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The semilinear parabolic partial differential equation 1 ∂ t u = Σ ∂ i a i j ( x ) ∂ j u − f ( u , x ) , x ∈ Ω , t ≥ 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/8f22f30f-aed9-454d-b29c-68586c7cd8b5/content/eq1018.tif"/>

A "hemispheres-in-cell" geometry is provided for prediction of colloid retention during transport in porous media. This new geometry preserves the utilities provided in the Happel sphere-in-cell geometry; namely, the ability to predict deposition for a range of porosities, and representation of the influence of neighboring collectors on the fluid flow field. The new geometry, which includes grain to grain contact, is justified by the eventual goal of predicting colloid deposition in the presence of energy barriers, which has been shown in previous literature to involve deposition within grain to grain contacts for colloid:collector ratios greater than approximately 0.005. In order to serve as a platform for predicting deposition in the presence of energy barriers, the model must be shown capable of quantitatively predicting deposition in the absence of energy barriers, which is a requirement that was not met by previous grain to grain contact geometries. This paper describes development of the fluid flow field and particle trajectory simulations for the hemispheres-in-cell geometry in the absence of energy barriers, and demonstrates that the resulting simulations compare favorably to existing models and experiments. A correlation equation for predicting collector efficiencies in the hemispheres-in-cell model in the absence of energy barriers was developed via regression of numerical results to dimensionless parameters.

Elements of the first-principles-based theory of Weiet al. (J. Fluid Mech., vol. 522, 2005, p. 303), Fifeet al. (Multiscale Model. Simul., vol. 4, 2005a, p. 936;J. Fluid Mech., vol. 532, 2005b, p. 165) and Fife, Klewicki & Wei (J. Discrete Continuous Dyn. Syst., vol. 24, 2009, p. 781) are clarified and their veracity tested relative to the properties of the logarithmic mean velocity profile. While the approach employed broadly reveals the mathematical structure admitted by the time averaged Navier–Stokes equations, results are primarily provided for fully developed pressure driven flow in a two-dimensional channel. The theory demonstrates that the appropriately simplified mean differential statement of Newton's second law formally admits a hierarchy of scaling layers, each having a distinct characteristic length. The theory also specifies that these characteristic lengths asymptotically scale with distance from the wall over a well-defined range of wall-normal positions,y. Numerical simulation data are shown to support these analytical findings in every measure explored. The mean velocity profile is shown to exhibit logarithmic dependence (exact or approximate) when the solution to the mean equation of motion exhibits (exact or approximate) self-similarity from layer to layer within the hierarchy. The condition of pure self-similarity corresponds to a constant leading coefficient in the logarithmic mean velocity equation. The theory predicts and clarifies why logarithmic behaviour is better approximated as the Reynolds number gets large. An exact equation for the leading coefficient (von Kármán coefficient κ) is tested against direct numerical simulation (DNS) data. Two methods for precisely estimating the leading coefficient over any selected range ofyare presented. These methods reveal that the differences between the theory and simulation are essentially within the uncertainty level of the simulation. The von Kármán coefficient physically exists owing to an approximate self-similarity in the flux of turbulent force across an internal layer hierarchy. Mathematically, this self-similarity relates to the slope and curvature of the Reynolds stress profile, or equivalently the slope and curvature of the mean vorticity profile. The theory addresses how, why and under what conditions logarithmic dependence is approximated relative to the specific mechanisms contained within the mean statement of dynamics.

The problem of discerning key features of steady turbulent flow adjacent to a wall has drawn the attention of some of the most noted fluid dynamicists of all time. Standard examples of such features are found in the mean velocity profiles of turbulent flow in channels, pipes or boundary layers. The aim of this article is to explain and further develop the recent concept of scaling patch for the time-averaged equations of motion of incompressible flow made highly turbulent by friction at a fixed boundary (introduced in recent papers by Wei et al, Fife et al, and Klewicki et al.) Besides outlining ways to identify the patches, which provide the scaling structure of mean profiles, a critical comparison will be made between that approach and more traditional ones. Our emphasis will be on the question of how and how well these arguments supply insight into the structure of the mean flow profiles. Although empirical results may initiate the search for explanations, they will be viewed simply as means to that end.

