Elliptic Umbilic Catastrophes Research Articles

The six roll mill: unfolding an unstable persistently extensional flow

The six roll mill produces steady two-dimensional flows with three incoming and three outgoing streams of fluid. Each flow is generated by a particular set of roller speeds represented by a point in ‘control space’, and is characterized by its ‘critical points’, at which the fluid velocity vanishes. A surface Σ separates control space into regions whose flows have different numbers of critical points; on Σ the critical points are degenerate. For the system studied, Σ is the ‘elliptic umbilic catastrophe’ in the classification of Thom. By using glycerol in the mill, a sequence of flows was explored, corresponding in control space to a loop intersecting Σ. The observed streamline patterns agree well with computer simulations. When the mill contained a 2% solution of polyethylene oxide in water the sequence of observed flow patterns was very different. This can be explained by the long chain molecules becoming persistently extended along the outgoing streamlines issuing from critical points, and the resulting sheets of high extension inhibiting the development of large strain rates in the outgoing fluid streams; the breaking of symmetry between inflows and outflows is shown to explain the observed flow patterns. The high extension of the polymer molecules was observed as intense localized flow birefringence. The diminution in this intensity near degenerate critical points can be used to give rapid estimates of macromolecular relaxation time.

On the analysis of diffraction catastrophes

The most important diffraction catastrophes (caustics) occurring in two-dimensional diffraction are analysed: the fold (elementary caustic), the cusp, the hyperbolic and the elliptic umbilic catastrophe. Special emphasis is given to the inverse problem: what information can be obtained from an observed diffraction catastrophe pattern? It is shown that the critical lines (rainbow lines) in the object plane and the behaviour of the phase function in their neighbourhood can be determined approximately from the characteristic vectors and the intensities of the main peaks in the pattern of these diffraction catastrophes.