HOME
OVERVIEW APPLICATIONS INTERACTIVE APPLETS HISTORY OF THE METHODS/FLOW CHART PUBLICATIONS EDUCATIONAL MATERIAL ACKNOWLEDGEMENTS ABOUT THE AUTHOR/CV Copyright: 19962010 J.A. Sethian 
ViscoElastic Flow"An inexpensive, nontoxic sample of a nonNewtonian fluid can be made quite easily by adding corn starch to a cup of water. Add the starch in small portions and stir in slowly. When the suspension nears the critical concentration  becoming like single cream in consistency  the so called "shear thickening" property of this nonNewtonian fluid becomes apparent. The application of force  for example by stabbing the surface with a finger, or rapidly inverting the container holding it  leads to the fluid behaving like a solid rather than a liquid. More gentle treatment, such as slowly inserting a spoon, will leave it in its liquid state.Trying to jerk the spoon back out again, however, will trigger the return on the temporary solid state." Applications involving viscoelastic fluid jets are quite broad, and include such areas as microdispensing of bioactive fluids through high throughput injection devices, creation of cell attachment sites, scaffolds for tissue engineering, coatings and drug delivery systems for controlled drug release, and viscoelastic blood flow flow past valves. Building viscoelastic flows using level set methods to track the interface is quite a challenge. Tracking the interface is not too difficult, and solving the fluid equations for two fluids can be done using an embedded level set method coupled to a projection scheme for incompressible flow. The challenge comes in dealing with something called the "high Weissenberg" problem: when the effects of viscoelastic flow become large, numerical schemes tend to go unstable and have trouble converging. Below, we show the results of a coupled level set/projection method scheme to compute viscoelastic flows in the context of ink jet plotters. Regular dyebased inks used in desktop printers are Newtonian, which means the relation between the stress tensor and the rate of deformation tensor at an instant is linear and not related to any other instant. The use of pigmentbased inks at the end of the 1990's improved the color durability of a ink jet printout, however, pigmentbased inks and inks used in industrial printing applications are usually viscoelastic. And there starts the problem. In the figures below, we imagine two fluids, similar to our ink jet simulation. The "ink" is taken as viscoelastic, while the air is a pure Newtonian fluid. As you see in the movies below, as we turn up the "viscoelastic" part of the equations as the figures move from right to left, the ejected bubble has a harder and harder time escaping, until, ultimately, the viscoelastic effects cause the jet to tract to its base.
Details We developed a coupled finite difference algorithm on rectangular grids 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. References

