## Quasi-incompressible Cahn–Hilliard fluids and topological transitions (1998)

Venue: | Proc. R. Soc. Lond. A |

Citations: | 53 - 4 self |

### BibTeX

@ARTICLE{Lowengrub98quasi-incompressiblecahn–hilliard,

author = {J. Lowengrub and L. Truskinovsky},

title = {Quasi-incompressible Cahn–Hilliard fluids and topological transitions},

journal = {Proc. R. Soc. Lond. A},

year = {1998}

}

### OpenURL

### Abstract

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non-locality (dispersion) associated with an internal length scale and localized dissipation due

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- LANDAU, LIFSHITZ
- 1984
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Free energy of a nonuniform system
- Cahn, Hilliard
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- Truesdell, Noll
- 1965
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- Chang, Hou, et al.
- 1996
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Dynamical critical phenomena
- Hohenberg, Halperin
- 1977
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- LAMB
- 1963
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- Landau, Lifshitz
- 1958
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- 1961
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- 1996
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- 1988
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- 1983
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- 1996
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- Joseph
- 1990
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- 1992
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- 1983
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- Davis, Scriven
- 1982
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- 1997
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- 1989
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- 1994
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- 1949
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- 1982
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- 1992
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- 1998
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- 1993
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- 1991
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