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

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

Human sperm accumulation near surfaces: a simulation study

A hybrid boundary integral/slender body algorithm for modelling flagellar cell motility is presented. The algorithm uses the boundary element method to represent the ‘wedge-shaped’ head of the human sperm cell and a slender body theory representation of the flagellum. The head morphology is specified carefully due to its significant effect on the force and torque balance and hence movement of the free-swimming cell. The technique is used to investigate the mechanisms for the accumulation of human spermatozoa near surfaces. Sperm swimming in an infinite fluid, and near a plane boundary, with prescribed planar and three-dimensional flagellar waveforms are simulated. Both planar and ‘elliptical helicoid’ beating cells are predicted to accumulate at distances of approximately 8.5–22 μm from surfaces, for flagellar beating with angular wavenumber of 3π to 4π. Planar beating cells with wavenumber of approximately 2.4π or greater are predicted to accumulate at a finite distance, while cells with wavenumber of approximately 2π or less are predicted to escape from the surface, likely due to the breakdown of the stable swimming configuration. In the stable swimming trajectory the cell has a small angle of inclination away from the surface, no greater than approximately 0.5°. The trapping effect need not depend on specialized non-planar components of the flagellar beat but rather is a consequence of force and torque balance and the physical effect of the image systems in a no-slip plane boundary. The effect is relatively weak, so that a cell initially one body length from the surface and inclined at an angle of 4°–6° towards the surface will not be trapped but will rather be deflected from the surface. Cells performing rolling motility, where the flagellum sweeps out a ‘conical envelope’, are predicted to align with the surface provided that they approach with sufficiently steep angle. However simulation of cells swimming against a surface in such a configuration is not possible in the present framework. Simulated human sperm cells performing a planar beat with inclination between the beat plane and the plane-of-flattening of the head were not predicted to glide along surfaces, as has been observed in mouse sperm. Instead, cells initially with the head approximately 1.5–3 μm from the surface were predicted to turn away and escape. The simulation model was also used to examine rolling motility due to elliptical helicoid flagellar beating. The head was found to rotate by approximately 240° over one beat cycle and due to the time-varying torques associated with the flagellar beat was found to exhibit ‘looping’ as has been observed in cells swimming against coverslips

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

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

Boundary conditions for interfaces of electromagnetic (photonic) crystals and generalized Ewald-Oseen extinction principle

The problem of plane-wave diffraction on semi-infinite orthorhombic electromagnetic (photonic) crystals of general kind is considered. Boundary conditions are obtained in the form of infinite system of equations relating amplitudes of incident wave, eigenmodes excited in the crystal and scattered spatial harmonics. Generalized Ewald-Oseen extinction principle is formulated on the base of deduced boundary conditions. The knowledge of properties of infinite crystal's eigenmodes provides option to solve the diffraction problem for the corresponding semi-infinite crystal numerically. In the case when the crystal is formed by small inclusions which can be treated as point dipolar scatterers with fixed direction the problem admits complete rigorous analytical solution. The amplitudes of excited modes and scattered spatial harmonics are expressed in terms of the wave vectors of the infinite crystal by closed-form analytical formulae. The result is applied for study of reflection properties of metamaterial formed by cubic lattice of split-ring resonators.Comment: 15 pages, 8 figures, submitted to PR

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.