[en] The slow dynamics of nearly stationary patterns in a FitzHugh-Nagumo model are studied using a phase dynamics approach. A Cross-Newell phase equation describing slow and weak modulations of periodic stationary solutions is derived. The derivation applies to the bistable, excitable, and Turing unstable regimes. In the bistable case stability thresholds are obtained for the Eckhaus and zigzag instabilities and for the transition to traveling waves. Neutral stability curves demonstrate the destabilization of stationary planar patterns at low wave numbers to zigzag and traveling modes. Numerical solutions of the model system support the theoretical findings. (c) 2000 The American Physical Society

[en] We study the stabilization of localized structures by discreteness in one-dimensional lattices of diffusively coupled nonlinear sites. We find that in an external driving field these structures may lose their stability by either relaxing to a homogeneous state or nucleating a pair of oppositely moving fronts. The corresponding bifurcation diagram demonstrates a cusp singularity. The obtained analytic results are in good quantitative agreement with numerical simulations. (c) 2000 The American Physical Society

[en] In the 1950s, D'yakov and Kontorovich predicted that under certain conditions perturbed shock waves in nonideal gases can become unstable by emitting undamped sound and entropy-vortex waves. For the last 45 years, though, little progress has been made in the identification and numerical modeling of physical conditions for which this phenomenon might occur. Using a van der Waals equation of state, we present for the first time a dynamical simulation of a D'yakov-Kontorovich instability. The two-dimensional emission pattern of acoustic waves appearing in the simulation agrees with the prediction of a linearized theory. (c) 2000 The American Physical Society

[en] A two-phase flow model for an acceleration-driven compressible fluid mixing layer is applied to an initially planar/cylindrical/spherical fluid configuration. A conservative form of the one-dimensional compressible equations is derived under the assumption that the fluid concentration is continuous. With a hyperbolic conservation law for the concentration gradient, the model supports traveling discontinuities in this quantity. The primary examples of this wave type are the moving boundaries of a finite mixing layer, which determine the instability growth rate. Constitutive laws for interfacial averages, previously derived for planar incompressible mixing, are reinterpreted and shown to be applicable to other one-dimensional mixing problems of interest. The equations of motion for an incompressible mixing layer in planar, cylindrical, or spherical geometry are solved exactly, up to a history integral of a function of the edge trajectories, and without assuming incompressible flow outside the layer. Full solutions are obtained by numerically integrating a coupled system of ordinary differential equations for the volume fraction characteristics. Results for self-similar Rayleigh-Taylor mixing in planar geometry are compared to the work of others. This comparison suggests that the shape of the fluid concentration profile is primarily a consequence of mass conservation, parameterized by the expansion ratio of the mixing zone edges. (c) 2000 American Institute of Physics

[en] An analysis is given of the effect of a slow uniform anisotropic compression or expansion on the linear stability of a normally accelerated planar interface between two fluids with different densities and tangential velocities, i.e., a combined Kelvin-Helmholtz and Rayleigh-Taylor instability, but generalized to an arbitrary time-dependent acceleration history. The compression is presumed to be sufficiently slow that the density remains uniform within each fluid and hence depends only on time. The perturbation is taken to be sinusoidal with amplitude h(t). The time evolution of h is determined by requiring pressure continuity across the interface in the usual way. The resulting linearized stability equation is a second-order linear ordinary differential equation for h(t). Compared to the corresponding well-known result for incompressible fluids, it is found that normal compression has the effect of reducing the perturbation growth rate h by an obvious geometrical correction, while transverse compression does not directly affect the net growth rate but rather has the dynamical effect of increasing its time derivative. When attention is focused on the masses transported across the initial interface rather than h, the purely geometrical effects of compression no longer appear explicitly, while the dynamical effects remain. It is thereby shown that both normal and transverse compression dynamically enhance the mixing of material masses, in spite of the corresponding purely geometrical reduction in h. (c) 2000 The American Physical Society

[en] We consider interaction of vortices in the vector complex Ginzburg-Landau equation (CVGLE). In the limit of small field coupling, it is found analytically that the interaction between well-separated defects in two different fields is long ranged, in contrast to the interaction between defects in the same field which falls off exponentially. In a certain region of parameters of CVGLE, we find stable rotating bound states of two defects--a ''vortex molecule.'' (c) 2000 The American Physical Society

[en] Fluid flowing down an inclined plane commonly exhibits a fingering instability in which the contact line corrugates. We show that below a critical inclination angle the base state before the instability is linearly stable. Several recent experiments explore inclination angles below this critical angle, yet all clearly show the fingering instability. We explain this paradox by showing that regardless of the long time linear stability of the front, microscopic scale perturbations at the contact line grow on a transient time scale to a size comparable with the macroscopic structure of the front. This amplification is sufficient to excite nonlinearities and thus initiate finger formation. The amplification is a result of the well-known singular dependence of the macroscopic profiles on the microscopic length scale near the contact line. Implications for other types of forced contact lines are discussed. copyright 1997 American Institute of Physics

[en] The effects of viscous heating on the stability of Taylor-Couette flow were investigated through flow visualization experiments for Newtonian and viscoelastic fluids. For highly viscous Newtonian fluids, viscous heating drives a transition to a new, oscillatory mode of instability at a critical Reynolds number significantly below that at which the inertial transition is observed in isothermal flows. The effects of viscous heating may explain the discrepancies between the observed and predicted critical conditions and the symmetry of the disturbance flow for viscoelastic instabilities. (c) 2000 The American Physical Society

[en] A simple model was recently described for predicting linear and nonlinear mixing at an unstable planar interface between two fluids of different density subjected to an arbitrary time-dependent variable acceleration history [J. D. Ramshaw, Phys. Rev. E 58, 5834 (1998)]. Here we generalize this model to include the Kelvin-Helmholtz (KH) instability resulting from a tangential velocity discontinuity Δu, as well as the effects of a uniform anisotropic compression or expansion of the mixing layer as a whole. The model consists of a second-order nonlinear ordinary differential equation of motion for the half-width h of the mixing layer. This equation is derived by combining the wavelength renormalization hypothesis used in the earlier model with a suitable expression for the rate of change of the kinetic energy of the mixing layer. The resulting generalized model contains no additional free parameters, and reduces to the previous model in the absence of tangential velocities and compression. It also reduces in the linear regime to the correct linearized stability equation for an accelerated shear layer with compression [J. D. Ramshaw, Phys. Rev. E 61, 1486 (2000)]. For a pure incompressible KH instability in the nonlinear regime, the model predicts that h=η|Δu|t, where η=[α(2-θ)/√(θ(1-θ))]√(ρ₁ρ₂)/(ρ₁+ρ₂), and α and θ are parameters appearing in the nonlinear Rayleigh-Taylor and Richtmyer-Meshkov growth laws. For equal densities and the same parameter values previously used to match variable-acceleration experimental data, we find η=0.10, in close agreement with experimental data for free shear layers. (c) 2000 The American Physical Society

[en] Boger fluids are used to study viscous fingering growth in viscoelastic fluids in channel Hele-Shaw flow. We have found that the viscous finger growing in the Boger fluid is unstable to tip splitting at high velocities, in a regime where a Newtonian viscous finger is stable. No fracturelike instabilities were observed. We show that the viscoelastic normal stress differences arising in shear and extensional flow reach very high values at shear and extensional rates comparable to those achieved at the tip of the finger at the onset of tip splitting, and the fluid becomes highly anisotropic. The viscoelastic stress could affect the dynamics of the finger and induce the tip-splitting instability. (c) 2000 The American Physical Society