Trajectory analysis for non-Brownian inertial suspensions in simple shear flow

We analyse pair trajectories of equal-sized spherical particles in simple shear flow for small but finite Stokes numbers. The Stokes number, \mbox{\textit{St}} \,{=}\, \dot{\gamma} \tau_p, is a dimensionless measure of particle inertia; here, $\tau_p$ is the inertial relaxation time of an individual particle and $\dot{\gamma}$ is the shear rate. In the limit of weak particle inertia, a regular small-\mbox{\textit{St}} expansion of the particle velocity is used in the equations of motion to obtain trajectory equations to the desired order in \mbox{\textit{St}}. The equations for relative trajectories are then solved, to O(\mbox{\textit{St}}), in the dilute limit, including only pairwise interactions. Particle inertia is found to destroy the fore–aft symmetry of the zero-Stokes trajectories, and finite-\mbox{\textit{St}} open trajectories suffer net transverse displacements in the velocity gradient and vorticity directions. The vorticity displacement remains O(\mbox{\textit{St}}), while the scaling of the gradient displacement increases from O(\mbox{\textit{St}}) for far-field open trajectories, to O(\mbox{\textit{St}}^{{1}/{2}}) for open trajectories with O(\mbox{\textit{St}}^{{1}/{2}}) upstream gradient offsets. The gradient displacement also changes sign, being negative close to the plane of the reference sphere (the shearing plane) on account of dominant lubrication interactions, and then becoming positive at larger off-plane separations. The transverse displacements accompanying successive pair interactions lead to a diffusive behaviour for long times. The shear-induced diffusivity in the vorticity direction is O(\mbox{\textit{St}}^2\phi \dot{\gamma} a^2), while that in the gradient direction scales as O(\mbox{\textit{St}}^2 \ln \mbox{\textit{St}}\,\phi \dot{\gamma} a^2) and O(\mbox{\textit{St}}^2 \phi \ln (1/\phi) \dot{\gamma} a^2) in the limits \phi \,{\ll}\, \mbox{\textit{St}}^{{1}/{3}} and \mbox{\textit{St}}^{{1}/{3}} \,{\ll}\, \phi \,{\ll}\, 1, respectively. Further, the region of zero-Stokes closed trajectories is destroyed, and there exists a new attracting limit cycle whose location in the shearing plane is, at leading order, independent of \mbox{\textit{St}}. The extension of the present analysis to include a generic linear flow, and the implications of the finite-\mbox{\textit{St}} trajectory modifications for coagulating systems are discussed

Complex oscillatory yielding of model hard sphere glasses

The yielding behaviour of hard sphere glasses under large amplitude oscillatory shear has been studied by probing the interplay of Brownian motion and shear-induced diffusion at varying oscillation frequencies. Stress, structure and dynamics are followed by experimental rheology and Browian Dynamics simulations. Brownian motion assisted cage escape dominates at low frequencies while escape through shear-induced collisions at high ones, both related with a yielding peak in\ $G^{\prime \prime}$. At intermediate frequencies a novel, for HS glasses, double peak in $G^{\prime \prime}$ is revealed reflecting both mechanisms. At high frequencies and strain amplitudes a persistent structural anisotropy causes a stress drop within the cycle after strain reversal, while higher stress harmonics are minimized at certain strain amplitudes indicating an apparent harmonic response.Comment: 4 figures placed at the end with following order: Figure 1, figure 3, figure 4 and figure

Hydrodynamic stress on fractal aggregates of spheres

We calculate the average hydrodynamic stress on fractal aggregates of spheres using Stokesian dynamics. We find that for fractal aggregates of force-free particles, the stress does not grow as the cube of the radius of gyration, but rather as the number of particles in the aggregate. This behavior is only found for random aggregates of force-free particles held together by hydrodynamic lubrication forces. The stress on aggregates of particles rigidly connected by interparticle forces grows as the radius of gyration cubed. We explain this behavior by examining the transmission of the tension along connecting lines in an aggregate and use the concept of a persistance length in order to characterize this stress transmission within an aggregate

Short- and intermediate-time behavior of the linear stress relaxation in semiflexible polymers

The linear viscoelasticity of semiflexible polymers is studied through Brownian Dynamics simulations covering a broad range of chain stiffness and time scales. Our results agree with existing theoretical predictions in the flexible and stiff limits; however, we find that over a wide intermediate-time window spanning several decades, the stress relaxation is described by a single power law t^(-alpha), with the exponent alpha apparently varying continuously from 1/2 for flexible chains, to 5/4 for stiff ones. Our study identifies the limits of validity of the t^(-3/4) power law at short times predicted by recent theories. An additional regime is identified, the "ultrastiff" chains, where this behavior disappears. In the absence of Brownian motion, the purely mechanical stress relaxation produces a t^(-3/4) power law for both short and intermediate times

