Flat Flow

1. 11 Flat Flow by Kamila Součková

2. Task Fill a thin gap between two large transparent horizontal parallel plates with a liquid and make a little hole in the center of one of the plates. Investigate the flow in such a cell, if a different liquid is injected through the hole.

6. Interfacial Pressure interfacial pressure due to curved surface must be overcome • pressures we work with are ⇒ negligible

7. Understood Phenomena • flow through a porous medium • Darcy’s Law: • flow in a small gap • Hele-Shaw equation: Figure from Hornberger et al. (1998) k:permeabilityofmedium h : gap between plates • : viscosity of liquid • ∇p : pressure gradient

8. Position, Velocity • measure how fast the interface moves • at various places – take the average weight on syringe removed

9. Formation of Instabilities

10. Formation of Instabilities small instability pressure differences instability grows “fingers”

11. Slow-Motion Video of Patterns 300 fps

12. What Affects Instabilities? promotes disturbances tries to dampen out disturbances

13. Low Viscosity → No Fingers low viscosity symmetrical situation easier to spread out evenly no “fingers”

14. Low Viscosity → No Fingers Ink (less viscous) Glycerol (more viscous)

15. Pattern Complexity Measurements

16. Equipment • Liquid: • more viscous: glycerol, motor oil • less viscous: water (colored), ink, ethanol • Plates – plexiglass (Hele–Shaw cell): • 25 x 25 cm • gap:distance setby weights • hole: size customized to the syringe

17. Equipment

18. Equipment

19. What to Measure quantify the instabilities • count perturbances • tells how “interesting” the pattern is • to make more objective:

20. Viscosity

21. Viscosity of Medium water → motor oil 5W water → motor oil 10W Viscosity < = 0.14Pa.s = 0.17 Pa.s

22. Viscosity of Medium water → glycerol water → motor oil 15W Viscosity < = 1.48 Pa.s = 0.41Pa.s

23. # of Fingers vs Viscosity of Medium

24. Viscosity of Injected Liquid oldink (more viscous) → glycerol • newink (lessviscous) → glycerol Viscosity >

25. Distance of the plates

26. Gap Size (Water → glycerol) 1mm ≈0.8 mm

27. ≈0.2mm ≈0.4mm ≈0.1mm

28. # of Fingers vs Gap Size

29. pressure in syringe

30. Pressure in Syringe changed by putting weights on syringe p = 15 kPa (300g) p = 22 kPa (450g) p = 49 kPa (1 kg)

31. # of Fingers vs Pressure

32. Interfacial Tension

33. Interfacial Tension Water: → glycerol Ethanol: → glycerol = 0.009Pa.s = 0.0225 N/m = 0.001 Pa.s = 0.072 N/m

34. Highersurface tension → more rounded Air → water ( = 0.072 N/m) Air → water with detergent (δ = 0.025 N/m) Lowersurfacetension → greater instability

35. Further investigation

36. An analogous experiment • Jose A. Miranda, Michael Widom: Radial Fingering in a Hele-Shaw Cell: a weakly nonlinear analysis, PhysicaD 120(1998) 315-328 Q = flow [m3/s] R = initial radius [m] viscosity of injected liquid [Pa.s] viscosity of medium [Pa.s] h = gap width [m] = interfacial surface tension [N.m]

37. Viscosity

38. Gap Size h

39. Pressure p

40. Interfacial Tension Water→ glycerol Ethanol→ glycerol = 0,0225 N/m = 0,072 N/m

41. Conclusion • observed the phenomenon • showedwhenitdoesnotwork • explainedthemechanismofemergingpatterns

42. Conclusion • Provedassumedinfluences on thephenomenon: • viscosity • gapbetweentheplates • pressure • surface tension < < < >

43. Thank you for your attention! • Provedassumedinfluences on thephenomenon: • viscosity • gapbetweentheplates • pressure • surface tension < < < >

44. 11. Flat Flow Kamila Součková

45. Appendix

46. Changingtemperature of glycerol 80°C → thesamepatterns

47. Dataforexperiments

48. Motor oil → glycerol(∆ ν = 1119,5 . 10-6 m2/s)