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

Kalogirou, A.; Papageorgiou, Demetrios T.

Authors

Demetrios T. Papageorgiou



Abstract

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 spatiooral evolution of the interface and its local surfactant concentration. The system is non-local 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 Peclet 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 becomes more complex and includes 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.

Citation

Kalogirou, A., & Papageorgiou, D. T. (2016). Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia. Journal of Fluid Mechanics, 802, 5-36. https://doi.org/10.1017/jfm.2016.429

Journal Article Type Article
Acceptance Date Jun 21, 2016
Online Publication Date Aug 1, 2016
Publication Date Sep 10, 2016
Deposit Date Jun 4, 2019
Publicly Available Date Mar 29, 2024
Journal Journal of Fluid Mechanics
Print ISSN 0022-1120
Electronic ISSN 1469-7645
Publisher Cambridge University Press
Peer Reviewed Peer Reviewed
Volume 802
Pages 5-36
DOI https://doi.org/10.1017/jfm.2016.429
Public URL https://nottingham-repository.worktribe.com/output/2138123
Publisher URL https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/nonlinear-dynamics-of-surfactantladen-twofluid-couette-flows-in-the-presence-of-inertia/CFD6699856258C3E080A2E4527AB5AB1

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