The effects of insoluble surfactants on the linear stability of a core–annular flow (original) (raw)
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On the flow-induced Marangoni instability due to the presence of surfactant
Journal of Fluid Mechanics, 2005
The flow-induced Marangoni instability due to the presence of surfactant is examined for long-wavelength perturbations. A unified view of the underlying mechanisms is provided through revisiting both falling film and two-fluid Couette flow systems. The analysis is performed by inspecting the corresponding coupled set of evolution equations for the interface and surfactant concentration perturbations. While both systems appear to have very similar sets of equations consisting of base flows and Marangoni effects, the origins of stability/instability are identified and illustrated from a viewpoint of vorticity. The base flow rearranges the surfactant distribution and the induced Marangoni flow tends to stimulate the interface's growth. But this destabilizing effect is reduced by effects combining the interface travelling motions and the Marangoni recoil. The competition between these opposing effects determines the system stability, and is elucidated using equations in concert with observations from initial value problems. Moreover, a criterion for the onset of instability can be established in line with the same rationale. The present work not only furnishes a lucid way to clarify the instability mechanisms, but also complements previous studies. Extension to the weakly nonlinear regime is also discussed.
Marangoni Instability in a Fluid Layer with Insoluble Surfactant
World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 2011
The Marangoni convective instability in a horizontal fluid layer with the insoluble surfactant and nondeformable free surface is investigated. The surface tension at the free surface is linearly dependent on the temperature and concentration gradients. At the bottom surface, the temperature conditions of uniform temperature and uniform heat flux are considered. By linear stability theory, the exact analytical solutions for the steady Marangoni convection are derived and the marginal curves are plotted. The effects of surfactant or elasticity number, Lewis number and Biot number on the marginal Marangoni instability are assessed. The surfactant concentration gradients and the heat transfer mechanism at the free surface have stabilizing effects while the Lewis number destabilizes fluid system. The fluid system with uniform temperature condition at the bottom boundary is more stable than the fluid layer that is subjected to uniform heat flux at the bottom boundary.
Nonlinear interfacial stability of core-annular film flows in the presence of surfactants
2002
NONLINEAR INTERFACIAL STABILITY OF CORE-ANNULAR FILM FLOWS IN THE PRESENCE OF SURFACTANTS by Said A. Kas-Danouche This work is an analytical and computational study of the nonlinear interfacial instabilities found in core-annular flows in the presence of surfactants. Core-annular flows arise when two immiscible fluids (for example water and oil) are caused to flow in a pipe under the action of an axial pressure gradient. In one typical type of flow regime, the fluids arrange themselves so that the less viscous (e.g. water) lies in the region of high shear near the pipe wall, with the more viscous fluid occupying the core region. Technologically, this arrangement provides an advantage since the highly viscous fluid is lubricated by the less viscous annulus and for a given pressure gradient the core-fluid flux can be greatly increased. The stability of these flows is of fundamental scientific and practical importance. The sharp interface between the two phases can become unstable by s...
Journal of Fluid Mechanics, 2003
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 , 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 (> 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 = 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.
Microgravity Science and Technology, 2017
The paper presents the analysis of the impact of vertical periodic vibrations on the long-wavelength Marangoni instability in a liquid layer with poorly conducting boundaries in the presence of insoluble surfactant on the deformable gas-liquid interface. The layer is subject to a uniform transverse temperature gradient. Linear stability analysis is performed in order to find critical values of Marangoni numbers for both monotonic and oscillatory instability modes. Longwave asymptotic expansions are used. At the leading order, the critical values are independent on vibration parameters; at the next order of approximation we obtained the rise of stability thresholds due to vibration.
Strongly nonlinear nature of interfacial-surfactant instability of Couette flow
2006
Nonlinear stages of the recently uncovered instability due to insoluble surfactant at the interface between two fluids are investigated for the case of a creeping plane Couette flow with one of the fluids a thin film and the other one a much thicker layer. Numerical simulation of strongly nonlinear longwave evolution equations which couple the film thickness and the surfactant concentration reveals that in contrast to all similar instabilities of surfactant-free flows, no amount of the interfacial shear rate can lead to a small-amplitude saturation of the instability. Thus, the flow is stable when the shear is zero, but with non-zero shear rates, no matter how small or large (while remaining below an upper limit set by the assumption of creeping flow), it will reach large deviations from the base values-of the order of the latter or larger. It is conjectured that the time this evolution takes grows to infinity as the interfacial shear approaches zero. It is verified that the absence of small-amplitude saturation is not a singularity of the zero surface diffusivity of the interfacial surfactant.
Journal of Non-Equilibrium Thermodynamics, 2000
We consider a system that consists of a layer of an incompressible binary liquid with a deformable free surface, and a solid substrate layer heated or cooled from below. The surface tension is assumed to depend linearly on both the temperature and the solute concentration. The Soret e¤ect is taken into account. We investigate the long-wave Marangoni instability in the case of asymptotically small Lewis and Galileo numbers for finite surface tension and Biot numbers. We find both long-wave monotonic and oscillatory modes of instability in various parameter domains of the Biot and the Soret number. The weakly nonlinear analysis is carried out in the case of a specified heat flux at the rigid substrate.
Nonlinear dynamics of core-annular film flows in the presence of surfactant
Journal of Fluid Mechanics, 2009
The nonlinear stability of two-phase core-annular flow in a cylindrical pipe is studied. A constant pressure gradient drives the flow of two immiscible liquids of different viscosities and equal densities, and surface tension acts at the interface separating the phases. Insoluble surfactants are included, and we assess their effect on the flow stability and ensuing spatio-temporal dynamics. We achieve this by developing an asymptotic analysis in the limit of a thin annular layer – which is usually the relevant regime in applications – to derive a coupled system of nonlinear evolution equations that govern the dynamics of the interface and the local surfactant concentration on it. In the absence of surfactants the system reduces to the Kuramoto–Sivashinsky (KS) equation, and its modifications due to viscosity stratification (present when the phases have unequal viscosities) are derived elsewhere. We report on extensive numerical experiments to evaluate the effect of surfactants on KS...
Journal of Colloid and Interface Science, 2005
This paper analyzes the effect of surfactant on the linear stability of an annular film in a capillary undergoing a time-periodic pressure gradient force. The annular film is thin compared to the radius of the tube. An asymptotic analysis yields a coupled set of equations with timeperiodic coefficients for the perturbed fluid-fluid interface and the interfacial surfactant concentration. Wei and Rumschitzki (submitted for publication) previously showed that the interaction between a surfactant and a steady base flow could induce a more severe instability than a stationary base state. The present work demonstrates that time-periodic base flows can modify the features of the steady-flow-based instability, depending on surface tension, surfactant activity, and oscillatory frequency. For an oscillatory base flow (with zero mean), the growth rate decreases monotonically as the frequency increases. In the low-frequency limit, the growth rate approaches a maximum corresponding to the growth rate of a steady base flow having the same amplitude. In the high-frequency limit, the growth rate reaches a minimum corresponding to the growth rate in the limit of a stationary base state. The underlying mechanisms are explained in detail, and extension to other time-periodic forms is further exploited. 2004 Elsevier Inc. All rights reserved.