arXiv subjects

J. Einarsson

Publications and source records attributed to J. Einarsson.

15 recordsLinked to original sources

arXiv2018

Intrinsic viscosity of a suspension of weakly Brownian ellipsoids in shear

We analyze the angular dynamics of triaxial ellipsoids in a shear flow subject to weak thermal noise. By numerically integrating an overdamped angular Langevin equation, we find the steady angular probability distribution for a range of triaxial particle shapes. From this distribution we compute the intrinsic viscosity of a dilute suspension of triaxial particles. We determine how the viscosity depends on particle shape in the limit of weak thermal noise. While the deterministic angular dynamics depends very sensitively on particle shape, we find that the shape dependence of the intrinsic viscosity is weaker, in general, and that suspensions of rod-like particles are the most sensitive to breaking of axisymmetry. The intrinsic viscosity of a dilute suspension of triaxial particles is smaller than that of a suspension of axisymmetric particles with the same volume, and the same ratio of major to minor axis lengths.

physics.flu-dyn↗

arXiv2017

Angular dynamics of small crystals in viscous flow

The angular dynamics of a very small ellipsoidal particle in a viscous flow decouples from its translational dynamics, and the particle angular velocity is given by Jeffery's theory. It is known that cuboid particles share these properties. In the literature a special case is most frequently discussed, namely that of axisymmetric particles with a continuous rotation symmetry. Here we compute the angular dynamics of crystals that possess a discrete rotation symmetry and certain mirror symmetries, but that do not have a continuous rotation symmetry. We give examples of such particles that nevertheless obey Jeffery's theory. But there are other examples where the angular dynamics is determined by a more general equation of motion.

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arXiv2016

Angular dynamics of a small particle in turbulence

We compute the angular dynamics of a neutrally buoyant nearly spherical particle immersed in an unsteady fluid. We assume that the particle is small, that its translational slip velocity is negligible, and that unsteady and convective inertia are small perturbations. We derive an approximation for the torque on the particle that determines the first inertial corrections to Jeffery's equation. These corrections arise as a consequence of local vortex stretching, and can be substantial in turbulence where local vortex stretching is strong and closely linked to the irreversibility of turbulence.

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arXiv2016

Spherical particle sedimenting in weakly viscoelastic shear flow

We consider the dynamics of a small spherical particle driven through an unbounded viscoelastic shear flow by an external force. We give analytical solutions to both the mobility problem (velocity of forced particle) and the resistance problem (force on fixed particle), valid to second order in the dimensionless Deborah and Weissenberg numbers, which represent the elastic relaxation time of the fluid relative to the rate of translation and the imposed shear rate. We find a shear-induced lift at $O({\rm Wi})$, a modified drag at $O({\rm De}^2)$ and $O({\rm Wi}^2)$, and a second lift that is orthogonal to the first, at $O({\rm Wi}^2)$. The relative importance of these effects depends strongly on the orientation of the forcing relative to the shear. We discuss how these forces affect the terminal settling velocity in an inclined shear flow. We also describe a new basis set of symmetric Cartesian tensors, and demonstrate how they enable general tensorial perturbation calculations such as the present theory. In particular this scheme allows us to write down a solution to the inhomogenous Stokes equations, required by the perturbation expansion, by a sequence of algebraic manipulations well suited to computer implementation.

physics.flu-dyn↗

arXiv2016

Angular velocity of a spheroid log rolling in a simple shear at small Reynolds number

We analyse the angular velocity of a small neutrally buoyant spheroid log rolling in a simple shear. When the effect of fluid inertia is negligible the angular velocity $ω$ equals half the fluid vorticity. We compute by singular perturbation theory how weak fluid inertia reduces the angular velocity in an unbounded shear, and how this reduction depends upon the shape of the spheroid (on its aspect ratio). In addition we determine the angular velocity by direct numerical simulations. The results are in excellent agreement with the theory at small but not too small values of the shear Reynolds number, for all aspect ratios considered. For the special case of a sphere we find $ω/s = -1/2+0.0540\, \mbox{Re}_\mbox{s}^{3/2}$ where $s$ is the shear rate, and $\mbox{Re}_\mbox{s}$ is the shear Reynolds number. This result differs from that derived by Lin et al. [J. Fluid Mech. 44 (1970) 1] who obtained a numerical coefficient roughly three times larger.

