The effects of additive Gaussian white noise on the onset of Rayleigh-Bénard convection are studied by means of a phenomenological model, the stochastic Swift-Hohenberg equation. The strength of the noise term arising from thermal fluctuations is given for both free-slip and rigid horizontal boundaries. As was already pointed out by previous authors this term contains the small parameter ${\mathit{k}}_{\mathit{B}}$T/ρd${\nu }^2$, where ρ is the mass density, d the plate separation, and ν the kinematic viscosity. For typical liquids this parameter is of order ${10}^{\mathrm{-}9}$. Experiments involving fluctuation effects may be interpreted in terms of this model if the noise strength is treated as an adjustable parameter, which turns out to be larger than the typical thermal value by four orders of magnitude. The effects of fluctuations on the bifurcation of an infinite system are studied, and the earlier arguments of the present authors leading to a first-order transition are reviewed [Swift and Hohenberg, Phys. Rev. A 15, 319 (1977)]. The conditions under which the multimode model can be approximated by a single-mode stochastic amplitude equation are investigated, and an earlier analytic approximation scheme for calculating the response to a time-dependent Rayleigh number is applied to the multimode model. A comparison with available experimental and numerical simulation data is presented.

We examine the appearance of surface waves governed by Burgers and Korteweg–de Vries equations in a shallow viscous heated fluid. We consider waves triggered by a surface-tension variation induced by both temperature and concentration gradients. We also establish the range of parameters for which the above-mentioned equations appear.

We explore the similarities between the dynamics of rough interfaces and fully developed hydrodynamical turbulence. In particular, we introduce a simple system of coupled mappings that (1) is chaotic with an attractor whose dimension grows proportionally to the system size, (2) generates small-scale structure, and (3) has structure functions that grow as power laws. We discuss the universality classes that determine the large-distance long-time behavior by computing the exponents for the scaling of the interface width.

The turbulent velocity field in wave-number space is decomposed into two distinct fields. One is a purely chaotic field; while the other is a correction field, and carries all the phase information. Application of this decomposition to a thin shell of wave numbers in the dissipation range allows the elimination of modes in that shell, with the usual mode-coupling problems being circumvented by the use of a conditional average. The (conditional) mean effect of the eliminated modes appears as an increment to the viscosity, with terms of order ${\lambda }^2$ being neglected, where λ is a dimensionless measure of bandwidth thickness, such that 0≤λ≤1. An iteration (with appropriate rescaling) to successively lower shells reaches a fixed point, corresponding to a renormalized turbulent viscosity. As previously reported [W. D. McComb and A. G. Watt, Phys. Rev. Lett. 65, 3281 (1990)], the spectrum of the purely chaotic field is found to take the Kolmogorov -5/3 power-law form, with a value for the Kolmogorov spectral constant of α=1.6, independent of λ over the range of bandwidths for which the theory is valid.

We study compressible fluid flow in narrow two-dimensional channels using a molecular-dynamics simulation method. In the simulation area, an upstream source is maintained at constant density and temperature while a downstream reservoir is kept at vacuum. The channel is sufficiently long in the direction of the flow that the finite length has little effect on the properties of the fluid in the central region. The simulated system is represented by an efficient data structure, whose internal elements are created and manipulated dynamically in a layered fashion. Consequently the computer code is highly efficient and manifests completely linear performance in simulations of large systems. We obtain the steady-state velocity, temperature, and density distributions in the system. The velocity distribution across the channel is very nearly a quadratic function of the distance from the center of the channel and reveals velocity slip at the boundaries; the temperature distribution is only approximately a quartic function of this distance from the center to the channel. The density distribution across the channel is nonuniform. We attribute this nonuniformity to the relatively high Mach number, approximately 0.5, in the fluid flow. An equation for the density distribution based on simple compressibility arguments is proposed; its predictions agree well with the simulation results. The validity of the concept of local dynamic temperature and the variation of the temperature along the channel are discussed.

