By Nick Ryan
Glass is endlessly recyclable with no loss of quality in the raw material and has a wide range of uses. Cutting-edge technologies using thin glass sheets, such as folding phone screens and fingerprint sensors, rely on high-quality manufacturing. However, despite the long history of glass manufacture, ripples (imperfections in the glass) still form in the molten glass during production, compromising quality and adding cost.
One method to make thin glass sheets that suffers from ripples is the redraw process, where prefabricated glass blocks are drawn through a furnace and stretched under tension to the desired thickness. To unpick whether it is the applied tension that is causing such buckling, Oxford mathematicians Nick Ryan, Chris Breward, Ian Griffiths, and Peter Howell have been considering the simpler case where no external tension is applied to the sheet.
The setup considered is as follows: a thin viscous blob of fluid, floating freely in the air (for we have neglected gravity), is retracting due to surface tension, with no inertia or other forces present. An example of the geometry is shown in Figure 1. Surface tension has a greater effect in regions of higher curvature. The region of highest curvature is at the lateral edge of the fluid, and we account for this curvature by imposing an effective boundary condition there. This boundary condition is found by solving a simpler, two-dimensional, fluid dynamics problem, and we find that this highly curved region can cause the fluid to be under a small compression.
To figure out whether this compression is enough to cause the sheet to buckle, we solve the governing fluid dynamics equations in two ways. The first approach is to solve the full Stokes equations that govern the flow, but this is computationally expensive and gives little insight into the solution. The second approach is to use asymptotic analysis to exploit the thinness of the sheet. This leads to equations that are much simpler and faster to solve, while also giving information about the behaviour of the fluid. For instance, we can isolate the role that nonlinearities in the geometry of the sheet play in limiting buckling. We find good agreement between the two methods, and we conclude that the glass sheet spontaneously buckles, as seen in Figure 2. In fact, in the setup considered, it is almost inevitable that the sheet will buckle!
We might try to control this buckling by introducing small perturbations in the thickness of the sheet that alter which regions of the disc are under compression. However, we find that although the thickness perturbations can alter the way the disc buckles (whether it forms a bowl or a Pringle shape), the buckling remains inevitable [1].
Despite it initially seeming unlikely that a slow, inertia-free fluid retracting under surface tension with no external forcing could buckle, we find that the highly curved edge of the sheet can induce compression in all the sheet, meaning that buckling becomes inevitable. Despite our setup being much simpler than that of glass redraw, we highlight the importance of the boundary effects applied, and that small levels of compression can have a large effect on whether ripples form in a glass sheet.