Lattice hyperbolic

Fig. 2. The star body Sf for a two-dimensional lattice subjected to a two-dimensional shear flow, (a) Elliptic streamlines, k < 0 (b) hyperbolic streamlines, k > 0 (c) simple shear flow, A = 0.

A similar analysis can be performed for three-dimensional lattices subjected to the same flow. The corresponding maximum concentration curve in three dimensions is shown in Fig. 3 as a function of the flow parameter X. This curve displays a discontinuous dependence on X in the neighborhood of X = 0, revealing a very special feature of simple shear flow. The saw-tooth property characterizing hyperbolic flows (X > 0) is derived from the best estimates... 

Finally, the self-reproducibility in time of the lattice configuration (for two-dimensional flows) must be addressed. In the elliptic streamline region (A < 0), the lattice necessarily replicates itself periodically in time owing to closure of the streamlines. For hyperbolic flows (A > 0), the lattice is not generally reproduced however, in connection with research on spatially periodic models of foams (Aubert et al., 1986 Kraynik, 1988), Kraynik and Hansen (1986, 1987) found a finite set of reproducible hexagonal lattices for the extensional flow case A = 1. It is not clear how this unique discovery can be extended, if at all. 

Recently, Binder et al. [118] considered the Ising lattice of a binary atomic (N=l) mixture confined in a very thin film by antisymmetric surfaces each attracting a different component. It was shown that the segregation of each blend component to opposite surfaces may create antisymmetric (with respect to the center of the film z=D/2) profiles ( >(z) even for temperatures above critical point T>TC, where flat profiles are expected when external interfaces are neglected. Such antisymmetric profiles would not be distinguished in experiments (with limited depth resolution) from coexisting profiles described by a hyperbolic tangent. 

SELF-AVOIDING WALKS AND POLYGONS ON HYPERBOLIC LATTICES... 

Note that one is not restricted to the unit disk, though that is the geometry utilised by Swierczak and Guttmann [66] in their study of SAW on the hyperbolic lattices. A conformal mapping would enable geometries other than the unit disk to be considered. 

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