Cosmic Censorship: As Strong As Ever

Spacetimes which have been considered counter-examples to strong cosmic censorship are revisited. We demonstrate the classical instability of the Cauchy horizon inside charged black holes embedded in de Sitter spacetime for all values of the physical parameters. The relevant modes which maintain the instability, in the regime which was previously considered stable, originate as outgoing modes near to the black hole event horizon. This same mechanism is also relevant for the instability of Cauchy horizons in other proposed counter-examples of strong cosmic censorship.Comment: 4 pages RevTeX style, 1 figure included using epsfi

Yielding of Hard-Sphere Glasses during Start-Up Shear

Concentrated hard-sphere suspensions and glasses are investigated with rheometry, confocal microscopy, and Brownian dynamics simulations during start-up shear, providing a link between microstructure, dynamics, and rheology. The microstructural anisotropy is manifested in the extension axis where the maximum of the pair-distribution function exhibits a minimum at the stress overshoot. The interplay between Brownian relaxation and shear advection as well as the available free volume determine the structural anisotropy and the magnitude of the stress overshoot. Shear-induced cage deformation induces local constriction, reducing in-cage diffusion. Finally, a superdiffusive response at the steady state, with a minimum of the time-dependent effective diffusivity, reflects a continuous cage breakup and reformation

Self-diffusion of Brownian particles in concentrated suspensions under shear

The self-diffusivity of Brownian hard spheres in a simple shear flow is studied by numerical simulation. Particle trajectories are calculated by Stokesian dynamics, with an accurate representation of the suspension hydrodynamics that includes both many-body interactions and lubrication. The simulations are of a monolayer of identical spheres as a function of the Péclet number: Pe =gamma-dot a^2/D0, which measures the relative importance of shear and Brownian forces. Here gamma-dot is the shear rate, a the particle radius, and D0 the diffusion coefficient of a single sphere at infinite dilution. In the absence of shear, using only hydrodynamic interactions, the simulations reproduce the pair-distribution function of the equivalent hard-disk system. Both short- and long-time self-diffusivities are determined as a function of the Péclet number. The results show a clear transition from a Brownian motion dominated regime (Pe10) with a dramatic change in the behavior of the long-time self-diffusivity

The rheology of Brownian suspensions

The viscosity of a suspension of spherical Brownian particles is determined by Stokesian dynamics as a function of the Péclet number. Several new aspects concerning the theoretical derivation of the direct contribution of the Brownian motion to the bulk stress are given, along with the results obtained from a simulation of a monolayer. The simulations reproduce the experimental behavior generally observed in dense suspensions, and an explanation of this behavior is given by observing the evolution of the different contributions to the viscosity with shear rate. The shear thinning at low Péclet numbers is due to the disappearance of the direct Brownian contribution to the viscosity; the deformation of the equilibrium microstructure is, however, small. By contrast, at very high Péclet numbers the suspension shear thickens due to the formation of large clusters

Anisotropic Diffusion Limited Aggregation

Using stochastic conformal mappings we study the effects of anisotropic perturbations on diffusion limited aggregation (DLA) in two dimensions. The harmonic measure of the growth probability for DLA can be conformally mapped onto a constant measure on a unit circle. Here we map $m$ preferred directions for growth of angular width $\sigma$ to a distribution on the unit circle which is a periodic function with $m$ peaks in $[-\pi, \pi)$ such that the width $\sigma$ of each peak scales as $\sigma \sim 1/\sqrt{k}$, where $k$ defines the ``strength'' of anisotropy along any of the $m$ chosen directions. The two parameters $(m,k)$ map out a parameter space of perturbations that allows a continuous transition from DLA (for $m=0$ or $k=0$) to $m$ needle-like fingers as $k \to \infty$. We show that at fixed $m$ the effective fractal dimension of the clusters $D(m,k)$ obtained from mass-radius scaling decreases with increasing $k$ from $D_{DLA} \simeq 1.71$ to a value bounded from below by $D_{min} = 3/2$. Scaling arguments suggest a specific form for the dependence of the fractal dimension $D(m,k)$ on $k$ for large $k$, form which compares favorably with numerical results.Comment: 6 pages, 4 figures, submitted to Phys. Rev.