Onset of Wave Drag due to Capillary- Gravity Surface Waves

1. Onset of Wave Drag due to Capillary- Gravity Surface Waves V. Steinberg in collaboration with T.Burghelea Department of Physics of Complex Systems

2. Dispersive (gravity) waves • Different nature of the drag-viscous, , eddy resistance (laminar and turbulent wakes), , and wave resistance, . For ships: , ,<< • Pattern of gravity waves generated by a ship, • Waves continuously remove momentum from a moving object to infinity-wave drag (DR) • Onset of drag at the threshold at , where and the Froude number is an analog of the Mach number. • Wave drag appears as a continuous bifurcation that was actually observed in experiment by Taylor(1908) (M. Shliomis & V. S., PRL 79, 4178 (1997).)

5. . Schematic sketch of a ship wake

6. Wave drag coefficient for a ship moving at speed V

7. Capillary-gravity surface waves Dispersion relation: For water: cm/s;cm

9. Detecting surface waves.Surface waves form at upstream and downstream sides of a fisher spider Dolomedes Triton (0.7 g) leg segment moving across the water surface at relative velocities greater than 20cm/s. The ovals in the images indicate the approximate locations of the shadow of the leg segment. (R. Suter et al, J. Exp. Biology, 200, 2523 (1997).

12. A disturbance moving with a constant velocity, V, creates surface waves only if -critical velocity • Examples of similar systems: • Superfluid helium • Cherenkov radiation

13. Superfluid helium -Landau critical velocity for breaking of superfluidity via generation of rotons-onset of drag. m/s Nonlinear Schreodinger equation as a model (Pomeau&Rica(1993)): Roton drag- Cherenkov radiation of rotons

14. Cherenkov radiation Cherenkov process Non-dispersive waves 1 At (light speed in a medium) Cherenkov cone is 2 Supersound at (sound wave speed) Mach cone Dispersive waves From stationary conditions in moving frame one gets angle between and There is no solution at and the Cherenkov opening cone appears at

15. Onset of wave drag in capillary-gravity surface waves E.Raphael & P.G. deGennes(1996) Kelvin’s model (suggested for gravity waves) P(x,y)-fixed point-like distribution For -distribution leads to , i.e. a finite jump

16. For finite width pressure distribution, p-is total vertical applied force. The wave resistance is -Fourier transform of the pressure distribution

17. 1 2 3 1. b/ : 1-10; 2-17; 3-25,4-inf 2. b/ : 1-0.89; 2-1.19; 3-1.79 3. b/ =1 Results of numerical calculations for finite depth distributions (type of transition remains (by jump) butfunctional behavior contradicts to experiment)

18. Capillary contribution to the measured force in the case of wetting, and the ball is allowed to move.

19. Ratio between capillary and viscous forces acting on a ball Vc=23 cm/s, R=0.157 cm, η=1cSt σ=73 dyn/cm for water for oil For a wire: d=0.7 mm, l=1mm this ratio is about 30 for oil and above 300 for water Balls of 1.57, 2.35, and 3.14 mm were used

20. Experimental setup: (a) wave visualization; (b) force measurements (Channels with R/gap: 12.4/3.6 and 17.5/ 15 cm)

21. Physical properties of fluids used in the experiments (at 25 C)

22. (from fit 8.5 cSt) The drag force vs velocity for silicone oilDC200/10 cStand3.14 mmball. The upper inset: the same data but for reduced force. The lower inset: the viscous drag force below the transition vs object velocity. Solid line is the second-order polynomial fit.

