Solutal Marangoni instability in layered two-phase flows

Picardo, Jason R.; Radhakrishna, T; Pushpavanam, S.

doi:10.1017/jfm.2016.135

Cited by 22 publications

(16 citation statements)

References 45 publications

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“…However it has been found by Halpern & Frenkel (2003) that when the surfactant is insoluble, it is possible for instability to exist in a finite interval of wavenumbers bounded below away from the origin (the so-called mid-wave instability), while long and short waves are stable. We find a similar result here for soluble surfactant (note that the mid-wave instability has also been reported for related systems in Picardo et al (2016) and Frenkel et al (2019b)). In figure 10 we take m = 17, n = 4 (i.e.…”

Section: Bulk Concentrations Below the Cmcsupporting

confidence: 90%

“…There have been a number of studies considering the effect of soluble surfactants on the linear stability of falling liquid films (Ji & Setterwall 1994;Karapetsas & Bontozoglou 2013 or two-layer channel flows (Sun & Fahmy 2006;Zaisha et al 2008;You et al 2014;Picardo et al 2016). In the latter case, the papers were either based on the simplifying assumption of a non-deformable interface or considered surfactant soluble in both phases, with the surface tension linearly dependent on either bulk concentration.…”

Section: Introductionmentioning

confidence: 99%

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The role of soluble surfactants in the linear stability of two-layer flow in a channel

Kalogirou

Blyth

2019

J. Fluid Mech.

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The linear stability of Couette–Poiseuille flow of two superposed fluid layers in a horizontal channel is considered. The lower fluid layer is populated with surfactants that appear either in the form of monomers or micelles and can also get adsorbed at the interface between the fluids. A mathematical model is formulated which combines the Navier–Stokes equations in each fluid layer, convection–diffusion equations for the concentration of monomers (at the interface and in the bulk fluid) and micelles (in the bulk), together with appropriate coupling conditions at the interface. The primary aim of this study is to investigate when the system is unstable to arbitrary wavelength perturbations, and in particular, to determine the influence of surfactant solubility and/or sorption kinetics on the instability. A linear stability analysis is performed and the growth rates are obtained by solving an eigenvalue problem for Stokes flow, both numerically for disturbances of arbitrary wavelength and analytically using long-wave approximations. It is found that the system is stable when the surfactant is sufficiently soluble in the bulk and if the fluid viscosity ratio $m$ and thickness ratio $n$ satisfy the condition $m<n^{2}$ . On the other hand, the effect of surfactant solubility is found to be destabilising if $m\geqslant n^{2}$ . Both of the aforementioned results are manifested for low bulk concentrations below the critical micelle concentration; however, when the equilibrium bulk concentration is sufficiently high (and above the critical micelle concentration) so that micelles are formed in the bulk fluid, the system is stable if $m<n^{2}$ in all cases examined.

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Section: Bulk Concentrations Below the Cmcsupporting

confidence: 90%

Section: Introductionmentioning

confidence: 99%

The role of soluble surfactants in the linear stability of two-layer flow in a channel

Kalogirou

Blyth

2019

J. Fluid Mech.

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“…In contrast to the well-known instability of two viscous fluids (Yih [1967]) which needs inertia effects for its existence, this instability may exist in the absence of fluid inertia. With regard to multi-fluid horizontal channel flows, this instability has been further studied in a number of papers, such as Blyth and Pozrikidis [2004b], Pozrikidis [2004], Blyth and Pozrikidis [2004a], Frenkel and Halpern [2005], Wei [2005], Frenkel and Halpern [2006], Halpern and Frenkel [2008], Bassom et al [2010], Peng and Zhu [2010], Kalogirou and Papageorgiou [2016], Picardo et al [2016], and Frenkel and Halpern [2017]. In the latter paper, we have added gravity to the long-wave considerations of FH.…”

Section: Introductionmentioning

confidence: 99%

Surfactant- and gravity-dependent instability of two-layer channel flows: linear theory covering all wavelengths. Part 2. Mid-wave regimes

