Evolution of an Elliptical Bubble in an Accelerating Extensional Flow Field

John Abbott, Corning, Inc.

Abstract
We are investigating, primarily for fundamental understanding, how a single bubble evolves in the dramatic extensional flow seen in the exponential taper of a high speed optical fiber draw process. There is an exponentially growing difference in velocity between the head and tail of the elongating bubble. Is the shape the same as a bubble distorted by simple shear in a steady low Re-flow?

Introduction & Background
In volcanic magma, gas bubbles can become distorted by the high viscosity/high shear flow [1] (see images). Theoretical work has been done to predict the effect of spherical bubbles on the shear viscosity of a liquid [2], and hence on of a shear flow [3]. However, the studies of the effect of large extensional flows on bubbles and the effect of long narrow bubbles have primarily been experimental [4] with at least some of the published data indicating a much larger effect on viscosity when the bubbles are long and near-cylindrical.

Bubbles occur and must be controlled in the melting of glass [5], for example in the production of thin glass sheets for LCD displays [6] or in the manufacture of blanks for the drawing of optical fibers [7].

We are interested in how a single bubble evolves during the drawing of optical fiber [8] [9]. In the optical fiber drawing process a blank is heated up to a melting temperature where the glass will flow either under its own weight or with a modest pulling force. As the root necks down, the problem involves radiative and convective cooling [9], [10].

Problem Statement
We are focusing on the evolution of a gas bubble in the exponential taper of the draw root, where the material is undergoing dramatic extensional flow while cooling. Does the analysis for the effect of shear flows on bubbles [3] apply with little change to the exponential extensional flow? What effect if any does the compressibility of the gas have?

Relevant work on Bubble Distortion
[1] Rust and Manga (2002) discuss measurements and theory for gas bubbles distorted in a viscous Newtonian fluid under a constant simple shear flow. Their references may be useful. They found good agreement with Hinch & Acrivos [11].

[11] Hinch and Acrivos (1980) study the deformation of long slender drops in a simple shear flow, where the drop has a lower viscosity than the shearing fluid. In section 4.5 they discuss the difference between a steady shear rate and the effect of increasing the shear rate, which is more analogous to the optical fiber draw problem. Note the draw problem is a continuous increase in shear rate. In addition some breakdown in their approach was noted. There are two other papers by Hinch [12], [13] on this topic.

[14] Wilmott (1989) looked at the stretching of a thin inclusion which is more viscous than the surrounding fluid. This analysis and that of Huang et al. [9] relate to the overall drawing of the fiber, rather than the bubble. [15] Howell and Siegel (2004) study the evolution of a (nonaxisymmetric!) bubble in a steady extensional flow, including the steady-state case of Hinch & Acrivos. The approach and references may be useful.

Extensions & Follow-up Problems
Should we include the compressibility of the gas—does a gas bubble stretch differently than an inviscid, incompressible bubble? Do surface tension, gas diffusion into the liquid, expansion/contraction of the gas due to thermal affects have any significant effect? Do we need to include the thermal history since the fiber ultimately finishes drawing at a constant speed?

As a follow-up problem, can the variation of viscosity with percent air fill seen experimentally by Bagassarov [3] and mentioned by Llewellin ([5], equations 3-10) for an extensional flow be given a theoretical basis? References

[1] Rust, A.C. & Manga, M., Bubble Shapes and Orientations in Low Re Simple Shear Flow, J. Colloid Interface Science, 249 (2002) 476-480.

[2] Llewellin, E.W., Mader, H.M. & Wilson, S.D.R., The rheology of a bubbly liquid, Proc. Roy Soc. Lond. A 458 (2002) 987-1016.

[3] Llewellin, E.W. & Manga, M., Bubble suspension rheology and implications for conduit flow, J. Volcanology and Geothermal Research, 143 (2005) 205-217.

[4] Bagdassarov, N. Sh. & Dingwell, D.B., A rheological investigation of vesicular rhyolite, J. Volcanology and Geothermal Research, 50 (1992) 307-322.

[5] Chang, S.L., Zhou, C.Q. & Golchert, B., Eulerian approach for multiphase flow simulation in a glass melter, Applied Thermal Engineering, 25 (2005) 3083-3103.

[6] Corning says new glass to cut LCD costs, 3/21/2006

[7] Prado, M.O. & Zanotto, Glass sintering with concurrent crystallization, Comptes Rendus Chimie, 5 Issue 11 (2002) 773-786.

[8] Wylie, J.J. & H. Huang, Extensional flows with viscous heating, Journal of Fluid Mechanics, 571 (2007) 359-370.

[9] Huang, H., Miura, R.M. & Wylie, J.J., Optical Fiber Drawing and Dopant Transport, submitted to SIAM J. Applied Math, 2008.

[10] Acquah, C., et al., Optimization of an Optical Fiber Drawing Process under Uncertainty, Ind. Eng. Chem. Res. 45 (2006) 8476-8483.

[11] Hinch, E.J. & Acrivos, A., Long slender drops in a simple shear flow, Journal of Fluid Mechanics 98 Issue 2 (1980) 305-328.

[12] Hinch, E.J., The evolution of slender inviscid drops in an axisymmetric straining flow, Journal of Fluid Mechanics 101 Issue 3 (1980) 545-553.

[13] Hinch, E.J. & Acrivos, A., Steady long slender droplets in two-dimensional straining motion, Journal of Fluid Mechanics 91 Issue 3 (1979) 401-414.

[14] Wilmott, P., The stretching of a thin viscous inclusion and the drawing of glass sheets, Phys. Fluids A 1 No. 7 (July 1989) 1098-1103.

[15] Howell, P.D., & Siegel, M., The evolution of a slender non-axisymmetric drop in an extensional flow, Journal of Fluid Mechanics 521 (2004) 155-180.