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We present a bifurcation study of the incompressible Navier-Stokes equations in a model complex geometry: a spatially periodic array of cylinders in a channel. The dynamics of the flow include a Hopf bifurcation from steady to oscillatory flow at an approximate Reynolds number R of 350 and the appearance of a second frequency at approximately R 890. The multiple frequency dynamics include a substantial increase in spatial and temporal scales with Reynolds number as compared with the simple limit cycle oscillation present close to R = 350. Numerical bifurcation studies of the dynamics are performed using three forms of global eigenfunction expansions. The first basis set is derived through principal factor analysis (Karhunen-Loève expansion) of snapshots from accurate direct spectral element numerical solutions of the Navier-Stokes equations. The second set is obtained from the eigenfunctions of the Stokes operator for this geometry. Finally eigenfunctions are derived from a singular Stokes operator, i.e., the Stokes operator modified to include a variable coefficient which vanishes at the domain boundaries. Truncated systems of (∼ 100) ODEs are obtained through projection of the Navier-Stokes equations onto the basis sets, and a comparative study of the resulting dynamical models is performed.

Recent experimental and model studies have revealed that the domain size may strongly influence the dynamics of rotating spirals in two-dimensional pattern forming chemical reactions. Hartmann et al. [Phys. Rev. Lett. 76, 1384 (1996)], report a frequency increase of spirals in circular domains with diameters substantially smaller than the spiral wavelength in a large domain for the catalytic NO+CO reaction on a microstructured platinum surface. Accompanying simulations with a simple reaction-diffusion system reproduced the behavior. Here, we supplement these studies by a numerical bifurcation and stability analysis of rotating spirals in a simple activator-inhibitor model. The problem is solved in a co-rotating frame of reference. No-flux conditions are imposed at the boundary of the circular domain. At large domain sizes, eigenvalues and eigenvectors very close to those corresponding to infinite medium translational invariance are observed. Upon decrease of domain size, we observe a simultaneous change in the rotation frequency and a deviation of these eigenvalues from being neutrally stable (zero real part). The latter phenomenon indicates that the translation symmetry of the spiral solution is appreciably broken due to the interaction with the (now nearby) wall. Various dynamical regimes are found: first, the spiral simply tries to avoid the boundary and its tip moves towards the center of the circular domain corresponding to a negative real part of the "translational" eigenvalues. This effect is noticeable at a domain radius of R

Spatiotemporal dynamics in reaction-diffusion systems can be altered through the properties (reactivity, diffusivity) of the medium in which they occur. We construct active heterogeneous media (composite catalytic surfaces with inert as well as active inclusions) using microelectronics fabrication techniques and study the spatiotemporal dynamics of heterogeneous catalytic reactions on these catalysts. In parallel, we perform simulations as well as numerical stability and bifurcation analysis of these patterns using mechanistic models. At the limit of large heterogeneity "grain size" (compared to the wavelength of spontaneously arising structures) the interaction of patterns with inert or active boundaries dominates (e.g., pinning, transmission, and boundary breakup of spirals, interaction of pulses with corners, "pacemaker" effects). At the opposite limit of very small or very finely distributed heterogeneity, effective behavior is observed (slight modulation of pulses, nearly uniform oscillations, effective spirals). Some representative studies of transitions between the two limits are presented. Composite Catalyst Surfaces

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