Bulletin of the American Physical Society
67th Annual Meeting of the APS Division of Fluid Dynamics
Volume 59, Number 20
Sunday–Tuesday, November 23–25, 2014; San Francisco, California
Session G17: Nonlinear Dynamics III: Chaos |
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Chair: Kevin A. Mitchell, University of Califonia, Merced Room: 2002 |
Monday, November 24, 2014 8:00AM - 8:13AM |
G17.00001: Weakly Nonlinear Analysis and Chaotic Growth in Nanofilm Flows Directed by Thermocapillary Forces Chengzhe Zhou, Sandra M. Troian We examine the nonlinear response of nanofilm flows subject to interface deformation and patterned growth by thermocapillary and capillary forces. The governing interface evolution equation describes growth induced by an initial uniform and tranverse thermal gradient in the long wavelength limit.\footnote{M. Dietzel and S. M. Troian, Phys. Rev. Lett. 103 (7), 074501 (2009)}$^,$\footnote{M. Dietzel and S. M. Troian, J. Appl. Phys. 108, 074308 (2010)} A bifurcation analysis via the method of multiple scales elucidates the influence of initial conditions, system geometry and material properties on the regions of stable and unstable flow. Investigation of the corresponding Ginzburg-Landau amplitude equation by finite element simulations reveals the existence of interesting spatiotemporal chaotic behavior during the later stages of patterned growth. Time permitting, we will discuss the possibility of tightly ordered symmetric growth by mode locking to spatially periodic external forcing,\footnote{N. Liu and S. M. Troian, preprint (2013)} in analogy to behavior recently reported for the spatially forced Swift-Hohenberg equation in 1- and 2- dimensions.\footnote{Y. Mau, L. Haim, A. Hagberg and E. Meron, Phys. Rev. E 88, 032917 (2013)} [Preview Abstract] |
Monday, November 24, 2014 8:13AM - 8:26AM |
G17.00002: Using tangles to quantify topological mixing of fluids Qianting Chen, Sulimon Sattari, Kevin Mitchell Topological mixing is important in understanding complex fluid problems, ranging from oceanic transport to the design of micro-mixers. Typically, topological entropy (TE), the exponential growth rate of material lines, is used to quantify topological mixing. Computing TE from the direct stretching rate is computationally expensive and sheds little light on the source of the mixing. Previous work has focused on braiding by ``ghost rods'' (See, e.g. works by Boyland, Aref, Stremler, Tiffeault, and Finn). Following Grover et al. [Chaos 22,043135 (2012)], we study topological mixing in a two-dimensional lid-driven cavity flow. For a certain parameter range, the TE is dominated by a period-3 braid. However, this braid alone cannot explain all the TE within this range, nor the TE outside the range of existence of the braid. By contrast, we explain TE through the topology of intersecting stable and unstable manifolds, i.e. heteroclinic tangles, using homotopic lobe dynamics (HLD). In the HLD approach, stirring originates from ``ghost rods'' placed on \emph{heteroclinic} orbits. We demonstrate that these heteroclinic orbits generate excess TE not accounted for in Grover et al. Furthermore, in the limit of utilizing arbitrarily long manifolds, the HLD technique converges to the true TE. [Preview Abstract] |
Monday, November 24, 2014 8:26AM - 8:39AM |
G17.00003: ABSTRACT WITHDRAWN |
Monday, November 24, 2014 8:39AM - 8:52AM |
G17.00004: Lagrangian transport characteristics of a class of three-dimensional inline-mixing flows with fluid inertia Michel Speetjens, Esubalew Demissie, Guy Metcalfe, Herman Clercx Laminar inline mixing is key to many industrial systems. However, insight into fundamental transport phenomena in case of 3D conditions and fluid inertia remains limited. This is studied for inline mixers with a cylindrical geometry. Said effects introduce three key features absent in simplified configurations: smooth transition between mixing cells; local upstream flow; symmetry breaking. Topological considerations imply a net throughflow region strictly separated from possible internal regions. The Lagrangian dynamics in this region admits representation by a 2D time-periodic Hamiltonian system. This establishes one fundamental kinematic structure for the present class of inline-mixing flows and implies universal behavior. All states follow from Hamiltonian breakdown of one common integrable state. Period-doubling bifurcation is the only way to eliminate transport barriers originating from the integrable state and thus necessary for global chaos. Important in a practical context is that a common simplification, i.e. cell-wise developed Stokes flow, retains these fundamental kinematic properties and deviates from the 3D inertial case essentially only in a quantitative sense. This substantiates its suitability for (at least first exploratory) studies on mixing properties. [Preview Abstract] |
Monday, November 24, 2014 8:52AM - 9:05AM |
G17.00005: Random fluctuations and resonances in near-integrable flows Dmitri Vainchtein Resonance phenomena, such as capture into resonance and scattering on resonance, are known to be major contributors to transport and mixing in near-integrable multi-scale flows. The long-time transport properties in such systems are described in terms of the evolution of the certain quantity, called adiabatic invariant. In the present talk we investigate the impact of different random fluctuations on adiabatic transport. This impact manifests itself in two ways: the statistical properties of the diffusion of the adiabatic invariant due to scattering are altered, and the fine properties of capture, such as the probability of capture and the input-output function, may change significantly. Using the Ekman pumping-driven flow in circular cells as example, we investigate the role these phenomena and obtain modifications to long-term diffusion equations derived before. [Preview Abstract] |
Monday, November 24, 2014 9:05AM - 9:18AM |