Moderately favourable pressure gradient turbulent boundary layers are investigated within a theoretical framework based on the unintegrated two-dimensional mean momentum equation. The present theory stems from an observed exchange of balance between terms in the mean momentum equation across different regions of the boundary layer. This exchange of balance leads to the identification of distinct physical layers, unambiguously defined by the predominant mean dynamics active in each layer. Scaling domains congruent with the physical layers are obtained from a multi-scale analysis of the mean momentum equation. Scaling behaviours predicted by the present theory are evaluated using direct measurements of all of the terms in the mean momentum balance for the case of a sink-flow pressure gradient generated in a wind tunnel with a long development length. Measurements also captured the evolution of the turbulent boundary layers from a non-equilibrium state near the wind tunnel entrance towards an equilibrium state further downstream. Salient features of the present multi-scale theory were reproduced in all the experimental data. Under equilibrium conditions, a universal function was found to describe the decay of the Reynolds stress profile in the outer region of the boundary layer. Non-equilibrium effects appeared to be manifest primarily in the outer region, whereas differences in the inner region were attributed solely to Reynolds number effects.

Mathematically, the problem considered here is that of heteroclinic connections for a system of two second-order differential equations of gradient type, in which a small parameter $$\epsilon$$ conveys a singular perturbation. The physical motivation comes from a multi-order-parameter phase field model developed by Braun et al. Trans. R. Soc. Lond. A, 355, 1997 and Tanoglu, Ph.D. Thesis, University of Delawere, (2000) for the description of crystalline interphase boundaries. In this, the smallness of $$\epsilon$$ is related to large anisotropy. The geometric theory of singular perturbations is employed.

The statistical properties of fully developed planar turbulent Couette–Poiseuille flow result from the simultaneous imposition of a mean wall shear force together with a mean pressure force. Despite the fact that pure Poiseuille flow and pure Couette flow are the two extremes of Couette–Poiseuille flow, the statistical properties of the latter have proved resistant to scaling approaches that coherently extend traditional wall flow theory. For this reason, Couette–Poiseuille flow constitutes an interesting test case by which to explore the efficacy of alternative theoretical approaches, along with their physical/mathematical ramifications. Within this context, the present effort extends the recently developed scaling framework of Weiet al. (2005a) and associated multiscaling ideas of Fifeet al. (2005a,b) to fully developed planar turbulent Couette–Poiseuille flow. Like Poiseuille flow, and contrary to the structure hypothesized by the traditional inner/outer/overlap-based framework, with increasing distance from the wall, the present flow is shown in some cases to undergo abalance breakingandbalance exchangeprocess as the mean dynamics transition from a layer characterized by a balance between the Reynolds stress gradient and viscous stress gradient, to a layer characterized by a balance between the Reynolds stress gradient (more precisely, the sum of Reynolds and viscous stress gradients) and mean pressure gradient. Multiscale analyses of the mean momentum equation are used to predict (in order of magnitude) the wall-normal positions of the maxima of the Reynolds shear stress, as well as to provide an explicit mesoscaling for the profiles near those positions. The analysis reveals a close relationship between the mean flow structure of Couette–Poiseuille flow and two internal scale hierarchies admitted by the mean flow equations. The averaged profiles of interest have, at essentially each point in the channel, a characteristic length that increases as a well-defined ‘outer region’ is approached from either the bottom or the top of the channel. The continuous deformation of this scaling structure as the relevant parameter varies from the Poiseuille case to the Couette case is studied and clarified.

Recent efforts by the present authors have focused on the fundamental multiscaling behaviors of the time averaged dynamical equations of wall turbulence. These efforts have generated a number of new results relating to dynamical structure, as well as a new mathematical foundation. Central to this has been the development of the so-called method of scaling patches. This method provides a formalism for determining scaling behaviors directly from the indeterminate equations. A general description of this methodology is provided herein, and in doing so its connections to well-established scaling notions are identified. Example problems for which the method has been successfully applied includes turbulent boundary layer, pipe and channel flows, turbulent Couette-Poiseuille flow, fully developed turbulent heat transfer in a channel, and favorable pressure gradient boundary layers.

Mathematically, the problem considered here is that of heteroclinic connections for a system of two second order differential equations of Hamiltonian type, in which a small parameter conveys a singular perturbation. The motivation comes from a multi-order-parameter phase field model developed by Braun et al. [5] and [22] for the description of crystalline interphase boundaries. In this model, the smallness of is related to large anisotropy. The existence of such a heteroclinic, and its dependence on , is proved. In addition, its robustness is investigated by establishing its spectral stability.

