Syncronization between tow hele-shaw cells

Resumo

#TITULO: SYNCHRONIZATION BETWEEN two hele-SHAW CELLS #RESUMEN: We have studied possible configurations to synchrorrize two Hele-shaw cells. A Hele-Shaw cell consists of two transparent plates separated by a small gap. In the interior a fluid is heated from below. If the gap between the plates is sufficiently small, the three dimensional flow is reduced to two dimensions. To model the fluid motion, we start with the Navier-stokes equations in Boussinesq approximation, reducing them to two dimensions, following a procedure exposed by C. Bizon. In particular, we choose a stream function formulation for the governing equations. The aim of the work has required first to study the dynamics of a single one. The integration of the Hele-Shaw cell for different values of the Rayleigh number has been done first by using finite difference scheme (2D uniform grid, 129x129 points). The results made clear the richness of the dynamics of the Hele-Shaw cell. Two types of transitions have been observed. The first one is a horizontal decrease of the aspect ratio of the convective rolls. The second one is from steady to unsteady patterns. For large value of the Rayleigh number, there is multi-stability between multi-cellular stationary solutions and the time dependent unicellular mode. We have decided to force the solution into the unicellular mode, which is unsteady for Rayleigh number larger than approximately 350. The pattern selection depends from the history of heating. By increasing the Rayleigh number, we have found the following sequence of solutions: stationary - periodic - quasi-periodic - periodic - chaotic. Thermal plumes are generated in the lower unstable boundary layer for large value of the Rayleigh number larger than 520. A weakly non linear analysis confirms the numerical results. Far from the threshold, the above amplitude equations need many modes to describe the dynamics. We provide qualitatively and quantitatively convergence of the bifurcation diagrams, showing our limitations when we increase the degrees of freedom. The bifurcation diagrams have been computed with AUTO, a bifurcation analysis package for ordinary differential equations. In the chaotic regime, synchronization is possible if the cells are coupled with all the internal points and a sufficient strong coupling. These requirements are quite demanding and in view of future experimental realization not very encouraging. Let us recall that synchronization through the lateral walls failed as well as when only every 4 grid points were connected in space (6% of the total number of mesh points). If we connect 25% of the total number of mesh points, synchronization is achieved, but the solution falls in the three-cellular mode. By using 2D non uniform grid (spectral method) the minimal number of connectors is outstandingly decreased. Al so chaotic behaviours are reproduced with only 36x36 Chebyshev points and again synchronization is obtained by coupling all the internal points