# Fluid Dynamics of Floating Particles

Mechanics

Oct 24, 2013 (4 years and 6 months ago)

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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

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

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

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
.

Assembly

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.