The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow

We propose a mathematical derivation of Brinkman's force for a cloud of particles immersed in an incompressible fluid. Our starting point is the Stokes or steady Navier-Stokes equations set in a bounded domain with the disjoint union of N balls of radius 1/N removed, and with a no-slip boundary condition for the fluid at the surface of each ball. The large N limit of the fluid velocity field is governed by the same (Navier-)Stokes equations in the whole domain, with an additional term (Brinkman's force) that is (minus) the total drag force exerted by the fluid on the particle system. This can be seen as a generalization of Allaire's result in [Arch. Rational Mech. Analysis 113 (1991), 209-259] who treated the case of motionless, periodically distributed balls. Our proof is based on slightly simpler, though similar homogenization techniques, except that we avoid the periodicity assumption and use instead the phase-space empirical measure for the particle system. Similar equations are used for describing the fluid phase in various models for sprays

Dynamics in supramolecular polymer networks formed by associating telechelic chains

We present hybrid molecular dynamics/Monte Carlo simulations of supramolecular networks formed by unentangled telechelic chains with sticky end monomers (or stickers) of finite functionality. The reversible bonding between sticky monomers leads to the formation of sticker clusters with well-defined size distribution, which in turn work as cross-links for transient polymer networks. We study the kinetics of sticky monomer association, the topological structure and the resulting dynamic and rheological behavior of the supramolecular systems as a function of the sticker bonding energy $\varepsilon$ and the parent polymer chain length. Percolated transient networks are formed above a threshold bonding energy around $4.3 k_B T$. At high bonding energies $\varepsilon \geq 10 k_B T$, the majority of the stickers are fully reacted and the fraction of open stickers is less than $1\%$. The conventional picture of a single sticker hopping from one cluster to another is energetically unfavorable. We find the dynamic and rheological behavior of such strongly associated supramolecular networks are dominated by a partner exchange mechanism in which the stickers exchange their associated partners, and so release the imposed topological constraints, through the association and disassociation of sticker clusters. The characteristic time of the partner exchange events grows exponentially with the bonding energy and is up to $2$ orders of magnitude longer than the average lifetime of the reversible bonds. As a result, three relaxation regimes can be clearly identified in the stress and chain end-to-end vector relaxation functions as well as the mean-squared displacements of the stickers, which are the initial Rouse regime, the intermediate rubbery regime and the terminal relaxation regime. A phantom chain hopping model based on the microscopic understanding is proposed to describe the chain relaxation dynamics in the supramolecular networks, which provides numerical predictions in reasonably good agreement with our simulation results

Structural, mechanical and thermodynamic properties of a coarse-grained DNA model

We explore in detail the structural, mechanical and thermodynamic properties of a coarse-grained model of DNA similar to that introduced in Thomas E. Ouldridge, Ard A. Louis, Jonathan P.K. Doye, Phys. Rev. Lett. 104 178101 (2010). Effective interactions are used to represent chain connectivity, excluded volume, base stacking and hydrogen bonding, naturally reproducing a range of DNA behaviour. We quantify the relation to experiment of the thermodynamics of single-stranded stacking, duplex hybridization and hairpin formation, as well as structural properties such as the persistence length of single strands and duplexes, and the torsional and stretching stiffness of double helices. We also explore the model's representation of more complex motifs involving dangling ends, bulged bases and internal loops, and the effect of stacking and fraying on the thermodynamics of the duplex formation transition.Comment: 25 pages, 16 figure

Boundary conditions associated with the Painlev\'e III' and V evaluations of some random matrix averages

In a previous work a random matrix average for the Laguerre unitary ensemble, generalising the generating function for the probability that an interval $(0,s)$ at the hard edge contains $k$ eigenvalues, was evaluated in terms of a Painlev\'e V transcendent in $\sigma$-form. However the boundary conditions for the corresponding differential equation were not specified for the full parameter space. Here this task is accomplished in general, and the obtained functional form is compared against the most general small $s$ behaviour of the Painlev\'e V equation in $\sigma$-form known from the work of Jimbo. An analogous study is carried out for the the hard edge scaling limit of the random matrix average, which we have previously evaluated in terms of a Painlev\'e \IIId transcendent in $\sigma$-form. An application of the latter result is given to the rapid evaluation of a Hankel determinant appearing in a recent work of Conrey, Rubinstein and Snaith relating to the derivative of the Riemann zeta function