The fractal nature of level sets and the multifractal nature of various scalar and vector fields in hydromagnetic and hydrodynamic turbulence are investigated using data of direct simulations. It turns out that fields whose evolution is governed by stretching terms (vortex stretching, magnetic-field line stretching) exhibit ‘‘near singularities’’ that result in a multifractal scaling. Such stretching terms can lead to a rapid increase in the local value of the field. Fields without rapid local increase have no multifractal scaling. Furthermore, the simulations support recent theoretical suggestions that the fractal properties of the level sets of various fields are quite insensitive to the existence of stretching. Indeed, all the fields under study (temperature, vorticity magnitude, magnetic-field magnitude) show rather universal behavior in the geometry of their level sets, consistent with a two-dimensional geometry at small scales, with a crossover to a universal fractal geometry at large scales. The dimension at large scales is compatible with the theoretical prediction of about 2.7. The most surprising result of the simulations is that it appears that the ‘‘near singularities’’ are not efficiently eliminated by viscous dissipation, but rather seem to be strongest at the Kolmogorov cutoff. The effects of the singularities do not quite penetrate into the inertial range. We offer a simple analytic model to account for this behavior. We conclude that our findings may be due to the relatively small Reynolds numbers, but may also be indicative of generic behavior at larger Reynolds numbers. We offer some thoughts about the expected scaling behavior in the inertial range in light of our findings.

We report results from a numerical study of a two-dimensional model for the surface-directed spinodal decomposition in a critical mixture. We explicitly add a long-range surface interaction term to the bulk free-energy functional leading to preferential attraction of one of the components of the mixtures to a free surface. We find that, for sufficiently long-range interactions, the thickness of the wetting layer l(t) varies as l(t)∼${\mathit{t}}^{1/3}$. We also find that the magnitudes of the average domain sizes in the parallel and perpendicular directions are different. However, the exponent (n) characterizing the asymptotic growth of domains in these two directions is found to be the same (n=1/3). As well, the density profile and the pair correlation functions satisfy dynamical scaling behavior.

A simple Ginzburg-Landau free-energy functional for oil-water-surfactant mixtures, with a single, scalar order parameter, is studied in detail. We show that the free energy of swollen spherical and cylindrical micelles obtained from this model can be described in terms of the Helfrich Hamiltonian of surfactant monolayers. The surface tension σ, the spontaneous curvature modulus λ, the saddle-splay modulus κ¯, and the bending rigidity κ of the surfactant sheet at the oil-water interface can all be calculated from the order-parameter profile of the planar oil-water interface. It is demonstrated that for stable droplets and cylinders, these expressions of κ and κ¯ give reliable predictions for the free energy only if the system is at oil-water coexistence. Off coexistence, the distortion of the profile due to the finite curvature of the interface has to be taken into account. The results are used to discuss the phase diagram of the Landau model. In addition to the bending elasticity, the interaction between monolayers plays an important role. This interaction is found to be always attractive in our model. We show that the simple Ginzburg-Landau theory describes various spatially modulated phases: the lamellar phase, the hexagonal phase of cylinders, a cubic crystal of spherical micelles, and bicontinuous cubic phases. Finally, we discuss the behavior of oil-water-surfactant mixtures near a wall. We find that under suitable conditions the lamellar phase wets the wall-oil (or wall-water) interface.

The temperature and frequency dependence of the complex dielectric susceptibility of a ferroelectric liquid crystal (FLC) near the smectic-${\mathit{C}}^{\mathrm{*}}$–smectic-A phase transition has been calculated using the generalized Landau model. It is shown that, although the dielectric response of the Sm-${\mathit{C}}^{\mathrm{*}}$ phase consists generally of four modes (soft, Goldstone, and two high-frequency polarization modes), only three bands appear in the dielectric-loss spectrum of FLC’s at the Sm-A–Sm-${\mathit{C}}^{\mathrm{*}}$ phase transition. The calculations based on the generalized Landau model show that the frequency split of the two high-frequency modes is too low to be detected as two separate relaxation processes. A special technique has to be used to split these modes. At the Sm-A–Sm-${\mathit{C}}^{\mathrm{*}}$ phase transition this process does not split or broaden. These results are in agreement with the recent experimental data. It is shown that in the light of these calculations a revision of the theory as proposed by Pleiner and Brand [Phys. Rev. A 43, 7064 (1991); Mol. Cryst. Liq. Cryst. Lett. 8, 11 (1991)] is rather unnecessary.

