Stationary disk assemblies in a ternary system with long range interaction

The free energy of a ternary system, such as a triblock copolymer, is a sum of two parts: an interface energy determined by the size of the interfaces separating the micro-domains of the three constituents, and a long range interaction energy that serves to prevent unlimited micro-domain growth. In two dimensions a parameter range is identified where the system admits stable stationary disk assemblies. Such an assembly consists of perturbed disks made from either type-I constituent or type-II constituent. All the type-I disks have approximately the same radius and all the type-II disks also have approximately the same radius. The locations of the disks are determined by minimization of a function. Depending on the parameters, the disks of the two types can be mixed in an organized way, or mixed in a random way. They can also be fully separated. The first scenario offers a mathematical proof of the existence of a morphological phase for triblock copolymers conjectured by polymer scientists. The last scenario shows that the ternary system is capable of producing two levels of structure. The primary structure is at the microscopic level where disks form near-perfect lattices. The secondary structure is at the macroscopic level forming two large regions, one filled with type-I disks and the other filled with type-II disks. A macroscopic, circular interface separates the two regions.

Impinging jet flow and hydraulic jump on a rotating disk

The free-surface flow formed by a circular jet impinging on a rotating disk is analysed theoretically. The study explores the effects of rotation and inertia on the thin-film flow. Both boundary-layer height and film thickness are found to diminish with rotation speed. A maximum film thickness develops in the supercritical region, which reflects the competition between the convective and centrifugal effects. Unlike the flow on a stationary disk, an increase in the wall shear stress along the radial direction is predicted, at a rate that strengthens with rotating speed. Our results corroborate well existing measurements. The location and height of the hydraulic jump are determined subject to the value of the thickness at the edge of the disk, which is established first for a stationary disk based on the capillary length, and then for a rotating disk using existing analyses and measurements in spin coating. The case of a stationary is revisited in an effort to predict the location and height of the jump uniquely. The formulated value of the height at the edge of the disk seems to give excellent results for a jet at moderately high flow rate (or low viscosity) where the jump structure is well identifiable in reality.