Faraday waves on a cylindrical fluid filament – generalised equation and simulations

2018 ◽  
Vol 857 ◽  
pp. 80-110 ◽  
Author(s):  
Sagar Patankar ◽  
Palas Kumar Farsoiya ◽  
Ratul Dasgupta
Keyword(s):  
Linear Theory ◽  
Body Force ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Mathieu Equation ◽  
Faraday Waves ◽  
Stability Chart ◽  
Fluid Filament ◽  
The Stability ◽  
Time Periodic

We perform linear stability analysis of an interface separating two immiscible, inviscid, quiescent fluids subject to a time-periodic body force. In a generalised, orthogonal coordinate system, the time-dependent amplitude of interfacial perturbations, in the form of standing waves, is shown to be governed by a generalised Mathieu equation. For zero forcing, the Mathieu equation reduces to a (generalised) simple harmonic oscillator equation. The generalised Mathieu equation is shown to govern Faraday waves on four time-periodic base states. We use this equation to demonstrate that Faraday waves and instabilities can arise on an axially unbounded, cylindrical capillary fluid filament subject to radial, time-periodic body force. The stability chart for solutions to the Mathieu equation is obtained through numerical Floquet analysis. For small values of perturbation and forcing amplitude, results obtained from direct numerical simulations (DNS) of the incompressible Euler equation (with surface tension) show very good agreement with theoretical predictions. Linear theory predicts that unstable Rayleigh–Plateau modes can be stabilised through forcing. This prediction is borne out by DNS results at early times. Nonlinearity produces higher wavenumbers, some of which can be linearly unstable due to forcing and thus eventually destabilise the filament. We study axisymmetric as well as three-dimensional perturbations through DNS. For large forcing amplitude, localised sheet-like structures emanate from the filament, suffering subsequent fragmentation and breakup. Systematic parametric studies are conducted in a non-dimensional space of five parameters and comparison with linear theory is provided in each case. Our generalised analysis provides a framework for understanding free and (parametrically) forced capillary oscillations on quiescent base states of varying geometrical configurations.

Quaternionic Neural Networks

2009 ◽  
pp. 411-439 ◽  
Author(s):  
Teijiro Isokawa ◽  
Nobuyuki Matsui ◽  
Haruhiko Nishimura
Keyword(s):  
Neural Networks ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Network Models ◽  
Recurrent Network ◽  
Affine Transformations ◽  
Neural Network Models ◽  
Fundamental Properties ◽  
Modern Physics ◽  
The Stability

Quaternions are a class of hypercomplex number systems, a four-dimensional extension of imaginary numbers, which are extensively used in various fields such as modern physics and computer graphics. Although the number of applications of neural networks employing quaternions is comparatively less than that of complex-valued neural networks, it has been increasing recently. In this chapter, the authors describe two types of quaternionic neural network models. One type is a multilayer perceptron based on 3D geometrical affine transformations by quaternions. The operations that can be performed in this network are translation, dilatation, and spatial rotation in three-dimensional space. Several examples are provided in order to demonstrate the utility of this network. The other type is a Hopfield-type recurrent network whose parameters are directly encoded into quaternions. The stability of this network is demonstrated by proving that the energy decreases monotonically with respect to the change in neuron states. The fundamental properties of this network are presented through the network with three neurons.

scholarly journals A Fractional-Order Discrete Noninvertible Map of Cubic Type: Dynamics, Control, and Synchronization

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang ◽  
Dafeng Tang
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Dimensional Space ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
State Variables ◽  
Discrete Maps ◽  
The Stability ◽  
Control And Synchronization ◽  
Simultaneous Variation

In this paper, a new fractional-order discrete noninvertible map of cubic type is presented. Firstly, the stability of the equilibrium points for the map is examined. Secondly, the dynamics of the map with two different initial conditions is studied by numerical simulation when a parameter or a derivative order is varied. A series of attractors are displayed in various forms of periodic and chaotic ones. Furthermore, bifurcations with the simultaneous variation of both a parameter and the order are also analyzed in the three-dimensional space. Interior crises are found in the map as a parameter or an order varies. Thirdly, based on the stability theory of fractional-order discrete maps, a stabilization controller is proposed to control the chaos of the map and the asymptotic convergence of the state variables is determined. Finally, the synchronization between the proposed map and a fractional-order discrete Loren map is investigated. Numerical simulations are used to verify the effectiveness of the designed synchronization controllers.

scholarly journals Faraday instability on a sphere: numerical simulation

2019 ◽  
Vol 870 ◽  
pp. 433-459 ◽  
Author(s):  
A. Ebo-Adou ◽  
L. S. Tuckerman ◽  
S. Shin ◽  
J. Chergui ◽  
D. Juric
Keyword(s):  
Body Force ◽  
Three Dimensional ◽  
Capillary Waves ◽  
Platonic Solids ◽  
Interface Motion ◽  
Spherical Drop ◽  
Low Viscosity ◽  
Faraday Instability ◽  
Viscous Drop ◽  
Time Periodic

