Preprint 21/2012

Crossover of the coarsening rates in demixing of binary viscous liquids

Felix Otto, Christian Seis, and Dejan Slepčev

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Submission date: 05. Apr. 2012
Pages: 41
published in: Communications in mathematical sciences, 11 (2013) 2, p. 441-464 
DOI number (of the published article): 10.4310/CMS.2013.v11.n2.a6
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Abstract:
We consider a model for phase separation in binary viscous liquids that allows for material transport due to cross-diffusion of unlike particles and convection by the hydrodynamic bulk flow. Typically, during the evolution, the average size of domains of the pure phases increases with time a phenomenon called coarsening. Siggia [24] predicts that at an initial stage, coarsening proceeds mainly by diffusion, which leads to the well-known evaporation-recondensation growth law t13, when denotes the average domains size and t denotes time. Furthermore, he argues that at a later stage, convection by the bulk flow becomes the dominant transport mechanism, leading to a crossover in the coarsening rates to t. Siggias predictions have been confirmed by experiments and numerical simulations. In this work, we prove the crossover in the coarsening rates in terms of time-averaged lower bounds on the energy, which scales like an inverse length. We use a method proposed by Kohn and the first author [15], which exploits the gradient flow structure of the dynamics. Our adaption uses techniques from optimal transportation. Our main ingredient is a dissipation inequality. It measures how the optimal transportation distance changes under the effects of convective and diffusive transport.

30.11.2017, 01:42