1,239 research outputs found
Strongly nonlinear convection in binary fluids: Minimal model for extended states using symmetry decomposed modes
Spatially extended stationary and traveling states in the strongly nonlinear
regime of convection in layers of binary fluid mixtures heated from below are
described by a few-mode-model. It is derived from the proper hydrodynamic
balance equations including experimentally relevant boundary conditions with a
non-standard Galerkin approximation that uses numerically obtained, symmetry
decomposed modes. Properties of the model are elucidated and compared with full
numerical solutions of the field equations.Comment: 16 pages, including 5 postscript figure
Homoclinic snaking of localized states in doubly diffusive convection
Numerical continuation is used to investigate stationary spatially localized states in two-dimensional thermosolutal convection in a plane horizontal layer with no-slip boundary conditions at top and bottom. Convectons in the form of 1-pulse and 2-pulse states of both odd and even parity exhibit homoclinic snaking in a common Rayleigh number regime. In contrast to similar states in binary fluid convection, odd parity convectons do not pump concentration horizontally. Stable but time-dependent localized structures are present for Rayleigh numbers below the snaking region for stationary convectons. The computations are carried out for (inverse) Lewis number \tau = 1/15 and Prandtl numbers Pr = 1 and Pr >> 1
Continuation for thin film hydrodynamics and related scalar problems
This chapter illustrates how to apply continuation techniques in the analysis
of a particular class of nonlinear kinetic equations that describe the time
evolution through transport equations for a single scalar field like a
densities or interface profiles of various types. We first systematically
introduce these equations as gradient dynamics combining mass-conserving and
nonmass-conserving fluxes followed by a discussion of nonvariational amendmends
and a brief introduction to their analysis by numerical continuation. The
approach is first applied to a number of common examples of variational
equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including
certain thin-film equations for partially wetting liquids on homogeneous and
heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal
equations. Second we consider nonvariational examples as the
Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard
equations and thin-film equations describing stationary sliding drops and a
transversal front instability in a dip-coating. Through the different examples
we illustrate how to employ the numerical tools provided by the packages
auto07p and pde2path to determine steady, stationary and time-periodic
solutions in one and two dimensions and the resulting bifurcation diagrams. The
incorporation of boundary conditions and integral side conditions is also
discussed as well as problem-specific implementation issues
Traveling Wavetrains in the Complex Cubic-Quintic Ginzburg-Laundau Equation
In this paper we use a traveling wave reduction or a so–called spatial approximation to comprehensively investigate the periodic solutions of the complex cubic–quintic Ginzburg–Landau equation. The primary tools used here are Hopf bifurcation theory and perturbation theory. Explicit results are obtained for the post–bifurcation periodic orbits and their stability. Generalized and degenerate Hopf bifurcations are also briefly considered to track the emergence of global structure such as homoclinic orbits
Instabilities and Patterns in Coupled Reaction-Diffusion Layers
We study instabilities and pattern formation in reaction-diffusion layers
that are diffusively coupled. For two-layer systems of identical two-component
reactions, we analyze the stability of homogeneous steady states by exploiting
the block symmetric structure of the linear problem. There are eight possible
primary bifurcation scenarios, including a Turing-Turing bifurcation that
involves two disparate length scales whose ratio may be tuned via the
inter-layer coupling. For systems of -component layers and non-identical
layers, the linear problem's block form allows approximate decomposition into
lower-dimensional linear problems if the coupling is sufficiently weak. As an
example, we apply these results to a two-layer Brusselator system. The
competing length scales engineered within the linear problem are readily
apparent in numerical simulations of the full system. Selecting a :1
length scale ratio produces an unusual steady square pattern.Comment: 13 pages, 5 figures, accepted for publication in Phys. Rev.
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