A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

/pdf/pattern-formation-outside-of-equilibrium-1601b9znum.pdf

Pattern formation outside of equilibrium

Fluid flows that are smooth at low speeds become unstable and then turbulent at higher speeds. This phenomenon has traditionally been investigated by linearizing the equations of flow and testing for unstable eigenvalues of the linearized problem, but the results of such investigations agree poorly in many cases with experiments. Nevertheless, linear effects play a central role in hydrodynamic instability. A reconciliation of these findings with the traditional analysis is presented based on the "pseudospectra" of the linearized problem, which imply that small perturbations to the smooth flow may be amplified by factors on the order of 105 by a linear mechanism even though all the eigenmodes decay monotonically. The methods suggested here apply also to other problems in the mathematical sciences that involve nonorthogonal eigenfunctions.

https://people.maths.ox.ac.uk/trefethen/publication/PDF/1993_57.pdf

Hydrodynamic Stability Without Eigenvalues

Chaotic time series data are observed routinely in experiments on physical systems and in observations in the field. The authors review developments in the extraction of information of physical importance from such measurements. They discuss methods for (1) separating the signal of physical interest from contamination ("noise reduction"), (2) constructing an appropriate state space or phase space for the data in which the full structure of the strange attractor associated with the chaotic observations is unfolded, (3) evaluating invariant properties of the dynamics such as dimensions, Lyapunov exponents, and topological characteristics, and (4) model making, local and global, for prediction and other goals. They briefly touch on the effects of linearly filtering data before analyzing it as a chaotic time series. Controlling chaotic physical systems and using them to synchronize and possibly communicate between source and receiver is considered. Finally, chaos in space-time systems, that is, the dynamics of fields, is briefly considered. While much is now known about the analysis of observed temporal chaos, spatio-temporal chaotic systems pose new challenges. The emphasis throughout the review is on the tools one now has for the realistic study of measured data in laboratory and field settings. It is the goal of this review to bring these tools into general use among physicists who study classical and semiclassical systems. Much of the progress in studying chaotic systems has rested on computational tools with some underlying rigorous mathematics. Heuristic and intuitive analysis tools guided by this mathematics and realizable on existing computers constitute the core of this review.

The analysis of observed chaotic data in physical systems

http://www.cdi-electrosorption.org/wp-content/uploads/2014/06/Porada-et-al.-2013-Review-on-the-Science-and-Technology-of-Water-Desalination-by-Capacitive-Deionization.pdf

Review on the science and technology of water desalination by capacitive deionization

A group theoretical approach to hydrodynamics considers hydrodynamics to be the differential geometry of diffeomorphism groups. The principle of least action implies that the motion of a fluid is described by the geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy. Investigation of the geometry and structure of such groups turns out to be useful for describing the global behavior of fluids for large time intervals.

Topological methods in hydrodynamics

On presente une theorie pour l'instabilite non visqueux d'une onde courte tridimensionnelle de l'ecoulement elliptique bidimensionnel simple

Three-dimensional instability of elliptical flow.

Developing an understanding of turbulent shear flow at high Reynolds numbers has been a central problem in the theory of fluid motion for over a century Like all turbulent flows, turbulent shear flow is a fluid motion of complex and irregular character whose exact behavior is very sensitive to small changes in initial or boundary conditions Turbulent shear flows are further characterized by a large range of length and time scales, with energy and momentum transfer predominantly affected by nonlinear (iner­ tial) processes between eddies of different scales In many situations of interest, the external conditions are geometrically and temporally simple, and the flow equations admit a correspondingly simple solution: one example is Hagen-Poiseuille flow in a pipe with a constant pressure gradi­ ent Simple, or laminar, flow is generally preferred at low or moderate Reynolds numbers, while turbulent flow is preferred at high Reynolds

Instability mechanisms in shear-flow transition

The DC response of an electrochemical thin film, such as the separator in a microbattery, is analyzed by solving the Poisson--Nernst--Planck equations, subject to boundary conditions appropriate for an electrolytic/galvanic cell. The model system consists of a binary electrolyte between parallel-plate electrodes, each possessing a compact Stern layer, which mediates Faradaic reactions with nonlinear Butler--Volmer kinetics. Analytical results are obtained by matched asymptotic expansions in the limit of thin double layers and compared with full numerical solutions. The analysis shows that (i) decreasing the system size relative to the Debye screening length decreases the voltage of the cell and allows currents higher than the classical diffusion-limited current; (ii) finite reaction rates lead to the important possibility of a reaction-limited current; (iii) the Stern-layer capacitance is critical for allowing the cell to achieve currents above the reaction-limited current; and (iv) all polarographic (cur...

Current-voltage relations for electrochemical thin films

The dc response of an electrochemical thin film, such as the separator in a micro-battery, is analyzed by solving the Poisson-Nernst-Planck equations, subject to boundary conditions appropriate for an electrolytic/galvanic cell. The model system consists of a binary electrolyte between parallel-plate electrodes, each possessing a compact Stern layer, which mediates Faradaic reactions with nonlinear Butler-Volmer kinetics. Analytical results are obtained by matched asymptotic expansions in the limit of thin double layers and compared with full numerical solutions. The analysis shows that (i) decreasing the system size relative to the Debye screening length decreases the voltage of the cell and allows currents higher than the classical diffusion-limited current; (ii) finite reaction rates lead to the important possibility of a reaction-limited current; (iii) the Stern-layer capacitance is critical for allowing the cell to achieve currents above the reaction-limited current; and (iv) all polarographic (current-voltage) curves tend to the same limit as reaction kinetics become fast. Dimensional analysis, however, shows that ``fast'' reactions tend to become ``slow'' with decreasing system size, so the nonlinear effects of surface polarization may dominate the dc response of thin films.

https://arxiv.org/pdf/physics/0406075.pdf

Current-Voltage Relations for Electrochemical Thin Films

Density variations in real fluids are related to both pressure and entropy variations, even in the incompressible limit. The behavior of density variations depends crucially on the relative sizes of the pressure, temperature, and entropy fluctuations. It is shown how this arises from a formal asymptotic expansion of the compressible Navier–Stokes equations about a uniform state. Direct numerical simulations of the full compressible equations verify the consistency of different asymptotic regimes. In the case of turbulent flow, the Kolmogorov–Obukhov theory allows one to predict inertial range scalings for the density fluctuation power spectra in various situations. The results may be relevant to observations of the interstellar medium.

B. J. Bayly

Papers

Three-dimensional instability of elliptical flow.

Instability mechanisms in shear-flow transition

Current-voltage relations for electrochemical thin films

Current-Voltage Relations for Electrochemical Thin Films

Density variations in weakly compressible flows