An analysis is given for fully developed thermal transport through a wall-bounded turbulent fluid flow with constant heat flux supplied at the boundary. The analysis proceeds from the averaged heat equation and utilizes, as principal tools, various scaling considerations. The paper first provides an accounting of the relative dominance of the three terms in that averaged equation, based on existing DNS data. The results show a clear decomposition of the turbulent layer into zones, each with its characteristic transport mechanisms. There follows a theoretical treatment based on the concept of a scaling patch that justifies and greatly extends these empirical results. The primary hypothesis in this development is the monotone and limiting Peclet number dependence (at fixed Reynolds number) of the difference between the specially scaled centerline and wall temperatures. This fact is well corroborated by DNS data. A fairly complete qualitative and order-of-magnitude quantitative picture emerges for a complete range in Peclet numbers. It agrees with known empirical information. In a manner similar to previous analyses of turbulent fluid flow in a channel, conditions for the existence or nonexistence of logarithmic-like mean temperature profiles are established. Throughout the paper, the classical arguments based on an assumed overlapping of regions where the inner and outer scalings are valid are avoided.

We investigate a model of anisotropic diffuse interfaces in ordered FCC crystals introduced recently by Braun et al and Tanoglu et al [3, 18, 19], focusing on parametric conditions which give extreme anisotropy. For a reduced model, we prove existence and stability of plane wave solutions connecting the disordered FCC state with the ordered $Cu_3Au$ state described by solutions to a system of three equations. These plane wave solutions correspond to planar interfaces. Different orientations of the planes in relation to the crystal axes give rise to different surface energies. Guided by previous work based on numerics and formal asymptotics, we reduce this problem in the six dimensional phase space of the system to a two dimensional phase space by taking advantage of the symmetries of the crystal and restricting attention to solutions with corresponding symmetries. For this reduced problem a standing wave solution is constructed that corresponds to a transition that, in the extreme anisotropy limit, is continuous but not differentiable. We also investigate the stability of the constructed solution by studying the eigenvalue problem for the linearized equation. We find that although the transition is stable, there is a growing number $0(\frac{1}{\epsilon})$, of critical eigenvalues, where $\frac{1}{\epsilon}$ » $1$ is a measure of the anisotropy. Specifically we obtain a discrete spectrum with eigenvalues $\lambda_n = \e^{2/3}\mu_n$ with $\mu_n$ ~ $Cn^{2/3}$, as $n \to + \infty$. The scaling characteristics of the critical spectrum suggest a previously unknown microstructural instability.

Experimental and numerical data of the Reynolds shear stress in turbulent channel and pipe flows under a mesonormalization are presented. The mesolength scale associated with this normalization is intermediate to the traditional inner and outer lengths. Justification for the mesoscales is provided by a direct analysis of the mean momentum equation. Specifically, the mesonormalization is revealed through a rescaling that appropriately reflects the physics of an internal mesolayer within which a balance breaking, and subsequent balance exchange of terms in the mean momentum equation takes place. Direct numerical simulation and experimental data are examined and shown to be in good agreement with the new scaling, supporting the new theory. Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Steady Couette and pressure-driven turbulent channel flows have large regions in which the gradients of the viscous and Reynolds stresses are approximately in balance (stress gradient balance regions). In the case of Couette flow, this region occupies the entire channel. Moreover, the relevant features of pressure-driven channel flow throughout the channel can be obtained from those of Couette flow by a simple transformation. It is shown that stress gradient balance regions are characterized by an intrinsic hierarchy of 'scaling layers' (analogous to the inner and outer domains), filling out the stress gradient balance region except for locations near the wall. The spatial extent of each scaling layer is found asymptotically to be proportional to its distance from the wall. There is a rigorous connection between the scaling hierarchy and the mean velocity profile. This connection is through a certain function A(y+) defined in terms of the hierarchy, which remains O(1) for all y+. The mean velocity satisfies an exact logarithmic growth law in an interval of the hierarchy if and only if A is constant. Although A is generally not constant in any such interval, it is arguably almost constant under certain circumstances in some regions. These results are obtained completely independently of classical inner/ outer/overlap scaling arguments, which require more restrictive assumptions. The possible physical implications of these theoretical results are discussed. © 2005 Cambridge University Press.

