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Application of Binary Sequences to Problems of Chaos

Oil and water in equal proportion are set into motion between horizontal concentric cylinders when the inner one rotates. In a range of speeds where the water is Taylor unstable and the oil Taylor stable, we get Taylor cells. The main focus of this paper is 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 define 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). EXPERIMENTS Experiments were carried out between two concentric cylinders with axis horizontal, perpendicular to gravity. The outer cylinder and end plates are plexiglass, while the inner cylinder is alumninum. The inside diameter of the outer cylinder is 2.495 inches; the outside diameter is 2.986 inches. The inner cylinder has a diameter of 1.985 inches, and a length of 11.985 inches. The outer cylinder is fixed and the inner one rotates with angular velocity Ω. Our Taylor apparatus uses two neoprene lip seals to prevent leakage. The shaft driving the inner cylinder is connected to a torque meter which has a provision for counting rpm. The torque meter is connected to a mechanical-digital converter which displays the value of the torque and rate of rotation. The laboratory is temperature controlled at 25° Celsius. We used Mobil heavy duty oil with density of 0.97 g/cm3 and viscosity 0.95 poise, and tap water. The interfacial tension between Mobil heavy-duty motor oil and tap water is 30.00 dyne/cm. The major effect of the density difference occurs in slow or lubricated flow in which the oil floats up. When the angular velocity is smaller than 500 rpm, the two fluids are arranged in alternating bands. When rpm is greater than 10 the water bands are Taylor unstable whereas the oil bands are Taylor stable. The Taylor instability leads to the formation of two counterrotating vortices in the water band. CHAOTIC TRAJECTORIES OF OIL BUBBLES IN AN UNSTABLE WATER CELL At sufficiently high values of the angular velocity prior to emulsification of motor oil, the secondary motion appears to be chaotic. In some situations we were able to get one small oil bubble into a Taylor cell. This oil bubble is carried round and round by water and is dragged around in the secondary motion due to Taylor instability. The accompanying videotape contains segments of a recording made of one such bubble. The small oil bubble on the left is the one for which the binary sequence is studied. Each time the oil drop goes around it is either in the left eddy or in the right eddy. We monitored about 3000 terms in the sequence LRLL... and assigned number minus one to left and one to right. It is difficult to get revealing still photographs of the motion of the small bubble in the leftmost water cell shown in Figure 10. However, the large bubble in the center water cell also executes a chaotic motion of a slightly different type as can be seen in the photographs. Binary sequences We are going to apply methods of estimation theory (see Singh and Joseph, 1989) to characterize the chaos in the binary number sequence generated by the bubble in our experiments. Consider a sequence of binary numbers. We assume that the sequence is ergodic so that time averages are the same as ensemble averages. In our experiment the average E u(n) [ ] = 1 N u(n) n =1 N → 0 when N is large, left and right or ±1 are equally probable. Singh and Joseph (1989) showed how to generate a binary sequence for chaotic trajectories of the Lorenz system for (σ, b, r) = (10, 3, 28) . The binary sequence is generated by projecting the trajectories into the xz plane, as shown in Figure 11, and monitoring the crossing points of trajectories on the segments AB and CD of the line AD. The crossing times are put into correspondence with the sequence n of integers, left crossings on AB are recorded as u(n) = 1 , and the right crossings of CD, as u(n) = −1 . The time averages of these sequences vanish for large N, independent of initial condition, so that left and right crossings are equally probably and we may assume that the sequences are ergodic. Autocorrelations An estimate of the autocorrelation function on an ergodic binary sequence can be obtained as follows: r(n) = 1 N u(k + n)u(k) k =1 N , n = 1, 2,... N >> n . (7) The value r(1) represents the correlation between immediate neighbors (1, 2), (2, 3), (3, 4), etc. Value r(2) gives the correlation between separated pairs (1, 3), (2, 4), etc. A chaotic response is one for which and for large n. For the oil bubble autocorrelation values r(n)’s, for large n, are not uniformly close to