PREPRINT. Paper has been published in Chemical Engineering Journal 64:1 (1996) 191-197. Spatio-temporal dynamics in a train of rising bubbles K. Nguyen a , C.S. Daw b , P. Chakka a , M. Cheng a , D.D. Bruns a , C.E.A. Finney a , M.B. Kennel a,b a University of Tennessee, Knoxville, Tennessee 37996, USA b Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8088, USA Abstract It has been suggested that rising bubbles in dense fluids resemble an inverted dripping faucet and that they undergo analogous period-doubling bifurcations to chaos. We present experimental results demonstrating that this analogy is weak because the dominant source of instability in the bubble train is inherently different mutual interactions between spatially separated bubbles as opposed to nozzle dynamics. Unlike the dripping faucet, the initial instability in a bubble train develops at a location far from the injection nozzle and progresses toward the nozzle with increasing gas flow. From both qualitative and rigorous quantitative observations, we conclude that rising-bubble dynamics is best described as "small-box spatio-temporal chaos" with a flow instability. Such dynamics can superficially appear to be simple temporal chaos when considering spatially localized measurements. We show similarity between our experimental results and a bubble- interaction model that accounts for drag and coalescence effects without considering any nozzle dynamics. Keywords: Bubbling; Chaos; Spatio-temporal systems 1. Introduction Bubbles are a key element of many basic engineering processes such as boiling, froth flotation, fermentation, fluidization, and gas-liquid extraction. In this study, we focus on understanding the patterns in a simple experiment where bubbles form at the bottom of a narrow container and rise in a single column. We present our experimental results from the perspective of recent developments in nonlinear dynamics. Our principal objective is to show the applicability of low- dimensional and spatio-temporal chaos and explain the most important characteristics of the underlying dynamics and its route to chaos. Irregular yet clearly non-random dynamic patterns were reported for rising bubbles over 30 years ago [1-4]. At the outset we conjectured that at least some of the irregularities result from deterministic nonlinear processes, motivated by two observations: first, there is some physical similarity between gas bubbles forming in a liquid and liquid drops forming in air; second, a recent model for bubble trains in gas-fluidized beds predicts chaotic behavior [5]. Previous investigators have shown that liquid drops in air (i.e., dripping faucets) clearly exhibit chaotic behavior [6-10], one of the seminal experimental investigations which sparked the modern interest in chaos. Since we began our investigation [11], others have independently reported period-doubling bifurcations and apparent chaos for gas bubbles in liquids [12,13]. Our study extends earlier experiments in the following ways: We make simultaneous measurements of bubble behavior at different spatial locations; We use the most recent chaotic signal-processing algorithms to confirm apparent low dimensionality in spatially localized measurements; We explain the essential dynamics as arising from a flow instability in the inter-bubble interaction; and We show that our experimental observations are consistent with a spatio-temporal model originally derived for bubbles in fluidized beds. 2. Experimental apparatus and procedure Our experimental apparatus consists of a plexiglass vessel 5×5 cm square in cross section and 26 cm high filled with pure glycerine to a depth of 20 cm [11]. Glycerine is used because of its high viscosity, which reduces bubble-rise rates and increases damping of external vibrations. Air is supplied at a constant flow from a compressed gas cylinder through a pressure regulator, a rotameter, a needle valve (for fine flow control), and a tapered injection tube at the bottom of the column. The air injector is constructed from a 7- mm-O.D. glass tube, tapered at the tip to form a 1-mm- I.D. orifice. An earlier reference [11] explains special details for draining the nozzle that are critical to its proper functioning. Other issues such as moisture absorption and foaming in the glycerine are also dis-