Breward, C. and Cumberbatch, E. and Cummings, L. J. and Please, C. and Richardson, G. and Schwendeman, D. W. and Tilley, B. S. and Ahmed, S. and Fosse, E. and Gewecke, N. and Guevara, A. and Xu, K. (2008) Evolution of an elliptical bubble in an accelerating extensional flow. [Study Group Report]

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Abstract
Mathematical models that describe the dynamical behavior of a thin gas bubble embedded in a glass fiber during a fiber drawing process have been discussed and analyzed.
The starting point for the mathematical modeling was the equations presented in [1] for a glass fiber with a hole undergoing extensional flow. These equations were reconsidered here with the additional reduction that the hole, i.e. the gas bubble, was thin as compared to the radius of the fiber and of finite extent. The primary model considered was one in which the mass of the gas inside the bubble was fixed. This fixedmass model involved equations for the axial velocity and fiber radius, and equations for the radius of the bubble and the gas pressure inside the bubble. The model equations assumed that the temperature of the furnace of the drawing tower was known.
The governing equations of the bubble are hyperbolic and predict that the bubble cannot extend beyond the limiting characteristics specified by the ends of the initial bubble shape. An analysis of pinchoff was performed, and it was found that pinchoff can occur, depending on the parameters of the model, due to surface tension when the bubble radius is small.
In order to determine the evolution of a bubble, a numerical method of solution was presented. The method was used to study the evolution of two different initial bubble shapes, one convex and the other nonconvex. Both initial bubble shapes had foreaft symmetry, and it was found that the bubbles stretched and elongated severely during the drawing process. For the convex shape, foreaft symmetry was lost in the middle of the drawing process, but the symmetry was regained by the end of the drawing tower. A small amount of pinchoff was observed at each end for this case, so that the final bubble length was slightly shorter than its theoretical maximum length. For the nonconvex initial shape, pinchoff occurred in the middle of the bubble resulting in two bubbles by the end of the fiber draw.
The two bubbles had different final pressures and did not have foreaft symmetry.
An extension of the fixedmass model was considered in which the gas in the bubble was allowed to diffuse into the surrounding glass. The governing equations for this leakymass model were developed and manipulated into a form suitable for a numerical treatment.
Item Type:  Study Group Report 

Problem Sectors:  Materials 
Study Groups:  US Workshop on Mathematical Problems in Industry > 24th MPI [Worcester 16/6/2008  20/6/2008] 
Company Name:  Corning, Inc. 
ID Code:  273 
Deposited By:  Dr Kamel Bentahar 
Deposited On:  25 Jan 2010 17:06 
Last Modified:  29 May 2015 19:53 
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