CAPILLARY COLLAPSE AND

RUPTURE

T.S. Lundgren & D.D. Joseph

The breakup of a liquid capillary �lament isanalyzed as a viscous potential ow near a stag-nation point on the centerline of the �lamenttowards which the surface collapses under theaction of surface tension forces. We �nd thatthe neck is of parabolic shape and its radiuscollapses to zero in a �nite time. During thecollapse the tensile stress due to viscosity in-creases in value until at a certain �nite radius,which is about 1.5 microns for water in air, thestress in the throat passes into tension, presum-ably inducing cavitation there.

[uz; ur] = a(t)[z;�r

2] Stagnation ow

' =1

2az2 �

1

4ar2 Potential ow

Bernoulli Equations:

@'

@t+

1

2(u2r + u2z) +

p

�=p0�

p� p0�

= �

�1

2_a +

1

2a2�z2 +

�1

4_a�

1

8a2�r2

Tzz = �p + 2�@uz@z

= �p + 2�a;

Trr = �p + 2�@ur@r

= �p� �aStresses

On r = R(z; t) = R0(t) +R2(t)z2 + O(z4)

�Tnn � pa = ��

Tnn = n2rTrr + n2

zTzz

� = �@2R

@z2�1+(@R@z )

2�32

+ 1

R�1+(@R@z )

2�12

= 1R0� 2R2 +O(Z2)

���������������

Normal

Stress

balance

It's zero at z = 0

because @uz=@r = 0

�������shear stress

ur =@R@t

+ uz@R@z

�

12aR = @R

@t+ az@R

@Z

�

12aR0 =

�

R0 Solve for a

�

52aR2 =

�

R2 R2 = CR50

��������������

Kinematic Condition

The parabola attens

as it collapses

a = �2

�

R0

R0

To lowest order in z2

pa � ��

�=Tnn�

=Trr�

= �p

a� va

pa�

+�

�

�1

R0� 2R2

�= �

p0��

�1

4

�

a�1

8a2�R2

0 � va

R2 = CR50 a = 2

�

R0=R0

\Rayleigh Plesset" equation:

p0 � pa�

�

1

2R0

�R0 �2v _R0

R0=�

�

�1

R0� 2CR5

0

�:

�

R0 =1

2

�

�v

Small R0, balances

viscosity against inertia

to leading

order in

t�� t

����������

R0 =�2�v(t� � t) R0 collapses to

a = 2t��t

zero in �nite time

p�= p0

��

3z2

(t��t)2

AXIAL STRESS (to leading order)

Tzz�

= �2p0 � pa

�+

4v

t�� t

= �2p0 � pa

�+

2�

�R0(t)

Tzz turns positive for small R0

Rocr =2�

2p0�pa

Estimating p0 = pa = 106 dynes/cm2

� = 75 dyne/cm

R0 = 1:5microns

At the collapse condition

the Reynolds number is

about 55 based on�

R0

The capillary thread cavitates before itcollapses to zero.

S

SS

S

11 22

1122

π

CAVITATIONOCCURS

pc2D: S22 S11=-

CAVITATION CRITERIA

Conventional

Maximum Tension:

MinimumTension:

π

pc

CAVITATION OCCURSπ

π −

S11

S11

pc

π −S11

CAVITATIONOCCURS

π + 11S

π

- π = (T + T + T )11 22 3313

A cavity opens when themean stress is below vaporpressure

θ

principal axesp is the "vaporpressure", thenucleationthreshold.

c

A cavity openswhen one of theprincipal stresses isbelow vaporpressure

A cavity opens whenall of the principalstresses are belowvapor pressure

CAVITATION IN SHEAR

L

x

x

1

2

U

The stress in this ow is given by24T11 T12 0T12 T22 00 0 T33

35 = ��

241 0 00 1 00 0 1

35 + �

24

0 U

L0

U

L0 0

0 0 0

35

where � = 1

3(T11 + T22 + T33) is determined

by the \pressurization" of the apparatus. Theangle which diagnonalizes T is given by

� = 45�

(In the break-up of viscous drop experiments inplane shear ow done by G.T. Taylor [1934], thedrops �rst extend at 45� from the direction ofshearing.)

In principal coordinates, we have24T11 + � 0 0

T22 + � 00 0 T33 + �

35 = �

U

L

24

1 0 00 �1 00 0 0

35

where

T11 + � = S11 = �U

L

is a tension.This tension is of the order of one atmosphere

of pressure if

�U

L= 106

dynes

cm2= 105Pa

If � = 1000 poise, U = 10 cm/sec and L =10�1cm, we may achieve such a stress. A shear

stress of this magnitude is enough to put the

liquid into tension.