We present a theoretical and experimental study of the molecular collective rotation induced in a nematic liquid crystal by a laser beam, when self-induced stimulated light scattering takes place. The model is accurate enough to obtain quantitative agreement with experimental results, including the observation of optical bistability and hysteresis. Exact plane-wave expressions for energy and angular-momentum exchange between light and matter are also derived.

As a model for natural convection in the Earth’s upper crust, we discuss the flow in a porous tilted layer of infinite extent in an impermeable surrounding of different thermal conductivity across which a vertical temperature gradient acts. We calculate the basic nonlinear flow solution; it is a large-scale along-slope flow with zero mean flow. Two important geophysical consequences are discussed: (a) the transport of passive tracers by applying Taylor’s theory of dispersion to our system, and (b) diagenetic changes of the porous matrix. For realistic geophysical parameters we estimate the order of magnitude of the effects and discuss the ranges of validity of the theory. The enhancement of diffusion of passive tracers due to the flow is quite significant and might have important implications for the migration of nuclear waste buried in the Earth.

Blue phases (BP’s) in liquid-crystalline systems of high chirality exhibiting a short cholesteric temperature interval are investigated. In a BP I supercooled with respect to the cholesteric phase, the orientation of the cubic lattice with the (1,1,0) wave vector perpendicular to the substrate is spontaneously turned to a [200] orientation within small areas of several tenths of micrometers in diameter. A pinning of BP I lattice temperature waves is observed on the [200] orientational inhomogeneities. The pinning effect explains the observed saturation of the BP I lattice constant on decreasing temperature and its dependence on the cooling rate observed in supercooled region. A different type of cubic blue phase, BP S (‘‘S’’ represents supercooled), is observed transforming reversibly from the supercooled BP I but metastable with respect to the cholesteric phase. The BP S has two scales of order: a long-range orientational blue-phase-like order and a short-range positional smecticlike order.

The method of the open-ended capillary tube has been used to determine the self-diffusion coefficient of the ${}_{}{}^{22}{}_{}{}^{}\mathrm{Na}_{}^{+}{}_{}{}^{}$ ion in an isobutyric acid, water, and a ${10}^{\mathrm{-}3}$M NaCl critical mixture. The experimental range of temperature is 0.360≤T-${\mathit{T}}_{\mathit{c}}$≤9.860 °C. This study is a preliminary step in the investigation of ion-transport phenomena in binary fluids. We find that in the critical region the Arrhenius law has an important deviation and the self-diffusion coefficient D (${\mathrm{cm}}^2$/s) can be governed by the mean-field theory.

A theory is developed for the dynamics of pattern formation in quasi-two-dimensional domains of magnetic fluids (ferrofluids) in transverse magnetic fields. The pattern formation is treated as a dissipative dynamical process, with the motion derived variationally from a static energy functional using minimal assumptions. This dynamics is one instance of a general formalism applicable to any system that can be modeled as a closed curve in a plane. In applying the formalism to ferrofluids, we present a calculation of the energy of a two-dimensional dipolar domain as a functional of the shape of its boundary. A detailed linear stability analysis of nearly circular shapes is presented, and pattern formation in the nonlinear regime, far from the onset of instability, is studied by numerical solution of the nonlinear, nonlocal evolution equations. The highly branched patterns obtained numerically bear a qualitative resemblance to those found experimentally. The time evolution exhibits sensitive dependence on initial conditions, suggesting the existence of many local minima in the space of accessible shapes. The analysis also provides a deterministic starting point for a theory of pattern formation in dipolar monolayers at the air-water interface, in which thermal fluctuations play a more dominant role.