This paper provides a multiscale analytical study of steady incompressible turbulent flow through a channel of either Couette or pressure-driven Poiseuille type. Mathematically, the paper’s two most novel features are that (1) the analysis begins with an underdetermined singular perturbation problem, namely the Reynolds averaged mean momentum balance equation, and (2) it leads to the existence of an infinite number of length scales. (These two features are probably linked, but the linkage will not be pursued.) The paper develops a credible assumption of a mathematical nature which, when added to the initial underdetermined problem, results in a knowledge of almost the complete layer (scaling) structure of the mean velocity and Reynolds stress profiles. This structure in turn provides a lot of other important information about those profiles. The possibility of almost-logarithmic sections of the mean velocity profile is given special attention. The sense in which the length scales are asymptotically propor...

The properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows are explored both experimentally and theoretically. Available highquality data reveal a dynamically relevant four-layer description that is a departure from the mean profile four-layer description traditionally and nearly universally ascribed to turbulent wall flows. Each of the four layers is characterized by a pre­ dominance of two of the three terms in the governing equations, and thus the mean dynamics of these four layers are unambiguously defined. The inner normalized physical extent of three of the layers exhibits significant Reynolds-number dependence. The scaling properties of these layer thicknesses are determined. Particular signi­ ficance is attached to the viscous/Reynolds-stress-gradient balance layer since its thickness defines a required length scale. Multiscale analysis (necessarily incomplete) substantiates the four-layer structure in developed turbulent channel flow. In parti­ cular, the analysis verifies the existence of at least one intermediate layer, with its own characteristic scaling, between the traditional inner and outer layers. Other information is obtained, such as (i) the widths (in order of magnitude) of the four layers, (ii) a flattening of the Reynolds stress profile near its maximum, and (iii) the asymptotic increase rate of the peak value of the Reynolds stress as the Reynolds num ber approaches infinity. Finally, on the basis of the experimental observation that the velocity increments over two of the four layers are unbounded with increasing Reynolds num ber and have the same order of magnitude, there is additional theore­ tical evidence (outside traditional arguments) for the asymptotically logarithmic character of the mean velocity profile in two of the layers; and (in order of magnitude) the mean velocity increments across each of the four layers are determined. All of these results follow from a systematic train of reasoning, using the averaged momentum balance equation together with other minimal assumptions, such as that the mean velocity increases monotonically from the wall.

The free boundary model of diffusion-induced grain boundary motion derived in Cahn et al. [3], Fife et al. [6] and Cahn & Penrose [4] is extended, in the case of thin metallic films, to account for bidirectional motion, together with the appearance of S-shapes and double seam configurations. These are often observed in the laboratory. Computer simulations based on the extended model are given to illustrate these and other features of bidirectional motion. More generally, the extension accounts for the motion of grain boundaries whose traces on the film's surface are curved. The new free boundary model is one of forced motion by curvature, the forcing term possibly changing sign due to the bidirectionality. The thin film model is derived systematically under explicit assumptions, and an adjustment for grooving is included.

On the basis of a phase field model previously proposed in Cahn et al. (Acta Mater. 45, 4397– 4413 (1997)) to describe the phenomenon of diffusion-induced grain boundary motion (DIGM), we present a formal asymptotic reduction to a moving free-boundary problem. This problem is one of enhanced motion by curvature. The enhancement depends on the local concentration jump, across the grain boundary, of a solute species that is diffusing along the boundary. The reduction depends on the material diffusivity vanishing outside the grain boundary; this also introduces important mathematical and conceptual difficulties which are addressed in detail. A rigorous theory is given for two-dimensional solutions of the free-boundary problem when there is steady motion of the grain boundary spanning a plate from face to face, solute being supplied at the faces of the plate. The motion is in the direction parallel to the face. This situation represents well known experiments designed to illustrate DIGM. The corresponding theory is also given for grain boundaries which do not span the plate, but rather trail behind the moving intersection with one of the faces, never reaching the other one.

