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 mg=Fc+Fp

Fp= ρlgvw + ρagva + (ρl + ρa)gh2A = 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.The left side is bounded by one.

Large, heavy particles Large, heavy particles p p gR gR 22// >> 1 cannot be suspended. >> 1 cannot be suspended.

Heavy particles can be suspended if they are small enough. Heavy particles can be suspended if they are small enough.

If sin If sin CC sin ( sin ( + + CC) ) 0, the particles sit on top of the fluid 0, the particles sit on top of the fluid CC = 0 = 0

or are held in place by capillarity or are held in place by capillarity + + C C = = . .

c

l

acc

l

acc

l

plcc R

hgR

2233

2

sin1cos3

1cos

3

2cos

3

1cos

3

2

3

4

2

1)sin(sin

FORCE BALANCE Fc=mg-Fp

1ψ 1ψ

2 1ψ >ψ 03ψ 90

W1 W1

W2

W3

(a) (b) (c) (d)

(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

nøn nø

n is not defined

ø=0, meniscus

Gibbs Inequality

180ˆ

The effective angle at a sharp corner is not determined by the Young-Dupré law; it is determined by dynamics.

The effective contact angle ̂

θ

γLGcosα=γSG-γSL

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) ρP =1.5, (c) ρP =1.2, and (d) ρP =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 A heavier-than-liquid particle will fall down a downward sloping meniscus while an upwardly buoyant particle will rise.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 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. 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. 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 We combine the method of distributed Lagrange multipliers (DLM) and level sets to study the motion of multipliers (DLM) and level sets to study the motion of floating solid particles.floating solid particles.

Both methods work on fixed grids.Both methods work on fixed grids. The Navier-Stokes equations are solved everywhere even The Navier-Stokes equations are solved everywhere even

in the region occupied by solid particles.in the region occupied by solid particles. The particles are represented by a field of Lagrange The particles are represented by a field of Lagrange

multipliers distributed on the places occupied by particles.multipliers distributed on the places occupied by particles. The multiplier fields are chosen so that the fluid moves as The multiplier fields are chosen so that the fluid moves as

a rigid body on the places occupied by particles.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 are moved by Newton’s laws for rigid particles.particles.

Fluid-fluid interface conditions are respected using Fluid-fluid interface conditions are respected using level sets.level sets.

A constant contact angle condition is enforced on the A constant contact angle condition is enforced on the three phase contact line by extending the level set three phase contact line by extending the level set into the particle (Sussman 2001)into the particle (Sussman 2001)

This is a direct numerical simulation of floating This is a direct numerical simulation of floating particles. Nothing is modeled. particles. Nothing is modeled.

Governing Equations Strong Form

region occupied by fluids and solidsregion occupied by fluids and solids

PP((tt)) region occupied by solids region occupied by solids

Equations in Equations in //PP((tt))

,div nσguuu

LLL t

div u = 0 in

,2 uD1σ LL p

,constinterfaceacross0]][[ u

set,level0

ut

).(on tP rωUu

Fluid in P(t), λ(x,t) is Lagrange Multiplier

The body force The body force - - aa222 2 is chosen so that is chosen so that uu = = UU + + rr, , ss = 0= 0 is is

a rigid motion on a rigid motion on PP((tt)) where where UU((tt)) and and ((tt)) satisfy satisfy

λλσguuu 22div

at SLL

)(on tPLS σσnn

P

L

P

L

spt

I

spMt

M

dd

d

dd

d

n1Xxω

n1gU

Pp

tta

L

L

on][2

d

d

d

d22

uDnnnσλn

grωωrωU

λλ

The multiplier field satisfiesThe multiplier field satisfies

Contact Angle and Contact Line(Sussman 2001)

Floating particles move under the constraint that the Floating particles move under the constraint that the contact angle contact angle is is fixed. The fixed. The contact line contact line must move.must move.

Extend the level set into the particle along the fixed angle Extend the level set into the particle along the fixed angle ..

nn

nnt

normal to n, t plane

0

extu

tn

tnn

2

uex is in , normal to t, points inward

uex=an + bn2

t • uex= 0 ,

n • uex= 0 ,

n • n = cos

n

n

Solution of Weak Equations

Marchuk-Yanenko splitting scheme decouplesMarchuk-Yanenko splitting scheme decouples The incompressibility condition and the related The incompressibility condition and the related

unknown pressureunknown pressure The nonlinear convection termThe nonlinear convection term The rigid body motion inside the particleThe rigid body motion inside the particle The interface problem and unknown level set The interface problem and unknown level set

distributiondistribution

The positions of the particles must be updated The positions of the particles must be updated at each time step.at each time step.

UX

td

d