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Inertial migration of rigid spherical particles in
Poiseuille ow
Jean-Philippe Matas, Jerey F. Morris, Elisabeth Guazzelli
To cite this version:
Jean-Philippe Matas, Jerey F. Morris, Elisabeth Guazzelli. Inertial migration of rigid spherical
particles in Poiseuille ow. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2003,
515, pp.171-195. �10.1017/S0022112004000254�. �hal-00252129�
Inertial migration of rigid spherical particles
in Poiseuille flow
By JEAN-PHILIPPE MATAS
1
, JEFFREY F. MORRIS
2
AND
´
ELISABETH GUAZZELLI
1
1
IUSTI – CNRS UMR 6595, Polytech’ Marseille, Technop
ˆ
ole de Ch
ˆ
ateau-Gombert,
13453 Marseille cedex 13, France
2
Halliburton, 3000 N. Sam Houston Parkway East, Houston, TX 77032, USA
An experimental study of the migration of dilute suspensions of particles in Poiseuille
flow at Reynolds numbers Re = 67–1700 was performed, with a few experiments
performed at Re up to 2400. The particles used in the majority of the experiments
were neutrally buoyant spheres with diameters d yielding a ratio of pipe to particle
diameter in the range D/d = 8–42. The volume fraction of solids was less than 1%
in all cases studied. The results of G. Segr
´
e & A. Silberberg (J. Fluid Mech. 14, 136,
1962) have been extended to show that the tubular pinch effect in which particles
accumulate on a narrow annulus is moved toward the wall as Re increases. A careful
comparison with asymptotic theory for Poiseuille flow in a channel was performed.
Another inner annulus closer to the centre, and not predicted by this asymptotic
theory, was observed at elevated Re.AsRe is increased, the distribution of particles
over the cross-section of the tube at the measurement location, lying at a distance
L
.
= 310D from the entrance, changes from one centred at the annulus predicted
by the theory to one with the particles primarily on the inner annulus. The case of
slightly non-neutrally buoyant particles was also investigated. A particle trajectory
simulation based on asymptotic theory was performed to facilitate the comparison of
theory and the experimental observations.
1. Introduction
A rigid sphere immersed in a spatially varying shear flow will undergo a lateral,
or cross-stream, motion in the presence of inertia. Inertia is necessary to break
the linearity of Stokes equations, under which lateral migration is forbidden, as
demonstrated by Bretherton (1962). In a study of the Poiseuille flow of a dilute
suspension of neutrally buoyant spheres, Segr
´
e & Silberberg (1962) observed that
a single rigid sphere in pipe flow migrated to an equilibrium position with its
centre located at r =0.6R, R being the pipe radius. The phenomenon of radial
migration driven by inertia was termed the tubular pinch effect to indicate that the
uniform distribution of particles over the pipe cross-section converges, or is ‘pinched’,
to a narrow annulus as the suspension moves downstream. These experiments
prompted a strong interest in the suspension community, because at the time of
the observations there was no theoretical explanation of this experimental result. A
series of experiments followed, which investigated lateral forces on a sphere in several
flow configurations: migration in Poiseuille flow in the absence of particle rotation
(Oliver 1962), for non-neutrally buoyant spheres in vertical flows (Repetti & Leonard
1
1964; Jeffrey & Pearson 1965;
Karnis, Goldsmith & Mason 1966; Aoki, Kurosaki &
Anzai 1979) and plane Poiseuille flow (Tachibana 1973). These experiments,
beyond confirming Segr
´
e & Silberberg’s observations, showed in particular that the
equilibrium position was shifted towards the axis when particles were lagging the
flow, and towards the wall when they were leading it. Experiments were performed
for pipe-scale Reynolds numbers Re =
¯
UD/ν larger than 30 (where
¯
U is the mean
axial velocity, D =2R and ν is the kinematic viscosity of the fluid). These experiments
indicated that the equilibrium position lies closer to the wall in the presence of
increased inertia. While inertially driven migration can be explained by consideration
of a single particle, Han et al. (1999) confirmed that it was a very robust phenomenon
which could be observed for volume fractions up to φ =0.2.
