1. A FemVariational approach to the droplet spreading over dry surfaces S.Manservisi Nuclear Engineering Lab. of Montecuccolino University of Bologna, Italy Department of mathematics Texas Tech University, USA

2. Simulations of droplets impacting orthogonally over dry surfaces at Low Reynolds Numbers OUTLINE OF THE PRESENTATION - Introduction to the impact problem - Front tracking method - Variational formulation of the contact problem - Numerical experiments

3. Depostion Prompt Splashd Corona Splashd INTRODUCTION

4. Depostion Partial reboundd Total reboundd INTRODUCTION

5. An experimental An experimental investigation..... C.D. Stow & M.G. Hadfield Spreading smooth surface v=3.65 m/s r=1.65mm INTRODUCTION

6. An experimental An experimental investigation..... C.D. Stow & M.G. Hadfield Splashing rough surface v=3.65 m/s r=1.65mm INTRODUCTION

7. An experimental INTRODUCTION 1) Problem : Numerical Representation of Interfaces Impact Dynamics : solid surface + liquid interface = drop surface Splash Dynamics : liquid interface -> more liquid interfaces 2) Problem : Correct Physics Impact Dynamics : solid surface + liquid interface = drop surface Splash Dynamics : liquid interface -> more liquid interfaces Hypoteses: No simulation of the impact No splash o total rebound (low Re numbers, no rough surfaces) Axisymmetric simulation Numerical Representation of Interfaces -> ok Correct Physics ?

8. Some features: Behavior of the impact for: Wettable-P/Wettable N/Wettable surfaces Deposition – Partial rebound – total rebound Surface capillary waves Spreading ratio and Max spreading ratio Static/Dynamic/apparent Contact angle INTRODUCTION

9. D=1.4mm v=0.77m/s Re=1000 We=10 Wettable Partially Wettable Non-Wettable Deposition Partially Wettable Non-Wettable

10. INTRODUCTION

11. τ= τ(μ) = Stress tensor Dynamics (incompressible. N.S.eqs) incompressible u = velocityp=pressure f_s = Surface tension f = Body force μ =viscosity = μ1 χ + (1-χ) μ2 ρ =viscosity = ρ1 χ + (1-χ) ρ2

12. Kinematics (Phase eq.) Equation for χ (phase indicator) χ =0 phase 1 χ =1 phase 2 Solution: 1)Weak form (method of characteristics) 2)Geometrical algorithm

13. Boundary conditions Static cos(  ) =cos(  s) v=0 no-slip boundary condition Non Static cos(  ) =cos(  s) ? v=0 no-slip boundary condition ?

14. V. FORM OF THE STOKES PROBLEM gives

15. CONTACT PROBLEM (NO INERTIAL FORCES) = Shape derivative in the direction u

16. CONTACT PROBLEM (NO INERTIAL FORCES) Minimization gives No angle condition

17. MINIMIZATION WITH PENALTY Remarks: Is a dissipation term Contact angle condition

18. CONTACT PROBLEM WITH PENALTY Minimization gives

19. Boundary condition over the solid surface Boundary condition Full slip boundary cond

20. V.F.OF THE CONTACT PROBLEM Near the contact point otherwise

21. Numerical solution Fem solution Weak form -> fem Advection equation -> integral form Density and viscosity are discountinuous -> weak f. Surface term singularity-> weak form

22. ADVECTION EQUATION Surface advection Integral form Advection equation

23. (2D) Reconstruction Advection ADVECTION EQUATION Markers= intersection (2markers) Conservation (2markers) Fixed mrks (if necessary)

24. ADVECTION EQUATION

25. Vortex tests ADVECTION EQUATION

28. Fem surface tension formulation Surface form Volume form Is extended over the droplet domain

29. Static: Laplace equation Solution for bubble v=0, p=p0 Spurious Currents Fem surface tension formulation

30. Static: Laplace equation Solution for bubble v=0, p=p0 1)Computation of the curvature 2)Computation of the singular term Solution v=0, v=0 p=0 outside p=P0=a/R inside Fem surface tension formulation

31. Casa A: exact curvature Solution Curvature=1/R Surface tens=σ V=0; p=p0 No parassitic currents Fem surface tension formulation

32. Case B: Numerical curvature With exact initial shape A t=0 B t=15 C t=50 Curvature Initial velocity Final velocity Fem surface tension formulation

33. Case C: Numerical curvature (ellipse) Shape time Fem surface tension formulation

34. Steady solution angle=120 angle=60 angle=90

35. Boundary condition over the solid surface

36. Full slip boundary cond

37. Re=100 We=20  =60 Deposition t=0 t=2.5 t=4 t=15 t=50 t=0 t=0.5 t=3 t=1.5 t=1

38. Re=100 We=20  =60 Deposition

39. Re=100 We=20  =90 partial rebound t=4 t=5 t=0 t=6 t=0 t=3 t=2 t=1.5 t=1 t=0.5 t=0

40. t=7 t=9 t=8 t=10 t=11 t=14

41. Re=100 We=20  =90 partial rebound

42. Re=100 We=20  =120 total rebound t=.5 t=1.5 t=3t=0 t=2 t=4 t=7 t=2.5t=1

43. Re=100 We=20  =120 total rebound

44. DIFFERENT WETTABILITY Wettable (60) ANon-wettable (120) C partially wettable (90) B

45. Re=100 We=100  =120 Re=100 We  =120 u0  =120 We= 100 A 50 B 20 C 10 D u0= 2 A 1 B.5 C Different impact velocity Different We

46. DYNAMICAL ANGLE Glycerin droplet impact v=1.4m/s D=1.4mm Wettable (18) Partially wettable (90)

47. DYNAMICAL ANGLE Friction over the solid surface Friction over the rotation

48. DYNAMICAL ANGLE MODEL Cox Blake Power law Jing

49. Non-Wettable Wettable D=1.4mm u0=1.4m/s glycerin droplet A=1 B=2 C=10 A=1 B=2 C=10

50. D/D0 h angle

51. Conclusions - Variational contact models can be used - Open question: Can we simulate large classes of droplet impacts with a unique setting of boundary conditions ?

52. Thanks