Summary Using the point-force model, we analyze how evolution of a suspension drop settling under gravity in a viscous fluid is influenced by the presence of a vertical wall near by. In particular, we show that a close drop moves away from the wall while settling along

Anna Myłyk

Maria L Ekiel-Jeżewska

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XXIII ICTAM, 19–24 August 2012, Beijing, China
EVOLUTION OF SUSPENSION DROPS SETTLING UNDER GRAVITY IN A VISCOUS FLUID
NEAR A VERTICAL WALL
Maria L. Ekiel-Jeżewskaa) & Anna Myłyk
Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawińskiego 5B, 02-106
Warsaw, Poland
Summary Using the point-force model, we analyze how evolution of a suspension drop settling under gravity in a viscous
fluid is influenced by the presence of a vertical wall near by. In particular, we show that a close drop moves away from the
wall while settling along.
INTRODUCTION
In the unbounded fluid, dynamics of settling suspension drops has been studied by many authors [1, 2, 3, 4].
In our previous publications [5, 6], we have analyzed, experimentally and numerically, destabilization of a
suspension drop of diameter D settling gravitationally in a viscous fluid close to a vertical planar solid wall.
We have shown that the destabilization time and length are linear functions of the inverse interparticle distance
h between the drop center and the wall, for h ≥ 1.5D. In this presentation, we use the point-particle model to
analyze earlier stages of the drop evolution. We determine as functions of time such quantities as the number
of particles, mean velocity, horizontal and vertical size, and the distance from the wall.
POINT-PARTICLE MODEL OF A SUSPENSION DROP
In a viscous fluid of viscosity η, bounded by a solid wall at z = 0, there are N0 identical point-forces, Fα=F =
(−F, 0, 0) (with F > 0), at positions rα, distributed with the uniform probability inside a spherical volume of
diameter D. At a low-Reynolds-number, the particle dynamics is governed by the following system of ODEs,
drα
dt
=
N0∑
β 6=α
[T0(rαβ) + T̃(rαβ′)] · F + T̃(rαα′) · F, α = 1, ..., N0. (1)
Here, rαγ = rα−rγ , and β′ is the mirror image of the particle β, obtained with the reflection operator P = 1−2ẑẑ,
where ẑ the unit vector perpendicular to the wall. The Oseen tensor, T0(rαβ), describes the interaction of a
particle α with the particle β in an unbounded fluid,
T0(rαβ) =
1
8πηrαβ
(
I+
rαβrαβ
r2αβ
)
, (2)
where r = |r| and I is the unit tensor. The difference T̃(rαβ′) between the Green tensors for the fluid bounded
by the wall and unbounded (i.e. the Blake tensor [7] and the Oseen tensor shown in Eq. (2)),
T̃(rαβ′) · F = −T0(rαβ′) · F − 2hβF · P ·∇rαβ′ T0(rαβ′) · ẑ+
2h2β
8πη
F · P ·∇rαβ′
(
rαβ′
r3αβ′
)
, (3)
describes the interaction of a particle α with the mirror image β′ of the particle β. Here hβ is the distance
between the particle β and the wall.
Velocity of a point-particle α, specified at the right-hand-side of Eq. (1), consists of two terms. The first one is
the sum of the fluid velocity fields generated by all the other particles and their images at the position rα where
the particle α is located. The second term is the self contribution: it specifies the velocity field generated by the
particle’s α own image α′. The wall is parallel to the force F, therefore
T̃(rαα′) · F = −
3F
32hαπη
. (4)
We have chosen the frame of reference moving with the Stokes velocity V0 of a single point-particle in an un-
bounded fluid. In this way, the dynamics of particles is independent of the particle radius.
We computed evolution of 30 different random initial configurations of particles inside a spherical volume of
diameter D, centered at 13 different distances h from the wall, for the initial number of particles inside the drop
N0=700 and 1000. For a given h, the quantities of interest were averaged over the initial configurations.
a)Corresponding author. E-mail: mekiel@ippt.gov.pl.
DROP EVOLUTION
Examples of our results are shown in Fig. 1. In the plots, time evolution of two characteristic parameters of the
drop is shown: the instantaneous number N of the particles inside the drop, normalized by the initial value,
N0 = 1000, and the increase κ of the drop distance from the wall, normalized by its diameter D,
κ/D =
[
1
N
N∑
α=1
xα − h
]
/D. (5)
Our simulations are shown until the moment of the first destabilization. The time unit
τ =
5πηD2
4N0F
. (6)
0 100 200 300 400
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
t/τ
N
/N
0
0 100 200 300 400
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t/τ
κ/
D
Figure 1. Evolution of a settling suspension drop. Left: the instantaneous numberN of particles inside the drop, normalized
by the initial value N0 = 1000. Right: the instantaneous shift away from the wall, κ/D, see Eq. (5). Colors denote the initial
distance from the wall, h/D = 0.75 , 1 , 1.5 , 2 , 2.5 , 3.5 , 5 , 7.5 , 10 , 15 ,
30 , 70 , ∞ .
CONCLUSIONS
Point-particle model close to a vertical wall has been used to model evolution of a suspension drop settling
under gravity in a viscous fluid near a vertical wall. Before destabilization, evolution of a drop consists of two
stages. The first one is shorter and characterized by a much larger rate of the particle loss and a faster drop
migration away from the wall than the second one. At the first stage, drops at a smaller distance from the wall
faster loose particles and faster migrate. At the second stage of the evolution, dN/dt is constant in time and
independent of the drop distance from the wall. Finally, in the presence of a vertical wall, suspension drops
destabilize faster and after traveling a shorter path than in an unbounded fluid, in consistence with our earlier
findings [5].
References
[1] J. M. Nitsche and G. K. Batchelor. Break-up of a falling drop containing dispersed particles. J. Fluid Mech., 340, 161-175 (1997).
[2] G. Machu, W. Meile, L. C. Nitsche, and U. Schaflinger. Coalescence, torus formation and break-up of sedimenting drops: experiments
and computer simulations. J. Fluid Mech., 447, 299-336 (2001).
[3] M. L. Ekiel-Jeżewska, B. Metzger, and É. Guazzelli. Spherical cloud of point particles falling in a viscous fluid. Phys. Fluids, 18, 038104,
(2006).
[4] B. Metzger, M. Nicolas, and É. Guazzelli. Falling clouds of particles in viscous fluids. J. Fluid Mech., 580, 283-301 (2007).
[5] A. Myłyk, W. Meile, G. Brenn, and M. L. Ekiel-Jeżewska, Break-up of suspension drops settling under gravity in a viscous fluid close
to a vertical wall, Phys. Fluids 23, 063302-1–14 (2011).
[6] A. Myłyk, M. L. Ekiel-Jeżewska, How walls influence destabilization of a suspension drop settling under gravity in a viscous fluid?
Colloids Surf. A 365, 109-111 (2010).
[7] J. R. Blake. A note on the image system for a Stokeslet in a no-slip boundary. Proc. Camb. Phil. Soc., 70, 303-310 (1971).


EVOLUTION OF SUSPENSION DROPS SETTLING UNDER GRAVITY IN A VISCOUS FLUID NEAR A VERTICAL WALL

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EVOLUTION OF SUSPENSION DROPS SETTLING UNDER GRAVITY IN A VISCOUS FLUID NEAR A VERTICAL WALL

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