Numerical Simulation of Spontaneous Capillary Penetration

 

PennStateTony FickComprehensive ExamOct. 27, 2004

Goal: Develop a first principle simulation to explore fluid uptake in capillaries

h(r,t)

r

01

V

h

UUh

h

U

),( trh

z

meqV

h

UVt

h

r

01

11

2 2

Vh

h

VU

hh

Uh

h

UV

hh rr

Motivation

• NASA Advanced Human Support Technology Capillarity critical in water recovery systems, thermal systems, and phase change processes

• Halliburton studying capillary flow Prevent losses in oil well drilling

• Paper products work by capillary motion Improved paper product fluid uptake New multi layered film with capillary gradient

Project Objectives

Proposed research to identify geometric effects on capillary rate

1) Compute equilibrium height/shape in cylindrical, conical, wedge shaped, elliptical cross sections, and periodic walled capillaries

2) Numerical simulation of capillary penetration in cylindrical, conical, and wedge shaped capillaries from infinite reservoir

3) Modeling kinetics of capillary penetration in cylindrical, conical, and wedge shaped capillaries from finite reservoir

4) Repeat steps 2 and 3 for elliptical cross section capillaries

5) Repeat step 2 for periodically corrugated capillaries

Literature Experimental Results

Region I Region II Region III

t*

h*

tI* tII*

hI*

hII*

M. Stange, M. E. Dreyer, and H. J. Rath, “Capillary driven flow in circular cylindrical tubes,” Physics of Fluids 15, 2587 (2003)

2th

Inertial forcedomain

Force from pressure drop at

entrance th

Friction forcedomain

th

System to Test Algorithm

Interface modeled as function h(r,t)

Dynamic contact angle

Need to transform system into simulation box

00 1

1

R

rz

z=h(r,t)

r

),( trh

z

00 R

r

),( trh

r

z

Co-ordinate Transformation

Use cylindrical capillary:easy systemexperimental results

00 1

1

Developing The Model

Governing equations for the transformed system:

Conservation of Mass

01

VUUU

h

h

h

rzrrrrrr

h

h

h

h

hh

h

ht

τττττPPUV

UUU

U 1

Re

11

Conservation of momentum direction

zzrzrzrz

h

h

hhh

h

ht

ττττPVV

VVU

V 1

Re

111direction

00 1

1

Boundary Conditions

Velocity

Pressure

VVUVU

Ph

hhh

h

h

hh

hCa

Bo

h

hhh

Ca rrr

r

rrrrII 11

1

2

Re

1

Re1

11

Re

1 22

2

32

2

Normal stress condition

Update height

01

11

2 2

VVUUUV h

hhh

h

hhh rr

Tangential stress condition

h

t

hUV

Kinematic conditionContact line velocityconstitutive equation

ma V

L. H. Tanner, “The spreading of silicone oil on horizontal surfaces,” J. Phys. D: Appl. Phys. 12, 1473 (1979).

Dimensionlessparameters

Numerical Method

Initial values for h, P, U, and V

Use h for factors in equations

Solve for U*Obtain P from div U*

Use P to get U from U*

Use U to get new h

Repeat until convergence

)()1( ihihh

22

2 )1()(2)1(

ihihihh ][

Re

1 nn

nnn

dtτUU

UU

Convective termsViscous terms

UPdt

nn

1]

1[ 1

][1 1

1

n

n

n

dtP

UU

Pressure terms

Preliminary Results

Static case

- test geometric effect on meniscus- determine improvement of conical capillary

Dynamic simulation of dodecane rise

- test model against earthbound experiment- match equilibrium height/shape

Dynamic simulation of microgravity rise

- match early time-height behavior- test effect of exponent in contact line velocity

0

0.01

0.02

0.03

0.04

0.05

0 2 4 6 8 10 12Time (10-4 s)

He

igh

t (m

m)

0

4

8

12

16

20

0 0.1 0.2 0.3 0.4 0.5Time (s)

Hei

gh

t (m

m)

Zhmud et al Paper

Dodecane run

Dynamic Rise of Dodecane

Region I

Region II

Region III

Data matches within 97.5% confidence

interval

B. V. Zhmud, F. Tiberg, and K. Hallstensson, “Dynamics of capillary rise,” J. Colloid Interface Sci. 228, 263 (2000)

Simulation shows behavior of all three regions

17.9

18

18.1

18.2

18.3

18.4

18.5

18.6

18.7

18.8

0 0.2 0.4 0.6 0.8 1

r/R

Hei

gh

t (m

m)

Equilibrium shapeEnd shape

Dodecane Equilibrium

0.1% error

Equilibrium shape calculated from static equations

Simulation end shape within 0.1% of equilibrium shape

Simulation matches dynamic and equilibrium behavior

Dynamic Simulation of Microgravity

ma V

• Use microgravity rise of Dow Corning Silicon fluid “SF 0.65” to match initial height behavior• Test effect of exponent in contact line equation

Previous work values 1.01, 2.73, 3.00, 3.76Test values

1.00, 3.00

Experiments carried out in jet producing free fall environment

-10

40

90

140

190

240

0 1 2 3 4 5

Time (s)

Hei

gh

t (m

m)

Stange Paper (7)

Microgravity run m=3Microgravity run m=1

m=1 simulation

m=3 simulation

Dynamic Simulation of Microgravity

M. Stange, M. E. Dreyer, and H. J. Rath, “Capillary driven flow in circular cylindrical tubes,” Physics of Fluids 15, 2587 (2003)

Simulations match experimental behavior

Static Case

Test geometric effect Model reduces to solving single height equation

22

32

2

2

11

1r

h

r

h

rr

hhBo

r

h

Two different capillaries: cylinder and cone

Static Case

Radius (mm)

Height (mm) Cone

Cylinder

Centerline WallCone wall

Same contact angle

h

Increased height for cone, also increased curvature

Conclusions

Static Case• height increase for conical capillary over cylindrical

Dynamic Dodecane Rise• end results within 0.1% of equilibrium• dynamic data within 97.5% confidence of experimental

Dynamic Microgravity Rise• simulation matched experimental results• exponent in constitutive equation only effects behavior in Region II

• Model for capillary flow developed based on first principle equations• Algorithm able to predict previous experimental results

Future Work

Dynamic simulation for capillaries with different geometries to determine geometric effect on capillary penetration (conical, wedge, ellipsoidal, periodic corrugated walls)

Experimental results for capillaries with different

geometries

Develop constitutive equation for contact line for multi phase systems (e.g. surfactants)

Acknowledgements

FundingPenn State Academic

Computing Fellowship

AcademicDr. Ali Borhan

Dr. Kit Yan Chan

PersonalDr. Kimberly Wain

Rory StineMichael Rogers

 

PennState

-10

40

90

140

190

240

290

340

390

440

0 1 2 3 4 5

Time (s)

Hei

gh

t (m

m)

Microgravity run m=3

Microgravity run m=1

Expanded microgravity graph

Constitutive contact line velocityplotted against contact angle

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

a

r