physics.flu-dyn↗

arXiv2015

Tumbling of asymmetric microrods in a microchannel flow

We describe results of measurements of the orientational motion of glass microrods in a microchannel flow, following the orientational motion of particles with different shapes. We determine how the orientational dynamics depends on the shape of the particle and on its initial orientation. We find that the dynamics depends so sensitively on the degree to which particle axisymmetry is broken that it is difficult to find particles that are sufficiently axisymmetric so that they exhibit periodic tumbling ("Jeffery orbits"). The results of our measurements confirm earlier theoretical analysis predicting sensitive dependence on particle shape and its initial orientation. Our results illustrate the different types of orientational dynamics for asymmetric particles predicted by theory.

physics.flu-dyn↗

arXiv2015

Numerical analysis of the angular motion of a neutrally buoyant spheroid in shear flow at small Reynolds numbers

We numerically analyse the rotation of a neutrally buoyant spheroid in a shear flow at small shear Reynolds number. Using direct numerical stability analysis of the coupled nonlinear particle-flow problem we compute the linear stability of the log-rolling orbit at small shear Reynolds number, ${\rm Re}_a$. As ${\rm Re}_a \to 0$ and as the box size of the system tends to infinity we find good agreement between the numerical results and earlier analytical predictions valid to linear order in ${\rm Re}_a$ for the case of an unbounded shear. The numerical stability analysis indicates that there are substantial finite-size corrections to the analytical results obtained for the unbounded system. We also compare the analytical results to results of lattice-Boltzmann simulations to analyse the stability of the tumbling orbit at shear Reynolds numbers of order unity. Theory for an unbounded system at infinitesimal shear Reynolds number predicts a bifurcation of the tumbling orbit at aspect ratio $λ_{\rm c} \approx 0.137$ below which tumbling is stable (as well as log rolling). The simulation results show a bifurcation line in the $λ$-${\rm Re}_a$ plane that reaches $λ\approx0.1275$ at the smallest shear Reynolds number (${\rm Re}_a=1$) at which we could simulate with the lattice-Boltzmann code, in qualitative agreement with the analytical results.

physics.flu-dyn↗

arXiv2015

The role of inertia for the rotation of a nearly spherical particle in a general linear flow

We analyse the angular dynamics of a neutrally buoyant nearly spherical particle immersed in a steady general linear flow. The hydrodynamic torque acting on the particle is obtained by means of a reciprocal theorem, regular perturbation theory exploiting the small eccentricity of the nearly spherical particle, and assuming that inertial effects are small, but finite.

physics.flu-dyn↗

arXiv2015

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 analyse how the linear stability of the \lq log-rolling\rq{} 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 non-linear terms in the Navier-Stokes equations are important.

physics.flu-dyn↗

arXiv2015

Rotation of a spheroid in a simple shear at small Reynolds number

We derive an effective equation of motion for the orientational dynamics of a neutrally buoyant spheroid suspended in a simple shear flow, valid for arbitrary particle aspect ratios and to linear order in the shear Reynolds number. We show how inertial effects lift the degeneracy of the Jeffery orbits and determine the stabilities of the log-rolling and tumbling orbits at infinitesimal shear Reynolds numbers. For prolate spheroids we find stable tumbling in the shear plane, log-rolling is unstable. For oblate particles, by contrast, log-rolling is stable and tumbling is unstable provided that the aspect ratio is larger than a critical value. When the aspect ratio is smaller than this value tumbling turns stable, and an unstable limit cycle is born.

physics.flu-dyn↗

arXiv2014

Shape-dependence of particle rotation in isotropic turbulence

We consider the rotation of neutrally buoyant axisymmetric particles suspended in isotropic turbulence. Using laboratory experiments as well as numerical and analytical calculations, we explore how particle rotation depends upon particle shape. We find that shape strongly affects orientational trajectories, but that it has negligible effect on the variance of the particle angular velocity. Previous work has shown that shape significantly affects the variance of the tumbling rate of axisymmetric particles. It follows that shape affects the spinning rate in a way that is, on average, complementary to the shape-dependence of the tumbling rate. We confirm this relationship using direct numerical simulations, showing how tumbling rate and spinning rate variances show complementary trends for rod-shaped and disk-shaped particles. We also consider a random but non-turbulent flow. This allows us to explore which of the features observed for rotation in turbulent flow are due to the effects of particle alignment in vortex tubes.

