Physics Letters A 372 (2008) 6155–6160 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Route to chaos in the rising dynamics of a bubble chain in a polymeric fluid Xavier Frank, Huai Zhi Li ∗ Laboratoire des Sciences du Génie Chimique, Nancy-Université, CNRS, 1 rue Grandville, BP 20451, 54000 Nancy cedex, France article info abstract Article history: Received 14 May 2008 Received in revised form 30 July 2008 Accepted 12 August 2008 Available online 22 August 2008 Communicated by C.R. Doering PACS: 47.50.-d 47.55.D- 47.52.+j Keywords: Bubble Polymeric fluid Competing mechanism Chaos Bifurcation We investigate the chaotic dynamics of a bubble chain rising in a polymeric fluid by the experimental temporary measurements, cognitive modelling and analytical analysis. The competition between the stress creation and the relaxation displays complex dynamical features and leads to chaos through a sequence of period doubling bifurcations.  2008 Published by Elsevier B.V. 1. Introduction Gas–liquid systems are widely encountered not only in natu- ral phenomena, such as volcanic eruptions [1] or decompression sickness [2], but also in many industrial processes like polymer devolatilization [3] or fermentation [4]. Multiphase flows are often very complex, as nonlinear interactions occur between numerous bubbles in fluid. To identify main governing mechanisms and gain an understanding of various phenomena involved step by step, in- vestigation should begin with relatively simple systems. Several authors observed very rich behaviours with quite simple bubbling or similar dripping phenomena. Mosdorf et al. investigated chaotic regime in free bubbling [5]. Da Silva et al. [6], Pinto et al. [7] and Sánchez-Oritz et al. [8] studied drop formation by a dripping faucet. Chaotic behaviour was found for the bubble formation in Newtonian fluids under sound wave perturbation by Tufaile et al. [9–11]. Recently, Tufaile et al. studied the bubbles’ agglomeration at the free surface and explored the relationship between the bub- bling regime and the two-dimensional emerging foam [12]. Nonlinear bubble–bubble interactions stemming from the fluid’s rheological properties can lead to chaotic dynamics too. Behaviour * Corresponding author. E-mail address: huai-zhi.li@ensic.inpl-nancy.fr (H.Z. Li). of a bubble chain rising in a polymeric non-Newtonian fluid is a typical example [13–17]. Two bubbling regimes can occur in such systems [15,18]. If the injection period between two consec- utive bubbles through a single orifice is long enough, bubbles rise steadily to reach free surface and the resulting bubble chain is per- fectly periodic in space and time [19]. On the contrary, if bubbles are periodically injected with a high frequency at the orifice, coa- lescence occurs between successive bubbles, and the bubble train exhibits then a complex dynamics: the bubble chain is regular near the orifice, as bubble formation is periodic [13,16], but above a crit- ical distance from the orifice, coalescence displays chaotic features [13–17]. Attractive interactions do take place between successive bubbles as the rising velocity of a periodic bubble chain is faster than that of an isolated bubble of the same volume [20]. However, the flow field measurements by Laser Doppler Anemometry (LDA) [21] and Particle Image Velocimetry (PIV) [22] reveal that just be- hind a bubble rising in a non-Newtonian fluid, a negative wake of viscoelastic origin [23] appears and could be supposed to result in repulsive interactions. The stress field around a bubble measured by the birefringence visualisation [15] rheological simulation [20] demonstrate that interactions in such media arise from a compet- ing mechanism between the creation of shear stresses due to the passage of bubbles and the temporary relaxation [24]. This physical picture was well confirmed through the computation of the veloc- ity of a periodic bubble chain rising in a polyacrylamide (PAAm) 0375-9601/$ – see front matter  2008 Published by Elsevier B.V. doi:10.1016/j.physleta.2008.08.031