Georgia Institute of Technology College of Sciences

This is a continuation of the previous talk.

In this work, we numerically studied the effect of the vorticity on the enhancement of heat transfer in a channel flow. Based on the model we proposed, we find that the flow exhibits different properties depending on the value of four dimensionless parameters. In particularly, we can classify the flows into two types, active and passive vibration, based on the sign of the incoming vortices. The temperature profiles according to the flow just described also show different characteristics corresponding to the active and passive vibration cases. In active vibration cases, we find that the heat transfer performance is directly related to the strength of the incoming vortices and the speed of the background flow. In passive vibration cases, the corresponding heat transfer process is complicated and varies dramatically as the flow changes its properties. Compared to the fluid parameters, we also find that the thermal parameters have much less effect on the heat transfer enhancement. Finally, we propose a more realistic optimization problem which is to minimize the maximum temperature of the solids with a given input energy. We find that the best heat transfer performance is obtained in the active vibration case with zero background flow.

We start studying open book foliations in this series of seminars. We will go through the theory and see how it is used in applications to contact topology.

Robertson and Seymour proved that graphs are well-quasi-ordered by the minor relation and the weak immersion relation. In other words, given infinitely many graphs, one graph contains another as a minor (or a weak immersion, respectively). An application of these theorems is that every property that is closed under deleting vertices, edges, and contracting (or "splitting off", respectively) edges can be characterized by finitely many graphs, and hence can be decided in polynomial time. In this thesis we are concerned with the topological minor relation. We say that a graph G contains another graph H as a topological minor if H can be obtained from a subgraph of G by repeatedly deleting a vertex of degree two and adding an edge incident with the neighbors of the deleted vertex. Unlike the relation of minor and weak immersion, the topological minor relation does not well-quasi-order graphs in general. However, Robertson conjectured in the late 1980's that for every positive integer k, the topological minor relation well-quasi-orders graphs that do not contain a topological minor isomorphic to the path of length k with each edge duplicated. This thesis consists of two main results. The first one is a structure theorem for excluding a fixed graph as a topological minor, which is analogous to a cornerstone result of Robertson and Seymour, who gave such structure for graphs that exclude a fixed minor. Results for topological minors were previously obtained by Grohe and Marx and by Dvorak, but we push one of the bounds in their theorems to the optimal value. This improvement is needed for the next theorem. The second main result is a proof of Robertson's conjecture. As a corollary, properties on certain graphs closed under deleting vertices, edges, and "suppressing" vertices of degree two can be characterized by finitely many graphs, and hence can be decided in polynomial time.