physics.flu-dyn↗

arXiv2013

Tumbling of small axisymmetric particles in random and turbulent flows

We analyse the tumbling of small non-spherical, axisymmetric particles in random and turbulent flows. We compute the orientational dynamics in terms of a perturbation expansion in the Kubo number, and obtain the tumbling rate in terms of Lagrangian correlation functions. These capture preferential sampling of the fluid gradients which in turn can give rise to differences in the tumbling rates of disks and rods. We show that this is a weak effect in Gaussian random flows. But in turbulent flows persistent regions of high vorticity cause disks to tumble much faster than rods, as observed in direct numerical simulations [Parsa et al., Phys. Rev. Lett. 109 (2012) 134501]. For larger particles (at finite Stokes numbers), rotational and translational inertia affects the tumbling rate and the angle at which particles collide, due to the formation of rotational caustics.

physics.flu-dyn↗

arXiv2013

Orientational dynamics of weakly inertial axisymmetric particles in steady viscous flows

The orientational dynamics of weakly inertial axisymmetric particles in a steady flow is investigated. We derive an asymptotic equation of motion for the unit axial vector along the particle symmetry axis, valid for small Stokes number St, and for any axisymmetric particle in any steady linear viscous flow. This reduced dynamics is analysed in two ways, both pertain to the case of a simple shear flow. In this case inertia induces a coupling between precession and nutation. This coupling affects the dynamics of the particle, breaks the degeneracy of the Jeffery orbits, and creates two limiting periodic orbits. We calculate the leading-order Floquet exponents of the limiting periodic orbits and show analytically that prolate objects tend to a tumbling orbit, while oblate objects tend to a log-rolling orbit, in agreement with previous analytical and numerical results. Second, we analyse the role of the limiting orbits when rotational noise is present. We formulate the Fokker-Planck equation describing the orientational distribution of an axisymmetric particle, valid for small St and general Peclet number Pe. Numerical solutions of the Fokker-Planck equation, obtained by means of expansion in spherical harmonics, show that stationary orientational distributions are close to the inertia-free case when Pe St << 1, whereas they are determined by inertial effects, though small, when Pe >> 1/St >> 1.

physics.flu-dyn↗

arXiv2013

Aperiodic tumbling of microrods advected in a microchannel flow

We report on an experimental investigation of the tumbling of microrods in the shear flow of a microchannel (40 x 2.5 x 0.4 mm). The rods are 20 to 30 microns long and their diameters are of the order of 1 micron. Images of the centre-of-mass motion and the orientational dynamics of the rods are recorded using a microscope equipped with a CCD camera. A motorised microscope stage is used to track individual rods as they move along the channel. Automated image analysis determines the position and orientation of a tracked rods in each video frame. We find different behaviours, depending on the particle shape, its initial position, and orientation. First, we observe periodic as well as aperiodic tumbling. Second, the data show that different tumbling trajectories exhibit different sensitivities to external perturbations. These observations can be explained by slight asymmetries of the rods. Third we observe that after some time, initially periodic trajectories lose their phase. We attribute this to drift of the centre of mass of the rod from one to another stream line of the channel flow.

physics.flu-dyn↗

arXiv2012

A microfluidic device for the study of the orientational dynamics of microrods

We describe a microfluidic device for studying the orientational dynamics of microrods. The device enables us to experimentally investigate the tumbling of microrods immersed in the shear flow in a microfluidic channel with a depth of 400 mu and a width of 2.5 mm. The orientational dynamics was recorded using a 20 X microscopic objective and a CCD camera. The microrods were produced by shearing microdroplets of photocurable epoxy resin. We show different examples of empirically observed tumbling. On the one hand we find that short stretches of the experimentally determined time series are well described by fits to solutions of Jeffery's approximate equation of motion [Jeffery, Proc. R. Soc. London. 102 (1922), 161-179]. On the other hand we find that the empirically observed trajectories drift between different solutions of Jeffery's equation. We discuss possible causes of this orbit drift.

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