The only theoretical evidence supporting lateral migration of a single rigid sphere
at the time of the experiments of Segr
´
e & Silberberg was that of Rubinow & Keller
(1961) who calculated the Magnus effect for a rigid sphere in a uniform flow. This
force, always directed towards the pipe centreline, could not predict the existence of
an equilibrium away from the axis. By a matched asymptotic expansion calculation,
Saffman (1965) demonstrated that a rigid sphere in a linear shear flow experienced
a lateral force proportional to its slip velocity relative to the fluid streamline going
through its centre. The departure from r =0.6R observed for non-neutrally buoyant
spheres in vertical flow, toward the wall for particles which lead the flow, and toward
the axis for particles which lag the flow (Jeffrey & Pearson 1965), suggested that
Saffman’s lift played a major role in the migration of non-neutrally buoyant spheres.
Ho & Leal (1974) calculated the force exerted on a rigid particle in a quadratic
bounded shear flow in the case of small Reynolds numbers by a regular perturbation
method, and were able to show that variation of the shear rate combined with the
presence of a wall acting to create a repulsion resulted in an equilibrium position at
r =0.6R consistent with the experimental results of Segr
´
e & Silberberg. Schonberg &
Hinch (1989) succeeded in lifting the low-Reynolds-number restriction by integrating
the solution of the differential equations yielded by the matched asymptotic expansion
method, and predicted the evolution of the equilibrium position for Re up to 150. Hogg
(1994) applied the same method to the problem of non-neutrally buoyant particles in
Poiseuille flow. Asmolov (1999) extended this method to Reynolds numbers as large
as 1500 by the use of an orthonormalization method to integrate the equations of the
matched asymptotic expansion problem. Note that, in all of the asymptotic theory,
the particle scale Reynolds number satisfies Re
p
1 where Re
p
≡ Re(d/D)
2
, with d
the diameter of the particles and D that of the pipe.
To date there has been little experimental study examining the validity of these
theories at elevated Re. The objective of the present work is to study the influence of
inertia on the radial migration of rigid neutrally buoyant spheres in Poiseuille flow. We
extended the Segr
´
e–Silberberg experiments up to Re = 1700 and considered particle
sizes yielding D/d = 8–42; a few experiments at larger Re will also be described. The
equilibrium radial position was examined and is termed the Segr
´
e–Silberberg annulus.
We also found a novel feature of inertial migration in the form of an inner annulus
and we examine whether it is another stable equilibrium position. Since the data were
collected at a single point at the end of the pipe, we did not study the evolution
of the concentration profile with position and therefore we did not check that the
distribution having an inner annulus reached a steady state within the length of the
tube. However, some observations such as the fact that the distribution switches from
one centred at the Segr
´
e–Silberberg position to one centred on the inner annulus
when Re is increased and that this occurs at the same Re for all particles provide
2
d (µm) 190 ± 10 450 ± 50 550 ± 50 750 ± 500 900 ± 50 1000 ± 50
D/d 42 ± 2.517± 2.515± 1.510.5 ± 19± 0.58± 0.5
Tab le 1 . Particle diameters and pipe to particle diameter ratios.
The pipe inner diameter is D = 8 mm.
alternative arguments in favour of the stability of the inner annulus that will be
discussed in detail. We have also examined the effect of a small buoyancy on the
migration process.
We first describe in § 2 the experimental techniques and the method used for
measurements of the particle distribution. The experimental results are presented in
§ 3. After reviewing the analysis and results of the matched asymptotic expansion
method in § 4, we compare and discuss the predictions of this theory with our
experimental results in § 5.