A free boundary problem due to Nishiura and Ohnishi is solved in one space dimension. That problem was derived, during their study of phase separation phenomena in diblock copolymers, as an asymptotic limit of pattern-forming PDEs generalizing that of Cahn and Hilliard. The free boundary problem in one dimension reduces to a linear system of ODEs for the lengths of the intervals between interfaces. This system also arises in a completely different context as the spatial discretization of a simple heat equation in a medium with periodic properties. (The medium is homogeneous in an important special case.) The initial-value problem for this system is completely solved, and global stability results for stationary solutions (in which the interfaces are regularly spaced) are obtained. Nucleation phenomena are briefly discussed.

Gradient flows for simple phase-field models based on a non-local free-energy functional (the one used in Bates et al (1997 Arch. Rat. Mech. Anal. 138 105-36) and elsewhere) of a scalar state variable are investigated. We also consider such functionals with no spatial interaction terms. It is known that in either case the state function may be discontinuous. We allow the discontinuity to migrate and treat its position as a primary ingredient in the dynamics, along with the state variable. This leads to an alternative type of gradient flow. The latter's initial-value problem suffers from a lack of uniqueness, and there are multiple travelling waves. One manifestation of this is the possibility of the spontaneous appearance of a new phase at a point, with the domain of the new phase then spreading outward. This ill-posedness can be alleviated by the addition, to the free-energy functional, of a term concentrated at the discontinuity. Similarly, so doing makes travelling waves unique.

Phase field models of solidification in binary alloys with order parameter, concentration and temperature as field variables are reconsidered. The free energy in these models includes the gradient contributions due to the phase field and the concentration field. Equilibrium properties of these models, such as the phase diagrams for curved interfaces, spatial structure of the phase and the composition fields and the dependence of surface tension on the concentration of impurities are analyzed. Solute trapping in planar solidification is studied for several growth regimes, accounting for a substantial difference in the diffusivities of solid and liquid. Owing to this difference there exists partial solute trapping at certain solidification speeds.

A general class of nonlinear evolution equations is described, which support stable spatially oscillatory steady solutions. These equations are composed of an indefinite self-adjoint linear operator acting on the solution plus a nonlinear function, a typical example of the latter being a double-well potential. Thus a Lyapunov functional exists. The linear operator contains a parameter ρ which could be interpreted as a measure of the pattern-forming tendency for the equation. Examples in this class of equations are an integrodifferential equation studied by Goldstein, Muraki, and Petrich and others in an activator-inhibitor context, and a class of fourth-order parabolic PDE's appearing in the literature in various physical connections and investigated rigorously by Coleman, Leizarowitz, Marcus, Mizel, Peletier, Troy, Zaslavskii, and others. The former example reduces to the real Ginzburg-Landau equation when ρ = 0 .

A discussion is given of recent advances in phase-field modeling of materials which change phase. On one hand, general models incorporating elasticity properties of the material, nonconserved and conserved order parameters, and nonlocal effects are now available. On the other hand, gradient theories for binary alloys have been developed which reflect such effects as the dependence of capillarity on the concentration of impurities, solute trapping in its dependence on velocity of solidification fronts, and other nonequilibrium phenomena.

A phase-field model with order parameter, concentration, and temperature as field variables is used to study the properties of solidification fronts in a binary alloy. As in previous papers, the model includes dependence of the free energy density not only on these field variables but also on the gradients of the order parameter and concentration. Terms with these gradients represent surface free energy associated with the phase interface and with the jump in concentration. We treat them as conceptually and physically different; in particular, the thicknesses of the two interfaces will generally be different. Based on the smallness of the coefficients of these gradient terms, and the largeness of the ratio of solute diffusivity in the liquid to that of the solid, asymptotic analyses in various parameter regimes are performed which reveal information on such things as the dependence of the discontinuity of concentration at the front on its velocity and on the above-mentioned parameters. More broadly, we in...

This paper addresses the issue of thermodynamically consistent derivations of field equations governing nonisothermal processes with diffuse interfaces and their implications for interface conditions in the associated sharp interface theories. We operate within the framework of extended irreversible thermodynamics, allowing gradient terms to be present not only in the free energy and entropy densities but also in the internal energy, for both nonconserved and conserved order parameter theories. These various gradient terms are shown to relate to the splitting of the surface tension into an energetic and an entropic part. It is shown that the principle of nondecreasing local entropy production does not single out uniquely the form of the governing field equations. Instead it leads naturally to a one-parameter family of alternative theories which do not contradict the Curie principle of irreversible thermodynamics. The principal applications are to the solidification of a pure material and of a binary alloy. In the case of a pure substance an asymptotic analysis is developed in the vicinity of the solidification front. The main effect of the alternative theories, as well as of the splitting of the surface tension, on the corresponding free boundary problem is manifested in the first order terms in the nonequilibrium contributions to the interface undercooling. We also consider the possibility of assigning from empirical data some specific values to the phenomenological parameters involved in these models.