23. Re numbers at the transition for different fluids and ball size vary from about 2 till 700. It meansthat below transition the drag consists of a viscous Stokes drag that changes linearly with and drag due to the eddy viscosity that is proportional to at

24. ■- DC200/50 cSt ●- glycerin-water-30cSt ▲- glycerin-water-46cSt 3.14 mm ball The inset: water Fit by the Ginzburg-Landau equation:

25. ■-water ●-silicone oil DC200/50 cSt ○-silicone oil DC200/50 cSt 2.35 mm ball The reduced scaled wave drag force vs the reduced velocity for two fluids and 2.35 mm ball (scaling with the factor ) (Dimple is an example of meniscus position instability in oil (non-wetting fluid))

26. ■-with feedback □-without feedback silicone oil DC200/50 cSt 3.14 mm ball The reduced wave drag force vs. the velocity for a silicone oil DC200/50 cSt and 3.14 mm ball.

27. The reduced drag force vs. the reduced velocity for water-silicone oil DC200/10 cSt interface and 3.14 mm ball. Bump is due to the interface instability; sometimes exchange of a bump to a dimple was observed, particularly with one wetting and another non-wetting fluids.

28. The reduced water-silicone drag force vs. the reduced velocity for oil DC200/50 cSt interface and 3.14 mm ball Experimental value of and that found from independent surface tension measurements agree well

29. Scaling relations 1. Scaling for the critical wave drag force 2. Together with another scaling factor found experimentally the scaling relation can be written is viscosity independent and is caused only by wave emission! 3. Finally, the scaling compatible with dimensionless analysis has the form (Buckingham -theorem and relevant parameters )

30. 3.14 mm ball ■-water ●-silicone oil DC200/50 cSt □-glycerin-water-30 cSt ▲-glycerin-water-46 cSt 2.35 mm ball ◊-silicone oil DC200/50 cSt The reduced wave drag force divided by non-dimensional parameter A vs. the reduced velocity for five different fluids with 3.14 and 2.35 mm ball. Solid line is the fit by the stationary Ginzburg-Landau equation.

31. Europhys.Lett. 53, 209 (2001) J. Browaeys, J. Bacri, R. Perzynski, M. Shliomis

32. Fiber of 0.7 mm in silicone oil DC200/10 cSt; circles-increasing velocity, squares-decreasing Wire of 0.3 mm in water; squares- increasing velocity, circles-decreasing

33. Comparison with theory • Raphael-de Gennes (1996): Kelvin’s model, 2D-singular solution at , 3D-discontinuous transition, for finite width 1/b-wrong functional dependence • Richard-Raphael (1999): 2D + viscosity-wave resistance increases continuously but steeply and remained bounded • Sun-Keller (2001): wave drag far away from transition (Kelvin’s model) • Chevy-Raphael (2003): constant force-discontinuous transition, constant depth-continuous one but very sharp (not of a second order) with different functional dependence; behavior of the wave drag above the transition also disagrees with experiment. It does not resolve contradiction with “French” experiment since it was not conducted at constant force. • Pham-Nore-Brachet (2005): first step in right direction-taking into account wetting and capillary force. However, the problem is 2D-shallow water problem, dispersion law without minimum, appearance of wave drag is similar to phonon excitation and onset of dissipation in BE condensate ( or generation of gravity waves by finite length ship-Shliomis&V.S.(1997)). Wetting is taking into account via boundary conditions. At subcritical speed unstable dynamics lead to dewetting instability. At critical speed continuous transition to the state with no stationary solution is observed.

34. V=25.33 cm/s

35. (1) (2) of the Cherenkov cone as a function of the velocity for the capillary ripples on a water in front of the ball. The solid line is calculation based on Eqs. (1) and (2) using the data on k(V), presented in the inset. At there are two values of k , at which :wave, for which, and, for which . Thus, short ripples are found upstream (in front of object) and long waves –downstream (behind object)-look picture!

36. Geometry of the wave crest

37. Data on the wave crests generated in water by 3.14 mm ball (squares). Calculated wave crests presented by full lines

38. Conclusions. • Transition to the wave drag state in gravity- capillary surface waves is continuous. • Ginzburg-Landau type model with a field describes well the data for various fluids and various sizes of a moving object. • Scaling of the parameters of G-L equation with physical parameters is found experimentally. • References: PRL, 86, 2557 (2001) and Phys. Rev. E 66, 051204 (2002).