COMPETITION BETWEENINERTIA & VISCOELASTICITY

• The shear rate is large where the velocityis large.

• The pressure of inertia is largest nearstagnation points and smallest where thevelocity is large.

• The viscoelastic pressure is large wherethe inertial pressure is small. Theviscoelastic pressure is small where theinertial pressure is large. This is thereason why the microstructure inNewtonian and viscoelastic fluids aremaximally different.

10

PARTICLES INWAKE

FLUIDIZEDRAFTPERPENDICULARTO

g

Two Phase Flow Models Do Not Predict

• Long bodies stable broad side on

• Drafting, Kissing, Tumbling

• Across the stream arrays of spheres• Nested wake structure, “Flying Birds”

• Doublets, Triplets, Quadraplets

• Fluidized Raft, Wake Aggregates

U

VIDEO 3 -TURBULENTPING PONG BALLS 8

PARTICLES INWAKE

FLUIDIZEDRAFTPERPENDICULARTO

g

Two Phase Flow Models Do Not Predict

• Long bodies stable broad side on

• Drafting, Kissing, Tumbling

• Across the stream arrays of spheres• Nested wake structure, “Flying Birds”

• Doublets, Triplets, Quadraplets

• Fluidized Raft, Wake Aggregates

U

VIDEO 3 -TURBULENTPING PONG BALLS 8

Mean normal stress and deviator

T = -p1 + S

p = -

S =

T + T2

11 22

S11

0

0

S22

, S + S = 011 22

θ

S

SS

S

11 22

1122

the fluid cannotaverage stresses

the extra stress isgood because ithas positive &negative components

STRESS, PRINCIPAL AXES,DEVIATOR

Stress in two dimensions

Principal coordinates

θT

TT

T

11

22

1122

T11

T12

T12

T22

"pressure" cannot berecognized in a liquid;it sees a state of stress

direction ofmaximum tension. Acavitation must openin the direction ;then it can rotateaway

θ

BREAKING STRENGTH OFPOLYMER STRANDS

(Wagner, Schulze and Gottfert [1996])

The strand breaks at the thinnest crosssection of the strand when the tensilestress

σ = ≈FV

A0

610 Pa =10 atmospheres

for many kinds of polymeric liquids. Theysay that the breaking stress is a materialconstant.

BREAKING TIME & FLOW TIMECavitation of ultrathin (nanometer) films of

Israelachvili, Chen, Kuhl & Ruths

Their experiments show that the formation of cavities

open in tension at a threshold value of the extensional

stress 2ηSo

which I estimate as

3.6 Pa < 2 Pa× < ×

10 3 6 105 6ηS .

o

and that the formation of cavities is analogous tothe fracture of solids except after fracture, vaporflows into the cavity “...When a cavity initiallyforms and grows explosively, it is essentially aVACUUM CAVITY since dissolved solutemolecules or gases have not had time to enter therapidly growing cavity.”

COMPUTATIONAL DOMAINSAND PROBLEMS

Finite representation of an infinitedomain

This is appropriate when tracking the motion of a cluster of particles, say in a pipe orchannel.

L L1 2 x

If we have Poiseuille flow at x = ± ∞, then we choose a finite domain, far away from theparticles say at L1 and L2. You must prescribe conditions at L1 and L2; this is subtle. Forexample, you could say that you must have Poiseuille flow at L1 and L2, which is notstrictly correct.

To do the computation, you must move L1 and L2 with the particles so that no particle willleave the computational domain.

Periodic flowEvery cell is the same as another. You compute everything by computing in one cell.

L L

Of course, the dynamics must give rise to periodic solutions under the conditions assumedin the calculations, otherwise you could have garbage. The problem of existence of periodicsolutions of particulate flows is deep, unsolved and virgin.

Finite DomainsWe prescribe conditions at the inlet and outlet and along the sides of thecomputational domain. This setup is straightforward. For example, we haveconsidered sedimentation of heavier than water particles against the solid bottom ofa container of water.

DIRECT NUMERICALSIMULATION (DNS)

Solves the dynamic problem of the motionof particles in a fluid exactly (in principle).The particles are moved by Newton’s lawsunder the action of hydrodynamic forcescomputed from the numerical solution ofthe fluid equations (Navier-Stokes equationsfor water, etc.)