2. Experimental techniques
2.1. Particles and fluid
We used as particles polystyrene beads with diameters in the range d = 190 µm–
1 mm, as indicated in table 1. The particles were supplied by Maxi-Blast (South Bend,
IN, USA) and were found to be spherical. However, the pump which circulated the
suspension tended to flatten a small fraction of the larger beads with each circulation
through the apparatus. Therefore after some time, some of the particles were no longer
perfectly spherical. These non-spherical particles were observed to have a different
migration behaviour from the spheres and this was found to be a significant source of
scatter in the data for the large particles. The density of the suspending fluid ρ
f
was
matched to the density of the particles ρ
p
by using a mixture of glycerol and water.
The density of the fluid was measured using a hydrometer, or float densimeter, from
ERTCO (West Paterson, NJ, USA). The particle density was determined by finding a
fluid of measured density in which no sedimentation of a batch of particles occurred.
The densities of the different particle sets were in the range ρ
p
=1.049 − 1.053 g cm
−3
,
for which the suspending fluid composition in glycerol : water fraction was between
0.21 : 0.79 and 0.23 : 0.77 at a temperature T =25
◦
C. The density of the particles
among a given set was homogeneous, except for the D/d =17 and D/d =15 sets,
for which slightly more and less dense particles were observed simultaneously at low
flow rates, and thus were used to study the influence of a dispersion in density upon
particle distribution. The suspension was maintained at T =25
◦
C, a temperature for
which the viscosity of the suspending fluid was in the range η =1.45 − 1.55 ± 0.03 cP.
2.2. Experimental apparatus
The experimental apparatus test section was a horizontal glass tube of inner diameter
D = 8 mm. The tube had a length L =2.6 m, longer than the entry length L
e
∼ 50 cm
necessary for the laminar flow to fully develop at Re ≈ 2000; above this Re,the
laminar flow becomes unstable. The measurement of particle distribution was made
at a position 2.5 m from the entrance. In order to ensure that the flow in the pipe
was undisturbed by perturbations from a pump, the flow was driven by gravity. The
suspension was delivered to the tube by overflow from a tank positioned at a fixed
height to an outlet of variable height, passing through the glass tube as indicated in
3
Glass tube
Pump
Suspension
Overflowing reservoir
L = 2.6 m
D = 8 mm
Figure 1. Diagram of the experimental apparatus used. The diameter noted on the figure,
D = 8 mm, is the inner diameter of the glass tube. The overflowing reservoir is elevated relative
to the remainder of the apparatus, with the flow rate controlled by the elevation of the
overflowing reservoir above the thermostated reservoir.
figure 1. A Moineau progressing cavity pump (PCM model MR2.6H24) carried the
suspension from a lower thermostated reservoir back to the overflowing tank. The
pump was, however, isolated from the flow through the glass tube. The flow rate Q
was determined by measuring a collected volume of the suspension at the outlet of
the tube in a given time.
2.3. Measurement of particle position
The position of particles in the pipe was measured by making a vertical section of
the tube with a laser sheet and recording the position of the particles intersecting the
sheet with a camera. The position was measured at a fixed location corresponding
to L/D
.
= 310 from the entrance of the tube. To limit the deformation of the image
caused by refraction effects, a Plexiglas vessel was placed around the test section of
the tube and filled with glycerol. The index of refraction of glycerol n
G
= 1.48 is close
to the index of refraction of glass. The air–glass refraction at the cylindrical outer
wall of the tube was then transferred to an air–glass refraction at the plane wall of the
Plexiglas box (see figure 2). There is also a deformation of the image associated with
the refraction at the inner wall of the tube. However, the index of the glycerol-water
mixture is n
F
=1.360 ± 0.002 at T =25
◦
C (measured with an Abbe refractometer),
rather close to the index of glass and thus this effect causes only limited deformation.
The images were obtained using a digital video camera at 10 images per second.
The frame rate is not increased beyond this value in order to avoid any particle being
present in two consecutive images and being counted twice. The laser sheet being
filmed under an angle of 40
◦
, is a three-dimensional object and we have to ensure
that its depth is smaller than the field depth in our optical conditions: the aperture
chosen is therefore the largest available with this camera, i.e. an aperture number
of 11. This aperture allows a good focus over the whole width of the laser sheet.
The shutter speed is fixed at an opening period of 1/1600 s. The images are then
4
