surface. The model is based on model-H supplemented by appropriate boundary conditions at the free surface and the solid substrate. The equations account for the coupled transport of the concentration of a component (convective Cahn-Hilliard equation) and the momentum (Korteweg-Navier-Stokes equation). The inclusion of convective motion makes surface deflections possible, i.e., the model allows to study couplings between the decomposition of the mixture and the evolving surface corrugations. We present selected steady layered film states for representative polymer mixtures, and show that convective motion favors their destabilization and qualitatively changes the linear instability modes in experimentally accessible ranges of parameters.     

Analysis of a combined influence of substrate wetting and surface electromigration on a thin film stability and dynamical morphologies

A PDE-based model combining surface electromigration and wetting is developed for the analysis of morphological stability of ultrathin solid films. Adatom mobility is assumed anisotropic, and two directions of the electric field (parallel and perpendicular to the surface) are discussed and contrasted. Linear stability analyses of small-slope evolution equations are performed, followed by computations of fully nonlinear parametric evolution equations that permit surface overhangs. The results reveal parameter domains of instability for wetting and non-wetting films and variable electric field strength, nonlinear steady-state solutions in certain cases, and interesting coarsening behavior for strongly wetting films.     

Free surface instability in a confined suspension jet

A suspension confined between two close parallel plates is studied in the Stokesian regime. The use of boundary integral equations and the lubrication approximation allows to compute the hydrodynamic forces acting on the particles. The forces are long ranged and depend on the orientation of the relative position and velocity of particles. This tensorial character predicts an “antidrag” that is observed in experiments. The effect of the computed hydrodynamic forces is studied in the dynamics of a jet of particles falling by a gravitational field, which shows a surface instability similar to the Kelvin–Helmholtz one. A theoretical model, based on hydrodynamic-like equations, is able to predict the instability that is produced by the interaction of the long-range forces and the free surface.     

Thermocapillary waves in a liquid film

This paper investigates two-dimensional waves in a heated liquid film in the presence of the thermocapillary effect. The waves in the film are described using the integral model. The first part of the paper considers film instability for cases of fixed temperature of the plate and fixed heat flux in the plate. The liquid temperature disturbance is calculated from the energy equation for arbitrary values of Peclet number. In the second part, the evolution of waves in a heated film is modeled based on the system of equations for film thickness, flow rate, and the energy equation. In numerical modeling of the wave evolution, the boundary of the region of growing disturbances agrees well with results of stability analysis. The calculations show that for a vertical film the thermocapillary effect leads to broadening of the instability region only at low Peclet numbers.     

Nonlinear dynamics of Aeolian sand ripples

We study the initial instability of flat sand surface and further nonlinear dynamics of wind ripples. The proposed continuous model of ripple formation allowed us to simulate the development of a typical asymmetric ripple shape and the evolution of a sand ripple pattern. We suggest that this evolution occurs via ripple merger preceded by several soliton-like interaction of ripples.     

Evolving complex dynamics in electronic models of genetic networks

Ordinary differential equations are often used to model the dynamics and interactions in genetic networks. In one particularly simple class of models, the model genes control the production rates of products of other genes by a logical function, resulting in piecewise linear differential equations. In this article, we construct and analyze an electronic circuit that models this class of piecewise linear equations. This circuit combines CMOS logic and RC circuits to model the logical control of the increase and decay of protein concentrations in genetic networks. We use these electronic networks to study the evolution of limit cycle dynamics. By mutating the truth tables giving the logical functions for these networks, we evolve the networks to obtain limit cycle oscillations of desired period. We also investigate the fitness landscapes of our networks to determine the optimal mutation rate for evolution.     

Dynamics of a creep-slip model of earthquake faults

Starting off from the relationship between time-dependent friction and velocity softening we present a generalization of the continuous, one-dimensional homogeneous Burridge–Knopoff (BK) model by allowing for displacements by plastic creep and rigid sliding. The evolution equations describe the coupled dynamics of an order parameter-like field variable (the sliding rate) and a control parameter field (the driving force). In addition to the velocity-softening instability and deterministic chaos known from the BK model, the model exhibits a velocity-strengthening regime at low displacement rates which is characterized by anomalous diffusion and which may be interpreted as a continuum analogue of self-organized criticality (SOC). The governing evolution equations for both regimes (a generalized time-dependent Ginzburg–Landau equation and a non-linear diffusion equation, respectively) are derived and implications with regard to fault dynamics and power-law scaling of event-size distributions are discussed. Since the model accounts for memory friction and since it combines features of deterministic chaos and SOC it displays interesting implications as to (i) material aspects of fault friction, (ii) the origin of scaling, (iii) questions related to precursor events, aftershocks and afterslip, and (iv) the problem of earthquake predictability. Moreover, by appropriate re-interpretation of the dynamical variables the model applies to other SOC systems, e.g. sandpiles.     

On the characteristic time for the realization of instability on a flat charged liquid surface

A nonlinear integral equation that describes the time evolution of the amplitude of a nonlinear unstable wave on the flat uniform charged surface of an ideal incompressible liquid has been derived and solved. The characteristic time for the realization of instability is found to be determined by the initial amplitude of a virtual wave initiating the instability and the supercritical increment in the Tonks-Frenkel parameter. At a zero supercritical increment, the characteristic time for the realization of instability is only determined by the initial amplitude and can be rather long (up to eight hours). This effect is characteristic of a flat charged liquid surface and does not occur in charged drops.