Incipient mixing by Marangoni effects in slow viscous flow of two immiscible fluid layers

Rickett, Lydia, Penfold, Robert, Blyth, Mark, Purvis, Richard and Cooker, Mark (2015) Incipient mixing by Marangoni effects in slow viscous flow of two immiscible fluid layers. IMA Journal of Applied Mathematics, 80 (5). pp. 1582-1618. ISSN 0272-4960

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Abstract

Ignoring inertia, a deformable interface separating two fluid films is considered, subject to non-uniform tension driven by the solutal Marangoni effect in the presence of a scalar concentration field. Detailed description of adsorption kinetics is abrogated by a simple ansatz directly relating interfacial tension and bulk solute concentration. Consequently, the formal mathematical treatment and some of the results share features in common with the Rayleigh–Bénard–Marangoni thermocapillary problem. Normal mode perturbation analysis in the limit of small interface deformations establishes the existence of an unstable response for low wavenumber excitation. In the classification of Cross & Hohenberg (1993, Pattern formation outside of equilibrium. Rev. Mod. Phys., 65, 851–1112), both type I and type II behaviour are observed. By considering the zero wavenumber situation exactly, it is proved that all eigenvalues are purely imaginary with non-positive imaginary part; hence, a type III instability is not possible. For characteristic timescales of mass diffusion much shorter than the relaxation time of interfacial fluctuations (infinite crispation number): the response growth rate is obtained explicitly; only a single excitation mode is available, and a complete stability diagram is constructed in terms of the relevant control parameters. Otherwise, from a quiescent base state, an infinite discrete spectrum of modes is observed that exhibit avoided crossing and switching phenomena, as well as exceptional points where stationary state pairs coalesce into a single oscillatory standing wave pattern. A base state plane Poiseuille flow, driven by an external pressure gradient, generally exaggerates the response: growth rates of instabilities are enhanced, and stable decay is further suppressed with increasing base flow speed, but the inherent symmetry breaking destroys stationary and standing wave response. Results are obtained in this most general situation by implementing a numerical Chebyshev collocation scheme. The model was motivated by hydrodynamic processes supposed to be involved in gastric digestion of humans.

Item Type:	Article
Additional Information:	This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
Uncontrolled Keywords:	liquid-liquid interface,gibbs elasticity,stokes flow,linear stability analysis,generalised eigenvalue problem,collocation method,avoided crossing,exceptional point
Faculty \ School:	Faculty of Science > School of Mathematics (former - to 2024) Faculty of Science > The Sainsbury Laboratory
UEA Research Groups:	Faculty of Science > Research Groups > Fluid and Solid Mechanics (former - to 2024) Faculty of Science > Research Groups > Fluids & Structures
Depositing User:	Pure Connector
Date Deposited:	26 Oct 2015 14:00
Last Modified:	07 Nov 2024 12:38
URI:	https://ueaeprints.uea.ac.uk/id/eprint/54809
DOI:	10.1093/imamat/hxv009

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