Fluid Dynamics of Floating Particles
Fluid dynamics of floating particles
(with experiments by Wang, Bai, and Joseph). J. Fluid Mech. Submitted.
D.D. Joseph, J. Wang, R. Bai and H. Hu. 2003.
Particle motion in a liquid film rimming the inside of a
rotating cylinder.
J. Fluid Mech.
496
, 139

163
•
Floating depth of a single heavier

than

liquid particle
•
Capillary attraction
•
Capillary attraction leads to self assembly
•
Nonlinear dynamics of pattern formation
•
Direct numerical simulation (DNS)
CONTACT ANGLE IS FIXED
CONTACT LINE MOVES
Floating Spheres
FORCE BALANCE
m
g
=
F
c
+
F
p
F
p
=
ρ
l
g
v
w
+
ρ
a
g
v
a
+
(
ρ
l
+
ρ
a
)gh
2
A
=
Pressure
Force
=
Weight of
displaced
fluids
Buoyant
weight of
liquid cylinder
above the
contact ring
Generalized
Archimedes
principle
=
= Floating depth. The more it sinks,
the more it is buoyed up.
The left side is bounded by one.
Large, heavy particles
r
p
gR
2
/
g
>> 1 cannot be suspended.
Heavy particles can be suspended if they are small enough.
If sin
q
C
sin (
a
+
q
C
)
0, the particles sit on top of the fluid
q
C
= 0
or are held in place by capillarity
a
+
q
C
=
p
.
FORCE BALANCE
F
c
=m
g

F
p
(a)
(b)
Teflon cylinder
pinned at the rim
FLOATING DISKS PINNED AT
SHARP EDGES
The contact line is fixed and the angle is determined by the force balance; just
the opposite.
The floating depth is not determined by wettability.
Glass
Aluminum
Sinks when
HYDROPHOBIC AND HYDROPHILIC
PARTICLES HANG AT THE SHARP RIM
Teflon
ψ > 90º
ψ = 90º
Equilibrium Contact Angle
Young

Dupré Law
n
ø
n
n
ø
n
is not defined
ø=0, meniscus
Gibbs Inequality
The effective angle at a sharp corner is not determined by
the Young

Dupr
é law; it is determined by dynamics.
The effective contact angle
θ
γ
LG
cosα=γ
SG

γ
SL
ranges over an interval 180º

θ
; 90º at a square corner
The
depth
to
which
a
cube
sinks
into
the
lower
fluid
increases
with
increasing
value
of
the
cylinder
density
.
The
contact
angle
on
the
plane
faces
is
120
degrees
and
the
interface
at
the
sharp
edges
AD
and
BC
is
fixed
.
(a)
Initial
state
.
(b)
ρ
P
=
1
.
5
,
(c)
ρ
P
=
1
.
2
,
and
(d)
ρ
P
=
1
.
1
.
Notice
that
in
(c)
and
(d)
the
interface
near
the
edges
AD
and
BC
rises,
as
for
these
cases
the
particle
position
is
higher
than
the
initial
position
.
Cubes can float in different ways.
This cube has an interface on a
sharp edge and smooth faces.
a
b
c
d
Capillary Attraction
When
there
are
two
or
more
particles
hanging
in
an
interface,
lateral
forces
are
generated
.
Usually,
these
forces
are
attractive
.
The
lateral
forces
arise
from
pressure
imbalance
due
to
the
meniscus
and
from
a
capillary
imbalance
.
Meniscus Effects Due to
Capillarity
After Poynting and Thompson 1913.
Horizontal Forces
A
heavier

than

liquid
particle
will
fall
down
a
downward
sloping
meniscus
while
an
upwardly
buoyant
particle
will
rise
.
If the contact angle doesn’t vary the particle must tilt causing an
imbalance of the horizontal component of capillary forces pulling
the spheres together.
If for any reason, the particle tilts with the two contact angles
equal, a horizontal force imbalance will result.
Neutrally buoyant copolymer spheres d = 1mm cluster in
an air/water interface.
DYNAMICS
(Gifford
and
Scriven
1971
)
“casual
observations
…
show
that
floating
needles
and
many
other
sorts
of
particles
do
indeed
come
together
with
astonishing
acceleration
.
The
unsteady
flow
fields
that
are
generated
challenge
analysis
by
both
experiment
and
theory
.
They
will
have
to
be
understood
before
the
common

place
‘capillary
attraction’
can
be
more
than
a
mere
label,
so
far
as
dynamic
processes
are
concerned
.
”
Capillary Attraction Leads to Self
Assembly
Free motions leading to self
assembly of floating particles
Sand in Glycerin
Sand in Water
Assembly of Floating Particles with
Sharp Edges
Circle Group
Square Group
Cube Group
Nonlinear Dynamics of Pattern
Formation
Free
floating
particles
self
assemble
due
to
capillarity
;
the
clusters
of
particles
can
be
forced
into
patterns
under
forced
oscillations
.
•
Patterns
formed
from
particle
clusters
on
liquid
surfaces
by
lateral
oscillations
•
Formation
of
rings
of
particles
in
a
thin
liquid
film
rimming
the
inside
of
a
rotating
cylinder
.
Pattern formation of particles under
forced tangential motion
Light Particles
in Water
Heavier

than

Water Particles
in Water
Frequency = 8 Hz
Particle segregation in a thin film
rimming a rotating cylinder
Aqueous Triton Mixture
Direct Numerical Simulation of
Floating Particles
We combine the method of distributed Lagrange
multipliers (DLM) and level sets to study the motion of
floating solid particles.
Both methods work on fixed grids.
The Navier

Stokes equations are solved everywhere even
in the region occupied by solid particles.
The particles are represented by a field of Lagrange
multipliers distributed on the places occupied by particles.
The multiplier fields are chosen so that the fluid moves as
a rigid body on the places occupied by particles.
Direct Numerical Simulation of
Floating Particles
Particles are moved by Newton’s laws for rigid
particles.
Fluid

fluid interface conditions are respected using
level sets.
A constant contact angle condition is enforced on the
three phase contact line by extending the level set
into the particle (Sussman 2001)
This is a direct numerical simulation of floating
particles. Nothing is modeled.
Governing Equations
Strong Form
W
region occupied by fluids and solids
P
(
t
)
region occupied by solids
Equations in
W
/
P
(
t
)
div
u
= 0 in
W
Fluid in
P(t),
λ
(x,t)
is
Lagrange Multiplier
The body force
l

a
2
2
l
is chosen so that
u
=
U
+
w
r
,
s
s
= 0
is a rigid motion on
P
(
t
)
where
U
(
t
)
and
w
(
t
)
satisfy
The multiplier field satisfies
Contact Angle and Contact Line
(Sussman 2001)
Floating particles move under the constraint that the
contact angle
is
fixed. The
contact line
must move.
Extend the level set into the particle along the fixed angle
a
.
normal to
n
,
t
plane
u
ex
is in
, normal to
t,
points inward
u
ex
=a
n
+
b
n
2
t • u
ex
=
0 ,
n
• u
ex
=
0 ,
n • n
=
cos
a
n
Solution of Weak Equations
Marchuk

Yanenko splitting scheme
decouples
The incompressibility condition and the related
unknown pressure
The nonlinear convection term
The rigid body motion inside the particle
The interface problem and unknown level set
distribution
The positions of the particles must be
updated at each time step.
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