40 research outputs found
Dynamic structure factor study of diffusion in strongly sheared suspensions
Diffusion of neutrally buoyant spherical particles in concentrated monodisperse suspensions under simple shear flow is investigated. We consider the case of non-Brownian particles in Stokes flow, which corresponds to the limits of infinite Péclet number and zero Reynolds number. Using an approach based upon ideas of dynamic light scattering we compute self- and gradient diffusion coefficients in the principal directions normal to the flow numerically from Accelerated Stokesian Dynamics simulations for large systems (up to 2000 particles). For the self-diffusivity, the present approach produces results identical to those reported earlier, obtained by probing the particles' mean-square displacements (Sierou & Brady, J. Fluid Mech. vol. 506, 2004 p. 285). For the gradient diffusivity, the computed coefficients are in good agreement with the available experimental results. The similarity between diffusion mechanisms in equilibrium suspensions of Brownian particles and in non-equilibrium non-colloidal sheared suspensions suggests an approximate model for the gradient diffusivity: {\textsfbi D}^\triangledown\,{\approx}\,{\textsfbi D}^s/S^{eq}(0), where {\textsfbi D}^s is the shear-induced self-diffusivity and is the static structure factor corresponding to the hard-sphere suspension at thermodynamic equilibrium
Collective diffusion in sheared colloidal suspensions
Collective diffusivity in a suspension of rigid particles in steady linear viscous flows is evaluated by investigating the dynamics of the time correlation of long-wavelength density fluctuations. In the absence of hydrodynamic interactions between suspended particles in a dilute suspension of identical hard spheres, closed-form asymptotic expressions for the collective diffusivity are derived in the limits of low and high Péclet numbers, where the Péclet number Pe = gamma-dot a^2/D0 with gamma-dot being the shear rate and D0 = kB T/6πη a is the Stokes–Einstein diffusion coefficient of an isolated sphere of radius a in a fluid of viscosity η. The effect of hydrodynamic interactions is studied in the analytically tractable case of weakly sheared (Pe « 1) suspensions.
For strongly sheared suspensions, i.e. at high Pe, in the absence of hydrodynamics the collective diffusivity Dc = 6 Ds∞, where Ds∞ is the long-time self-diffusivity and both scale as φ gamma-dot a^2$, where φ is the particle volume fraction. For weakly sheared suspensions it is shown that the leading dependence of collective diffusivity on the imposed flow is proportional to D0 φPe Ê, where Ê is the rate-of-strain tensor scaled by gamma-dot, regardless of whether particles interact hydrodynamically. When hydrodynamic interactions are considered, however, correlations of hydrodynamic velocity fluctuations yield a weakly singular logarithmic dependence of the cross-gradient-diffusivity on k at leading order as ak → 0 with k being the wavenumber of the density fluctuation. The diagonal components of the collective diffusivity tensor, both with and without hydrodynamic interactions, are of O(φPe2), quadratic in the imposed flow, and finite at k = 0.
At moderate particle volume fractions, 0.10 ≤ φ ≤ 0.35, Brownian Dynamics (BD) numerical simulations in which there are no hydrodynamic interactions are performed and the transverse collective diffusivity in simple shear flow is determined via time evolution of the dynamic structure factor. The BD simulation results compare well with the derived asymptotic estimates. A comparison of the high-Pe BD simulation results with available experimental data on collective diffusivity in non-Brownian sheared suspensions shows a good qualitative agreement, though hydrodynamic interactions prove to be important at moderate concentrations
Lamb-type solution and properties of unsteady Stokes equations
We derive the general solution of the unsteady Stokes equations for an
unbounded fluid in spherical polar coordinates, in both time and frequency
domains. The solution is an expansion in vector spherical harmonics and given
as a sum of a particular solution, proportional to pressure gradient exhibiting
power-law spatial dependence, and a solution of vector Helmholtz equation
decaying exponentially in far field, the decomposition originally introduced by
Lamb. The solution can be applied to construct the transient exterior flow
induced by an arbitrary velocity distribution at the spherical boundary, such
as arising in the squirmer model of a microswimmer. It can be used to construct
solutions for transient flows driven by initial conditions, unbounded flows
driven by volume forces or disturbance to the unsteady flow due to a stationary
spherical particle. The long-time behavior of solution is controlled by the
flow component corresponding to average (or collective) motion of the boundary.
This conclusion is illustrated by the study of decay of transversal wave in the
presence of a fixed sphere. We further show that the general representation
reduces to the well-known solutions for unsteady flow around a sphere
undergoing oscillatory rigid-body (translation and rotation) motion. The
proposed solution representation provides an explicit form of the velocity
potential far from an oscillating body ("generalized" Darcy's law) and high-
and low-frequency expansions. The leading-order high-frequency expansion yields
the well-known ideal (inviscid) flow approximation, and the leading-order
low-frequency expansion yields the steady Stokes equations. We derive the
higher-order corrections to these approximations and discuss d'Alembert
paradox. Continuation of the general solution to imaginary frequency yields the
general solution of the Brinkman equations describing viscous flow in porous
medium.Comment: 30 pages, revised versio
The role of symmetry in driven propulsion at low Reynolds number
We theoretically and experimentally investigate low-Reynolds-number
propulsion of geometrically achiral planar objects that possess a dipole moment
and that are driven by a rotating magnetic field. Symmetry considerations
(involving parity, , and charge conjugation, )
establish correspondence between propulsive states depending on orientation of
the dipolar moment. Although basic symmetry arguments do not forbid individual
symmetric objects to efficiently propel due to spontaneous symmetry breaking,
they suggest that the average ensemble velocity vanishes. Some additional
arguments show, however, that highly symmetrical (-even) objects
exhibit no net propulsion while individual less symmetrical
(-even) propellers do propel. Particular magnetization
orientation, rendering the shape -odd, yields
unidirectional motion typically associated with chiral structures, such as
helices. If instead of a structure with a permanent dipole we consider a
polarizable object, some of the arguments have to be modified. For instance, we
demonstrate a truly achiral (- and -even)
planar shape with an induced electric dipole that can propel by
electro-rotation. We thereby show that chirality is not essential for
propulsion due to rotation-translation coupling at low Reynolds number.Comment: 5 pages, 5 figure