Numerical Simulation of Spontaneous Capillary Penetration

1. Numerical Simulation of Spontaneous Capillary Penetration PennState r h(r,t) Goal: Develop a first principle simulation to explore fluid uptake in capillaries Tony Fick Comprehensive Exam Oct. 27, 2004

2. 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 Motivation

3. 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

4. Force from pressure drop at entrance Friction force domain Inertial force domain Literature Experimental Results Region I Region II Region III h* hII* hI* tI* tII* t* M. Stange, M. E. Dreyer, and H. J. Rath, “Capillary driven flow in circular cylindrical tubes,” Physics of Fluids 15, 2587 (2003)

5. R q z=h(r,t) z r 1 z 0 0 1 0 0 r 1 0 1 0 System to Test Algorithm Use cylindrical capillary: easy system experimental results Interface modeled as function h(r,t) Dynamic contact angle q Need to transform system into simulation box Co-ordinate Transformation

6. 1 0 Conservation of Mass 1 0 Conservation of momentum h direction e direction Developing The Model Governing equations for the transformed system:

7. Dimensionless parameters Normal stress condition L. H. Tanner, “The spreading of silicone oil on horizontal surfaces,” J. Phys. D: Appl. Phys. 12, 1473 (1979). Tangentialstresscondition Contact line velocity constitutive equation Kinematic condition Boundary Conditions Velocity Pressure Update height

8. Use h for factors in equations Obtain P from div U* Solve for U* Use P to get U from U* Convective terms Viscous terms Pressure terms Repeat until convergence Use U to get new h Numerical Method Initial values for h, P, U, and V

9. Preliminary Results 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 Static case - test geometric effect on meniscus - determine improvement of conical capillary

10. Region I Region II Data matches within 97.5% confidence interval Region III Dynamic Rise of Dodecane Simulation shows behavior of all three regions B. V. Zhmud, F. Tiberg, and K. Hallstensson, “Dynamics of capillary rise,” J. Colloid Interface Sci. 228, 263 (2000)

11. 0.1% error Dodecane Equilibrium Equilibrium shape calculated from static equations Simulation end shape within 0.1% of equilibrium shape Simulation matches dynamic and equilibrium behavior

12. Previous work values 1.01, 2.73, 3.00, 3.76 Test values 1.00, 3.00 Experiments carried out in jet producing free fall environment Dynamic Simulation of Microgravity • Use microgravity rise of Dow Corning Silicon fluid “SF 0.65” to match initial height behavior • Test effect of exponent in contact line equation

13. 240 m=1 simulation 190 140 Height (mm) Stange Paper (7) 90 Microgravity run m=3 Microgravity run m=1 m=3 simulation 40 -10 0 1 2 3 4 5 Time (s) Dynamic Simulation of Microgravity Simulations match experimental behavior M. Stange, M. E. Dreyer, and H. J. Rath, “Capillary driven flow in circular cylindrical tubes,” Physics of Fluids 15, 2587 (2003)

14. Static Case Test geometric effect Model reduces to solving single height equation Two different capillaries: cylinder and cone

15. Same contact angle Centerline Wall D h Height (mm) Cone Cylinder Radius (mm) Cone wall Static Case Increased height for cone, also increased curvature

16. Conclusions • Model for capillary flow developed based on first principle equations • Algorithm able to predict previous experimental results • Dynamic Dodecane Rise • end results within 0.1% of equilibrium • dynamic data within 97.5% confidence of experimental • Static Case • height increase for conical capillary over cylindrical • Dynamic Microgravity Rise • simulation matched experimental results • exponent in constitutive equation only effects behavior in Region II

17. 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)

18. PennState Acknowledgements Funding Penn State Academic Computing Fellowship Academic Dr. Ali Borhan Dr. Kit Yan Chan Personal Dr. Kimberly Wain Rory Stine Michael Rogers

20. Expanded microgravity graph

21. Constitutive contact line velocity plotted against contact angle qa qr