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OVERVIEW APPLICATIONS INTERACTIVE APPLETS HISTORY OF THE METHODS/FLOW CHART PUBLICATIONS EDUCATIONAL MATERIAL ACKNOWLEDGEMENTS ABOUT THE AUTHOR/CV Copyright: 1996, 1999, 2006 J.A. Sethian 
Applications to Fluid Mechanics


Movie of Rising Thermal  Ink Jet Plotter 
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A special page on inkjet plotters 
A special page on ViscoElastic Flow 
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AbstractIn this paper, we merge modern techniques for computing the solution to the viscous NavierStokes equations with modern techniques for computing the motion of interfaces propagating with curvaturedependent speeds. The resulting algorithm tracks the motion of an evolving interface in a complex flow field, and easily handles complex changes in the front, including the development of spikes and cusps, topological changes and breaking/merging. As examples, we apply the resulting algorithm to interface boundaries in a driven cavity and in a shear layer, and cold flame propagation in a hydrodynamic field.
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AbstractLevel set techniques are numerical techniques for tracking the evolution of interfaces. They rely on two central embeddings; first the embedding of the interface as the zero level set of a higher dimensional function, and second, the embedding (or extension) of the interface's velocity to this higher dimensional level set function. This paper applies Sethian's Fast Marching Method, which is a very fast technique for solving the Eikonal and related equations, to the problem of building fast and appropriate extension velocities for the neighboring level sets. Our choice and construction of extension velocities serves several purposes. First, it provides a way of building velocities for neighboring level sets in the cases where the velocity is defined only on the front itself. Second, it provides a subgrid resolution in some cases not present in the standard level set approach. Third, it provides a way to update an interface according to a given velocity field prescribed on the front in such a way that the signed distance function is maintained, and the front is never reinitialized; this is valuable in many complex simulations. In this paper, we describe the details of such implementations, together with speed and convergence tests, and applications to problems in visibility relevant to semiconductor manufacturing and thin film physics.
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AbstractA large collection of fluid problems involve moving interfaces. Applications include airwater dynamics, breaking surface waves, solidification melt dynamics, and combustion and reacting flows. In many such applications, the interplay between the interface dynamics and the surrounding fluid motion is subtle, with factors such as density ratios and temperature jumps across the interface, surface tension effects, topological connectivity and boundary conditions playing significant roles in the dynamics.
Over the past fifteen years, a class of numerical techniques known as level set methods have been built to tackle some of the most complex problems in fluid interface motion. Level set methods, introduced by Osher and Sethian, are computational techniques for tracking moving interfaces; they rely on an implicit representation of the interface whose equation of motion is numerically approximated using schemes built from those for hyperbolic conservation laws. The resulting techniques are able to handle problems in which the speed of the evolving interface may sensitively depend on local properties such as curvature and normal direction, as well as complex physics off the front and internal jump and boundary conditions determined by the interface location. Level set methods are particularly designed for problems in multiple space dimensions in which the topology of the evolving interface changes during the course of events, and problems in which sharp corners and cusps are present.
In this review, we discuss the numerical development of these techniques and their application to a collection of problems in fluid mechanics, including incompressible and compressible flow, and applications to bubble dynamics, ship hydrodynamics, and inkjet printhead design. We note that there already exists a collection of review articles and books on these techniques, and refer the interested reader to works by Osher and Fedkiw (2001) and Sethian (1996b 1996c, 1999a, 1999b, 2001).Download publications
AbstractA finite difference level setprojection method on rectangular grid is developed for piezoelectric ink jet simulation. The model is based on the NavierStokes equations for incompressible twophase 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 airinkwall dynamics. We simulate the axisymmetric flow using a combination of second order projection methods to solve the fluid equations and level set methods to track the air/ink interface. The numerical method can be used to analyze the motion of the interface, breakoff and formation of satellites, and effect of nozzle geometry on droplet size and motion. We focus on close comparison of our numerical ink jet simulation with experimental data.
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AbstractA coupled level setprojection method on quadrilateral grids is developed for piezoelectric ink jet simulations. The model is based on the NavierStokes equations for incompressible twophase 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 airinkwall dynamics. We simulate the axisymmetric flow on quadrilateral grids using a combination of secondorder 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 reinitialization on quadrilateral grids. The numerical method is used to analyze the motion of the interface, droplet pinch off, formation of satellites, effect of nozzle geometry on droplet size and motion, and the dynamics for droplet landing. The simulations are faithful to the dimensions and physics of a particular class of inkjet devices.
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AbstractA coupled finite difference algorithm on rectangular grids is developed for viscoelastic ink ejection simulations. The ink is modeled by the OldroydB viscoelastic fluid model. The coupled algorithm seamlessly incorporates several things: (1) a coupled level setprojection method for incompressible immiscible twophase fluid flows; (2) a higherorder Godunov type algorithm for the convection terms in the momentum and level set equations; (3) a simple firstorder upwind algorithm for the convection term in the viscoelastic stress equations; (4) central difference approximations for viscosity, surface tension, and upperconvected derivative terms; and (5) an equivalent circuit model to calculate the inflow pressure (or flow rate) from dynamic voltage.
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