Two Mathematical Problems Related to Drawing of Glass Sheets

 

John Abbott

Corning Incorporated

 

[A] Dripping of Viscous Fluids

 

One way to make large glass sheets is with the overflow fusion process [1].  “Overflow” of viscous fluids and “dripping” [2] are general phenomena. Recent work by Lin and Kondic [3] covers the stability of thin films flowing down inverted substrates, with a particular assumption of the flow at the leading edge.  In other cases of physical interest [4], the flow at the leading edge is different with a “tank tread type motion” and particles on the free surface end up on the wall.  Lava can flow in a similar way [5]

 

The proposed problems:

 

(1) What are the non-dimensional parameters that predict leading edge ‘dripping’ of a viscous fluid on an inverted substrate with a “tank-tread” type leading edge flow?

 

(2) Can we analyze the problem if the leading edge is not straight (i.e. slightly wavy)?

 

 

[B] Deformation of Thin Viscous Sheets

 

Howell has presented [6], [7] a rigorous derivation of the equations governing the stretching of thin viscous sheets. Further work in this area has been done by Filippov & Zheng [8] in an application to the drawing of thin glass sheets for TFT LCD displays.

Sheets as large as 2880x3130x0.70mm [9] are formed in manufacturing.


The proposed problems:

 

(1) When can bending stresses be neglected?  Edge effects: How close to the edge can bending stresses be neglected?  How important are these relative to capillary forces near the edge?

 

(2) If there is a viscosity gradient through the thickness (for example a small thermal gradient) how should that affect the evolution of the shape and Howell’s conjecture [6] about isometric shapes?

 

 

REFERENCES

 

Problem [A]:

 

[1] Lin, H-J., and Chang, W-K., “Design of a sheet forming apparatus for overflow fusion process by numerical simulation”, J. Non-Crystalline Sol. (2007) Vol. 353 pp.2817-2825.

[2] Wilson, S.D.R, “The slow dripping of a viscous fluid”, J. Fluid Mech. (1988) Vol. 190 pp.561-570. 

[3] Lin, T., and Kondic, L, “Thin films flowing down inverted substrates”, submitted to Phys. Fluids, pre-print available at

http://arxiv.org/PS_cache/arxiv/pdf/0912/0912.3306v2.pdf

References at

http://eprintweb.org/s/article/physics/0912.3306/refs

[4] Radler, J., Strey, H., and Sackmann, E., “Phenomology and Kinetics of Lipid Bilayer Spreading on Hydrophilic Surfaces”, Langmuir (1995) Vol. 11. pp.4539-4548.

http://pubs.acs.org/doi/abs/10.1021/la00011a058

[5] Osiptsov, A.A., “Three-Dimensional Isothermal Lava Flows over a Non-Axisymmetric Conical Surface”, Fluid Dynamics (2006) Vol. 41 No. 2

http://www.springerlink.com/content/v001n3935252n61j/

 

Problem [B]:

 

[6] Howell, P.D., “Models for thin viscous sheets”, Eur. J. Appl. Math. (1996) Vol. 7 pp.321-343.

Also (2008)  http://eprints.maths.ox.ac.uk/106/1/thin_viscous_sheets.pdf

[7] Howell, P.D., “Extensional thin layer flows”, Ph.D. thesis, Univ. Oxford 1994.

[8] Filippov, A., and Zheng, Z., “Dynamics and shape instability of thin viscous sheets”, Phys. Fluids (2010) Vol. 22, p. 023601

[9] “Corning Gives Tour of New LCD Glass Substrate Facility”, Ceramic Industry 3/18/2010

http://www.ceramicindustry.com/Articles/Todays_Headlines/BNP_GUID_9-5-2006_A_10000000000000781914