G17.00006: Construction of an Optimal Background Profile for the Kuramoto-Sivashinsky Equation using Semidefinite Programming Andrew Wynn, Giovanni Fantuzzi The Kuramoto-Sivashinsky (KS) equation has been derived in many physical contexts to describe systems whose dynamics are characterised by long-wavelength instability, for example flame-front instabilities or flow stability for thin liquid films on inclined planes. It is known that the KS equation has chaotic solutions if the governing parameter (typically the length $L$ of the domain) is sufficiently large. Furthermore, numerical evidence suggests that the asymptotic energy of the solution scales according to $L^{\frac{1}{2}}$ although, despite much effort, it has not yet been possible to prove such a bound analytically. We present a novel method of proving bounds on the asymptotic energy of the KS equation, by constructing a `background profile' using Semidefinite Programming. The advantage of the method is that the background profile may be searched for automatically by solving a standard optimization problem, while coupling the numerics to a careful mathematical analysis of the PDE ensures that the bounds hold analytically. The obtained scaling of $L^{\frac32}$ agrees with the previous best results using the background profile method. Interestingly, the obtained profiles closely resemble the `viscous shock' solutions which are known to exist for destabilized KS equations. [Preview Abstract] |
Monday, November 24, 2014 9:18AM - 9:31AM |
G17.00007: Burning invariant manifolds in time-periodic and time-aperiodic vortex flows Savannah Gowen, Tom Solomon We present experiments that study reaction fronts in a flow composed of a single, translating vortex. The fronts are produced by the excitable Belousov-Zhabotinsky (BZ) chemical reaction, and the vortex flow is driven magnetohydrodynamically by a radial current in a thin fluid layer above a Nd-Fe-Bo magnet. The magnet is mounted on a pair of perpendicular translation stages, allowing for controlled, two-dimensional movement of the magnet and the resulting vortex. We study reaction fronts that pin to the vortex for time-independent flows (produced by moving the vortex with a constant velocity) and for time-periodic and time-aperiodic flows produced by oscillating the vortex laterally. The steady-state front shape is analyzed in terms of {\em burning invariant manifolds}\footnote{J. Mahoney, D. Bargteil, M. Kingsbury, K. Mitchell and T. Solomon, Europhys. Lett. {\bf 98}, 44005 (2012).} (BIMs) that act as one-way barriers against any propagating reaction fronts. For time independent and time-periodic flows, the location of the BIMs are calculated numerically and are compared with experimental images of the pinned reaction fronts. We investigate extensions of this BIM approach for analyzing fronts in time-aperiodic flows. [Preview Abstract] |
Monday, November 24, 2014 9:31AM - 9:44AM |
G17.00008: Defining Lagrangian coherent structures for reactions in time-aperiodic flows Kevin Mitchell, John Mahoney Recent theoretical and experimental investigations have highlighted the role of invariant manifolds, termed \emph{burning invariant manifolds} (BIMs), as one-way barriers to reaction fronts propagating through a flowing medium. Originally, BIM theory was restricted to time-independent or time-periodic flows. The present work extends these ideas to flows with a general time-dependence, thereby constructing coherent structures that organize and constrain the propagation of reaction fronts through general flows. This permits a much broader and physically realistic class of problems to be addressed. Our approach follows the recent work of Farazmand, Blazevski, and Haller [Physica D 278-279, 44 (2014)], in which Lagrangian coherent structures (LCSs), relevant to purely advective transport, are characterized as curves of minimal Lagrangian shear. [Preview Abstract] |
Monday, November 24, 2014 9:44AM - 9:57AM |
G17.00009: Chaotic mixing and front propagation in a three-dimensional flow Sarah Holler, Tom Solomon We present experiments on passive mixing and on the behavior of the excitable Belousov-Zhabotinsky (BZ) chemical reaction in a time-independent, three-dimensional (3D) flow. The flow is composed of nested horizontal and vertical chains of vortices, a flow that has been shown\footnote{M.A. Fogleman, M.J. Fawcett and T. H. Solomon, Phys. Rev. E {\bf 63}, 02101(R) (2001).} numerically to produce chaotic mixing with a complicated structure of ordered and chaotic regions. We study mixing experimentally by tracking neutrally-buoyant tracer particles in 3D and by imaging the evolution of a fluorescent dye with a scanning laser technique. The same scanning technique enables us to image fronts of the Ruthenium-catalyzed BZ reaction in the same flow. We analyze the behavior of these fronts with an extension of a theory of {\em burning invariant manifolds} \footnote{J. Mahoney, D. Bargteil, M. Kingsbury, K. Mitchell and T. Solomon, Europhys. Lett. {\bf 98}, 44005 (2012).} that has been shown to predict accurately the locations of barriers that impede reaction fronts in 2D flows. [Preview Abstract] |
Monday, November 24, 2014 9:57AM - 10:10AM |
G17.00010: Front pinning in single vortex flows John Mahoney, Kevin Mitchell We study fronts propagating in 2D fluid flows and show that there exist stable invariant front configurations for fairly generic flows. Here we examine the simple flow which combines a single vortex with an overall ``wind.'' We discuss how the invariant front can be derived from a simple 3D ODE. Existence of this front can then be understood in terms of bifurcations of fixed points, and the behavior of the invariant ``sliding front'' submanifold. Interestingly, the front bifurcation precedes the saddle-node bifurcation which gives rise to the vortex. This elementary structure has application in chemical reactor beds and laminar combustion in well-mixed fluids. [Preview Abstract] |
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