A system of partial differential equations modelling a van der Waals fluid or an elastic medium with nonmonotone pressure-density relation is studied. As the system changes type, regularisations are considered. The existence of one-dimensional periodic travelling waves, with prescribed average density in a certain range, average velocity and wavelength, is proved. They exhibit layer structure when the regularisation parameter is small. Similarities with the Cahn–Hilliard equation are explored.

Existence, uniqueness and regularity properties are established for monotone travelling waves of a convolution double-obstacle problemut =J*u−u−f (u),the solution u(x, t) being restricted to taking values in the interval [−1, 1]. When u=±1, the equation becomes an inequality. Here the kernel J of the convolution is nonnegative with unit integral and f satisfies f(−1)>0>f(1). This is an extension of the theory in Bates et al. (1997), which deals with this same equation, without the constraint, when f is bistable. Among many other things, it is found that the travelling wave profile u(x−ct) is always ±1 for sufficiently large positive or negative values of its argument, and a necessary and sufficient condition is given for it to be piecewise constant, jumping from −1 to 1 at a single point.

We model diffusion-induced grain boundary motion (DIGM) with a pair of differential equations: Download : Download high-res image (9KB) Download : Download full-size image Here u represents the concentration of solute atoms, ∅ takes the values + 1 and − 1 in the two perfect crystal grains and intermediate values in the boundary between them, τ, δ and e are constants characterizing the material, p(∅, u) is an interaction energy density, and the diffusivity D(∅) is large in the grain boundary (−1

The existence, uniqueness, stability and regularity properties of traveling-wave solutions of a bistable nonlinear integrodifferential equation are established, as well as their global asymptotic stability in the case of zero-velocity continuous waves. This equation is a direct analog of the more familiar bistable nonlinear diffusion equation, and shares many of its properties. It governs gradient flows for free-energy functionals with general nonlocal interaction integrals penalizing spatial nonuniformity.

Properties of the solidification front in a hypercooled liquid, so called because the temperature of the resulting solid is below the melting temperature, are derived using a phase field (diffuse interface) model. Certain known properties for hypercooled fronts in specific materials are reflected within our theories, such as the presence of thin thermal layers and the trend towards smoother fronts (with less pronounced dendrites) when the undercooling is increased within the hypercooled regime. Both an asymptotic analysis, to derive the relevant free boundary problems, and a rigorous determination of the inner profile of the diffusive interface are given. Of particular interest is the incorporation of anisotropy and general microscale interactions leading to higher order differential operators. These features necessitate a much richer mathematical analysis than previous theories. Anisotropic free boundary problems are derived from our models, the simplest of which involves determining the evolution of a set (a solid particle) whose boundary moves with velocity depending on its normal vector. Considerable attention is given to the identification of surface tension, to comparison with previous theories and to questions of stability.

This paper establishes the existence and, in certain cases, the uniqueness of travelling wave solutions of both second-order and higher-order phase-eld systems. These solutions describe the propagation of planar solidication fronts into a hypercooled liquid. The equations are scaled in the usual way so that the relaxation time is " 2 , where " is a nondimensional measure of the interfacial thickness. The equations for the transition layer separating the two phases form a system identical to that for the travelling-wave problem, in which the temperature is strongly coupled with the order parameter. Thus there is no longer a well-dened temperature at the inteface, as is the case in the more frequently studied situation in which the liquid phase is undercooled but not hypercooled. For phase-eld systems of two second-order equations, we prove a general existence theorem based upon topological methods. A second, constructive proof based upon invariant-manifold methods is also given when the parameter is either suciently small or suciently large. In either regime, it is also proved that the wave and the wave velocity are globally unique. Analogous results are also obtained for generalized phase-eld systems in which the order pa- rameter solves a higher-order dierential equation. In this paper, the higher-order tems occur as a singular peturbation of the standard (isotropic) second-order equation. The higher-order terms are useful in modelling anisotropic interfacial motion.

