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

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 IRegion IIRegion III t*t* h*h* t I* t II* h I* h II* M. Stange, M. E. Dreyer, and H. J. Rath, “Capillary driven flow in circular cylindrical tubes,” Physics of Fluids 15, 2587 (2003) Inertial force domain Force from pressure drop at entrance Friction force domain

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

Developing The Model Governing equations for the transformed system: Conservation of Mass Conservation of momentum  direction  direction

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

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 Convective terms Viscous terms 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

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

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 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, 3.76 Test values 1.00, 3.00 Experiments carried out in jet producing free fall environment

Time (s) Height (mm) Stange Paper (7) Microgravity run m=3 Microgravity 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 Two different capillaries: cylinder and cone

Static Case Radius (mm) Height (mm) Cone Cylinder Centerline Wall Cone 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 Funding Penn State Academic Computing Fellowship Academic Dr. Ali Borhan Dr. Kit Yan Chan Personal Dr. Kimberly Wain Rory Stine Michael Rogers PennState

Expanded microgravity graph

Constitutive contact line velocity plotted against contact angle aa rr