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–Benard–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.
Highlights
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
To investigate necessary conditions for the onset of turbulent mixing in a two-layer fluid system, the present work considers the influence of a scalar concentration field on the linear stability of a fluid interface subject to small deformations, as a means to study incipient mixing under the regime of Stokes flow
The bulk concentration field is convected by the fluid flow and acts to nonuniformly alter the interfacial tension that, in turn, induces flow by the solutal Marangoni mechanism
Summary
Food) (Kong & Singh, 2008). The stomach offers a prominent line of defense against pathogenic microorganisms, but more importantly it is the primary site of mechanical action where ingested material is subjected to a complicated unsteady shear flow, dominated by frictional dissipation rates with relatively negligible inertial forces (Pal et al, 2007). Miscibility influences stability by extending the interface into a three dimensional domain with finite width and is typically modelled (Sahu et al, 2009a,b) by a smooth viscosity distribution coupled to a convective-diffusion equation for a scalar concentration field of “friction inducing solute” This is very similar to our present treatment of Marangoni effects (Johnson & Narayanan, 1997) where spatial variations of interfacial tension are produced, for example, by a temperature field or by a nonuniform distribution of surfactants. After observing that linear analysis of Stokes flow has established insoluble surfactant is unable to destabilise a sheared interface between two semi-infinite fluids, Pozrikidis & Hill (2011) have recently questioned the necessity of a bounded fluid domain to realise the Marangoni instability They concluded that one wall is required to engage the Marangoni mechanism, but the presence of a second wall may stabilise the flow.
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