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We observed in numerical simulations that the interaction of striped-pattern-forming instability and a neutrally stable zero mode induces patterns of domains of upflow hexagons coexisting with domains of downflow hexagons. They appear only when hydrodynamic flow is present.
A previously formulated smoothed particle hydrodynamics model for a phase separating mixture is tested for the case when viscous processes are negligible and only mass and energy diffusive processes take place. We restrict ourselves to the case of a binary mixture that can exhibit liquid-liquid phase separation. The thermodynamic consistency of the model is assessed and the potential of the model to study complex pattern formation in the presence of various thermal boundaries is illustrated.
We investigate the slip boundary condition for flows past a chemically patterned surface. Molecular dynamics simulations show that fluid forces and stresses vary laterally along the patterned surface. A subtraction scheme is developed to verify the validity of the Navier slip boundary condition, locally, for the patterned surface. A continuum hydrodynamic model is formulated using the Navier-Stokes equation and the Navier boundary condition, with a slip length varying along the patterned surface. Steady-state velocity fields from continuum calculations are in quantitative agreement with those from molecular simulations.
A two-dimensional numerical model based on smoothed particle hydrodynamics (SPH) was used to simulate unsaturated (multiphase) flow through fracture junctions. A combination of standard SPH equations with pairwise fluid-fluid and fluid-solid particle-particle interactions allowed surface tension and three-phase contact dynamics to be simulated. The model was validated by calculating the surface tension in four different ways: (i) from small-amplitude oscillations of fluid drops, (ii) from the dependence of the capillary pressure on drop radius, (iii) from capillary rise simulations, and (iv) from the behavior of a fluid drop confined between parallel walls under the influence of gravity. All four simulations led to consistent values for the surface tension. The dependence of receding and advancing contact angles on droplet velocity was studied. Incorporation of surface tension and fluid-solid interactions allowed unsaturated flow through fracture junctions to be realistically simulated, and the simulation results compare well with the laboratory experiments of Dragila and Weisbrod.
We develop a hydrodynamic theory for flows of incompressible blends of flexible polymers and rodlike nematic polymers (RNPs) or rodlike nematic liquid crystal polymers (RNLCPs) extending the thermodynamical theory of Muratov and E [J. Chem. Phys. 116, 4723 (2002)] for phase separation kinetics of the blend. We model the flexible polymer molecules in the polymer matrix as Rouse chains and assume the translational diffusion of the molecules is predominantly through the volume fraction of the flexible polymer and the molecules of rodlike nematic liquid crystal polymers. We then (i) derive the translational flux for the rodlike nematic liquid crystal polymers to ensure the incompressibility constraint; (ii) derive the elastic stress tensor, accounting for the contribution from both the rodlike nematic polymer and the flexible polymer matrix, as well as the extra elastic body force due to the nonlocal intermolecular potential for long range molecular interaction; (iii) show that the theory obeys positive entropy production and thereby satisfies the second law of thermodynamics. By applying the gradient expansion technique on the number density function of RNLCPs, we present an approximate, weakly nonlocal theory in differential form in which the intermolecular potential is given by gradients of the number density function of the RNLCP and the volume fraction of the flexible polymer. In the approximate theory, the elastic stress is augmented by an extra stress tensor due to the spatial convection of the macroscopic material point and long range interaction, whose divergence yields the analogous extra elastic body force with respect to the nonlocal intermolecular potential. Finally, we compare the model in steady simple shear with the Doi theory for bulk monodomains of rodlike nematic polymers.
In this paper, numerical and analytical results are presented which indicate that hydrodynamic Lyapunov modes (HLMs) also exist for coupled map lattices (CMLs). The dispersion relations for the HLMs of CMLs are found to fall into two different universality classes. It is characterized by lambda approximately k for coupled standard maps and lambda approximately k2 for coupled circle maps. The conditions under which HLMs can be observed are discussed. The role of the Hamiltonian structure, conservation laws, translational invariance, and damping is elaborated. Our results are as follows: (1) The Hamiltonian structure is not a necessary condition for the existence of HLMs. (2) Conservation laws or the translational invariance alone cannot guarantee the existence of HLMs. (3) Including a damping term in the system of coupled Hamiltonian maps does not destroy the HLMs. The lambda-k dispersion relation of HLMs, however, changes to the universality class with lambda-k2 under damping. In contrast, no HLMs survives in the system of coupled circle maps under damping. (4) An on-site potential destroys the HLMs. (5) The study of zero-value Lyapunov exponents (LEs) and associated Lyapunov vectors (LVs) shows that translational invariance and conservation laws play different roles in the tangent space dynamics. (6) The dynamics of the coordinate and momentum parts of LVs in Hamiltonian systems are related but different. Furthermore, numerical results for a two-dimensional system show that the appearance of HLMs in CMLs is not restricted to the one-dimensional case.
