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# Communications in Mathematical Sciences

## Volume 10 (2012)

### Number 1

### Special Issue on the Occasion of C. David Levermore’s Sixtieth Birthday

### Diffuse interface surface tension models in an expanding flow

Pages: 387 – 418

DOI: http://dx.doi.org/10.4310/CMS.2012.v10.n1.a16

#### Authors

#### Abstract

We consider a diffusive interface surface tension model under compressible flow. The equation of interest is the Cahn-Hilliard or Allen-Cahn equation with advection by a non-divergence free velocity field. These are two reduced models which show important properties of the full-scale surface tension model. We prove that both model problems are well-posed. We are especially interested in the behavior of solutions with respect to droplet breakup phenomena. Numerical simulations of 1, 2, and 3D all illustrate that the Cahn-Hilliard model is much more effective for droplet breakup. Using asymptotic methods we correctly predict the breakup condition for the Cahn-Hilliard model. Moreover, we prove that the Allen-Cahn model will not break up under certain circumstances due to a maximum principle.

#### Keywords

diffuse interface, surface tension, Cahn-Hilliard equation, numerical simulatio

#### 2010 Mathematics Subject Classification

35B32, 35B50, 76T10