• Geometry
  • Soap Bubbles
  • Medical Imaging
  • Robotics
  • Fluids
  • Semiconductors
  • Wave Propagation
  • Image Denoising
  • Optimal Design
  • Seismic Analysis
  • Tumor Modeling
  • Optimal Control
  • InkJet Plotters
  • Traveling Salesmen
  • ViscoElastic Flow
  • Pinching Droplets
  • Chemical Pathways







    J.A. Sethian
  • Ink Jet Plotters

    The goal of this work is to develop a numerical simulation tool for fluid flow phenomena associated with ink jet printers. The physical goal is to analyze the motion of the boundary, pinch off of droplets, formation of satellites, and the effect of nozzle geometry on ink ejection size and motion. These extra satellites break off because of the role of surface tension along the air/ink boundary: the ejected bubble elongates and then pinches off.

    In order to accurately simulate this process, the underlying algorithms should be able to faithfully discretize non-rectangular geometries, accurately capture two-phase flows through an axisymmetric nozzle, handle complicated topological change of ink droplets, conserve mass to a good approximation, and couple to external models which simulate the ink cartridge, supply channel, pressure chamber, and piezoelectric actuator.


    You might ask, "why bother" with inkjets?"---they cost about $49.00, laser writers are of higher quality, and companies make money off the ink itself (it costs about $750,000 to fill your car's gas tank with ink).

    The answer is that inkjet plotters are used in a wide spectrum of applications beyond the printing of documents. Vast arrays of inkjet plotters working together are used in the manufacture of LCD/plasma displays, fine-scale printing of circuit boards, and pharmaceutical testing.

    In one type of printing process, called "drop-on-demand", ink drops are expelled one by one against a background.
    Single Drop per instant
    Surface tension holds droplets
    Limited throw distance

    The inkjet head itself is axisymmetric. Ink is stored in a bath reservoir (cartridge), and driven through the nozzle in response to a dynamic pressure at the lower boundary. The dynamics of incompressible flow through the nozzle, coupled to surface tension effects along the ink-air interface and boundary conditions along the wall, act to determine the shape of the interface as it moves. A negative pressure at the lower boundary induces a backflow which causes a bubble to pinch off.
    Our model assumes the axi-symmetric Navier-Stokes equations for two-phase immiscible incompressible flow with surface tension and density jumps across an infinitely thin immiscible interface separating ink and air, each with constant viscosity and density. A contact model is devised to capture air-ink-wall dynamics. We use a combination of level set methods to characterize air/ink boundary, projection methods to help track the motion of the fluid boundaries and model the surface tension by evaluating the axisymmetric curvature, which acts a great deal like curvature flow.

    Below, show an experimental photograph on the left and the numerical simulation at the same time on the right. You can see the first bubble being ejected from the chamber, followed by the pinch-off and separation into a main part and a residual satellite bubble.
    Movie of the above process:

    Details The calculation was made using a coupled level set-projection method on quadrilateral grids is developed for piezoelectric ink jet simulations. The model is based on the Navier-Stokes equations for incompressible two-phase flows in the presence of surface tension and density jump across the interface separating ink and air, coupled to an electric circuit model which describes the driving mechanism behind the process, and a macroscopic contact model which describes the air-ink-wall dynamics. We simulate the axisymmetric flow on quadrilateral grids using a combination of second-order finite difference projection methods to solve the fluid equations and level set methods to track the air/ink interface. To improve the mass conservation performance of the coupled level set method, a bicubic interpolation is combined with the fast marching method for level set re-initialization on quadrilateral grids. to track the motion of the fluid interface.

    • A Coupled Level Set Projection Method Applied to Ink Jet Simulation, ,
            Yu, J-D., Sakai, S., and Sethian, J.A., Interfaces and Free Boundaries, 193, No. 1, pp 275-305, 2003,
      This paper List of downloadable publications

    • A Coupled Quadrilateral Grid Level Set Projection Method Applied to Ink Jet Simulation, ,
            Yu, J.D., Sakai, S., and Sethian, J.A., J. Computational Physics, 206, 1, pp. 227-251, 2005.
      This paper List of downloadable publications