Instability and freezing in a solidifying melt conduit

Previous works have shown that when liquid flows in a pipe whose boundary temperature is below freezing, a tubular drainage conduit forms surrounded by solidified material that freezes shut under the appropriate combination of forcing conditions. We conduct laboratory experiments with wax in which the tube freezes shut below a certain value of flux from a pump. As the flux is gradually decreased to this value, the total pressure drop across the length of the tube first decreases to a minimum value and then rises before freezing. Previous theoretical models of a tube driven by a constant pressure drop suggest that once the pressure minimum is reached, the states for a lower flux should be unstable and the tube should therefore freeze-up. In our experiments, flux and pressure drop were coupled, and this motivates us to extend the theory for low Reynolds number flow through a tube with solidification to incorporate a simple pressure-drop-flux relationship. Our model predicts a steady-state relationship between flux and pressure drop that has a minimum pressure as the flux is varied. The stability properties of these steady states depend on the boundary conditions: for a fixed flux, they are all stable, whereas for fixed pressure drop, only those with a flux larger than that at the pressure drop minimum are stable. For a mixed pressure-flux condition, the stability threshold of the steady states lies between these two end members. This provides a possible mechanism for the experimental observations.     

Calculating the Symmetry Number of Flexible Sphere Clusters

We present a theoretical and computational framework to compute the symmetry number of a flexible sphere cluster in \({\mathbb {R}}^3\), using a definition of symmetry that arises naturally when calculating the equilibrium probability of a cluster of spheres in the sticky-sphere limit. We define the sticky symmetry group of the cluster as the set of permutations and inversions of the spheres which preserve adjacency and can be realized by continuous deformations of the cluster that do not change the set of contacts or cause particles to overlap. The symmetry number is the size of the sticky symmetry group. We introduce a numerical algorithm to compute the sticky symmetry group and symmetry number, and show it works well on several test cases. Furthermore, we show that once the sticky symmetry group has been calculated for indistinguishable spheres, the symmetry group for partially distinguishable spheres (those with nonidentical interactions) can be efficiently obtained without repeating the laborious parts of the computations. We use our algorithm to calculate the partition functions of every possible connected cluster of six identical sticky spheres, generating data that may be used to design interactions between spheres so they self-assemble into a desired structure.