1. Granular Jets Alexander BarnaveliGeorgia If a steel ball is dropped onto a bed of dry sand, a "splash" will be observed that may be followed by the ejection of a vertical column of sand. Reproduce and explain this phenomenon.

2. Presentation Plan 1.Experiments 2. Jet formation mechanisms  Void Collapse  Density waves in the sand 3. Ball penetration into granular media  Forces acting on the ball  Estimation of penetration depth 4.Estimation of granular jet height 5.Granular jet clustering 6. Conclusion 1.Experiments 2. Jet formation mechanisms  Void Collapse  Density waves in the sand 3. Ball penetration into granular media  Forces acting on the ball  Estimation of penetration depth 4.Estimation of granular jet height 5.Granular jet clustering 6. Conclusion

3. Granular media - powder of sodium hydrogen carbonate NaHCO 3 Granular Density: ρ g =2.16g/cm 3 Particle Size: D g =5·10 -3 cm Ball diameters: D b1 =2.9cm ; D b2 =3.9cm Ball Density: ρ b =11g/cm 3 Ball Mass: m b1 =100g ; m b2 =230g

4. Thin Thick

5. The impacting ball creates a void Void is pressed together through the “hydrostatic” pressure p(z) = ρ s gz, from the side. The sand converges towards the cavity center. At the middle depth the collapse is finished first. Radial velocity diverges as 1/r as the cavity closes up Singularity leads to the formation of two jets: Upwards and Downwards into air pocket Upward jet forms "thin granular jet" Pressurized air pocket drives the "thick granular jet"

7. Penetrating ball pushes the granules downwards and to the sides. This creates “Density Waves”. “Density Waves” are being reflected by the walls of the container. Reflected waves support the granular jet burst The shape of the container walls and the place of ball penetration can play the sufficient role in the jet generation. Pushing granules Hemisphere penetration Hemisphere

8. h jet H jet Non - DampedDamped H jet >> h jet

9. Inverse Angle

10. Energy redistribution channels: Lifting grains during crater formation Ball/grain, grain/grain Inelastic collisions; Particle-particle sliding, ball-grain sliding friction; Motion of grain mass pushed by penetrating ball. Total force on the ball: Gravity, Depth-dependent resistance force, Velocity-dependent drag force

11. α - scaling factor, F 0 - parameter Velocity-dependent drag force Interpretation: Inertial force required for the ball to mobilize a volume D b 3 of granular media. Poncelet force law F(v) = F 0 + cv 2

12. Depth-dependent force Experimental measurements. Granular Density: ρ g =2.16g/cm 3 Ball parameters: D b1 =2.9cm ; m b1 =100g ; D b2 =3.9cm ; m b2 =230g Resistance force is parabolic. Also: Using dimensional considerations: Comparing with experimental measurements η≈25. η - scaling factor

13. 1. Only ρ b = 11 g/cm 3, ρ g = 2,16 g/cm 3, η≈25 fit the experiments. h - dropping height, d - penetration depth., where H=h+d

14. 2. Only α ≈ 2 ; F 0 ≈ m b g + 0.65N fit the experiments. Solution penetration depth:

15. Jet height vs drop height

17. Possible mechanisms for Jet Clustering: Hydrodynamic interactions with the surrounding gas, Inelastic grain–grain collisions, Cohesive forces. Cohesion can arise from : Electrostatic charging, Van der Waals forces Capillary forces.

18. Conclusions 1.Main mechanisms of granular jet formation are: Inertial focusing of the void due to “hydrostatic” pressure in the granular media. “Density waves” induced by the impacting wall. 2.Density waves are very important. They support jet formation. We showed that container shape crucially affects the Jet height 3.Forces acting on the ball are: Depth-dependent resistance force Velocity-dependent drag force Estimated the shape of these forces and calculated penetration depth. 4.Bursting jets are clusterized Mentioned the reasons of this clusterization. It needs further investigation. 1.Main mechanisms of granular jet formation are: Inertial focusing of the void due to “hydrostatic” pressure in the granular media. “Density waves” induced by the impacting wall. 2.Density waves are very important. They support jet formation. We showed that container shape crucially affects the Jet height 3.Forces acting on the ball are: Depth-dependent resistance force Velocity-dependent drag force Estimated the shape of these forces and calculated penetration depth. 4.Bursting jets are clusterized Mentioned the reasons of this clusterization. It needs further investigation.

19. Thank you for Attention!

20. References: [1]. Raymond Bergmann. “Impact on Sand and Water”. Physics of Fluids, University of Twente, The Netherlands 2007; [2]. John R. Royer et Al, “Birth and growth of a granular jet”. PHYSICAL REVIEW E 78, 011305 (2008); [3]. S. T. Thoroddsen and Amy Q. Shen. “Granular Jets”. PHYSICS OF FLUIDS VOL. 13, No 1, JAN 2001. [4]. H. Katsuragi and D. J. Durian. “Unified force law for granular impact cratering”. Nature Physics 3 (6), 420-423 JUN 2007.

21. Additional Slides:

22. Jet height increases with the drop height; Rough Considerations Hj - Jet Height; D b - Ball diameter; ρ b - ball density, D g - grain diameter, ρ g - grain density, Energy acquired by jet by the collapse Jet energy S void - Void crossection; S jet - Jet crossection; g - gravity, d - void depth, ; E j ~ E coll so

23. 2. Only α ≈ 2 ; F 0 ≈ m b g + 0.65N fit the experiments. Boundary condition: Motion equation: Solution for ball velocity: Boundary condition: Solution penetration depth:

24. 1. Only ρ b = 11 g/cm 3, ρ g = 2,16 g/cm 3, η≈25 fit the experiments. h - dropping height, d - penetration depth. Here from: Energy conservation law:, where H=h+d

25. Exit