Roundabout

1. Roundabout Belarusian team Presents Reporter: Alexandr Mamoiko Tianjin, China2009

2. Formulation of the problem 6. Roundabout Put a plastic cupon a thinlayerofliquidon a flatsolidsurface. Make thecuprotate. Onwhatparametersdoestherotational deceleration of the cupdepend?

3. Contents: • 1)The main forces of deceleration • 2)Determination of the depth of liquid under cup • A. Research of flows under cup • B. Mathematical model of laminar mode • C. Turbulence mode • 3)Parameters of racing • A. Initial angular velocity • B. Duration of racing • 4)Properties of cup and liquid Conclusions

4. I. The forcesof deceleration Viscosity of water and air:

5. The main forces of deceleration

6. m – mass of the cup R – radius of cup ρ – density of liquid θ – angle of contact σ – surface tension of liquid Subsidence of static cup

7. Experimental setup “Roundabout” Device Surface

8. Demonstration

9. D II. Dependence of deceleration against depth of layer of liquid

10. D=1mm Dependence ω against time When cup’s angular velocity is enough high liquid can outflow from under “roundabout”. Herewith the cup will down and fast decelerate.

11. D=1mm D=1.2mm D=1.7mm D=2.2mm D=2.7mm D=4.2mm D=8.7mm The approximating curves Dependence cup’s angular velocity against time

12. Movie laminar flow

13. The linearization(prediction) The basic equation of rotational motion’s dynamics: If flow under cup is laminar moment of viscous forces is proportionalto the angular velocity. ω– angular velocity of cup t – time of rotation ω0– initial angular velocity τ– characteristic time

14. Z ω Elementary force (Newton’s theory): Mathematic model (laminar flow) Gradient of liquid’s velocityon z direction

15. Mathematic model (prediction) The Couette flow Liquid’s velocity on z direction Simple solution of Navier-Stokes’ equation Moment of viscosity friction inversely layer’s depth therefore characteristic time must be proportional D

16. Z ω Dependence characteristic time against depth layer for the “laminar” part of deceleration Tau against D The linear dependence characteristic time against depth layer under cup allows to expect gradient velocity liquid on z direction and determine moment of viscosity forces .

17. Moment of viscosity forces A bottom of cup Velocity of liquid under cup on z direction: r dφ Elementary force: dr

18. The law of cup’s motion ω– angular velocity of cup t – time of rotation ω0– initial angular velocity τ– characteristic time J – moment of inertia R – cup’s radius D – layer’s depth of liquid µ - viscosity

19. D=8.7mm D=4.2mm D=2.2mm Comparison theory with experiment for the “laminar” part of deceleration Theory Effect rotationally subsidence

20. D=1.7mm D=1.2mm Initial depth of layer Correction of depth due to down More thin layer

21. Movie with down 21

22. Subsidence due to the rotation Condition for Hrot: Forces of inertia = Hydrostatic forces 22

23. D=1.7mm D=4.2mm D=1.2mm The “laminar” part of deceleration Linearization ω(t) – LN(ω) against time Linearization of graphics The nonlinear part

24. Movie with Mg

25. The linearization(turbulence) When flow under cup is turbulence moment of viscous forces is proportional a square angular velocity. M ~ ω2 The law of cup’s motion : ω– angular velocity of cup t – time of rotation ω0– initial angular velocity τ– characteristic time

26. D=8.7mm D=4.2mm D=2.2mm D=1.7mm D=1.2mm Linearization turbulence Laminar deceleration + dynamic subsidence Turbulence part

27. The kinds of deceleration: • 1)Gluing • 2)M~w , laminar flow • 3)M~w2 , turbulence

28. III. Parameters of racing • 1) Initial angular velocity • 2)Time of racing

29. Dependence ω against time with various initial ω0 Dependence ω against time with various initial ω0 ω0= 5.3rad/s ω0= 8.5rad/s ω0= 12.5rad/s ω0= 18.5rad/s

30. LINEARIZATION (1/w against time)

31. Dependence ω against time with various time of racing Dependence ω against time with various time of racing t= 0.2s (Push) t= 1.0s t= 1.5s t= 3.1s

32. IV. Parameters of cup 1)Mass 2)Radius(square of bottom) 3)Moment of inertia 4)Angle of contact

33. Experiment with radius disc

34. Dependence against radius disc Dependence against radius disc

35. V. Parameters of liquid • 1) Viscosity • 2) Density • 3)Surface tension

36. Parameters of the roundabout 1) Depth of liquid under cup 2) Initial angular velocity 3) Duration of racing 4) A cup: a) Mass of cup b) Radius of cup c) Moment of inertia d) Contact’s angle 5) A liquid: a) Viscosity b) Density c) Surface tension

37. Conclusions: • 1)Depth of liquid layer determines character of deceleration • 2)There are 3 different modes of cup deceleration: a) gluing b) laminar flows c) turbulence • 3)Turbulent and laminar deceleration may be obtained during a single experiment. • 4)Laminar mode is explained by model of Couette flow • 5)The characteristic time doesn’t depend on depth of liquid layer for turbulence mode.

38. Summary: • The experimental base of deceleration was built for explanation phenomenon. • The most important parameters were marked. • The experimental investigation some

39. References: Wikipedia: Couette flow. A. D. Glinkin and V. A. Rukavishnikov. Laminar flow of a viscous incompressible liquid over the surface of solids of revolution. J. Eng. Phys. Thermophys., 39, 1 (1980) Slezkin. Hydrodynamics.

40. THANK YOU FORATTENTION!

41. The conditions for laminar flow ρ– density of liquid μ– viscosity v– characteristic velocity ( ω*R ) l– characteristic size ( D ) Boundary value in our experiments: