A FemVariational approach to the droplet spreading over dry surfaces

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. INTRODUCTION Depostion Prompt Splashd Corona Splashd

4. INTRODUCTION Depostion Partial reboundd Total reboundd

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

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

7. 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 An experimental 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. INTRODUCTION • 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

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

10. INTRODUCTION

11. Dynamics (incompressible. N.S.eqs) incompressible τ= τ(μ) = Stress tensor u = velocity p=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: Weak form (method of characteristics) 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 Advection equation Integral form Surface advection

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

24. ADVECTION EQUATION

25. ADVECTION EQUATION Vortex tests

26. ADVECTION EQUATION

27. ADVECTION EQUATION

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

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

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

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

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

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

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

35. Boundary condition over the solid surface

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

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

38. Re=100 We=20 =60 Deposition

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

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

41. Re=100 We=20 =90 partial rebound

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

43. Re=100 We=20 =120 total rebound

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

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

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

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

48. DYNAMICAL ANGLE MODEL Cox Blake Power law Jing

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

50. D/D0 h angle