We investigate the attenuation and dispersion of sound waves in suspensions and emulsions caused by the thermal-transport process. They combine to constitute the effective compressibility of the system. We begin with an attempt to justify the Isakovich formula for calculating the effective compressibility. The formula is then rewritten in terms of the interfacial heat flux. Isakovich’s analysis is simply an independent-particle approximation. It is the purpose of this paper to consider the effect of interparticle interactions. The effective compressibility is calculated for an array of spherical particles or droplets centered at the points of a periodic lattice, immersed in a fluid of different species. Ewald’s method of fast-convergent lattice sums in electrostatics is extended to a technique for the heat-conduction problem in a periodic emulsion. The computation for cubic lattices reveals that the interparticle interactions act to reduce, in the lower-frequency range, both the attenuation coefficient and the departure of the sound velocity from its high-frequency limit. The striking feature is that a drastic change in attenuation occurs when the thermal conductivity of the particle is substantially larger than that of the ambient fluid.

An instability of the pair-distribution function of the fluid with respect to the diagrammatic iteration loop has been discovered recently [Phys. Rev. A 43, 6526 (1991)], which correlates well with the freezing density of the soft inverse-power potentials. The present work extends that study to a large variety of pair interactions and establishes the general validity and accuracy of this instability as a semiempirical freezing indicator. Preliminary results for the instability lines for mixtures are also presented that are in qualitative agreement with freezing densities from thermodynamic calculations and from recent experiments on the phase behavior of mixtures of colloids. The stability analysis of the iteration loop is complicated by the need to go beyond linear order.

We use Monte Carlo simulation methods to study the structure of many polymer chains end-grafted on a flat interacting surface. We consider both attractive and repulsive interactions between the grafting surface and the monomers. We also study the effect of short-range and long-range surface interactions. In all cases, the density profiles of the grafted polymer layer obtained from Monte Carlo studies are in good agreement with various recent self-consistent-field calculations. For repulsive potential wells we find the presence of a definite exclusion zone from which the free ends of the chains are repelled. This result is in excellent agreement with the theoretical prediction.

For a surface interaction quadratic in the magnitude of the order parameter and favoring an in-plane molecular orientation, it is shown that a nematic surface state with quasi-two-dimensional order can exist in a narrow temperature range above the bulk transition point. This phase is characterized by fully biaxial order and the transition between it and the fully disordered one is via the Berezinskii-Kosterlitz-Thouless (BKT) mechanism. The BKT transition temperature is calculated as are the elastic and inelastic light-scattering intensities. The latter is a possible technique for observing the transition and confirming its BKT character. Numerical estimates indicate that systems having the surface interactions required to exhibit this type of nematic phase transition can be prepared.

We present a generalized mode-coupling theory for a dense binary fluid mixture. The theory is used to calculate molecular-scale renormalizations to the stress-tensor autocorrelation function (STAF) and to the long-wavelength zero-frequency shear viscosity. As in the case of a dense simple fluid, we find that the STAF appears to decay as ${\mathit{t}}^{\mathrm{-}3/2}$ over an intermediate range of time. The coefficient of this long-time tail is more than two orders of magnitude larger than that obtained from conventional mode-coupling theory. Our study focuses on the effect of compositional disorder on the decay of the STAF in a dense mixture.

We report experimental results on the stability diagram of the normal-roll pattern in electroconvection in nematic liquid crystals. The stable wave-number bands of the normal rolls are determined for various frequencies of the applied voltage. We find that the mode destabilizing the normal rolls is of the zigzag type at low frequencies of the applied voltage and at higher frequencies mainly of the skewed varicose type. The change of the destablizing mode of the normal rolls is closely related to the change of the transition scenarios to weak turbulence. With increasing amplitude of the external voltage, at low frequencies the normal rolls are followed by a zigzaglike pattern and at higher frequencies they are followed by a breathing pattern consisting of a two-dimensional lattice of defects. Both secondary patterns are therefore on a different route to weak turbulence to which they finally bifurcate at higher voltages.