The Structure of Internal Layers for Unstable Nonlinear Diffusion Equations

Witelski, Thomas P.

doi:10.1002/sapm1996973277

Cited by 35 publications

(31 citation statements)

References 33 publications

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“…The excess free energy of each phase are due to compositional effects; here we adopt the Wilson model [45]: f l (c) = c log c + β l c) for the liquid, and f g (c) = c log c + β g c) for the gas. The equilibrium concentrations within each phase are then obtained by the common tangent construction of f l and f g [3,46] (Fig. 1, top).…”

Section: Mathematical Modelmentioning

confidence: 99%

Thermodynamic coarsening arrested by viscous fingering in partially miscible binary mixtures

Cueto‐Felgueroso

Juanes

2016

Phys. Rev. E

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We study the evolution of binary mixtures far from equilibrium, and show that the interplay between phase separation and hydrodynamic instability can arrest the Ostwald ripening process characteristic of nonflowing mixtures. We describe a model binary system in a Hele-Shaw cell using a phase-field approach with explicit dependence of both phase fraction and mass concentration. When the viscosity contrast between phases is large (as is the case for gas and liquid phases), an imposed background flow leads to viscous fingering, phase branching, and pinch off. This dynamic flow disorder limits phase growth and arrests thermodynamic coarsening. As a result, the system reaches a regime of statistical steady state in which the binary mixture is permanently driven away from equilibrium.

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Section: Mathematical Modelmentioning

confidence: 99%

Thermodynamic coarsening arrested by viscous fingering in partially miscible binary mixtures

Cueto‐Felgueroso

Juanes

2016

Phys. Rev. E

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“…Equation (3.9) can be interpreted as an equal-area rule for F(U ; ω) (Rubinstein and Sternberg [1992], Witelski [1996], ), see Figure 6. For n = 1, (3.9) trivially reduces to an equalarea rule for F (U ; ω).…”

Section: Integral Formulation Of the Radial Problemmentioning

confidence: 99%

On Axisymmetric Traveling Waves and Radial Solutions of Semi‐linear Elliptic Equations

Witelski¹,

Ono²,

Kaper³

2000

Natural Resource Modeling

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ABSTRACT. Combining analytical techniques from perturbation methods and dynamical systems theory, we present an elementary approach to the detailed construction of axisymmetric diffusive interfaces in semi-linear elliptic equations. Solutions of the resulting non-autonomous radial differential equations can be expressed in terms of a slowly varying phase plane system. Special analytical results for the phase plane system are used to produce closed-form solutions for the asymptotic forms of the curved front solutions. These axisymmetric solutions are fundamental examples of more general curved fronts that arise in a wide variety of scientific fields, and we extensively discuss a number of them, with a particular emphasis on connections to geometric models for the motion of interfaces. Related classical results for traveling waves in one-dimensional problems are also reviewed briefly. Many of the results contained in this article are known, and in presenting known results, it is intended that this article be expository in nature, providing elementary demonstrations of some of the central dynamical phenomena and mathematical techniques. It is hoped that the article serves as one possible avenue of entree to the literature on radially symmetric solutions of semilinear elliptic problems, especially to those articles in which more advanced mathematical theory is developed.

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“…The finite domain version of this problem gives Vi behaviour for the initial stages until the front hits the far boundary, and then the time behaviour switches to a linear behaviour in t [28,29].…”

Section: The Mathematical Modelmentioning

confidence: 99%