Bulletin of the American Physical Society
68th Annual Meeting of the APS Division of Fluid Dynamics
Volume 60, Number 21
Sunday–Tuesday, November 22–24, 2015; Boston, Massachusetts
Session E29: Nonlinear Dynamics and Waves I |
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Chair: Oscar Velasco Fuentes, CICESE Room: 310 |
Sunday, November 22, 2015 4:50PM - 5:03PM |
E29.00001: Motion of multiple helical vortices Oscar Velasco Fuentes In 1912 Joukowsky deduced that in an unbounded ideal fluid a set of helical vortices ---when these are equal, coaxial and symmetrically arranged--- would translate and rotate steadily while the vortices preserve their form and relative position. Each vortex is an infinite tube whose cross-section is circular (with radius $a$) and whose centerline is a helix of pitch $L$ and radius $R$. The motion is thus determined by three non-dimensional parameters only: the number of vortices $N$, the vortex radius $\alpha=a/R$ and the vortex pitch $\tau=L/2\pi R$. Here, we express the linear and angular velocities of the vortices as the sum of the mutually induced velocities found by Okulov (2004) and the self-induced velocities found by Velasco Fuentes (2015). We verified that our results are accurate over the whole range of values of the vortices' pitch and radius by numerically computing the vortex motion with two smoothed versions of the Biot-Savart law. It was found that the translation velocity $U$ grows with the number of vortices ($N$) but decreases as the vortices' radius and pitch ($a$ and $\tau$, respectively) increase; in contrast, the rotation velocity $\Omega$ grows with $N$ and $a$ but has a local minimum around $\tau=1$ for fixed values of $N$ and $a$. [Preview Abstract] |
Sunday, November 22, 2015 5:03PM - 5:16PM |
E29.00002: The Method of Decomposition in Invariant Structures: Exact Solutions for \textit{N} Internal Waves in Three Dimensions Victor Miroshnikov The Navier-Stokes system of PDEs is reduced to a system of the vorticity, continuity, Helmholtz, and Lamb-Helmholtz PDEs. The periodic Dirichlet problems are formulated for conservative internal waves vanishing at infinity in upper and lower domains. Stationary kinematic Fourier (SKF) structures, stationary kinematic Euler-Fourier (SKEF) structures, stationary dynamic Euler-Fourier (SDEF) structures, and SKEF-SDEF structures of three spatial variables and time are constructed to consider kinematic and dynamic problems of the three-dimensional theory of the Newtonian flows with harmonic velocity. Exact solutions for propagation and interaction of $N$ internal waves in the upper and lower domains are developed by the method of decomposition in invariant structures and implemented through experimental and theoretical programming in Maple. Main results are summarized in a global existence theorem for the strong solutions. The SKEF, SDEF, and SKEF-SDEF structures of the cumulative flows are visualized by two-parametric surface plots for six fluid-dynamic variables. [Preview Abstract] |
Sunday, November 22, 2015 5:16PM - 5:29PM |
E29.00003: Spectral analysis of approximations of Dirichlet-Neumann operators and nonlocal shallow water wave models Rosa Vargas-MagaĆ±a, Panayotis Panayotaros We study the problem of wave propagation in a long-wave asymptotic regime over variable bottom of an ideal irrotational fluid in the framework of the Hamiltonian formulation in which the non-local Dirichlet-Neumann (DtN) operator appears explicitly in the Hamiltonian. We propose a non-local Hamiltonian model for bidirectional wave propagation in shallow water that involves pseudodifferential operators that approximate the DtN operator for variable depth. These models generalize the Boussinesq system as they include the exact dispersion relation in the case of constant depth. We present results for the normal modes and eigenfrequencies of the linearized problem. We see that variable topography introduces effects such as steepening of normal modes with increasing variation of depth, as well as amplitude modulation of the normal modes in certain wavelength ranges. Numerical integration shows that the constant depth nonlocal Boussinesq model with quadratic nonlinearity can capture the evolution obtained with higher order approximations of the DtN operator. In the case of variable depth we observe certain oscillations in width of the crest and also some interesting textures in the evolution of wave crests during the passage from obstacles. [Preview Abstract] |
Sunday, November 22, 2015 5:29PM - 5:42PM |
E29.00004: Pattern formation in thin film evolution equations for complex fluids Markus Wilczek, Svetlana V. Gurevich, Uwe Thiele The description of thin layers of complex fluids like suspensions and solutions is often based on so-called thin film evolution equations which are derived from basic hydrodynamic equations by a long-wave approximation. We present a systematic approach to construct such models in a gradient dynamics formulation for a free energy accounting for wettability and capillarity. We propose extensions in this framework and apply it to dewetting and dip-coating problems. Using these models, we study pattern formation phenomena in Langmuir-Blodgett transfer experiments. [Preview Abstract] |
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