INTERFACE · Integrable Structures in theory of multi-phase flows
6РП — Действия „Мария Кюри“
- Период
- 2005-01-01 → 2006-12-31
- Финансиране от ЕС
- 160 294 €
- Участници
- 1
- Схема
- IIF
Линиите свързват координатора с партньорите.
Накратко на български
Динамиката на границата между две течности, като например движението на петрол в пластове земна кора, се анализира чрез математически модели. Разбирането на тези процеси помага за оптимизиране на добива на нефт и развитието на теорията за солитоните.
Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.
Резултати накратко
Final Activity Report Summary - INTERFACE (Integrable Structures in Theory of Multiphase Flows)
In recent years, integrable structures were found in a class of hydrodynamics' problems, leading to a pattern formation in a regime far from equilibrium. Growth problems of this type are unified by the name Laplacian growth (LG). Also known as the Hele-Shaw problem (HSP), it refers to dynamics of a moving front, i.e. an interface, between two distinct phases driven by a harmonic scalar field, which is a potential for the growth velocity field. The HSP appears in different physical and mathematical contexts and has a number of practical applications, e.g. in oil industry. The most interesting and most studied dynamics occurs in the two-dimensional problem. In experiments, the two-dimensional 2d geometry is realised in a Hele-Shaw cell, i.e. a narrow gap between two parallel plates. In this version the problem is also known as the Saffman-Taylor problem or viscous fingering. The problem recently acquired a new facet, due to its connection with modern soliton theory and matrix models. Further development in this direction led to the observation that the equations describing HSP turned out to be constraints, known as 'string' equations in soliton theory, imposed on dispersionless limits of solutions of integrable hierarchies. During the initial stage of our project we dealt with finite-dimensional reductions of LG or related solutions of dispesionless hierarchies. In particular, we established the Hamiltonian structure of such reductions, describing, for instance, multi-finger patterns in Hele-Shaw cell. During the later stages we studied important generalisations of the LG describing free-boundary flows in nonhomogeneous media, such as oil flow in stratified medium, and encountered new exciting links with the theory of quantum integrable systems of the Calogero-Moser type. Our observations led to new developments in fluid mechanics, theory of quadrature domains and analysis of partial differential equations, such as the classical Hadamard's problem. The above outlined directions led to understanding the mathematical aspects of the Hele-Shaw type problems and advanced the theory of partial differential equations.
Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз
Цел на проекта
In recent years, integral structures have been found in a class of hydrodynamics problem leading to pattern formation in a regime far from equilibrium. Growth problems of this type, Laplacian Growth or Hele-Shaw problems, refer to dynamics of a moving front between two distinct phases driven by a harmonic scalar field. These problems have been primarily approached using asymptotic and complex variable methods to address both their stable and the notoriously unstable versions. The relation between the Hel e-Shaw problems and theory of integral systems is an exciting emerging research area. We also mention that it has recently been shown that there is a connection between the two free- boundary problems above and that of the support of Eigen values of an ensemble of large random normal matrices. We intend to explore this aspect of the problem in conjunction with the integral system approach. The aim of the project is two-fold: to address basic questions in the theory of unstable interfaces and multiphase flows, combining recent achievements in the theory of integrable systems with asymptotic and complex-analytic approaches. Connecting and generalizing three subjects of Hele- Shaw problems, theories of integral hierarchies and matrix models, we expect to advance under- standing of interface dynamics and account for its basic physical features. Vice versa, using the theory of interfaces, we illustrate and develop physically important integral models.
Оригинален текст от CORDIS (на английски).
Участници
- THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD · OXFORDКоординаторОбединеното кралство
Връзки
Данни: CORDIS, © Европейски съюз