The Cahn-Allen model for the motion of phase-antiphase boundaries is generalized to account for nonlinearities in the kinetic coefficient (relaxation velocity) and the coefficient of the gradient free energy. The resulting equation is $$\varepsilon ^2 u_l = \alpha (u)(\varepsilon [\kappa (u)]^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} \nabla \cdot \{ [\kappa (u)]^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} \nabla u\} - f(u))$$ wheref is bistable. Hereu is an order parameter and κ and α are physical quantities associated with the system's free energy and relaxation speed, respectively. Grain boundaries, away from triple junctions, are modeled by solutions with internal layers when e≪1. The classical motion-by-curvature law for solution layers, well known when κ and α are constant, is shown by formal asymptotic analysis to be unchanged in form under this generalization, the only difference being in the value of the coefficient entering into the relation. The analysis is extended to the case when the relaxation time for the process vanishes for a set of values ofu. Then α is infinite for those values.

A further comparison is made between the standard phase-field equations αφ t =▽ 2 φ+( 1 ξ 2 )[g(φ)−u] , u t =▽ 2 u+ 1 2 lφ t , and the relevant “thermodynamically consistent model of phase transitions” proposed by the authors [Physica D 43 (1990) 44–62]. Here we concentrate on the usual case where g(φ)=φ−φ3, and for comparison purposes retain this expression for the analogous nonlinear functions in the latter model. It is brought out, among other things, that the standard model is thermodynamically consistent in the sense of being derivable from a free-energy functional. However, this free-energy functional is of a somewhat unusual kind: it implies that, at constant temperature, the energy density varies linearly with the order parameter φ and the entropy density is a non-concave function of φ. The example of a hard sphere system indicates that such behaviour is not impossible, but in most other models the energy density and the entropy are both strictly concave in φ.

When a constant metastable solution of the Cahn–Hilliard equation is subjected to a spatially localized large-amplitude perturbation, a transition process may be triggered leading to a globally stable stationary solution. In one space dimension, the existence and instability of a third stationary solution with the same mass is proved: A spike-like solution called a canonical nucleus. Within the class of solutions which are even with respect to the center of the spike, it has a one-dimensional unstable manifold. In addition, the process of nucleation by formal arguments using two space scales and two timescales is described. The last stage in the process can be approximated by a nonlinear Stefan free boundary problem.

Fully developed turbulent flow through straight ducts is considered. An analysis in a Reynolds stress principal-axis coordinate frame is performed with the aim of clarifying the main mechanisms responsible for turbulence-driven secondary (transverse) mean flow in noncircular ducts. Together with this, some basic properties of that flow are deduced.

Stationary solutions of the bistable Cahn-Allen diffusion equation in the plane are constructed, which are positive in quadrants 1 and 3 and negative in the other two quadrants. They are unique and obey certain monotonicity and limiting properties. Solutions of the dynamic problem near this stationary solution generally split the cross-shaped nullset into two disconnected parts which remain bounded away from each other.

1 P.e. Fife et aI., An asymptotic and rigorous study of flames with reversible chain branching, Asymptotic Analysis 5 (1991) 1-26. An asymptotic analysis is given for the system of nonlinear second order ordinary differential equations modeling the propagation of a flame with chemical reactions typical of reversible chain branching kinetics. The basic small parameter is the inverse of the activation energy of first forward reaction, but 2 other parameters enter prominently into the analysis. A thorough formal analysis is given for this problem, and the asymptotics is justified, in its essential features, through a rigorous analysis.

A general framework is given for the phenomenological kinetics of phase transitions in which not only the order parameter but also the temperature may vary in time and space. Instead of a Ginzburg-Landau free energy functional, as used in formulating the Cahn-Hilliard equation, we use the analogous entropy functional. Model entropy functionals, and the kinetic equations resulting from them, are constructed for various cases: phase transitions with and without a critical point, and (in the former case) with or without a latent heat. The class considered is general enough to include the entropy functionals for the Ising model in mean-field approximation, the van der Waals fluid, and a simplified version of the density-functional theory of freezing. A case without critical point, for which the energy is conserved but the order parameter is not, provides a thermodynamically consistent derivation of the phase-field equations studied by Caginalp, Fix and others, and also leads in a natural way to the Lyapunov functional given by Langer for these equations; but the treatment also suggests that a modified version of the phase-field equations might provide a more realistic model of freezing.

