Couette Flows, Rollers, Emulsions, Tall Taylor Cells, Phase Separation and Inversion, and a Chaotic Bubble in Taylor-Couette Flow of Two Immiscible Liquids

Abstract

Oil and water in equal proportion are set into motion between horizontal concentric cylinders when the inner one rotates. Many different flows are realized and described. In one regime many large bubbles of oil are formed. In a range of speeds where the water is Taylor unstable and the oil Taylor stable, we get strange Taylor cells of emulsified fluids whose length may be three or even four times larger than normal. The length of cells appears to be associated with effective properties of a non-uniform emulsion, so the cell sizes vary along the cylinder. At much higher speeds we get a fine grained emulsion which behaves like a pure fluid with normal Taylor cells. A second focus of the paper is on the mathematical description of the apparently chaotic trajectory of a small oil bubble moving between an eddy pair in a single Taylor cells trapped between the oil bands of a banded Couette flow. We defined a discrete autocorrelation sequence on a binary sequence associated with left and right transitions in the cell to show that the motion of the bubble is chaotic. A formula for a macroscopic Lyapunov exponent for chaos on binary sequences is derived and applied to the experiment and to the Lorenz equation to show how binary sequences can be used to discuss chaos in continuous systems. We use our results and recent results of Feeny and Moon (1989) to argue that Lyapunov exponents for switching sequences are not convenient measures for distinguishing between chaos (short range predictability) and white noise (no predictability).

The flows which develop between our rotating cylinders depend strongly on the material properties of the two liquids. A third focus of the paper is on dynamically maintained emulsions of two immiscible liquids with nearly matched density. The two fluids are 20 cp silicone oil and soybean oil with a very small density difference and small interfacial tension. The two fluids are vertically stratified by weight when the angular velocity is small. Then one fluid fingers into another. The fingers break into small bubbles driven by capillary instability. The bubbles may give rise to uniform emulsions which are unstable and break up into bands of pure liquid separated by bands of emulsified liquid. We suggest that the mechanics of band formation is associated with the pressure deficit in the wake behind each microbubble.