scholarly journals Pattern formation in weakly damped parametric surface waves driven by two frequency components

1997 ◽  
Vol 341 ◽  
pp. 225-244 ◽  
Author(s):  
WENBIN ZHANG ◽  
JORGE VIÑALS
Keyword(s):  
Pattern Formation ◽  
Surface Waves ◽  
Frequency Ratio ◽  
Navier Stokes ◽  
Parametric Surface ◽  
Navier Stokes Equation ◽  
Low Viscosity ◽  
Frequency Components ◽  
Subharmonic Response ◽  
Arbitrary Frequency

A quasi-potential approximation to the Navier–Stokes equation for low-viscosity fluids is developed to study pattern formation in parametric surface waves driven by a force that has two frequency components. A bicritical line separating regions of instability to either of the driving frequencies is explicitly obtained, and compared with experiments involving a frequency ratio of 1/2. The procedure for deriving standing wave amplitude equations valid near onset is outlined for an arbitrary frequency ratio following a multiscale asymptotic expansion of the quasi-potential equations. Explicit results are presented for subharmonic response to a driving force of frequency ratio 1/2, and used to study pattern selection. Even though quadratic terms are prohibited in this case, hexagonal or triangular patterns are found to be stable in a relatively large parameter region, in qualitative agreement with experimental results.

scholarly journals Pattern formation in weakly damped parametric surface waves

1997 ◽  
Vol 336 ◽  
pp. 301-330 ◽  
Author(s):  
WENBIN ZHANG ◽  
JORGE VIÑALS
Keyword(s):  
Asymptotic Expansion ◽  
Pattern Formation ◽  
Surface Waves ◽  
Fluid Motion ◽  
Wave Pattern ◽  
Single Frequency ◽  
Explicit Calculation ◽  
Parametric Surface ◽  
Low Viscosity ◽  
Resonant Interactions

We present a theoretical study of nonlinear pattern formation in parametric surface waves for fluids of low viscosity, and in the limit of large aspect ratio. The analysis is based on a quasi-potential approximation to the equations governing fluid motion, followed by a multiscale asymptotic expansion in the distance away from threshold. Close to onset, the asymptotic expansion yields an amplitude equation which is of gradient form, and allows the explicit calculation of the functional form of the cubic nonlinearities. In particular, we find that three-wave resonant interactions contribute significantly to the nonlinear terms, and therefore are important for pattern selection. Minimization of the associated Lyapunov functional predicts a primary bifurcation to a standing wave pattern of square symmetry for capillary-dominated surface waves, in agreement with experiments. In addition, we find that patterns of hexagonal and quasi-crystalline symmetry can be stabilized in certain mixed capillary–gravity waves, even in this case of single-frequency forcing. Quasi-crystalline patterns are predicted in a region of parameters readily accessible experimentally.

ORIENTATION-INDUCED PATTERN FORMATION: SWARM DYNAMICS IN A LATTICE-GAS AUTOMATON MODEL

1996 ◽  
Vol 06 (09) ◽  
pp. 1735-1752 ◽  
Author(s):  
ANDREAS DEUTSCH
Keyword(s):  
Pattern Formation ◽  
Lattice Gas ◽  
Collective Motion ◽  
Particle Density ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Interaction Operator ◽  
Swarm Dynamics ◽  
Lattice Gas Models ◽  
Non Local

Swarming patterns might arise not just at organismic levels (bird and fishes exhibiting particularly striking examples) but even at cellular and intracellular scales whenever “collective motion” of biological or chemical entities is involved. Examples are the swarming of myxobacteria and ants, aggregation and slug pattern formation of the slime mold Dictyostelium discoideum, or intracellular network dynamics of actin filaments. Here a stochastic process — discrete in space and time — is developed, the “swarm lattice-gas automaton”. For some lattice-gas models (in physics and chemistry) it was demonstrated that the limit behavior resembles known master equations by means of expectation values of suitably chosen microscopic variables. In particular, for the Navier–Stokes equation the derivation of a continuous macroscopic description from discrete microdynamic equations was shown. The “swarm lattice-gas automaton” possesses a non-local integral-like interaction operator. Particles (cells, organisms) are assigned some orientation (and fixed absolute velocity) which might change by means of interaction with other members of the swarm within a given “region of perception”. The corresponding microdynamical equation is given and results of numerical experiments are shown. Simulations exhibit a variety of aggregation patterns which are distinguished by means of microscopic and macroscopic variables. The influence of a sensitivity parameter and particle density on pattern formation is examined systematically.

