Effect of inertia on the Marangoni instability of two-layer channel flow, Part II: normal-mode analysis

Blyth, Mark; Pozrikidis, C.

doi:10.1007/s10665-004-3691-z

Cited by 32 publications

(38 citation statements)

References 15 publications

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“…An increase in the Reynolds number stabilises the flow if Λ < 0 and destabilises the flow if Λ > 0. This result is consistent with previous studies that demonstrate that inertial effects increase the growth of the Marangoni instability to produce new regions of instability, as long as the film is more viscous than the overlying fluid (Pozrikidis 2004;Blyth & Pozrikidis 2004b;Frenkel & Halpern 2005). This is also confirmed by figure 3(a) for the unstable case Λ = 0.25, Γ 0 = 1, η = 1, where the growth rates are shown as functions of the wavenumber for R e = 0, 25, 50, 100.…”

Section: Linear Stability Analysissupporting

confidence: 93%

“…The effect of inertia on the Marangoni instability of the interface was studied numerically by Pozrikidis (2004) and by a linear stability analysis in Blyth & Pozrikidis (2004b) for arbitrary wavelength perturbations based on the Orr-Sommerfeld equation. The role of inertia was further investigated by Frenkel & Halpern (2005) for perturbations of long wavelength.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia

Kalogirou

Papageorgiou

2016

J. Fluid Mech.

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The nonlinear stability of immiscible two-fluid Couette flows in the presence of inertia is considered. The interface between the two viscous fluids can support insoluble surfactants and the interplay between the underlying hydrodynamic instabilities and Marangoni effects is explored analytically and computationally in both two and three dimensions. Asymptotic analysis when one of the layers is thin relative to the other yields a coupled system of nonlinear equations describing the spatiotemporal evolution of the interface and its local surfactant concentration. The system is nonlocal and arises by appropriately matching solutions of the linearised Navier-Stokes equations in the thicker layer to the solution in the thin layer. The scaled models are used to study different physical mechanisms by varying the Reynolds number, the viscosity ratio between the two layers, the total amount of surfactant present initially and a scaled Péclet number measuring diffusion of surfactant along the interface. The linear stability of the underlying flow to two-and three-dimensional disturbances is investigated and a Squire's type theorem is found to hold when inertia is absent. When inertia is present, three-dimensional disturbances can be more unstable than two-dimensional ones and so Squire's theorem does not hold. The linear instabilities are followed into the nonlinear regime by solving the evolution equations numerically; this is achieved by implementing highly accurate linearly implicit schemes in time with spectral discretisations in space. Numerical experiments for finite Reynolds numbers indicate that for two-dimensional flows the solutions are mostly nonlinear travelling waves of permanent form, even though these can lose stability via Hopf bifurcations to time-periodic travelling waves. As the length of the system (that is the wavelength of periodic waves) increases, the dynamics become more complex and include time-periodic, quasi-periodic as well as chaotic fluctuations. It is also found that one-dimensional interfacial travelling waves of permanent form can become unstable to spanwise perturbations for a wide range of parameters, producing three-dimensional flows with interfacial profiles that are two-dimensional and travel in the direction of the underlying shear. Nonlinear flows are also computed for parameters which predict linear instability to three-dimensional disturbances but not two-dimensional ones. These are found to have a one-dimensional interface in a rotated frame with respect to the direction of the underlying shear and travel obliquely without changing form.

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Section: Linear Stability Analysissupporting

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia

Kalogirou

Papageorgiou

2016

J. Fluid Mech.

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“…Wei 9 provided a discussion of the underlying instability mechanism taken from the viewpoint of vorticity. The role of inertia on the interfacial instability was investigated by Pozrikidis 10 and Blyth and Pozrikidis 11 for perturbations of arbitrary wavelength at arbitrary Reynolds number and by Frenkel and Halpern 12 for long wavelength perturbations. The results showed that inertia tends to widen the range of unstable wavenumbers.…”

Section: Yihmentioning

confidence: 99%

Nonlinear development of two-layer Couette–Poiseuille flow in the presence of surfactant

2010

Self Cite

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The two-dimensional nonlinear evolution of the interface between two superposed layers of viscous fluid moving in a channel in the presence of an insoluble surfactant is examined. A pair of coupled weakly nonlinear equations is derived describing the interfacial and surfactant dynamics when one of the two fluid layers is very thin in comparison to the other. In contrast to previous work, the dynamics in the thin film are coupled to the dynamics in the thicker layer through a nonlocal integral term. For asymptotically small Reynolds number, the flow in the thicker layer is governed by the Stokes equations. A linearized analysis confirms the linear instability identified by previous workers and it is proven that the film flow is linearly unstable if the undisturbed surfactant concentration exceeds a threshold value. Numerical simulations of the weakly nonlinear equations reveal the existence of finite amplitude traveling-wave solutions. For order one Reynolds number, the flow in the thicker layer is governed by the linearized Navier-Stokes equations. In this case the weakly nonlinear film dynamics are more complex and include the possibility of periodic traveling-waves and chaotic flow.

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“…Since Frenkel and Halpern's discovery, there has been a number of studies investigating the problem further: Blyth & Pozrikidis 3,13 analysed the destabilising effect of inertia and performed numerical simulations in short domains, Frenkel & Halpern 14 solved the problem under the assumption of both layers being thin, while Bassom et al 4 presented a local model valid in the case where one of the fluid layers is much thinner than the other. The work of Bassom et al 4 was extended by Kalogirou et al 15 and Kalogirou & Papageorgiou 5 , who solved the full nonlocal system and also examined the effect of inertia on nonlinear saturated solutions.…”

Section: Introductionmentioning

confidence: 99%

Instability of two-layer film flows due to the interacting effects of surfactants, inertia, and gravity

Kalogirou

2018

Physics of Fluids

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We consider a two-fluid shear flow where the interface between the two fluids is coated with an insoluble surfactant. An asymptotic model is derived in the thin-layer approximation, consisting of a set of nonlinear PDEs describing the evolution of the film and surfactant disturbances at the interface. The model includes important physical effects such as Marangoni forces (caused by the presence of surfactant), inertial forces arising in the thick fluid layer, as well as gravitational forces. The aim of this study is to investigate the effect of density stratification or gravity -represented through the Bond number Bo -on the flow stability, and the interplay between the different (de)stabilisation mechanisms. It is found that gravity can either stabilise or destabilise the interface (depending on fluid properties), but not always as intuitively anticipated. Different travelling-wave branches are presented for varying Bo and the destabilising mechanism associated with the Marangoni forces is discussed.

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Effect of inertia on the Marangoni instability of two-layer channel flow, Part II: normal-mode analysis

Cited by 32 publications

References 15 publications

Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia

Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia

Nonlinear development of two-layer Couette–Poiseuille flow in the presence of surfactant

Instability of two-layer film flows due to the interacting effects of surfactants, inertia, and gravity

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