1. Coarsening/Scavenging of Drops Numerical simulations Stability Analytic predictions Coarsening of Jet and other PDEs Dynamics and Stability of a Network of Coupled Drop Elements: Coarsening by Capillarity Henrik van Lengerich, Mike Vogel, Paul Steen Rutland and Jameson, JFM 46 267 (1971)

2. Probability of Winning

3. Formulation

4. Stability 1.) For the ‘star’ network with Hagen-Poiseuille flow: 2.) The eigenvalues of the Hessian: Number of sub-hemispherical drops = stable directions Number of super-hemispherical drops less one = unstable directions 3.) For an arbitrary conduit network with and arbitrary flow rate: 4.) As the conduit network and flow type is varied the signs of the eigenvalues stay unchanged.

5. Structure of the dynamical system The stability result along with a Lyaponov function can be used to organize the structure of the phase space. N=2N=3 N=4

6. Prediction of Winning

7. For a linear array of drops with Hagen-Poiseuille flow This is just a discretization of the PDE A similar process of discretization can be used for any arbitrary PDE Generalization/Future Work The stability of a stationary state is determined by the stability of a Jacobian The dynamics are predicted using domains of attraction and fixed points Rutland and Jameson, JFM 46 267 (1971)

8. The end

9. Probability of Winning

10. Prediction of Winning From center manifold theory J. Carr, “Applications of Centre Manifold Theory” (1984)