I. Since solutions are exact, they revealthe properties that control transport:

A. Lubrication layersB. Aggregation or dispersionC. Lift offD. Slip velocityE. Concentration gradientsF. Microstructural features

• These simulations complement andcompete with experiments. Experimentstell the truth in all detail. Thesimulations, though potentially exact,have limitations of various kindsG. Number of particlesH. Modeling of collisionsI. TurbulenceJ. High Deborah numbersK. 3D effects for large numbers ofparticles

Advantages of simulations:

I. They are much cheaperII. You can isolate effects

A. Turn shearing on or offB. Turn on normal stresses with orwithout shear thinningC. Test scale up

Aims of simulation:

I. Produce understanding of mechanismsunderlying propant transport

II. Provide data for correlation whichformerly were obtained by experiments.For example, we can make a program togenerate Richardson-Zaki correlations fortransport models from DNS

III. Provide a standard for validatingcontinuum models of propant transport

The production of cavitation in pure shearappears to have been realized recently

(1997)

“FRACTURE” PHENOMENA IN SHEARING FLOW OF VISCOUSLIQUIDS

L.A. Archer, D. Ternet and R. Larson

ABSTRACT

In startup of steady shearing flow of two viscous unentangled liquids, namely low-

molecular-weight polystyrene and α-D-glucose, the shear stress catastrophically

collapses if the shear rate is raised above a value corresponding to a critical initial

shear stress of around 0.1 --- 0.3 Mpa. The time-dependence of the shear stress

during this process is similar for the two liquids, but visualization of samples in situ

and after quenching reveals significant differences. For α-D-glucose, the

stress collapse evidently results from debonding of the

sample from the rheometer tool, while in polystyrene,

bubbles open up within the sample; as occurs in cavitation.Some similarities are pointed out between these phenomena and that of “lubrication

failure” reported in the tribology literature.

We have adhesive and cohesive fracture, 0.1--0.3 Mpa =1--3 atm. This is enough to put the sample into tension 45°from the direction of shearing.

PRINCIPLES OF CAVITATION IN AFLOWING LIQUID

• Conventional cavitation based on pressureI. What is pressure?

A. Newtonian fluidsB. Non-Newtonian fluids

• Nonconventional cavitation based onprincipal stresses

II. Maximum tension

• Mean normal stress and deviator

• An angle is always involved

• Cavitation in shear

• Tensile strength of liquids.A. Polymer strandsB. Breaking time and flow time, vacuum

cavitiesC. Capillary collapse and rupture

III. Outgassing is cavitation of liquid gas insolution

All books on cavitation havesections on the “tensile strengthof liquids.” A stress tensor isnever introduced.

Knapp et al. [1970] note that:

“If a cavity is to be created in ahomogeneous liquid, the liquidmust be ruptured, and the stressrequired to do this is notmeasured by the vapor pressurebut is the tensile strength of theliquid at that temperature. Thequestion naturally arises then asto the magnitudes of tensilestrengths and the relation thesehave to experimental findingsabout inception.

CONVENTIONAL CAVITATION

A fluid will cavitate when the local pressurefalls below the cavitation pressure

• The cavitation pressure is the vaporpressure in a pure liquid

• Natural liquids have nucleation sitesdefined by impurities and may cavitate athigher pressures

WHAT IS PRESSURE?

• In an incompressible Newtonian fluid“pressure” is the mean normal stress. Afluid cannot average its stresses, eventhough you can. The fluid knows its stateof stress at a point.

• In non-Newtonian fluids the pressure is anunknown flow variable, usually not eventhe mean normal stress, and the definitionof it is determined by the constitutiveequation. This “pressure” has no intrinsicsignificance. The fluid doesn’t recognizesuch a “pressure” and knows its state ofstress.

NONCONVENTIONALCAVITATION BASED ONPRINCIPAL STRESSES

Look at the state of stress at each point inthe fluid in principal axis coordinates.Identify the largest of the stresses. Supposea static fluid cavitates at zero pressure. Itwill cavitate in flow wherever themaximum tensile stress is positive.

If it cavitates statically when the pressurefalls below the vapor pressure, it willcavitate in flow even when the maximumtensile stress is only slightly negative.

Suppose you do an experiment in your labwhere the ambient pressure is

one atmosphere = 105 Pa

Then if you get tensile stresses due to flowlarger than this, the fluid will cavitate.

MAXIMUM TENSION

All books on cavitation have sections on“tensile strength of liquids.” A stress tensoris never introduced.

“If a cavity is to be created in ahomogeneous liquid, the liquid must beruptured, and the stress required to do this isnot measured by the vapor pressure but isthe tensile strength of the liquid at thattemperature.” (Knapp et al. 1970)

FIRST THE FLUID RUPTURES THEN VAPOR FILLS THE CAVITY