We consider a spherical variant of the Faraday problem, in which a spherical drop is subjected to a time-periodic body force, as well as surface tension. We use a full three-dimensional parallel front-tracking code to calculate the interface motion of the parametrically forced oscillating viscous drop, as well as the velocity field inside and outside the drop. Forcing frequencies are chosen so as to excite spherical harmonic wavenumbers ranging from 1 to 6. We excite gravity waves for wavenumbers 1 and 2 and observe translational and oblate–prolate oscillation, respectively. For wavenumbers 3 to 6, we excite capillary waves and observe patterns analogous to the Platonic solids. For low viscosity, both subharmonic and harmonic responses are accessible. The patterns arising in each case are interpreted in the context of the theory of pattern formation with spherical symmetry.

scholarly journals Onset of centrifugal filtration convection: departure from thermal equilibrium

2013 ◽  
Vol 469 (2154) ◽  
pp. 20120655 ◽  
Author(s):  
S. Saravanan ◽  
D. Brindha
Keyword(s):  
Linear Theory ◽  
Thermal Equilibrium ◽  
Convective Instability ◽  
Energy Exchange ◽  
Body Force ◽  
Field Model ◽  
Centrifugal Filtration ◽  
Alternating Direction ◽  
Filtration Convection ◽  
The Stability

This paper deals with convective instability in a fluid-saturated, rotating porous layer subject to alternating direction of the centrifugal body force, when the layer fails to exhibit thermal equilibrium. The Darcy model is used to describe the flow, and a two-field model is used to take care of the energy exchange. The normal mode approach of the linear theory and the energy approach of the nonlinear theory are used to find the stability characteristics. Unconditional and sharp nonlinear thresholds are found. The study brings out the failure of the linear theory in describing the instability in most parts of the parameter space of interest where possible subcritical instabilities may arise. The stability boundaries are presented graphically and it is found that the inter-phase heat transfer coefficient has a significant effect in the thermal non-equilibrium regime.

scholarly journals Optimal Reliable Point-in-Polygon Test and Differential Coding Boolean Operations on Polygons

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 477
Author(s):  
Jianqiang Hao ◽  
Jianzhi Sun ◽  
Yi Chen ◽  
Qiang Cai ◽  
Li Tan
Keyword(s):  
Parallel Implementation ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Geometric Probability ◽  
Boolean Operations ◽  
Straight Line ◽  
Speed Up ◽  
The Stability ◽  
Serial Algorithm ◽  
Degenerate Cases

This paper provides a full theoretical and experimental analysis of a serial algorithm for the point-in-polygon test, which requires less running time than previous algorithms and can handle all degenerate cases. The serial algorithm can quickly determine whether a point is inside or outside a polygon and accurately determine the contours of input polygon. It describes all degenerate cases and simultaneously provides a corresponding solution to each degenerate case to ensure the stability and reliability. This also creates the prerequisites and basis for our novel boolean operations algorithm that inherits all the benefits of the serial algorithm. Using geometric probability and straight-line equation F ( P ) = ( y i − y i + 1 ) ( x p − x i ) − ( y i − y p ) ( x i + 1 − x i ) , it optimizes our two algorithms that avoid the division operation and do not need to compute any intersection points. Our algorithms are applicable to any polygon that may be self-intersecting or with holes nested to any level of depth. They do not have to sort the vertices clockwise or counterclockwise beforehand. Consequently, they process all edges one by one in any order for input polygons. This allows a parallel implementation of each algorithm to be made very easily. We also prove several theorems guaranteeing the correctness of algorithms. To speed up the operations, we assign each vector a number code and derive two iterative formulas using differential calculus. However, the experimental results as well as the theoretical proof show that our serial algorithm for the point-in-polygon test is optimal and the time complexities of all algorithms are linear. Our methods can be extended to three-dimensional space, in particular, they can be applied to 3D printing to improve its performance.

