1. John Abbott Mathematics in Industry Workshop 06/16/2008 MTE-AMA Elongating Bubble in an Accelerating Extensional Flow

2. 2 June 2008 Mathematical Problems in Industry June 2008 Outline Introduction to Corning! Brief Problem Statement Background, Introduction, and Context –Pictures –References & Previous work Restatement of Problem Extensions, further work which might be possible

3. 3 June 2008 Mathematical Problems in Industry June 2008 Corning Incorporated R&D labs celebrating 100 th year anniversary Corporate member of IMA at Univ. of Minnesota Emphasis on innovation and new products –Light bulb: worked with Edison, all incandescent light bulbs in the world are made by the high speed ‘ribbon machine’ developed by Corning –Glass for Cathode Ray TVs (CRTs); color TVs –Silicones – spun off to Dow-Corning (private company) –Fiber glass insulation – spun off to Owens-Corning Fiberglas –Optical Fiber – low cost, low attenuation optical fiber for telecommunications –Precision glass for LCD flat panel displays –Extruded Ceramics for catalytic converter substrates –Specialty glass & ceramics (mirror blank for Hubble telescope)

4. 4 June 2008 Mathematical Problems in Industry June 2008 Environmental Products Corning makes ceramic ‘substrates’ catalytic converters for cars The substrate is made by extruding wet ceramic through a die with hundreds of holes – the holes need to be uniform. In drilling the holes the drills used to wear out..

5. 5 June 2008 Mathematical Problems in Industry June 2008 Display – Large Glass Sheets for LCD displays Corning invented a process for making sheet glass where the glass flows over two sides of a ‘weir’, so that the final sheet has two pristine surfaces. Gen 8 2160x2460mm Gen10 3000x3000m 0.7mm thick

6. 6 June 2008 Mathematical Problems in Industry June 2008 Optical Fibers One of Corning’s big businesses is optical fiber – essential to the internet. The combustion synthesis process used to make pure SiO2 was developed by Franklin Hyde at the Corning R&D labs, the same fellow who invented silicones. The same process is used to make windows for the space shuttle and the glass for the Hubble space telescope. GeO2 &SiO2 – optical fibers TiO2 & SiO2 – low thermal expansion

7. 7 June 2008 Mathematical Problems in Industry June 2008 Optical Fiber

8. 8 June 2008 Mathematical Problems in Industry June 2008 Outline Introduction to Corning! Brief Problem Statement Background, Introduction, and Context –Pictures –References & Previous work Restatement of Problem Extensions, further work which might be possible

9. 9 June 2008 Mathematical Problems in Industry June 2008 Problem Statement How is a single spherical bubble or ‘seed’ stretched as it moves down the tapering root during a high speed optical fiber draw process? The leading tip of the bubble sees an exponentially increasing velocity (and extensional rate). How (if at all) does the shape differ from the distortion in either simple shear flow or simple extensional flow.

10. 10 June 2008 Mathematical Problems in Industry June 2008 Outline Introduction to Corning! Brief Problem Statement Background, Introduction, and Context –Pictures –References & Previous work Restatement of Problem Extensions, further work which might be possible

11. 11 June 2008 Mathematical Problems in Industry June 2008 “Blisters” in volcanic magma

12. 12 June 2008 Mathematical Problems in Industry June 2008 Experimental measurement of bubble shapes Rust, Manga 2002 An optical fiber airline has a large aspect ratio l/a

13. 13 June 2008 Mathematical Problems in Industry June 2008 Previous Analysis (Hinch & Acrivos papers) [11] Hinch, E.J., and Acrivos, A., “Long slender drops in a simple shear flow”, Journal of Fluid Mechanics Vol. 98 Issue 2 (1980) pp. 305-328. Some differences between ‘simple shear flow’ and ‘simple extensional flow’

14. 14 June 2008 Mathematical Problems in Industry June 2008 Previous Analysis (Howell & Siegel papers) [15] Howell, P.D., and Siegel, M., “The evolution of a slender non-axisymmetric drop in an extensional flow”, Journal of Fluid Mechanics Vol. 521 (2004) pp. 155-180.. Both Howell-Siegel and Hinch-Acrivos predict the pointed ends to the elongating bubbles which is seen in practice.

15. 15 June 2008 Mathematical Problems in Industry June 2008 Generic Draw Model –use Huang/Miura/Wylie where possible. [9] Huang, H., Miura, R.M., and Wylie, J.J., “Optical Fiber Drawing and Dopant Transport”, submitted to SIAM J. Applied Math 2008 www.math.yorku.ca/Who/Faculty/hhuang/preprints/DopantFinal.pdf Where log R vs. z is convex, heat is going into blank. Where log R vs. z is concave, heat is coming out. Because of cylindrical geometry 2D plot does not show cross-sectional area. dT/dz=0 determines ‘inflection’ log R

16. 16 June 2008 Mathematical Problems in Industry June 2008 Velocity U(z) and extension rate dU/dz z (du/dz) log u(z) z We can approximate u(z) by something convenient: exponential heating, exponential cooling.

17. 17 June 2008 Mathematical Problems in Industry June 2008 Input to problem Need assumed V(z) and dV/dz. Blank shape D(z), velocity V(z), and temperature T(z) all need to be consistent. JSA to provide base model.

18. 18 June 2008 Mathematical Problems in Industry June 2008 Outline Introduction to Corning! Brief Problem Statement Background, Introduction, and Context –Pictures –References & Previous work Restatement of Problem Extensions, further work which might be possible

19. 19 June 2008 Mathematical Problems in Industry June 2008 Problem Statement How is a single spherical bubble or ‘seed’ stretched as it moves down the tapering root during a high speed optical fiber draw process?

20. 20 June 2008 Mathematical Problems in Industry June 2008 Outline Introduction to Corning! Brief Problem Statement Background, Introduction, and Context –Pictures –References & Previous work Restatement of Problem Extensions, further work which might be possible

21. 21 June 2008 Mathematical Problems in Industry June 2008 Thank you!