Abstract
Ignoring inertia, a deformable interface separating two fluid films is considered, subject to nonuniform 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 nonpositive 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.
Original language  English 

Pages (fromto)  15821618 
Number of pages  37 
Journal  IMA Journal of Applied Mathematics 
Volume  80 
Issue number  5 
Early online date  9 May 2015 
DOIs  
Publication status  Published  Oct 2015 
Keywords
 liquidliquid interface
 Gibbs elasticity
 Stokes flow
 linear stability analysis
 generalised eigenvalue problem
 collocation method
 avoided crossing
 exceptional point
Profiles

Mark Blyth
 School of Mathematics  Professor of Applied Mathematics
 Fluid and Solid Mechanics  Member
Person: Research Group Member, Academic, Teaching & Research

Mark Cooker
 School of Mathematics  Honorary Associate Professor
 Fluid and Solid Mechanics  Member
Person: Honorary, Research Group Member

Richard Purvis
 School of Mathematics  Associate Professor
 Fluid and Solid Mechanics  Member
Person: Research Group Member, Academic, Teaching & Research