(124c) The Evolution of Bubble Strings in Taylor Vortices of a Couette Device

Bubbles trapped inside a stable Taylor vortex in a Couette device, at the first shear flow instability, tend to repulse each other. The long time migration of equal-sized bubbles results in "bubble crystals", which have the form of ordered circular strings with equal separation distances between the neighbors. Hence, two bubbles assume opposite positions, three bubbles form a triangle, four a square and so on. The evolution of the initially shortest separation distance appears to be monotonic, except for the case when only two bubbles are present, where an overshoot above the final separation distance and some oscillations were experimentally evident.

We assume that inertia forces are the driving mechanism for this observed dynamics. The interaction of two bubbles embedded in a laminar simple shear flow has been solved at low and high Reynolds numbers. These solutions were used as a basis for the analysis of the experimental findings. For the case of low Re we adapted the method of Ho & Leal (1974) where the Lorenz reciprocal theorem is used to calculate the small inertia induced repulsive force, thereby bypassing the need to explicitly employ singular perturbations. For the case of high Re, when inertia effects are dominant, we solved the time-dependent inviscid problem, accounting for the added mass, and introduced viscous effects when calculating the stress in the fluid layer near the interface. The asymptotic solutions for the small and large inertia effects were obtained by employing conformal mapping techniques. These solutions were corroborated by the approximate method of reflections.      

The two-bubble solutions were super imposed to describe the long time dynamic change of the separation distances between multiple bubbles in the Couette-Taylor vortex. For the low Re cases, when viscous forces are dominant over inertia, there appears to be a difference in the evolution pattern of the various separation distances within the string of bubbles. Distances that are initially shortest or longest remain such along the entire evolution process, and they approach their final magnitude monotonically. However, other separation distances may exhibit non monotonic evolution. When the inertia is dominant, oscillations are present and accompany the long term process. The shape, frequency, amplitude and rate of decay of the oscillations depend on the initial separations and on the relative level of viscous effects.

Ho, B. P. and Leal, L. G., 1974, "Inertial migration of rigid spheres in two-dimensional unidirectional flows", J. Fluid Mech.  65(2), 365-400.