749 research outputs found
Stability boundaries of roll and square convection in binary fluid mixtures with positive separation ratio
Rayleigh-B\'{e}nard convection in horizontal layers of binary fluid mixtures
heated from below with realistic horizontal boundary conditions is studied
theoretically using multi-mode Galerkin expansions. For positive separation
ratios the main difference between the mixtures and pure fluids lies in the
existence of stable three dimensional patterns near onset in a wide range of
the parameter space. We evaluated the stationary solutions of roll, crossroll,
and square convection and we determined the location of the stability
boundaries for many parameter combinations thereby obtaining the Busse balloon
for roll and square patterns.Comment: 19 pages + 15 figures, accepted by Journal of Fluid Mechanic
Wave-number dependence of the transitions between traveling and standing vortex waves and their mixed states in the Taylor-Couette system
Previous numerical investigations of the stability and bifurcation properties
of different nonlinear combination structures of spiral vortices in a
counterrotating Taylor-Couette system that were done for fixed axial
wavelengths are supplemented by exploring the dependence of the vortex
phenomena waves on their wavelength. This yields information about the
experimental and numerical accessability of the various bifurcation scenarios.
Also backwards bifurcating standing waves with oscillating amplitudes of the
constituent traveling waves are found.Comment: 4 pages, 5 figure
Roll convection of binary fluid mixtures in porous media
We investigate theoretically the nonlinear state of ideal straight rolls in
the Rayleigh-B\'enard system of a fluid layer heated from below with a porous
medium using a Galerkin method. Applying the Oberbeck-Boussinesq approximation,
binary mixtures with positive separation ratio are studied and compared to
one-component fluids. Our results for the structural properties of roll
convection resemble qualitatively the situation in the Rayleigh--B\'enard
system without porous medium except for the fact that the streamlines of binary
mixtures are deformed in the so-called Soret regime. The deformation of the
streamlines is explained by means of the Darcy equation which is used to
describe the transport of momentum. In addition to the properties of the rolls,
their stability against arbitrary infinitesimal perturbations is investigated.
We compute stability balloons for the pure fluid case as well as for a wide
parameter range of Lewis numbers and separation ratios which are typical for
binary gas and fluid mixtures. The stability regions of rolls are found to be
restricted by a crossroll, a zigzag and a new type of oscillatory instability
mechanism, which can be related to the crossroll mechanism
Traveling Wave Fronts and Localized Traveling Wave Convection in Binary Fluid Mixtures
Nonlinear fronts between spatially extended traveling wave convection (TW)
and quiescent fluid and spatially localized traveling waves (LTWs) are
investigated in quantitative detail in the bistable regime of binary fluid
mixtures heated from below. A finite-difference method is used to solve the
full hydrodynamic field equations in a vertical cross section of the layer
perpendicular to the convection roll axes. Results are presented for
ethanol-water parameters with several strongly negative separation ratios where
TW solutions bifurcate subcritically. Fronts and LTWs are compared with each
other and similarities and differences are elucidated. Phase propagation out of
the quiescent fluid into the convective structure entails a unique selection of
the latter while fronts and interfaces where the phase moves into the quiescent
state behave differently. Interpretations of various experimental observations
are suggested.Comment: 46 pages, 11 figures. Accepted for publication in Phys. Rev.
Bifurcation of standing waves into a pair of oppositely traveling waves with oscillating amplitudes caused by a three-mode interaction
A novel flow state consisting of two oppositely travelling waves (TWs) with
oscillating amplitudes has been found in the counterrotating Taylor-Couette
system by full numerical simulations. This structure bifurcates out of axially
standing waves that are nonlinear superpositions of left and right handed
spiral vortex waves with equal time-independent amplitudes. Beyond a critical
driving the two spiral TW modes start to oscillate in counterphase due to a
Hopf bifurcation. The trigger for this bifurcation is provided by a nonlinearly
excited mode of different symmetry than the spiral TWs. A three-mode coupled
amplitude equation model is presented that captures this bifurcation scenario.
The mode-coupling between two symmetry degenerate critical modes and a
nonlinearly excited one that is contained in the model can be expected to occur
in other structure forming systems as well.Comment: 4 pages, 5 figure
Truncated Variational Sampling for "Black Box" Optimization of Generative Models
We investigate the optimization of two probabilistic generative models with
binary latent variables using a novel variational EM approach. The approach
distinguishes itself from previous variational approaches by using latent
states as variational parameters. Here we use efficient and general purpose
sampling procedures to vary the latent states, and investigate the "black box"
applicability of the resulting optimization procedure. For general purpose
applicability, samples are drawn from approximate marginal distributions of the
considered generative model as well as from the model's prior distribution. As
such, variational sampling is defined in a generic form, and is directly
executable for a given model. As a proof of concept, we then apply the novel
procedure (A) to Binary Sparse Coding (a model with continuous observables),
and (B) to basic Sigmoid Belief Networks (which are models with binary
observables). Numerical experiments verify that the investigated approach
efficiently as well as effectively increases a variational free energy
objective without requiring any additional analytical steps
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