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Basic assumptions of the model

Exploring basic assumptions of circuit models with nodes, channels, and bubble-sources. Investigating stationary flow in single and two-channel systems, and analyzing regular and chaotic bands in loop configurations. Observing the impact of different flow rates on system behavior and patterns.

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Basic assumptions of the model

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  1. Basic assumptions of the model Example of circuit : • Circuits consist of nodes, channels, pumps and bubble-sources • Channels are 1-dimensional, nodes are 0-dimensional, as in a planar graph • Bubbles are point charges of resistance, added to the constant resistance of the channel they occupy • Bubbles flow at the superficial speed of flow • Flow is driven by a pressure drop applied to the inlet and outlet of the system

  2. Stationary flow – single channel

  3. Stationary flow – single channel

  4. Approaching the stationary flow – single channel

  5. Approaching the stationary flow – single channel

  6. In search of a stationary flow – two channel loop Input Outlet The stationary flow is not unique - different patterns can be repeated !!

  7. What happens for f other than fsync ?

  8. What happens for f other than fsync ? The Fourier transform of n(t) for two different cases :

  9. Regular and chaotic bands for an asymmetric loop (L1/L2 = 1.002)

  10. Regular and chaotic band (magnified fragment)

  11. Within the regular band pattern resists perturbations (Frame distance in the movie is equal to the average period of the pattern)

  12. A network with three possible paths • Three possible trajectories: • RB RC  RE • RB RD  RE • RA

  13. Summary • A simple loop exhibits non-trivial behavior such as regular/chaotic bands • A unique stationary state does not exist (memorizing patterns) • Some systems exhibit spontaneous oscillations of large amplitudes • Systems with large number of bubbles are equivalent as long as the flux of resistance (z = f r ) is kept constant • Possibly there exist a continuous limit r  0 with constantz • Such a limit represents waves of flowing resistance density, instead of discrete bubbles

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