By Michael Pegg
The problem I (Mike) worked on involved investigating the creation and growth of bubbles inside an ink printer cartridge. The problem was brought by Dr William Lee from the University of Portsmouth who also acted as our group leader. Our international group had members from as far afield as Pakistan and academic backgrounds ranging from fluid mechanics to differential geometry.
We were presented with a new kind of ink cartridge which uses high frequency vibrations to create small ink droplets. The cartridge consists of 3 components. An ink reservoir with a depth of 1 centimetres. A plate at the bottom of the reservoir vibrating ultrasonically at 10,000 hertz. A conical nozzle embedded in the centre of the vibrating plate with a maximum diameter of 50 micrometers, aminimum diameter of 5 micrometers and a thickness of 70 micrometers. The ink cartridge operates normally for some time, after which it fails. When the cartridges are cut open bubbles of around 0.2 millimetres are found inside the reservoir. We were tasked with investigating how the bubbles form with the overall goal to prevent bubble formation.
Our group leader offered three potential bubble formation mechanisms. Free surface recoil where asan ink droplet is shed the thread of liquid joining it to the bulk of ink recoils rapidly, like a snapped elastic band, which pulls bubbles inside the nozzle. Bubbles shedding off a gas pocket which forms inside the nozzle. Air-ink interface migrating to the reservoir side of the nozzle which sheds large bubbles. We added an additional potential mechanism in cavitation, where a vapour cavity is formed along the plate because of the vibrations.
By the end of the week we had investigated a mechanism by which small bubbles, such as those formed by recoil, can grow and move up the pressure gradient. We compared the pressure due to plate oscillation relative to hydrostatic and atmospheric pressure and found it to be dominant. From this we focused on the effects of oscillations and looked into rectified diffusion, a mechanism by which bubbles in an oscillating pressure field can grow beyond their equilibrium radius. We also looked into the Bjerknes force, which is a lift force generated because of an oscillating bubble. The oscillations of a bubble were modelled numerically using the Rayleigh-Plesset equation. Finally we suggested a few novel ideas which could alleviate the bubble problem, such as adding a small amount of porous media on the walls of the cartridge to allow air to drain without letting ink through.
The workshop provided insight into the world of industrial mathematics and a rare opportunity to work with peers on a joint problem. It was an excellent week which has pushed me towards searching for a job in mathematical modelling and provided valuable experience. I would like to thank Dawn Wasley and everyone else involved in the organisation of this event.
Michael Pegg is a Research Student and associate tutor in the School of Mathematics at the University of East Anglia.