Papageorgiou This project will study several nonlinear problems involving free surfaces separating fluids with different properties and where surface tension is important. We will study flows modelled by nonlinear evolution equations which encounter finite-time singularities as interfaces touch and a change in topology takes place. An example is the breakup of a liquid jet into droplets. A paradigm problem of physical interest is the compound jet comprised of a core fluid surrounded by a second immiscible annular fluid. Breakup under the capillary instability usually produces compound spherical particles and analysis of the local singular phenomenon is of value in both designing experiments and incorporation into and testing of hybrid large scale simulations. In addition we will study the motion and management of bubbles in surfactant solutions (fluid phases with impurities). Of particular theoretical importance is the prediction of parameter ranges where the bubble is remobilized after its bouyant or thermocapillary motion is arrested or retarded by surfactant build-up at the back end of the spherical bubble. In many cases system parameters are large and asymptotic solutions of the singular perturbation type will be constructed and analyzed. We study two related classes of fundamental problems encountered in several key technological and manufacturing situations. A common feature is the presence of interfaces separating fluids. These have important effects on how the fluids move and the mathematical and computational analysis of such models is extremely useful in diverse manufacturing processes as well as on-going experiments in microgravity environments (for instance parabolic flights as well as in the space shuttle). Using tools of mathematical analysis and modern computational methods will enable us to identify important physical regimes which would perhaps be prohibitively expensive in the laboratory or in space. In addition, the use of mathematical analy sis is imperative in extreme situations (which are more the rule rather than the exception in many processes - for instance the breakup of a liquid jet into droplets). Besides their intrinsic usefulness, such mathematical solutions can be used to make large scale computations more accurate and efficient as well as test the accuracy of existing codes. Liquid jet breakup has many modern applications in addition to the more traditional ones of printing and fuel injection systems. Compound jets are useful in the manufacture of compound particles used for slow drug release, the manufacture of fibres used to reinforce mechanical or aerodynamic components, and color printing with environmentally friendly inks and dyes. Our studies of bubble dynamics should lead to identification of efficient operational regimes in large scale purification systems (contaminated liquids are often purified by passing large numbers of gas bubbles through them), provide theoretical guidelines for the design of safe agricultural sprays, and produce theories that can be compared to experiments done in space aiming at the production of sophisticated materials in containerless microgravity environments.
