Paradoxes in Capillary Flows

James SprittlesYulii Shikhmurzaev

What Is A Capillary Flow?!

• One in which surface tension is of importance

– An effect caused by asymmetry of intermolecular forces.

– Acts as a stretched elastic membrane at a surface.

– Gradients in surface tension drive

bulk motion (Marangoni effects).

– Present at both liquid-gas and

liquid-solid boundaries.

Introduction

• With:– A Big Computer

– A Good Textbook

– An Endless Supply of (Good) Coffee

– A Lack of Social Life

• Can we describe all capillary flows?

1. A Classical Approach

The Classical Recipe

• In The Bulk:– (Incompressible) Navier–Stokes

• At Fluid Boundaries:– Balance of Stress With Capillary Forces.

– Particles On a Free Surface, f(x,y,z,t)=0, Move With It (Kinematic Condition)

• At Solid Boundaries:– No Slip

Tuu = 0; P f ; P = I + u u

Dp

Dt

0n + n P = n np

0D f

Dt

u.n u.t 0

Ink Jet Printing: Breakup of Liquid Threads

• A drop of ink is pushed from the nozzle.

Breakup of Liquid Threads

• Predictions of Classical Model:

– Infinite Axial Velocity at Breakup.

– Infinite Pressure at Breakup.

– Rate of Fresh Free Surface Area Creation Becomes Infinite

• Main Problem:

– Solution Required After Breakup.

Ink Jet Printing: Spreading of Liquids

Drop In Equilibrium

No Solution!

Drop Out of Equilibrium

• Ink drops land and then spread on solid.

Coalescence Of Liquid Volumes

• Ink drops coalesce with adjacent ones on the paper.

Infinite Stresses!

Ink Jet Printing: Impact On Chemically Patterned Surfaces

• Pattern a surface to correct deposition

Flow Over Chemically Patterned Surfaces

Solid 1 Solid 2

Predictions of The Classical Recipe

Molecular Dynamics Simulations of Flow Over Chemically Patterned Surfaces

Courtesy of Professor N.V. Priezjev

More wettable CompressedMore wettable CompressedLess wettable RarefiedLess wettable Rarefied

No – Slip = No Effect!No – Slip = No Effect!

Also - Flow Generated By Rotating Cylinders

Formation of a

Cusp/Corner

The Free Surface Is a Streamline.

Summary• No Solution

– Flow of Liquids Over Solids

• Singular Solution– Coalescence of Drops– Cusps– Breakup Of Liquids

• Wrong Solution– Flow Over Chemically Patterned Surfaces

A Big Computer Can’t Handle These.

2. A Standard Approach

Flow Of Liquids Over Solids

Two Issues:– Allow For A Solution

– Describe The Angle Between The Free Surface and the Solid (The Contact Angle).

dθ U

Standard Solution:– Allow Slip Between Solid

and Liquid

– Let dθ = ( U )f

Q) Does The Standard Model Work? Impact of a Microdrop

Radius = 25 m, Impact Speed = 12.2 m/sRe=345, We=51, β = 100, .67s

Experiments of Dong 06. My Simulation

Q) Does The Standard Model Work?

My

Simulation

Experiment: Renardy et al

A) Yes and No!

dθ ( U )f

ExperimentallyPrediction of

Standard Model

Standard Model’s Problems:

• Incorrect Kinematics

• Pressure Singularity at Contact Line

• Contact Angle Depends on Flow

U, m/s

dθ

dθU

3. A New (ish) Approach

Breakup Of Liquid Threads• New free surface

is created.

• New free surface particles are initially out of equilibrium.

Spreading of Liquids on Solids

lg

slSolid

Gas

Liquid

In Frame Moving With Drop

Interfaces are shown with finite thickness for representation only.

Coalescence Of Liquid Volumes

• Particles on the surface become trapped in the bulk.

lg

ll

Coalescence Of Liquid Volumes

Near Cusp/Corner

lg

ll

Gas

Liquid

Corner/CuspInterfaces are shown with finite thickness for representation only.

Surface Tension Relaxation

Summary

• All are associated with transition from one surface tension to another

• Relaxation of surface tension takes finite time/distance

• Mass, momentum and energy exchange between surface and bulk

• Gradients in surface tension (Marangoni effect)

Simplest Model of Interface Formation

s1

*1

*1

s 1 11

s 1 111 1

1 1|| ||

v 0

n [( u) ( u) ] n n

n [( u) ( u) ] (I nn) 0

(u v ) n

( v )

(1 4 ) 4 (v u )

s se

s sss e

s

ff

t

p

t

s s1 1 1 2 2 2

1 3 2

v e v e 0

cos

s s

d

2

* 12 || ||2

s 2 22

s 2 222 2

12|| || || 2 22

21,2 1,2 1,2

u 1u 0, u u u

n [ u ( u) ] (I nn) (u U )

(u v ) n

( v )

v (u U ) , v U

( )

s se

s sss e

s s

s s

pt

t

a b

In the bulk:

On liquid-solid interfaces:On free surfaces:

At contact lines:

θd

e2

e1

n

n

f (r, t )=0 • Generalisation of standard/classical model

Predictions/Propaganda

• Generalises standard/classical recipe.

• Removes singularities inherent in both classical and standard approaches.

• Predicts contact angle depends on flow field.

• Ensures one can (numerically) apply a unified approach to all these problems (=easier!).

• Agrees with experiment.

Conclusion/Sales Pitch

• With:– A Big Computer

– An Endless Supply of (Good) Coffee

– A Lack of Social Life

– The RIGHT Textbook..

• We can describe capillary flows!

Chemically Patterned Surface• Surfaces With Wettability Gradients