Experiments reported by Blake [Phys. Fluids., 11, 1995 (1999)] suggest that the dynamic contact angle formed between the free surface of a liquid and a moving solid boundary at a fixed contact-line speed depends on the flow field and geometry near the moving contact line. We examine quantitatively whether or not it is possible to attribute this effect to the bending of the free surface due to hydrodynamic stresses acting upon it and hence interpret the results in terms of the so-called "apparent" contact angle. It is shown that this is not the case. Numerical analysis of the problem demonstrates that, at the spatial resolution reported in the experiments, the variations of the "apparent" contact angle (defined in two different ways) caused by variations in the flow field, at a fixed contact-line speed, are too small to account for the observed effect. The results clearly indicate that the actual (macroscopic) dynamic contact angle--i.e., the one used in fluid mechanics as a boundary condition for the equation determining the free surface shape--must be regarded as dependent not only on the contact-line speed but also on the flow field and geometry in the vicinity of the moving contact line.
We investigate the nonlinear oscillations of heat-conductive, viscous, liquid drops in vacuum with zero gravity, using smoothed particle hydrodynamics (SPH). The liquid drops are modeled as a van der Waals fluid in two dimensions so that the models apply to flat, disklike drops. Attention is focused on small- to large-amplitude oscillations of drops that are released from a static elliptic shape. We find that for small-amplitude motions the combined dissipative effects of finite viscosity and heat conduction induce rapid decay of the oscillations after a few periods, while for large-amplitude motions wave damping is governed by the action of both viscous dissipation and surface tension forces. The transition from periodic to aperiodic decay at Re approximately 1 as well as the quadratic decrease of the frequency with the initial aspect ratio at large Re are reproduced in good agreement with previous theoretical predictions and experimental results.
We investigate shear-induced crystallization in a very dense flow of monodisperse inelastic hard spheres. We consider a steady plane Couette flow under constant pressure and neglect gravity. We assume that the granular density is greater than the melting point of the equilibrium phase diagram of elastic hard spheres. We employ a Navier-Stokes hydrodynamics with constitutive relations all of which (except the shear viscosity) diverge at the crystal-packing density, while the shear viscosity diverges at a smaller density. The phase diagram of the steady flow is described by three parameters: an effective Mach number, a scaled energy loss parameter, and an integer number m: the number of half-oscillations in a mechanical analogy that appears in this problem. In a steady shear flow the viscous heating is balanced by energy dissipation via inelastic collisions. This balance can have different forms, producing either a uniform shear flow or a variety of more complicated, nonlinear density, velocity, and temperature profiles. In particular, the model predicts a variety of multilayer two-phase steady shear flows with sharp interphase boundaries. Such a flow may include a few zero-shear (solidlike) layers, each of which moving as a whole, separated by fluidlike regions. As we are dealing with a hard sphere model, the granulate is fluidized within the "solid" layers: the granular temperature is nonzero there, and there is energy flow through the boundaries of the solid layers. A linear stability analysis of the uniform steady shear flow is performed, and a plausible bifurcation diagram of the system, for a fixed m, is suggested. The problem of selection of m remains open.
The existence of a strong stochasticity threshold (SST) has been detected in many Hamiltonian lattice systems, including the Fermi-Pasta-Ulam (FPU) model, which is characterized by a crossover of the system dynamics from weak to strong chaos with increasing energy density epsilon. Correspondingly, the relaxation time to energy equipartition and the largest Lyapunov exponent exhibit different scaling behavior in the regimes below and beyond the threshold value. In this paper, we attempt to go one step further in this direction to explore further changes in the energy density dependence of other Lyapunov exponents and of hydrodynamic Lyapunov modes (HLMs). In particular, we find that for the FPU-beta and FPU-alpha(beta) models the scalings of the energy density dependence of all Lyapunov exponents experience a similar change at the SST as that of the largest Lyapunov exponent. In addition, the threshold values of the crossover of all Lyapunov exponents are nearly identical. These facts lend support to the point of view that the crossover in the system dynamics at the SST manifests a global change in the geometric structure of phase space. They also partially answer the question of why the simple assumption that the ambient manifold representing the system dynamics is quasi-isotropic works quite well in the analytical calculation of the largest Lyapunov exponent. Furthermore, the FPU-beta model is used as an example to show that HLMs exist in Hamiltonian lattice models with continuous symmetries. Some measures are defined to indicate the significance of HLMs. Numerical simulations demonstrate that there is a smooth transition in the energy density dependence of these variables corresponding to the crossover in Lyapunov exponents at the SST. In particular, our numerical results indicate that strong chaos is essential for the appearance of HLMs and those modes become more significant with increasing degree of chaoticity.
