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.

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Pattern formation outside of equilibrium

Conceptions for heat transfer correlation of nanofluids

Stokes flow through a rigid porous medium is analyzed in terms of the method of volume averaging. The traditional averaging procedure leads to an equation of motion and a continuity equation expressed in terms of the volume-averaged pressure and velocity. The equation of motion contains integrals involving spatial deviations of the pressure and velocity, the Brinkman correction, and other lower-order terms. The analysis clearly indicates why the Brinkman correction should not be used to accommodate ano slip condition at an interface between a porous medium and a bounding solid surface.

Flow in porous media I: A theoretical derivation of Darcy's law

The development of regional metamorphism in areas of thickened continental crust is investigated in terms of the major controls on regional-scale thermal regimes. These are: the total radiogenic heat supply within the thickened crust, the supply of heat from the mantle, the thermal conductivity of the medium and the length and time scales of erosion of the continental crust. The orogenic episode is regarded as consisting of a relatively rapid phase of crustal thickening, during which little temperature change occurs in individual rocks, followed by a lengthier phase of erosion, at the end of which the crust is at its original thickness. The principal features of pressure-temperature-time (PTt) paths followed by rocks in this environment are a period of thermal relaxation, during which the temperature rises towards the higher geotherm that would be supported by the thickened crust, followed by a period of cooling as the rock approaches the cold land surface. The temperature increase that occurs is governed by the degree of thickening of the crust, its conductivity and the time that elapses before the rock is exhumed sufficiently to be affected by the proximity of the cold upper boundary. For much of the parameter range considered, the heating phase encompasses a considerable portion of the exhumation (decompression) part of the PTt path. In addition to the detailed calculation of PTt paths we present an idealized model of the thickening and exhumation process, which may be used to make simple calculations of the amount of heating to be expected during a given thickening and exhumation episode and of the depth at which a rock will start to cool on its ascent path. An important feature of these PTt paths is that most of them lie within 50 °C of the maximum temperature attained for one third or more of the total duration of their burial and uplift, and for a geologically plausible range of erosion rates the rocks do not begin to cool until they have completed 20 to 40 per cent of the total uplift they experience. Considerable melting of the continental crust is a likely consequence of thickening of crust with an average continental geotherm. A companion paper discusses these results in the context of attempts to use metamorphic petrology data to give information on tectonic processes. © 1984 Oxford University Press.

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Pressure-Temperature-Time Paths of Regional Metamorphism I. Heat Transfer during the Evolution of Regions of Thickened Continental Crust

Boundary-layer flow of a nanofluid past a stretching sheet

This introduction to convection in porous media assumes the reader is familiar with basic fluid mechanics and heat transfer, going on to cover insulation of buildings, energy storage and recovery, geothermal reservoirs, nuclear waste disposal, chemical reactor engineering and the storage of heat-generating materials like grain and coal. Geophysical applications range from the flow of groundwater around hot intrusions to the stability of snow against avalanches. The book is intended to be used as a reference, a tutorial work or a textbook for graduates.

Convection in Porous Media

Natural convective boundary-layer flow of a nanofluid past a vertical plate

The Cheng–Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid

The problem of the onset of convection, induced by buoyancy effects resulting from vertical thermal and solute concentration gradients, in a horizontal layer of a saturated porous medium, is treated by linear perturbation analysis. It is shown that oscillatory instability may be possible when a strongly stabilizing solute gradient is opposed by a destabilizing thermal gradient, but attention is concentrated on monotonic instability. The eigenvalue equation, which involves a thermal Rayleigh number R and an analogous solute Rayleigh number S, is obtained, by a Fourier series method, for a general set of boundary conditions. Numerical solutions are found for some special limiting cases, extending existing results for the thermal problem. When the thermal and solute boundary conditions are formally identical, the net destabilizing effect is expressed by the sum of R and S.

Onset of Thermohaline Convection in a Porous Medium

The cells observed by Benard (1901) when a horizontal layer of fluid is heated from below were explained by Rayleigh (1916) in terms of buoyancy, and by Pearson (1958) in terms of surface tension. These rival theories are now combined. Linear perturbation techniques are used to derive a sixth-order differential equation subject to six boundary conditions. A Fourier series method has been used to obtain the eigenvalue equation for the case where the lower boundary surface is a rigid conductor and the upper free surface is subject to a general thermal condition. Numerical results are presented. It was found that the two agencies causing instability reinforce one another and are tightly coupled. Cells formed by surface tension are approximately the same size as those formed by buoyancy. Benard's experiments are briefly discussed.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112064000763

Donald A. Nield

Papers

Convection in Porous Media

Natural convective boundary-layer flow of a nanofluid past a vertical plate

The Cheng–Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid

Onset of Thermohaline Convection in a Porous Medium

Surface tension and buoyancy effects in cellular convection