Cubic Hybrid Mechanism Based on S[T] Output Base and P/R Input Base

2012 ◽  
Vol 430-432 ◽  
pp. 1725-1728
Author(s):  
Jian Guo Luo ◽  
Mao Yan He
Keyword(s):  
Parallel Mechanism ◽  
Dimensional Space ◽  
Three Dimensional ◽  
One Dimensional ◽  
Hybrid Mechanism ◽  
Hybrid Manipulator ◽  
New Type ◽  
The Stability ◽  
Serial Mechanism ◽  
Three Dimensional Space

Based on the flexibility of single couple of serial mechanism and the stability of multi couples of parallel mechanism, a new type of S[T] output base of hybrid mechanism presented, component of sphere joint run through the tiger joint, this component still the output one with the capability of rotate in three dimensional space. Add serial branch including three translation couple P or/and rotation couple R to the new type of S[T] output base, put these members on one cubic frame, twenty seven configurations obtained with 3-DOF(degree of freedom) allow of three dimensional rotation, twenty seven configurations belong to three conditions obtained with 4-DOF allow of three dimensional rotation and one dimensional translation, nine configurations belong to three conditions obtained with 5-DOF allow of three dimensional rotation and two dimensional translation, one configuration obtained with 6-DOF allow of three dimensional rotation and three dimensional translation, all those sixty four configurations have no more than six translation couple or rotation couple, and the sum of two kind of couple is equal to six. Developing new type of hybrid manipulator based on the hybrid cubic mechanism constructed with S[T] output base and P/R input base will be possible in theory and useful.

Queueing in space

1994 ◽  
Vol 26 (04) ◽  
pp. 1095-1116 ◽  
Author(s):  
Eitan Altman ◽  
Hanoch Levy
Keyword(s):  
Convex Space ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Finite Size ◽  
System Stability ◽  
Single Server ◽  
Greedy Method ◽  
Service Policies ◽  
The Stability ◽  
Greedy Policy

We consider a problem in which a single server must serve a stream of customers whose arrivals are distributed over a finite-size convex space. Under the assumption that the server has full information on the customer location, obvious service policies are the FCFS and the greedy (serve-the-closest-customer) approaches. These algorithms are, however, either inefficient (FCFS) or ‘unfair' (greedy). We propose and study two alternative algorithms, the gated-greedy policy and the gated-scan policy, which are more ‘fair' than the pure greedy method. We show that the stability conditions of the gated-greedy are p < 1 (where p is the expected rate at which work arrives at the system), implying that the method is at least as efficient (in terms of system stability) as any other discipline, in particular the greedy one. For the gated-scan policy we show that for any p < 1 one can design a stable gated-scan policy; however, for any fixed gated-scan policy there exists p < 1 for which the policy is unstable. We evaluate the performance of the gated-scan policy, and present bounds for the performance of the gated-greedy policy. These results are derived for systems in which the arrivals occur on a two-dimensional space (a square) but they are not limited to this configuration; rather they hold for more complex N-dimensional spaces, in particular for serving customers in (three-dimensional) convex space and serving customers on a line.

Stability bands for moving solitons in uniform and inhomogeneous damped ac-driven circular Toda lattices

Open Physics ◽  
2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Lautaro Vergara ◽  
Boris Malomed
Keyword(s):  
Periodic Boundary Conditions ◽  
Initial Position ◽  
Mass Defect ◽  
Parametric Plane ◽  
Stability Chart ◽  
The Stability ◽  
Time Periodic ◽  
Discrete Solitons ◽  
Toda Lattices ◽  
Stability Area

AbstractBy means of systematic simulations, we study the motion of discrete solitons in weakly dissipative Toda lattices (TLs) with periodic boundary conditions, resonantly driven by a spatially staggered time-periodic (ac) force. A complex set of alternating stability bands and instability gaps, including scattered isolated stability points, is revealed in the parametric plane of the soliton’s velocity and forcing amplitude for a given size of the circular lattice. The analysis is also reported for the circular TL including a single light- or heavymass defect. The stability chart as a whole shrinks and eventually disappears with the increase of the lattice’s size and strength of the mass defect. Qualitative explanations to these findings are proposed. We also report the dependence of the stability area on the initial position of the soliton, finding that the area is largest for some intersite position. For a pair of solitons traveling in opposite directions, there exist regimes where both solitons survive periodic collisions in small-size lattices.

scholarly journals Quantum Bose-Fermi droplets

2019 ◽  
Vol 6 (6) ◽  
Author(s):  
Debraj Rakshit ◽  
Tomasz Karpiuk ◽  
Miroslaw Brewczyk ◽  
Mariusz Gajda
Keyword(s):  
Dimensional Space ◽  
Mean Field ◽  
Three Dimensional ◽  
Field Energy ◽  
Liquid Droplets ◽  
Spin Polarized ◽  
The Stability ◽  
Three Body ◽  
Higher Order Corrections ◽  
Three Dimensional Space

We study the stability of a zero temperature mixture of attractively interacting degenerate bosons and spin-polarized fermions in the absence of confinement. We demonstrate that higher order corrections to the standard mean-field energy can lead to a formation of Bose-Fermi liquid droplets – self-bound systems in three-dimensional space. The stability analysis of the homogeneous case is supported by numerical simulations of finite systems by explicit inclusion of surface effects. We discuss the experimental feasibility of formation of quantum droplets and indicate the main obstacle – inelastic three-body collisions.

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