Rheology of gelling polymers in the Zimm model

In order to study rheological properties of gelling systems in dilute solution, we investigate the viscosity and the normal stresses in the Zimm model for randomly crosslinked monomers. The distribution of cluster topologies and sizes is assumed to be given either by Erd\H os-R\'enyi random graphs or three-dimensional bond percolation. Within this model the critical behaviour of the viscosity and of the first normal stress coefficient is determined by the power-law scaling of their averages over clusters of a given size $n$ with $n$. We investigate these Mark--Houwink like scaling relations numerically and conclude that the scaling exponents are independent of the hydrodynamic interaction strength. The numerically determined exponents agree well with experimental data for branched polymers. However, we show that this traditional model of polymer physics is not able to yield a critical divergence at the gel point of the viscosity for a polydisperse dilute solution of gelation clusters. A generally accepted scaling relation for the Zimm exponent of the viscosity is thereby disproved.Comment: 9 pages, 2 figure

Generalized Helmholtz-Kirchhoff model for two dimensional distributed vortex motion

The two-dimensional Navier-Stokes equations are rewritten as a system of coupled nonlinear ordinary differential equations. These equations describe the evolution of the moments of an expansion of the vorticity with respect to Hermite functions and of the centers of vorticity concentrations. We prove the convergence of this expansion and show that in the zero viscosity and zero core size limit we formally recover the Helmholtz-Kirchhoff model for the evolution of point-vortices. The present expansion systematically incorporates the effects of both viscosity and finite vortex core size. We also show that a low-order truncation of our expansion leads to the representation of the flow as a system of interacting Gaussian (i.e. Oseen) vortices which previous experimental work has shown to be an accurate approximation to many important physical flows [9]

The short-time self-diffusion coefficient of a sphere in a suspension of rigid rods

The short--time self diffusion coefficient of a sphere in a suspension of rigid rods is calculated in first order in the rod volume fraction. For low rod concentrations the correction to the Einstein diffusion constant of the sphere is a linear function of the rod volume fraction with the slope proportional to the equilibrium averaged mobility diminution trace of the sphere interacting with a single freely translating and rotating rod. The two--body hydrodynamic interactions are calculated using the so--called bead model in which the rod is replaced by a stiff linear chain of touching spheres. The interactions between spheres are calculated numerically using the multipole method. Also an analytical expression for the diffusion coefficient as a function of the rod aspect ratio is derived in the limit of very long rods. We show that in this limit the correction to the Einstein diffusion constant does not depend on the size of the tracer sphere. The higher order corrections depending on the applied model are computed numerically. An approximate expression is provided, valid for a wide range of aspect ratios.Comment: 11 pages, 6 figure

Analytic results for the three-sphere swimmer at low Reynolds number

The simple model of a low Reynolds number swimmer made from three spheres that are connected by two arms is considered in its general form and analyzed. The swimming velocity, force--velocity response, power consumption, and efficiency of the swimmer are calculated both for general deformations and also for specific model prescriptions. The role of noise and coherence in the stroke cycle is also discussed.Comment: 7 pages, 3 figure

Direct measurement of the flow field around swimming microorganisms

Swimming microorganisms create flows that influence their mutual interactions and modify the rheology of their suspensions. While extensively studied theoretically, these flows have not been measured in detail around any freely-swimming microorganism. We report such measurements for the microphytes Volvox carteri and Chlamydomonas reinhardtii. The minute ~0.3% density excess of V. carteri over water leads to a strongly dominant Stokeslet contribution, with the widely-assumed stresslet flow only a correction to the subleading source dipole term. This implies that suspensions of V. carteri have features similar to suspensions of sedimenting particles. The flow in the region around C. reinhardtii where significant hydrodynamic interaction is likely to occur differs qualitatively from a "puller" stresslet, and can be described by a simple three-Stokeslet model.Comment: 4 pages, 4 figures, accepted for publication in PR

Brownian Dynamics of a Sphere Between Parallel Walls

We describe direct imaging measurements of a colloidal sphere's diffusion between two parallel surfaces. The dynamics of this deceptively simple hydrodynamically coupled system have proved difficult to analyze. Comparison with approximate formulations of a confined sphere's hydrodynamic mobility reveals good agreement with both a leading-order superposition approximation as well as a more general all-images stokeslet analysis.Comment: 4 pages, 3 figures, REVTeX with PostScript figure

Noisy swimming at low Reynolds numbers

Small organisms (e.g., bacteria) and artificial microswimmers move due to a combination of active swimming and passive Brownian motion. Considering a simplified linear three-sphere swimmer, we study how the swimmer size regulates the interplay between self-driven and diffusive behavior at low Reynolds number. Starting from the Kirkwood-Smoluchowski equation and its corresponding Langevin equation, we derive formulas for the orientation correlation time, the mean velocity and the mean square displacement in three space dimensions. The validity of the analytical results is illustrated through numerical simulations. Tuning the swimmer parameters to values that are typical of bacteria, we find three characteristic regimes: (i) Brownian motion at small times, (ii) quasi-ballistic behavior at intermediate time scales, and (iii) quasi-diffusive behavior at large times due to noise-induced orientation flipping. Our analytical results can be useful for a better quantitative understanding of optimal foraging strategies in bacterial systems, and they can help to construct more efficient artificial microswimmers in fluctuating fluids.Comment: minor changes/additions in the text, references added/updated, to appear in Phys. Rev.

Simple Viscous Flows: from Boundary Layers to the Renormalization Group

The seemingly simple problem of determining the drag on a body moving through a very viscous fluid has, for over 150 years, been a source of theoretical confusion, mathematical paradoxes, and experimental artifacts, primarily arising from the complex boundary layer structure of the flow near the body and at infinity. We review the extensive experimental and theoretical literature on this problem, with special emphasis on the logical relationship between different approaches. The survey begins with the developments of matched asymptotic expansions, and concludes with a discussion of perturbative renormalization group techniques, adapted from quantum field theory to differential equations. The renormalization group calculations lead to a new prediction for the drag coefficient, one which can both reproduce and surpass the results of matched asymptotics

Director configuration of planar solitons in nematic liquid crystals

The director configuration of disclination lines in nematic liquid crystals in the presence of an external magnetic field is evaluated. Our method is a combination of a polynomial expansion for the director and of further analytical approximations which are tested against a numerical shooting method. The results are particularly simple when the elastic constants are equal, but we discuss the general case of elastic anisotropy. The director field is continuous everywhere apart from a straight line segment whose length depends on the value of the magnetic field. This indicates the possibility of an elongated defect core for disclination lines in nematics due to an external magnetic field.Comment: 12 pages, Revtex, 8 postscript figure