The application code which is the focus of the MPF project's research program is a numerical method for modeling the interface between two immiscible fluids in an incompressible, viscous flow in microscale jetting devices. Two illustrations of computations made with this code are shown in Table1 and Table 2. They show that the current generation of the RZ and 3D Multi-PhaseFlow code without geometry converges at the predicted design rate of global second-order accuracy for flows in which there is no jump in density and viscosity across the interface, and first-order accuracy for the more realistic flows in which there is a jump in density and viscosity across the interface. (The case shown in Table 2 and Figure 1.  is the classical Rayleigh capillary instability of a perturbed cylindrically symmetric column of water in air.) It is very important to note that although the design rate of the commercial code that is most commonly used to compute jetting flows in industry (Flow3D) is first-order, its rate of convergence will also decrease (to roughly 0.5) when used to compute flows in which there is a discontinuity in density and viscosity across the interface. This is analogous to the convergence rates of the formally second-order accurate methods designed to model discontinuous solutions of the the compressible Euler equations (i.e., shock waves) such as PPM and PLMDE when compared to the first-order accurate methods such as the first order Godunov method.

Table 1: Convergence rate in the L¹ norm for the Rayleigh capillary instability with a density ratio of _w/_a = 1 and a viscosity ratio of µ_w/µ_a =1. Note that the method is second-order accurate.

Table 2: Convergence rate in the L¹ norm for the classical Rayleigh capillary instability (a slightly perturbed cylindrical stream of water in air) with a density ratio of  _w/_a  = 816 and a viscosity ratio of µ_w/µ_a = 64. Note that as a result of the discontinuity in density and viscosity at the interface, the method is now globally first-order accurate.

The the capillary instability computation represented in Table 2 is shown in a sequence of four pictures in time in the postscript file, Figure 1. The same computation, but made with the adaptive mesh refinement  turned on, is shown in Figure 2. Here the coarse grid is 16x32 and there are two levels of  finer grids each a factor of two finer than the previous coarser grid, so that the effective fine grid is 64x128, which is the same as the single grid computation.

Professor Puckett and his team are currently working to empirically determine the convergence rate of the RZ code with an embedded boundary description of the device geometry. Preliminary results from this work are shown in Figures 3-6. In particular, note that the number of satellite drops and the velocity with which they are traveling appear to converge under grid refinement. We expect to have an accurate quantitative measure of the convergence rate for problems involving Xerox's ink jet devices by early fall 1998.