Effect of weak fluid inertia upon Jeffery orbits

We consider the rotation of small neutrally buoyant axisymmetric particles in a viscous steady shear flow. When inertial effects are negligible the problem exhibits infinitely many periodic solutions, the "Jeffery orbits." We compute how inertial effects lift their degeneracy by perturbatively solving the coupled particle-flow equations. We obtain an equation of motion valid at small shear Reynolds numbers, for spheroidal particles with arbitrary aspect ratios. We analyze how the linear stability of the "log-rolling" orbit depends on particle shape and find it to be unstable for prolate spheroids. This resolves a puzzle in the interpretation of direct numerical simulations of the problem. In general, both unsteady and nonlinear terms in the Navier-Stokes equations are important.

Rheological study of two-dimensional very anisometric colloidal particle suspensions: from shear-induced orientation to viscous dissipation

In the present study, we investigate the evolution with shear of the viscosity of aqueous suspensions of size-selected natural swelling clay minerals for volume fractions extending from isotropic liquids to weak nematic gels. Such suspensions are strongly shear-thinning, a feature that is systematically observed for suspensions of nonspherical particles and that is linked to their orientational properties. We then combined our rheological measurements with small-angle X-ray scattering experiments that, after appropriate treatment, provide the orientational field of the particles. Whatever the clay nature, particle size, and volume fraction, this orientational field was shown to depend only on a nondimensional Péclet number (Pe) defined for one isolated particle as the ratio between hydrodynamic energy and Brownian thermal energy. The measured orientational fields were then directly compared to those obtained for infinitely thin disks through a numerical computation of the Fokker-Plank equation. Even in cases where multiple hydrodynamic interactions dominate, qualitative agreement between both orientational fields is observed, especially at high Péclet number. We have then used an effective approach to assess the viscosity of these suspensions through the definition of an effective volume fraction. Using such an approach, we have been able to transform the relationship between viscosity and volume fraction (ηr = f(φ)) into a relationship that links viscosity with both flow and volume fraction (ηr = f(φ, Pe)).

Chaotic particle sedimentation in a rotating flow with time-periodic strength

Particle sedimentation in the vicinity of a fixed horizontal vortex with time-dependent intensity can be chaotic, provided gravity is sufficient to displace the particle cloud while the vortex is off or weak. This "stretch, sediment, and fold" mechanism is close to the so-called blinking vortex effect, which is responsible for chaotic transport of perfect tracers, except that in the present case the vortex motion is replaced by gravitational settling. In the present work this phenomenon is analyzed for heavy Stokes particles moving under the sole effect of gravity and of a linear drag. The vortex is taken to be a fixed isolated point vortex, the intensity of which varies under the effect of either boundary conditions or volume force. When the unsteadiness of the vortex is weak and the free-fall velocity is of the order of the fluid velocity, and the particle response time is small, the particle motion equation can be written asymptotically as a perturbed Hamiltonian system, the phase portrait of which displays a homoclinic trajectory. A homoclinic bifurcation is therefore likely to occur, and the contribution of particle inertia to the occurrence of this bifurcation is analyzed asymptotically by using Melnikov's method.

Analytical investigation of the combined effect of fluid inertia and unsteadiness on low-Re particle centrifugation

We analyze the explicit contribution of fluid inertia and fluid unsteadiness to the force acting on a solid sphere moving in a vertical solid-body rotation flow, in the limit of small Reynolds and Taylor numbers. This problem can be thought of as a test case where the flow induced by the particle is both unsteady (in the laboratory frame) and convected by the unperturbed flow. Many authors assume that the contributions of these two effects can be approximately superposed, and postulate that the particle motion equation is composed of the classical Boussinesq-Basset-Oseen equation (obtained by neglecting the fluid inertia) plus an additive lift force. In the present paper the simplicity of the unperturbed flow enables one to calculate analytically the explicit contribution of each term appearing in the perturbed flow equation (by using matched asymptotic expansions). Our results show how the convective terms and the unsteady term do contribute to the particle drag and lift coefficients in a very complex and nonadditive manner.

Geometrical analysis of chaotic mixing in a low Reynolds number magnetohydrodynamic quadripolar flow

A mixing device for highly viscous fluids with finite electrical conductivity is investigated theoretically. Stirring is performed by means of electromagnetic forces provided by inductor wires located outside the flow domain. The flow shows hyperbolic and elliptic singular points. Inductors are displaced in a periodic manner, leading to an efficient stretching and folding mechanism. The goodness of mixing is quantified by means of a geometrical analysis based on box-counting techniques. This analysis gives valuable information about advection of a spot of dye injected in the flow, in the limit of infinite Peclet numbers. A spatiotemporal criterion for mixing efficiency is derived, and characteristic scales are analyzed. The influence of various parameters on mixing efficiency is discussed by making use of both the geometrical analysis and Poincaré sections.

Time-dependent geometry and energy distribution in a spiral vortex layer

The purpose of this paper is to study how the geometry and the spatial distribution of energy fluctuations of different length scales in a spiral vortex layer are related to each other in a time-dependent way. The numerical solution of Krasny [J. Comput. Phys. 65, 292 (1986)], corresponding to the development of the Kelvin-Helmholtz instability, is analyzed in order to determine some geometrical features necessary for the analysis of Lundgren's unstrained spiral vortex. The energy distribution of the asymptotic solution of Lundgren characterized by a similar geometry is investigated analytically (1) in the wavelet radius-scale space, with a wavelet selective in the radial direction, and (2) in the wavelet azimuth-scale space, with a wavelet selective in the azimuthal direction. Energy in the wavelet radius-scale space is organized in "blobs" distributed in a way determined by the Kolmogorov capacity of the spiral D(K) in [1,2] (which determines the rate of accumulation of spiral turns). As time evolves these blobs move towards the small scale region of the wavelet radius-scale space, until their scale is of the order of the diffusive length scale square root[nut], where t is the time and nu is the kinematic viscosity. In contrast, energy in the wavelet azimuth-scale space is not localized, and is characterized by a shear-augmented viscous cutoff proportional to sqaure root[nut(3)]. An accelerated viscous dissipation of the enstrophy and energy of Lundgren's spiral vortex is found for D(K) > 1.75, but not for D(K) < or =1.75.