Bubble Penetration Through a Single Layer Sphere Bed

1. School of Aeronautics and Astronautics Bubble Penetration Through a Single Layer Sphere Bed Robert E. Manning Steven H. Collicott Purdue University Supported in part by NASA research grant NNC05GA25G, Dr. Walter Duval, Program Manager

2. Capillary Phenomena in a Sphere Packed Bed • Sphere packed beds are used as a model for porous media. • Porosity can vary from approximately 48 to 26%. • Used for infiltration problems. • Mayer & Stowe (2006) • Hilden & Trumble (2003) • Slobozhanin, Alexander, Collicott, & Gonzalez (2005, unpublished) • Here we examine a non-wetting liquid droplet passing through the pore (or a gas bubble surrounded by wetting liquid)

3. Problem Formulation • Single layer with four spheres of fixed radius r • Droplet can impinge on any number of spheres • Hexagonal (δ=90º) to square (δ=90º) packing angles • Droplet is above the pore under a uniform gravitational field (g). • Under what conditions will the droplet pass through the pore? • What is the topology progression of the droplet? -g

4. Topologies • Droplet will impinge on spheres and form contact lines. • Seven unique topologies exist. • Clearly unique topologies with one contact line and four contact lines exist. • Due to symmetry, the two and three contact line cases are noted below. • Below, gray denotes wetted sphere.

5. Solution Methodology • Assume quasi-static. Both gravitational and capillary energies are considered. • Use Surface Evolver to solve for minimum energy based on volume, packing angle, and liquid contact angle. • Program seven different topologies and find the Bond number intervals where droplet is stable. • Two and one contact line solutions cannot exist for negative Bond numbers (driving droplet into pore).

6. Convergence Study 1 • Examined how energy and center of mass are affected by refinement. For accuracy of 0.01 Bond, 8000 facets needed. For 0.001, over 12000 needed.

7. Convergence Study 2 • Examined criteria for convergence. • If both energy and center of mass are within 1e-4, then we assume convergence.

8. Convergence Study 3 & 4 • Considered how many facets are needed to resolve a droplet impinging on another dry sphere. • As few as 1000 facets resolved this to within 0.001 Bond number. • Also approximately 40-50 iterations were needed to detect instability due to contact line collapse (droplet pulling away from a sphere). • For results presented today, 15000 facets were used to model the liquid-gas interface of the droplet.

9. A Single Packing Angle • A droplet with unity volumeand liquid contact angle of 135º. The sphere layer isnear square packed with δ=88º. • The vertical distance to the droplet’s center of mass (z) was calculated for a range of Bond numbers. • Repeated for all seven topologies. • Interestingly this packing exhibits all seven possible droplet types. z

10. 88º Packing Angle

11. 88º Packing Angle

12. 88º Packing Angle

13. Four Contact Lines Stability Regions 90º packing angle 70º packing angle

14. Four Contact Lines Three A Stability Regions 70º Packing Angle

15. Four Contact Lines Three A Two B Stability Regions 70º Packing Angle

16. Contact Lines One Two A Stability Regions

17. Four Contact Lines Three A Two B One Two A Stability Regions

18. Four Contact Lines Three A Two B One Two A Near Square Packed Stability Regions

19. Contact Lines Four Three A Two B One Two A Three B Two C Near Square Packed Stability Regions

20. Conclusion • Necessary criteria were computed for both droplet penetration of the pore and “dripping” from the sphere layer. • A variety of droplet shapes exist for different regions. • Future work will examine different contact angles and volumes.