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]. ÒOverflowÓ of viscous fluids and ÒdrippingÓ [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 Òtank tread type motionÓ 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 ÔdrippingÕ of a viscous fluid on an inverted substrate with a Òtank-treadÓ 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 HowellÕs conjecture [6] about isometric shapes?

__REFERENCES__

Problem [A]:

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

[2] Wilson, S.D.R, ÒThe slow dripping of a viscous fluidÓ, ** J.
Fluid Mech.** (1988) Vol. 190 pp.561-570.

[3] Lin, T., and Kondic, L, ÒThin films flowing down inverted substratesÓ, 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., ÒPhenomology
and Kinetics of Lipid Bilayer Spreading on Hydrophilic SurfacesÓ,**
Langmuir** (1995) Vol. 11. pp.4539-4548.

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

[5] Osiptsov, A.A., ÒThree-Dimensional Isothermal Lava Flows
over a Non-Axisymmetric Conical SurfaceÓ, ** Fluid Dynamics** (2006) Vol. 41 No. 2

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

Problem [B]:

[6] Howell, P.D., ÒModels for thin viscous sheetsÓ, ** 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., ÒExtensional thin layer flowsÓ, ** Ph.D.
thesis**, Univ. Oxford 1994.

[8] Filippov, A., and Zheng, Z., ÒDynamics and shape
instability of thin viscous sheetsÓ, ** Phys. Fluids** (2010) Vol. 22, p.
023601

[9] ÒCorning Gives Tour of New LCD Glass Substrate
FacilityÓ, ** Ceramic Industry** 3/18/2010

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