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Surfactant- and gravity-dependent instability of two-layer channel flows: linear theory covering all wavelengths. Part 2. Mid-wave regimes
- Alexander L. Frenkel, David Halpern, Adam J. Schweiger
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- Journal:
- Journal of Fluid Mechanics / Volume 863 / 25 March 2019
- Published online by Cambridge University Press:
- 23 January 2019, pp. 185-214
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The joint effects of an insoluble surfactant and gravity on the linear stability of a two-layer Couette flow in a horizontal channel are investigated. The inertialess instability regimes are studied for arbitrary wavelengths and with no simplifying requirements on the system parameters: the ratio of thicknesses of the two fluid layers; the viscosity ratio; the base shear rate; the Marangoni number $Ma$; and the Bond number $Bo$. As was established in the first part of this investigation (Frenkel, Halpern & Schweiger, J. Fluid Mech., vol. 863, 2019, pp. 150–184), a quadratic dispersion equation for the complex growth rate yields two, largely continuous, branches of the normal modes, which are responsible for the flow stability properties. This is consistent with the surfactant instability case of zero gravity studied in Halpern & Frenkel (J. Fluid Mech., vol. 485, 2003, pp. 191–220). The present paper focuses on the mid-wave regimes of instability, defined as those having a finite interval of unstable wavenumbers bounded away from zero. In particular, the location of the mid-wave instability regions in the ($Ma$, $Bo$)-plane, bounded by their critical curves, depending on the other system parameters, is considered. The changes of the extremal points of these critical curves with the variation of external parameters are investigated, including the bifurcation points at which new extrema emerge. Also, it is found that for the less unstable branch of normal modes, a mid-wave interval of unstable wavenumbers may sometimes coexist with a long-wave one, defined as an interval having a zero-wavenumber endpoint.
Surfactant- and gravity-dependent instability of two-layer channel flows: linear theory covering all wavelengths. Part 1. ‘Long-wave’ regimes
- Alexander L. Frenkel, David Halpern, Adam J. Schweiger
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- Journal:
- Journal of Fluid Mechanics / Volume 863 / 25 March 2019
- Published online by Cambridge University Press:
- 23 January 2019, pp. 150-184
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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 $Ma\downarrow 0$. Regimes have been uncovered with crossings of the two dispersion curves, their reconnections at the point of crossing and separations as $Bo$ changes. Due to the availability of the explicit forms for the growth rates, in many instances the numerical results are corroborated with analytical asymptotics.
Surfactant and gravity dependent instability of two-layer Couette flows and its nonlinear saturation
- Alexander L. Frenkel, David Halpern
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- Journal:
- Journal of Fluid Mechanics / Volume 826 / 10 September 2017
- Published online by Cambridge University Press:
- 03 August 2017, pp. 158-204
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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 $-\infty$ to $\infty$. The other parameters are the top-to-bottom thickness ratio $n$, which is restricted to $n\geqslant 1$ 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.
Nonlinear evolution, travelling waves, and secondary instability of sheared-film flows with insoluble surfactants
- DAVID HALPERN, ALEXANDER L. FRENKEL
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- Journal:
- Journal of Fluid Mechanics / Volume 594 / 10 January 2008
- Published online by Cambridge University Press:
- 14 December 2007, pp. 125-156
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The nonlinear development of the interfacial-surfactant instability is studied for the semi-infinite plane Couette film flow. Disturbances whose spatial period is close to the marginal wavelength of the long-wave instability are considered first. Appropriate weakly nonlinear partial differential equations (PDEs) which couple the disturbances of the film thickness and the surfactant concentration are obtained from the strongly nonlinear lubrication-approximation PDEs. In a rescaled form each of the two systems of PDEs is controlled by a single parameter C, the ‘shear-Marangoni number’. From the weakly nonlinear PDEs, a single Stuart–Landau ordinary differential equation (ODE) for an amplitude describing the unstable fundamental mode is derived. By comparing the solutions of the Stuart–Landau equation with numerical simulations of the underlying weakly and strongly nonlinear PDEs, it is verified that the Stuart–Landau equation closely approximates the small-amplitude saturation to travelling waves, and that the error of the approximation converges to zero at the marginal stability curve. In contrast to all previous stability work on flows that combine interfacial shear and surfactant, some analytical nonlinear results are obtained. The Hopf bifurcation to travelling waves is supercritical for C < Cs and subcritical for C > Cs, where Cs is approximately 0.29. This is confirmed with a numerical continuation and bifurcation technique for ODEs. For the subcritical cases, there are two values of equilibrium amplitude for a range of C near Cs, but the travelling wave with the smaller amplitude is unstable as a periodic orbit of the associated dynamical system (whose independent variable is the spatial coordinate). By using the Bloch (‘Floquet’) disturbance modes in the linearized PDEs, it transpires that all the small-amplitude travelling-wave equilibria are unstable to sufficiently long-wave disturbances. This theoretical result is confirmed by numerical simulations which invariably show the large-amplitude saturation of the disturbances. In view of this secondary instability, the existence of small-amplitude periodic solutions (on the real line) bifurcating from the uniform flow at the marginal values of the shear-Marangoni number does not contradict the earlier conclusions that the interfacial-surfactant instability has a strongly nonlinear character, in the sense that there are no small-amplitude attractors such that the entire evolution towards them is captured by weakly-nonlinear equations. This suggests that, in general, for flowing-film instabilities that have zero wavenumber at criticality, the saturated disturbance amplitudes do not always have to decrease to zero as the control parameter approaches its value at criticality.