scholarly journals Thermal and hydrodynamic effects in the ordering of lamellar fluids

2011 ◽  
Vol 369 (1945) ◽  
pp. 2592-2599 ◽  
Author(s):  
G. Gonnella ◽  
A. Lamura ◽  
A. Tiribocchi
Keyword(s):  
High Viscosity ◽  
Navier Stokes ◽  
Finite Difference Schemes ◽  
Hydrodynamic Modes ◽  
Dynamical Equations ◽  
Navier Stokes Equation ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Low Viscosity ◽  
Hydrodynamic Effects

Phase separation in a complex fluid with lamellar order has been studied in the case of cold thermal fronts propagating diffusively from external walls. The velocity hydrodynamic modes are taken into account by coupling the convection–diffusion equation for the order parameter to a generalized Navier–Stokes equation. The dynamical equations are simulated by implementing a hybrid method based on a lattice Boltzmann algorithm coupled to finite difference schemes. Simulations show that the ordering process occurs with morphologies depending on the speed of the thermal fronts or, equivalently, on the value of the thermal conductivity ξ . At large values of ξ , as in instantaneous quenching, the system is frozen in entangled configurations at high viscosity while it consists of grains with well-ordered lamellae at low viscosity. By decreasing the value of ξ , a regime with very ordered lamellae parallel to the thermal fronts is found. At very low values of ξ the preferred orientation is perpendicular to the walls in d =2, while perpendicular order is lost moving far from the walls in d =3.

scholarly journals An Incompressible Polymer Fluid Interacting with a Koiter Shell

2021 ◽  
Vol 31 (1) ◽  
Author(s):  
Dominic Breit ◽  
Prince Romeo Mensah
Keyword(s):  
Moving Boundary ◽  
Classical Model ◽  
Elastic Shell ◽  
Planck Equation ◽  
Navier Stokes ◽  
Flexible Shell ◽  
Navier Stokes Equation ◽  
Shell System ◽  
Forward Equation ◽  
Polymer Fluid

AbstractWe study a mutually coupled mesoscopic-macroscopic-shell system of equations modeling a dilute incompressible polymer fluid which is evolving and interacting with a flexible shell of Koiter type. The polymer constitutes a solvent-solute mixture where the solvent is modelled on the macroscopic scale by the incompressible Navier–Stokes equation and the solute is modelled on the mesoscopic scale by a Fokker–Planck equation (Kolmogorov forward equation) for the probability density function of the bead-spring polymer chain configuration. This mixture interacts with a nonlinear elastic shell which serves as a moving boundary of the physical spatial domain of the polymer fluid. We use the classical model by Koiter to describe the shell movement which yields a fully nonlinear fourth order hyperbolic equation. Our main result is the existence of a weak solution to the underlying system which exists until the Koiter energy degenerates or the flexible shell approaches a self-intersection.

scholarly journals Effects of Salt Tracer Volume and Concentration on Residence Time Distribution Curves for Characterization of Liquid Steel Behavior in Metallurgical Tundish

Metals ◽  
2021 ◽  
Vol 11 (3) ◽  
pp. 430
Author(s):  
Changyou Ding ◽  
Hong Lei ◽  
Hong Niu ◽  
Han Zhang ◽  
Bin Yang ◽  
...  
Keyword(s):  
Residence Time ◽  
Time Distribution ◽  
Residence Time Distribution ◽  
Mass Concentration ◽  
Distribution Curve ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Residence Time Distribution Curve ◽  
Tracer Solution ◽  
Time Distribution Curve

The residence time distribution (RTD) curve is widely applied to describe the fluid flow in a tundish, different tracer mass concentrations and different tracer volumes give different residence time distribution curves for the same flow field. Thus, it is necessary to have a deep insight into the effects of the mass concentration and the volume of tracer solution on the residence time distribution curve. In order to describe the interaction between the tracer and the fluid, solute buoyancy is considered in the Navier–Stokes equation. Numerical results show that, with the increase of the mass concentration and the volume of the tracer, the shape of the residence time distribution curve changes from single flat peak to single sharp peak and then to double peaks. This change comes from the stratified flow of the tracer. Furthermore, the velocity difference number is introduced to demonstrate the importance of the density difference between the tracer and the fluid.

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