Recent studies by the present authors have empirically and analytically explored the properties and scaling behaviours of the Reynolds averaged momentum equation as applied to wall-bounded flows. The results from these efforts have yielded new perspectives regarding mean flow structure and dynamics, and thus provide a context for describing flow physics. A physical model of the turbulent boundary layer is constructed such that it is consonant with the dynamical structure of the mean momentum balance, while embracing independent experimental results relating, for example, to the statistical properties of the vorticity field and the coherent motions known to exist. For comparison, the prevalent, well-established, physical model of the boundary layer is briefly reviewed. The differences and similarities between the present and the established models are clarified and their implications discussed.

A framework is developed for constructing a large class of discrete generation, continuous space models of evolving single species populations and finding their bifurcating patterned spatial distributions. Our models involve, in separate stages, the spatial redistribution (through movement laws) and local regulation of the population; and the fundamental properties of these events in a homogeneous environment are found. Emphasis is placed on the interaction of migrating individuals with the existing population through conspecific attraction (or repulsion), as well as on random dispersion. The nature of the competition of these two effects in a linearized scenario is clarified. The bifurcation of stationary spatially patterned population distributions is studied, with special attention given to the role played by that competition.

Stable and metastable patterned solutions of nonlinear evolution equations of gradient type are discussed. Examples include classes of higher order parabolic PDEs and integrodifferential equations. A governing theme is that patterns can arise as a result of a competition between opposing influences such as destabilizing and stabilizing mechanisms. The discussion is within the context of a general framework although in the case of conserved evolutions, most attention is given to the Cahn-Hilliard equation and metastable patterns. The focus is on rigorous results; however some important formal stability and modulational theories are also reviewed.

In this paper we prove the existence of doubly periodic solutions of certain nonlinear elliptic problems on R 2 and study the geometry of their nodal domains. In particular, we will show that if we perturb a nonlinear elliptic equation exhibiting a small amplitude doubly periodic solution whose nodal domains form a checkerboard pattern, then the perturbed equation will have a unique nearby solution which is still doubly periodic, but for which the nodal line structure breaks up. Moreover, we indicate what can happen if we start with a large amplitude doubly periodic solution whose nodal domains form a checkerboard pattern, and we relate these solutions to the Cahn-Hilliard equation and spinodal decomposition.

The world of combustion phenomena is rich in problems intriguing to the mathematical scientist, offering challenges on several fronts: mathematical modelling, devising appropriate asymptotic and computational methods and developing sound mathematical theories. Papers in this volume describe how all of these challenges have been met for particular examples within a number of common combustion scenarios: reactive shocks, low mach number premixed reactive flow, non-premixed phenomena and solid propellants. The types of phenomena they examine are also diverse: properties of interfaces and shocks, including curvature effects, the stability and other properties of steady structures, the long time dynamics of evolving solutions and spatio-temporal patterns. The papers collected here provide a representative view of current activity in this field.

This paper presents a new numerical method for computation of solutions of prototypical equations of isotachophoresis. Numerical computation is complicated because the Poisson equation, which relates electrostatic potential to space charge density, contains a small parameter. This parameter is usually assumed to have the value of zero. Under this assumption the Poisson differential equation is replaced by an algebraic equation, which is often called the equation of electroneutrality, because it indeed states that the electrolyte is electrically neutral this assumption were not studied in the past. Here we propose an iterative procedure which allows for computation of solutions without the assumption of electroneutrality. The accuracy is controlled by a number of iterations and is limited by a computer round-off error only. The method is based on our previously published theory of existence and uniqueness of solutions of isotachophoretic equations. Details of the computational algorithm for prototypical equations of isotachophoresis are given. A numerical example and comparison with previously published data are also provided.

Correction for ‘A physical model of the turbulent boundary layer consonant with mean momentum balance structure’ by Joe Klewicki, Paul Fife, Tei Wei and Pat McMurtry (Phil. Trans. R. Soc. A 365 , 823–839. (doi: 10.1098/rsta.2006.1944 )). Line 19 of §3( f ) is incorrect in the print version but is correct as follows. It follows that an exactly logarithmic mean profile will occur when the locally normalized second derivative of the Reynolds stress (gradient of the Lamb vector) remains invariant over a range of y (i.e. for a range of β).