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Effect of driving on coarsening dynamics in phase-separating systems

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posted on 26.05.2020, 09:18 authored by Dmitri TseluikoDmitri Tseluiko, Meshari Alesemi, Te-Sheng Lin, Uwe Thiele
We consider the Cahn-Hilliard (CH) equation with a Burgers-type convective term that is used as a model of coarsening dynamics in laterally driven phase-separating systems. In the absence of driving, it is known that solutions to the standard CH equation are characterized by an initial stage of phase separation into regions of one phase surrounded by the other phase (i.e., clusters or drops/holes or islands are obtained) followed by the coarsening process, where the average size of the structures grows in time and their number decreases. Moreover, two main coarsening modes have been identified in the literature, namely, coarsening due to volume transfer and due to translation. In the opposite limit of strong driving, the well-known Kuramoto-Sivashinsky (KS) equation is recovered, which may produce complicated chaotic spatio-temporal oscillations. The primary aim of the present work is to perform a detailed and systematic investigation of the transitions in the solutions of the convective CH (cCH) equation for a wide range of parameter values, and, in particular, to understand in detail how the coarsening dynamics is affected by an increase of the strength of the lateral driving force. Considering symmetric two-drop states, we find that one of the coarsening modes is stabilized at relatively weak driving, and the type of the remaining mode may change as driving increases. Furthermore, there exist intervals in the driving strength where coarsening is completely stabilized. In the intervals where the symmetric two-drop states are unstable they can evolve, for example, into one-drop states, two-drop states of broken symmetry or even time-periodic two-drop states that consist of two traveling drops that periodically exchange mass. We present detailed stability diagrams for symmetric two-drop states in various parameter planes and corroborate our findings by selected time simulations.

Funding

Coherent Structures in non-local active-dissipative equations: theory and computations : EP/J001740/1.

EPSRC under grant No. EP/J001740/1.

Ministry of Science and Technology of Taiwan under research grant MOST-107-2115-M-009-008- MY2.

Deutsche Forschungsgemeinschaft (DFG; Grant No. TH781/8-1) and the German-Israeli Foundation for Scientific Research and Development (GIF, Grant No. I-1361-401.10/2016).

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Nonlinearity

Volume

33

Issue

9

Pages

4449 - 4483

Publisher

IOP Publishing

Version

VoR (Version of Record)

Rights holder

© The Authors

Publisher statement

Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence (https://creativecommons.org/licenses/by/3.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Acceptance date

21/04/2020

Publication date

2020-07-21

Copyright date

2020

ISSN

0951-7715

eISSN

1361-6544

Language

en

Depositor

Dr Dmitri Tseluiko. Deposit date: 22 May 2020

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