John Abbott

Mathematics in Industry Workshop 06/16/2008

MTE-AMA

Elongating Bubble in an Accelerating Extensional Flow

2June 2008Mathematical 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

3June 2008Mathematical Problems in Industry June 2008

Corning Incorporated

• R&D labs celebrating 100th 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)

4June 2008Mathematical 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..

5June 2008Mathematical 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

6June 2008Mathematical 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

7June 2008Mathematical Problems in Industry June 2008

Optical Fiber

8June 2008Mathematical 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

9June 2008Mathematical 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.

10June 2008Mathematical 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

11June 2008Mathematical Problems in Industry June 2008

“Blisters” in volcanic magma

Rust, Manga, Cashman 2003

12June 2008Mathematical Problems in Industry June 2008

Experimental measurement of bubble shapes

Rust, Manga 2002

An optical fiber airline has a large aspect ratio l/a

13June 2008Mathematical 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’

14June 2008Mathematical 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.

15June 2008Mathematical 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

16June 2008Mathematical 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.

17June 2008Mathematical 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.

18June 2008Mathematical 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

19June 2008Mathematical 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?

20June 2008Mathematical 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

21June 2008Mathematical Problems in Industry June 2008

Thank you!