The project encompasses the application of mathematics to four important physical problems: (1) the development of optimal protocols for the removal of glycerol from frozen blood; (2) the development of optimal strategies for bending and stretching one shape into another (for example, bending metal plates into curved structural surfaces or morphing in digital imaging); (3) the stability of water wave patterns for flow over plates; and (4) the modern physics two-body problem (that is, the theory of motion for two bodies whose motions produce waves, which travel with finite speeds, that influence the motion of the other body in the system). Glycerol must be added to whole blood to prevent cell damage when it is frozen and must be removed before the blood is transfused. Current removal strategies require procedures that are too lengthy for emergencies or military applications. Using mathematical modeling (which includes models that couple fluid flow, convection diffusion, and transport across semipermeable membranes), optimization theory and numerical methods, the project will scale up laboratory tested removal protocols for medical use. The research on optimal morphing uses differential geometry and elasticity theory to obtain cost functionals for minimal strain morphing. Minima over admissible sets of deformations (diffeomorphisms between compact hypersurfaces) will be proved to exist using the calculus of variations, and numerical methods will be developed to approximate these minima. The existence and stability of surface waves produced by flow over flat plates will be proved using fluid dynamical modeling and the theory of free boundary problems. In addition, the mathematical analysis will be complemented with numerical simulations of a physical system. Mathematical models of two-body interaction, where forces propagate with finite speeds, will be derived. Such models consist of systems of hyperbolic partial differential equations coupled with ordinary differential equations, or hybrid systems of differential equations and delay equations. The well-posedness and qualitative dynamics of the two types of models will be determined and models of this type will be applied to understand synchronization phenomena for acoustical surface wave sensors.
The research project has two main purposes: the application of mathematics to advance understanding in four fields of physical science and the training of four Ph.D. students in these important areas of applied mathematics. The first project has direct applications in the health sciences and medicine. Living cells, for example blood cells, can be preserved for long periods of time by freezing; and, they can later be thawed for use in transfusions. To freeze and thaw living cells requires a complex technology including the addition of chemicals called cryoprotectants to the insides of the cells so they will not be damaged during this process. These chemicals must be removed before blood is transfused. Methods are available for doing this, but current methods require a long time to implement. Thus, frozen blood is not available for emergency use (for example, in military conflicts or natural disasters). Fast removal of cryoprotectants has been proved to be possible at the level of single cells; the purpose of the research in to design (optimal) methods that can be used for fast removal on the larger scale necessary for practical application in medicine. The second project has applications in materials science. A sheet of metal is to be bent and stretched into a desired shape. The desired shape can be achieved by many different deformation processes (for example, the sheet can first be stretched then bent or it can be stretched in one direction while being bent in another, among many other possibilities). Which method uses the least energy? This problem and others of the same type will be solved using mathematical analysis. The third project will determine the stability of the surface waves often formed by a fluid (for example water) moving over a plate. Surface waves are known to exist and have a particular shape. Also, experiments show that when the waves are disturbed, they return rapidly to that particular shape. But, the mechanism for this remarkable stability property is not known at present. The research project will determine the reason for the observed stability. The fourth research project is concerned with the theory of two-body interactions in physics and engineering. A practical example occurs in applied acoustics. Electrical currents applied to electrodes embedded in the surface of a silicon wafer causes acoustic surface waves to propagate across the wafer. A second set of electrodes (transducers), embedded in the wafer, can transform these waves back to electrical currents. When the electrodes and transducers are connected to an appropriate electrical circuit, such surface waves can be constantly produced and amplified by the circuit. If a second set of electrodes and transducers are embedded in the surface on the same wafer, a second set of waves can be produced. The two circuits can be tuned so that the waves oscillate in perfect synchrony. A very sensitive detector can be constructed by covering half of the wafer (with one pair of electrodes and transducers) and exposing the other half to the environment. If the surface of the second half is disturbed (for example, by the deposition of a hazardous biological agent) the synchrony is broken and this occurrence can be instantly detected by a computer connected to the circuits. A goal of this research is to provide the mathematical theory for understanding the behavior of these devices.
