Accurate measurement of hydrodynamic interactions
between a particle and walls
K. Masmoudi, N. Lecoq, R. Anthore, F. Bostel, F. Feuillebois
Abstract A new experimental setup allows the measure-
ment of hydrodynamic interactions between the walls of a
vessel and a particle moving along a three-dimensional
trajectory in a viscous ¯uid. The vertical motion of the
particle is measured with an accuracy of 50 nm using laser
interferometry, while its horizontal movements are con-
trolled with an accuracy of 20 lm by displacing the vessel
in the horizontal direction so as to keep the sphere in the
®xed vertical laser beam. Three axisymmetric closed con-
tainers are used as examples: two vertical cylinders (with
¯at and convex lens-shaped lower walls) and a cone.
Various effects of combined creeping ¯ow hydrodynamic
interactions between the particle and walls are observed.
The motion of a particle in a viscous ¯uid at low Reynolds
number (Re 1) creates long-range perturbations. The
motion of a particle then may depend on its hydrodynamic
interactions with the container walls through the ¯uid.
These interactions are relevant in various applications
where small particles are encountered, such as in biology,
chemical engineering, civil engineering, etc., typical appli-
cations being ®ltration, sediment transport and deposition.
The ¯ow ®eld between a particle and walls may in
principle be calculated numerically by solving the Stokes
equations for ¯uid motion with appropriate boundary
conditions. However, the complexity of the walls' geome-
try may be a hindrance. Experimental modelling using a
reasonably large container and a very viscous ¯uid is then
appropriate. Alternatively, such an experiment may be
used to validate numerical codes. The experimental tech-
nique should allow the measurement of three-dimensional
particle motion in various geometries, and the experi-
mental accuracy should be suf®ciently high.
Earlier experiments providing the possibility of follow-
ing particles in a viscous ¯uid were based on imaging. For
instance, Adamczyk et al. (1983) observed the motion of a
sphere settling onto a horizontal wall and towards another
identical ®xed sphere in an axisymmetrical con®guration.
The movement of the sphere was measured from multiple
photographic images taken with ¯ashes of a stroboscope.
This technique only gave the vertical motion of the sphere.
In the same group, using a similar technique, Malysa et al.
(1986) observed the sedimentation of a sphere past a sec-
ond one attached to a wall. Their experimental results are in
agreement with a model that is valid provided the spheres
are not too close. More recently, Zeng et al. (1996) studied
with a video camera system the settling motion of a sphere
while in hydrodynamic interaction with another sphere.
This visualization technique allows the projection of the
movement of the two particles onto a vertical plane to be
followed. The main dif®culty with imaging systems is that
the particle should be kept in focus while moving along a
three-dimensional trajectory. Moreover, for present video
systems, the accuracy is restricted due to the limited
number of pixels. In another technique demonstrated by
Ambari et al. (1984), a particle was held ®xed by a magnetic
®eld and a wall was moved in its vicinity; the force on the
particle was then measured at constant velocity. The dis-
tance between the particle and the wall was limited by the
required intensity of the magnetic ®eld.
This paper presents a new experimental setup designed
to follow the 3D motion of a settling sphere at various
distances from walls in a complicated geometry. It is based
on an interferometer providing an accurate measurement
of the vertical velocity of the particle. This system is
coupled to two micro-displacement motors for the mea-
surement of horizontal displacements. Typical accuracies
of the vertical and horizontal displacements are 50 nm and
20 lm respectively, a signi®cant improvement over earlier
The experimental setup and procedure are presented in
Sect. 2. The technique then is applied to the study of hy-
drodynamic interactions between a settling sphere and the
walls of an axisymmetric closed container. Cylindrical
containers are considered in Sect. 3, viz: (1) a closed cyl-
inder with a ¯at lower wall; (2) a closed cylinder with a
large radius optical lens glued on its bottom. A closed
conical container is considered in Sect. 4. In all cases, 3D
Experiments in Fluids 32 (2002) 55±65 Ó Springer-Verlag 2002
Received: 12 March 1999 / Accepted: 7 June 2001
Published online: 29 November 2001
K. Masmoudi, N. Lecoq, R. Anthore, F. Bostel
UMR 6634, Universite
F-76821 Mont Saint Aignan, France
F. Feuillebois (&)
PMMH, ESPCI, 10 rue Vauquelin
F-75231 Paris cedex 05, France
The authors acknowledge the contribution of J. Bayard (IUT
AlencËon, France) and R. Gre
ge (IUT Ge
lectrique de Rouen,
France) in the construction of the electronic part of the setup, and
of E. Lecoq (CNAM, Paris) in the technical design of the system
for horizontal displacements.