Analysis of transients for binary mixture convection in cylindrical geometry

Abstract

We present experimental results for early transients near the onset of convection of an ethanol-water mixture in cylindrical containers heated from below. The separation ratio of the mixture was $\ensuremath{\psi}\ensuremath{\approx}\ensuremath{-}0.08,$ and the aspect ratios $\ensuremath{\Gamma}\ensuremath{\equiv}r/d (r$ is the radius and d the height of the sample cell) of two different containers were 10.91 and 11.53. For this system the onset of convection occurs via a subcritical Hopf bifurcation to traveling waves. Beyond the bifurcation we found transient radially traveling waves whose amplitude grew in time. We decomposed the transient patterns into azimuthal modes of the form $\mathrm{cos}m\ensuremath{\theta}.$ The azimuthal symmetry of the pattern depended strongly on $\ensuremath{\Gamma}.$ For $\ensuremath{\Gamma}=10.91$ odd azimuthal modes were preferred, while for $\ensuremath{\Gamma}=11.53$ even modes dominated. We measured the spatial and temporal growth rates at various $\ensuremath{\epsilon}\ensuremath{\equiv}\ensuremath{\Delta}T/\ensuremath{\Delta}{T}_{c}\ensuremath{-}1$ for different azimuthal modes and compared the results for the two aspect ratios. We found the temporal growth rates to be proportional to $\ensuremath{\epsilon},$ but the spatial growth rates were essentially independent of $\ensuremath{\epsilon}.$ Reflection coefficients deduced from the spatial growth rates agree with theory reasonably well. As convection evolved, the patterns collapsed onto one or more diameters, during which time higher-order azimuthal modes grew significantly in amplitude.