Destabilization of a creeping flow by interfacial surfactant: linear theory extended to all wavenumbers
- DAVID HALPERN, ALEXANDER L. FRENKEL
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- Journal:
- Journal of Fluid Mechanics / Volume 485 / 25 May 2003
- Published online by Cambridge University Press:
- 24 June 2003, pp. 191-220
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Creeping flow of a two-layer system with a monolayer of an insoluble surfactant on the interface is considered. The linear-stability theory of plane Couette–Poiseuille flow is developed in the Stokes approximation. To isolate the Marangoni effect, gravity is excluded. The shear-flow instability due to the interfacial surfactant, uncovered earlier for long waves only (Frenkel & Halpern 2002), is studied with inclusion of all wavelengths, and over the entire parameter space of the Marangoni number $M$, the viscosity ratio $m$, the interfacial velocity shear $s$, and the thickness ratio $n$ (${\ge}\,1$). The complex wave speed of normal modes solves a quadratic equation, and the growth rate function is continuous at all wavenumbers and all parameter values. If $M\,{>}\,0$, $s\,{\ne}\,0$, $m\,{<}\,n^2$, and $n\,{>}\,1$, the small disturbances grow provided they are sufficiently long wave. However, the instability is not long wave in the following sense: the unstable waves are not necessarily much longer than the smaller of the two layer thicknesses. On the other hand, there are parametric regimes for which the instability has a mid-wave character, the flow being stable at both sufficiently large and small wavelengths and unstable in between. The critical (instability-onset) manifold in the parameter space is investigated. Also, it is shown that for certain parametric limits the convergence of the dispersion function is non-uniform with respect to the wavenumber. This is used to explain the parametric discontinuities of the long-wave growth-rate exponents found earlier.
Saturated Rayleigh–Taylor instability of an oscillating Couette film flow
- DAVID HALPERN, ALEXANDER L. FRENKEL
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- Journal:
- Journal of Fluid Mechanics / Volume 446 / 10 November 2001
- Published online by Cambridge University Press:
- 23 October 2001, pp. 67-93
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The nonlinear stability of a two-fluid system consisting of a viscous film bounded above by a heavier and thicker layer, between two horizontal plates, with one of the plates oscillating horizontally about a fixed position, is investigated. An evolution equation governing the thickness of the viscous film is derived. Numerical simulations of this equation on extended spatial intervals demonstrate nonlinear small-amplitude saturation of the Rayleigh–Taylor instability in certain parametric regimes. In the low-frequency time-asymptotic regimes, the averaged properties of the extensive spatio-temporal chaos are not steady, but rather oscillate in time. A quasi-equilibrium theory is proposed in which the low-frequency results are interpreted by building upon the notions developed earlier for the simpler case of a non-oscillatory film governed by the classical, constant-coefficient Kuramoto–Sivashinsky equation. In contrast, the higher-frequency solutions exhibit piecewise linear profiles that have never been encountered in simulations of non-oscillatory films. The amplitude as a function of frequency has a single minimum point which is of order one. Also, preliminary results of numerical simulations of film evolution are given for the large-amplitude parametric regimes. At some parameter values, rupture is observed, similar to the case with no base flow; in other regimes the basic flows succeeds in preventing rupture. The complete characterization of the factors responsible for the particular asymptotic fate of the film, rupture or no rupture, remains an open question.