2019

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A linear stability analysis of a two-layer plane Couette flow of two immiscible fluid layers with different densities, viscosities and thicknesses, bounded by two infinite parallel plates moving at a constant relative velocity to each other, with an insoluble surfactant monolayer along the interface and in the presence of gravity is carried out. The normal modes approach is applied to the equations governing flow disturbances in the two layers. These equations, together with boundary conditions at the plates and the interface, yield a linear eigenvalue problem. When inertia is neglected the velocity amplitudes are the linear combinations of certain hyperbolic functions, and a quadratic dispersion equation for the increment, that is the complex growth rate, is obtained where coefficients depend on the aspect ratio, the viscosity ratio, the basic velocity shear, the Marangoni number Ma that measures the effects of surfactant, and the Bond number Bo that measures the influence of gravity. An extensive investigation is carried out that examines the stabilizing or destabilizing influences of these parameters. Since the dispersion equation is quadratic in the growth rate, there are two continuous branches of the normal modes: a robust branch that exists even with no surfactant, and a surfactant branch that, to the contrary, vanishes when Ma ↓ 0. Due to the availability of explicit forms for the growth rates, in many instances the numerical results are corroborated with analytical asymptotics. For the less unstable branch, a mid-wave interval of unstable wavenumbers (Halpern and Frenkel [2003]) sometimes co-exists with a long-wave one. We study the instability landscape, determined by the threshold curve of the long-wave instability and the critical curve of the mid-wave instability in the (Ma, Bo)-plane. The changes of the extremal points of the critical curves with the variation of the other parameters, such as the viscosity ratio, and the extrema bifurcation points are investigated.

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“…In contrast to the well-known instability of two viscous fluids (Yih (1967)) which needs inertia effects for its existence, the new instability may exist in the absence of fluid inertia. With regard to multi-fluid channel flows, this instability has been further studied in such papers as Blyth and Pozrikidis (2004b), Pozrikidis (2004), Blyth and Pozrikidis (2004a), Frenkel and Halpern (2005), Wei (2005), Frenkel and Halpern (2006), Halpern and Frenkel (2008), Bassom et al (2010), Peng and Zhu (2010), Kalogirou et al (2012), Samanta (2013), Kalogirou and Papageorgiou (2016) and Picardo et al (2016).…”

Section: Introductionmentioning

confidence: 99%

Surfactant and gravity dependent instability of two-layer Couette flows and its nonlinear saturation

Frenkel

Halpern

2017

J. Fluid Mech.

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A horizontal channel flow of two immiscible fluid layers with different densities, viscosities and thicknesses, subject to vertical gravitational forces and with an insoluble surfactant monolayer present at the interface, is investigated. The base Couette flow is driven by the uniform horizontal motion of the channel walls. Linear and nonlinear stages of the (inertialess) surfactant and gravity dependent long-wave instability are studied using the lubrication approximation, which leads to a system of coupled nonlinear evolution equations for the interface and surfactant disturbances. The (inertialess) instability is a combined result of the surfactant action characterized by the Marangoni number Ma and the gravitational effect corresponding to the Bond number Bo that ranges from −∞ to ∞. The other parameters are the top-to-bottom thickness ratio n, which is restricted to n ≥ 1 by a reference frame choice, the top-to-bottom viscosity ratio m, and the base shear rate s.The linear stability is determined by an eigenvalue problem for the normal modes, where the complex eigenvalues (determining growth rates and phase velocities) and eigenfunctions (the amplitudes of disturbances of the interface, surfactant, velocities, and pressures) are found analytically by using the smallness of the wavenumber. For each wavenumber, there are two active normal modes, called the surfactant and the robust modes. The robust mode is unstable when Bo/Ma falls below a certain value dependent on m and n. The surfactant branch has instability for m < 1, and any Bo, although the range of unstable wavenumbers decreases as the stabilizing effect of gravity represented by Bo increases. Thus, for certain parametric ranges, even arbitrarily strong gravity cannot completely stabilize the flow.The correlations of vorticity-thickness phase differences with instability, present when gravitational effects are neglected, are found to break down when gravity is important. The physical mechanisms of instability for the two modes are explained with vorticity playing no role in them. This is in marked contrast to the dynamical role of vorticity in the mechanism of the well-known Yih instability due to effects of inertia, and is contrary to some earlier literature.Unlike the semi-infinite case that we previously studied, a small-amplitude saturation of the surfactant instability is possible in the absence of gravity. For certain (m, n)-ranges, the interface deflection is governed by a decoupled Kuramoto-Sivashinsky equation, which provides a source term for a linear convection-diffusion equation governing the surfactant concentration. When the diffusion term is negligible, this surfactant equation has an analytic solution which is consistent with the full numerics. Just like the interface, the surfactant wave is chaotic, but the ratio of the two waves turns out to be constant.

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Solutal Marangoni instability in layered two-phase flows

Cited by 22 publications

References 45 publications

The role of soluble surfactants in the linear stability of two-layer flow in a channel

The role of soluble surfactants in the linear stability of two-layer flow in a channel

Surfactant- and gravity-dependent instability of two-layer channel flows: linear theory covering all wavelengths. Part 2. Mid-wave regimes

Surfactant and gravity dependent instability of two-layer Couette flows and its nonlinear saturation

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