The diffusivity of spherical colloidal particles close to a planar hard wall is studied by dynamic light scattering with evanescent illumination. A novel setup allows us to independently vary the scattering vector components parallel Q parallel and normal Q perpendicular to the wall. An expression for the initial decay rate Gamma of the time autocorrelation functions is derived as a function of both Q parallel and Q perpendicular, as well as the penetration depth of the evanescent wave, where hydrodynamic interactions of particles with the wall are included. This makes it possible to study the viscous wall drag effect quantitatively for particles as small as 85 nm in radius.
We study analytically the emergence of spontaneous collective motion within large bidimensional groups of self-propelled particles with noisy local interactions, a schematic model for assemblies of biological organisms. As a central result, we derive from the individual dynamics the hydrodynamic equations for the density and velocity fields, thus giving a microscopic foundation to the phenomenological equations used in previous approaches. A homogeneous spontaneous motion emerges below a transition line in the noise-density plane. Yet, this state is shown to be unstable against spatial perturbations, suggesting that more complicated structures should eventually appear.
We study the statistical properties of many Brownian particles under the influence of both a spatially homogeneous driving force and a periodic potential with period l in a two-dimensional space. In particular, we focus on two asymptotic cases lint< >l , where lint represents the interaction length between two particles. We derive fluctuating hydrodynamic equations describing the evolution of a coarse-grained density field defined on scales much larger than l for both cases. Using the obtained equations, we calculate the equal-time correlation functions of the density field to the lowest order of the interaction strength. We find that the system exhibits long-range correlation of the type r-d (d=2) for the case lint>>l , while no such behavior is observed for the case lint<<l .
We have used equilibrium and nonequilibrium molecular dynamics simulations to study the solute self-diffusion coefficient and the shear rate dependence of the solution viscosity in solutions of model nanocolloidal particles that range in mass ratio from mu=1 up to mu=50 and size ratio from s=1 up to s=4.03 at various concentrations. The zero shear rate viscosities and the initial rates of shear thinning were determined from data in the shear rate region in which the suspension is strongly shear thinning while the solvent remains Newtonian or is weakly shear thinning. The rate of shear thinning increased dramatically with solute volume fraction, regardless of whether the increase was due to increasing solute size or increasing the solute concentration. In a series of simulations in which the mass ratio was varied while keeping the size ratio fixed at s=1, we found that the approach of the viscosities and self-diffusion coefficients to their limiting mass ratio independent values was well described by a rather simple exponential dependence on mass ratio. The concentration dependence of the limiting infinite mass ratio values of the self-diffusion coefficients and zero shear rate viscosities were determined, and used to compute the hydrodynamic radius RH of the solute particles by various methods. The values of RH that were obtained by the different methods were reasonably consistent with each other, and indicated that the radius at which the slip boundary condition holds is slightly smaller than the cross-interaction radius between the solute and solvent particles.
By suitable interpretation of results from the linear analysis of interface dynamics, it is found that the hydrodynamic growth of the size L of domains that follow spinodal decomposition in fluid mixtures scales with time as L approximately t(alpha), with alpha = 4/7 in the inertial regime. The previously proposed exponent alpha = 2/3 is shown to indicate only the scaling of the oscillatory frequency omega(-2/3) approximately L of the largest structures of the system. The viscous dissipation in the system occurs within a layer of thickness L(d) that also follows a power law of the form L(d) approximately L3/4 in the inertial regime. In the viscous regime the growth is linear in time L approximately t and the dissipative region remains constant L(d) approximately L0.
We have performed extensive numerical studies of Coulomb explosion of small molecules such as CO2 and N2O with the help of a hydrodynamic model, which allows one to deal with many-electron systems in intense laser fields. The predicted kinetic energy releases of the fragmentation channels are in good agreement with the measured energies, using realistic molecular and laser excitation condition parameters.
Analytic solutions are presented for the hydrodynamic collimation of a relativistic fireball by a surrounding baryonic wind emanating from a torus. The opening angle is shown to be the ratio of the power output of the inner fireball to that of the exterior baryonic wind. The gamma ray burst 990123 might thus be interpreted as a baryon-poor jet (BPJ) with an energy output of order 10(50) erg or less, collimated by a baryonic wind from a torus with an energy output of order 10(52.5) erg, roughly the geometric mean of the BPJ and its isotropic equivalent.
We demonstrate that the attractive interaction measured between like-charged colloidal spheres near a wall can be accounted for by a nonequilibrium hydrodynamic effect. We present both analytical results and Brownian dynamics simulations which quantitatively capture the one-wall experiments of Larsen and Grier [Nature (London) 385, 230 (1997)], using a single unmeasured parameter.