16 research outputs found
Hysteretic and chaotic dynamics of viscous drops in creeping flows with rotation
It has been shown in our previous publication
(Blawzdziewicz,Cristini,Loewenberg,2003) that high-viscosity drops in two
dimensional linear creeping flows with a nonzero vorticity component may have
two stable stationary states. One state corresponds to a nearly spherical,
compact drop stabilized primarily by rotation, and the other to an elongated
drop stabilized primarily by capillary forces. Here we explore consequences of
the drop bistability for the dynamics of highly viscous drops. Using both
boundary-integral simulations and small-deformation theory we show that a
quasi-static change of the flow vorticity gives rise to a hysteretic response
of the drop shape, with rapid changes between the compact and elongated
solutions at critical values of the vorticity. In flows with sinusoidal
temporal variation of the vorticity we find chaotic drop dynamics in response
to the periodic forcing. A cascade of period-doubling bifurcations is found to
be directly responsible for the transition to chaos. In random flows we obtain
a bimodal drop-length distribution. Some analogies with the dynamics of
macromolecules and vesicles are pointed out.Comment: 22 pages, 13 figures. submitted to Journal of Fluid Mechanic
Equilibrium and nonequilibrium thermodynamics of particle-stabilized thin liquid films
Our recent quasi-two-dimensional thermodynamic description of thin-liquid
films stabilized by colloidal particles is generalized to describe nonuniform
equilibrium states of films in external potentials and nonequilibrium transport
processes produced in the film by gradients of thermodynamic forces. Using a
Monte--Carlo simulation method, we have determined equilibrium equations of
state for a film stabilized by a suspension of hard spheres. Employing a
multipolar-expansion method combined with a flow-reflection technique, we have
also evaluated the short-time film-viscosity coefficients and collective
particle mobility.Comment: 16 pages, 10 figure
Swapping trajectories: a new wall-induced cross-streamline particle migration mechanism in a dilute suspension of spheres
Binary encounters between spherical particles in shear flow are studied for a
system bounded by a single planar wall or two parallel planar walls under
creeping flow conditions. We show that wall proximity gives rise to a new class
of binary trajectories resulting in cross-streamline migration of the
particles. The spheres on these new trajectories do not pass each other (as
they would in free space) but instead they swap their cross-streamline
positions. To determine the significance of the wall-induced particle
migration, we have evaluated the hydrodynamic self-diffusion coefficient
associated with a sequence of uncorrelated particle displacements due to binary
particle encounters. The results of our calculations quantitatively agree with
the experimental value obtained by \cite{Zarraga-Leighton:2002} for the
self-diffusivity in a dilute suspension of spheres undergoing shear flow in a
Couette device. We thus show that the wall-induced cross-streamline particle
migration is the source of the anomalously large self-diffusivity revealed by
their experiments.Comment: submited to JF
Many-particle hydrodynamic interactions in parallel-wall geometry: Cartesian-representation method
This paper describes the results of our theoretical and numerical studies of
hydrodynamic interactions in a suspension of spherical particles confined
between two parallel planar walls, under creeping-flow conditions. We propose a
novel algorithm for accurate evaluation of the many-particle friction matrix in
this system--no such algorithm has been available so far.
Our approach involves expanding the fluid velocity field into spherical and
Cartesian fundamental sets of Stokes flows. The interaction of the fluid with
the particles is described using the spherical basis fields; the flow scattered
with the walls is expressed in terms of the Cartesian fundamental solutions. At
the core of our method are transformation relations between the spherical and
Cartesian basis sets. These transformations allow us to describe the flow field
in a system that involves both the walls and particles.
