# Mathematics Colloquia and Seminars

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### Mathematical Modeling and Numerical Simulation of Microstructured Materials

**Colloquium**

Speaker: | John Lowengrub, University of Minnesota |

Location: | 693 Kerr |

Start time: | Wed, Apr 17 2002, 4:10PM |

Microstructured materials, such as emulsions and polymer blends, crystals and metallic alloys, blood and biological tissues, are fundamental to many industrial and biomedical applications. These diverse materials share the common feature that the microscale and macroscale are linked. The phenomena at microscopic scale, such as the morphological instability of crystalline precipitates and drop deformation, break-up and coalescence determine the microstructure and its time evolution; thus affecting the rheology and mechanical properties of the materials on the macroscale. In this talk, I will focus on mathematical and numerical modeling at the microscale. In particular, I will present a class of physically-based models of complex (multicomponent) fluid flows which incorporate buoyancy, viscosity, compressibility and surface tension at interfaces. The models are capable of describing systems with both miscible and immiscible components. In addition, the models allow topological transitions such as pinchoff and reconnection of interfaces to occur without relying on ad hoc 'cut and connect' or smoothing procedures. Results will be presented for a variety of physically interesting flows. To validate the model and numerical algorithms, we examine the pinchoff of liquid/liquid threads (Rayleigh instability) and compare the numerical results to theory and experiments. We then consider the development of a complex, three dimensional microstructure in which the flow components fully interpenetrate one another to yield a sponge-like microstructure. Such co-continuous microstructures have many important industrial applications. Finally, I will demonstrate how these numerical methods can be made adaptive to more efficiently and accurately simulate flows with wide-ranges of length-scales. Time permitting, I will also demonstrate how the models and numerical techniques can be modified to investigate the behavior of other complex materials and biological systems.

(Coffee & cookies @ 3:45 in 550 Kerr )