Infinite-cluster geometry in central-force networks

We show that the infinite percolating cluster (with density P_inf) of central-force networks is composed of: a fractal stress-bearing backbone (Pb) and; rigid but unstressed ``dangling ends'' which occupy a finite volume-fraction of the lattice (Pd). Near the rigidity threshold pc, there is then a first-order transition in P_inf = Pd + Pb, while Pb is second-order with exponent Beta'. A new mean field theory shows Beta'(mf)=1/2, while simulations of triangular lattices give Beta'_tr = 0.255 +/- 0.03.Comment: 6 pages, 4 figures, uses epsfig. Accepted for publication in Physical Review Letter

Critical behaviour of the Rouse model for gelling polymers

It is shown that the traditionally accepted "Rouse values" for the critical exponents at the gelation transition do not arise from the Rouse model for gelling polymers. The true critical behaviour of the Rouse model for gelling polymers is obtained from spectral properties of the connectivity matrix of the fractal clusters that are formed by the molecules. The required spectral properties are related to the return probability of a "blind ant"-random walk on the critical percolating cluster. The resulting scaling relations express the critical exponents of the shear-stress-relaxation function, and hence those of the shear viscosity and of the first normal stress coefficient, in terms of the spectral dimension $d_{s}$ of the critical percolating cluster and the exponents $\sigma$ and $\tau$ of the cluster-size distribution.Comment: 9 pages, slightly extended version, to appear in J. Phys.

Short Time Behavior in De Gennes' Reptation Model

To establish a standard for the distinction of reptation from other modes of polymer diffusion, we analytically and numerically study the displacement of the central bead of a chain diffusing through an ordered obstacle array for times $t < O(N^2)$. Our theory and simulations agree quantitatively and show that the second moment approaches the $t^{1/4}$ often viewed as signature of reptation only after a very long transient and only for long chains (N > 100). Our analytically solvable model furthermore predicts a very short transient for the fourth moment. This is verified by computer experiment.Comment: 4 pages, revtex, 4 ps file

Phase-based video motion processing

We introduce a technique to manipulate small movements in videos based on an analysis of motion in complex-valued image pyramids. Phase variations of the coefficients of a complex-valued steerable pyramid over time correspond to motion, and can be temporally processed and amplified to reveal imperceptible motions, or attenuated to remove distracting changes. This processing does not involve the computation of optical flow, and in comparison to the previous Eulerian Video Magnification method it supports larger amplification factors and is significantly less sensitive to noise. These improved capabilities broaden the set of applications for motion processing in videos. We demonstrate the advantages of this approach on synthetic and natural video sequences, and explore applications in scientific analysis, visualization and video enhancement.Shell ResearchUnited States. Defense Advanced Research Projects Agency. Soldier Centric Imaging via Computational CamerasNational Science Foundation (U.S.) (CGV-1111415)Cognex CorporationMicrosoft Research (PhD Fellowship)American Society for Engineering Education. National Defense Science and Engineering Graduate Fellowshi

Dynamics of gelling liquids: a short survey

The dynamics of randomly crosslinked liquids is addressed via a Rouse- and a Zimm-type model with crosslink statistics taken either from bond percolation or Erdoes-Renyi random graphs. While the Rouse-type model isolates the effects of the random connectivity on the dynamics of molecular clusters, the Zimm-type model also accounts for hydrodynamic interactions on a preaveraged level. The incoherent intermediate scattering function is computed in thermal equilibrium, its critical behaviour near the sol-gel transition is analysed and related to the scaling of cluster diffusion constants at the critical point. Second, non-equilibrium dynamics is studied by looking at stress relaxation in a simple shear flow. Anomalous stress relaxation and critical rheological properties are derived. Some of the results contradict long-standing scaling arguments, which are shown to be flawed by inconsistencies.Comment: 21 pages, 3 figures; Dedicated to Lothar Schaefer on the occasion of his 60th birthday; Changes: added comments on the gel phase and some reference

On the Decoupling of Layered Superconducting Films in Parallel Magnetic Field

The issue of the decoupling of extreme type-II superconducting thin films ($\lambda_L\rightarrow\infty$) with weakly Josephson-coupled layers in magnetic field parallel to the layers is considered via the corresponding frustrated $XY$ model used to describe the mixed phase in the critical regime. For the general case of arbitrary field orientations such that the perpendicular magnetic field component is larger than the decoupling cross-over scale characteristic of layered superconductors, we obtain independent parallel and perpendicular vortex lattices. Specializing to the double-layer case, we compute the parallel lower-critical field with entropic effects included, and find that it vanishes exponentially as temperature approaches the layer decoupling transition in zero-field. The parallel reversible magnetization is also calculated in this case, where we find that it shows a cross-over phenomenon as a function of parallel field in the intermediate regime of the mixed phase in lieu of a true layer-decoupling transition. It is argued that such is the case for any finite number of layers, since the isolated double layer represents the weakest link.Comment: 29 pages of plain TeX, 2 postscript figures, improved discussio