We used our accurate numerical results to test the single-wall superposition
approximation for the hydrodynamic friction matrix. The approximation yields
fair results for quantities dominated by single particle contributions, but it
fails to describe collective phenomena, such as a large transverse resistance
coefficient for linear arrays of spheres
Lubrication approximation for micro-particles moving along parallel walls
Lubrication expressions for the friction coefficients of a spherical particle
moving in a fluid between and along two parallel solid walls are explicitly
evaluated in the low-Reynolds-number regime. They are used to determine
lubrication expression for the particle free motion under an ambient Poiseuille
flow. The range of validity and the accuracy of the lubrication approximation
is determined by comparing with the corresponding results of the accurate
multipole procedure. The results are applicable for thin, wide and long
microchannels, or quasi-two-dimensional systems.Comment: 4 pages, 5 figure
An analysis of the far-field response to external forcing of a suspension in Stokes flow in a parallel-wall channel
The leading-order far-field scattered flow produced by a particle in a
parallel-wall channel under creeping flow conditions has a form of the
parabolic velocity field driven by a 2D dipolar pressure distribution. We show
that in a system of hydrodynamically interacting particles, the pressure
dipoles contribute to the macroscopic suspension flow in a similar way as the
induced electric dipoles contribute to the electrostatic displacement field.
Using this result we derive macroscopic equations governing suspension
transport under the action of a lateral force, a lateral torque or a
macroscopic pressure gradient in the channel. The matrix of linear transport
coefficients in the constitutive relations linking the external forcing to the
particle and fluid fluxes satisfies the Onsager reciprocal relation. The
transport coefficients are evaluated for square and hexagonal periodic arrays
of fixed and freely suspended particles, and a simple approximation in a
Clausius-Mossotti form is proposed for the channel permeability coefficient. We
also find explicit expressions for evaluating the periodic Green's functions
for Stokes flow between two parallel walls.Comment: 23 pages, 12 figure
Hydrodynamic interactions of spherical particles in Poiseuille flow between two parallel walls
We study hydrodynamic interactions of spherical particles in incident
Poiseuille flow in a channel with infinite planar walls. The particles are
suspended in a Newtonian fluid, and creeping-flow conditions are assumed.
Numerical results, obtained using our highly accurate Cartesian-representation
algorithm [Physica A xxx, {\bf xx}, 2005], are presented for a single sphere,
two spheres, and arrays of many spheres. We consider the motion of freely
suspended particles as well as the forces and torques acting on particles
adsorbed at a wall. We find that the pair hydrodynamic interactions in this
wall-bounded system have a complex dependence on the lateral interparticle
distance due to the combined effects of the dissipation in the gap between the
particle surfaces and the backflow associated with the presence of the walls.
For immobile particle pairs we have examined the crossover between several
far-field asymptotic regimes corresponding to different relations between the
particle separation and the distances of the particles from the walls. We have
also shown that the cumulative effect of the far-field flow substantially
influences the force distribution in arrays of immobile spheres. Therefore, the
far-field contributions must be included in any reliable algorithm for
evaluating many-particle hydrodynamic interactions in the parallel-wall
geometry.Comment: submitted to Physics of Fluid
Far-field approximation for hydrodynamic interactions in parallel-wall geometry
A complete analysis is presented for the far-field creeping flow produced by
a multipolar force distribution in a fluid confined between two parallel planar
walls. We show that at distances larger than several wall separations the flow
field assumes the Hele-Shaw form, i.e., it is parallel to the walls and varies
quadratically in the transverse direction. The associated pressure field is a
two-dimensional harmonic function that is characterized by the same multipolar
number m as the original force multipole. Using these results we derive
asymptotic expressions for the Green's matrix that represents Stokes flow in
the wall-bounded fluid in terms of a multipolar spherical basis. This Green's
matrix plays a central role in our recently proposed algorithm [Physica A xx,
{\bf xxx} (2005)] for evaluating many-body hydrodynamic interactions in a
suspension of spherical particles in the parallel-wall geometry. Implementation
of our asymptotic expressions in this algorithm increases its efficiency
substantially because the numerically expensive evaluation of the exact matrix
elements is needed only for the neighboring particles. Our asymptotic analysis
will also be useful in developing hydrodynamic algorithms for wall-bounded
periodic systems and implementing acceleration methods by using corresponding
results for the two-dimensional scalar potential.Comment: 28 pages 5 figure