Benjamin-feir and eckhaus instabilities with galilean invariance: the case of interfacial waves in viscous shear flows

This paper presents an interface tracking method for modelling the flow of immiscible metallic liquids in mixing processes. The methodology can provide an insight into mixing processes for studying the fundamental morphology development mechanisms for immiscible interfaces. The volume-of-fluid (VOF) method is adopted in the present study, following a review of various modelling approaches for immiscible fluid systems. The VOF method employed here utilises the piecewise linear for interface construction scheme as well as the continuum surface force algorithm for surface force modelling. A model coupling numerical and experimental data is established. The main flow features in the mixing process are investigated. It is observed that the mixing of immiscible metallic liquids is strongly influenced by the viscosity of the system, shear forces and turbulence. The numerical results show good qualitative agreement with experimental results, and are useful for optimisating the design of mixing casting processes.

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“…The process of drop formation displays a good qualitative agreement with hydrodynamic characteristics observed experimentally [58][59][60].…”

Section: Tracking Droplet Formationsupporting

confidence: 81%

Modelling the interfacial flow of two immiscible liquids in mixing processes

Tang

Wrobel

2005

International Journal of Engineering Science

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“…Previous work has required complex coefficients (Riecke 1992a, 1992b, Sakaguchi 1993), a subcritical bifurcation (Herrero andRiecke 1995, Sakaguchi andBrand 1996) or both. Our amplitude equations apply when the bifurcation from the basic state is stationary; the case of oscillatory bifurcation to travelling-wave solutions has been analysed in a series of papers by Riecke and co-workers (Riecke 1992a, 1992b, 1996, Herrero and Riecke 1995 and by others (Sakaguchi 1993, Barthelet andCharru 1998), who find a variety of localised and front-type solutions. They find that the coupling to a mean field (our B) can have a significant effect on solutions and their stability, and can stabilise localised travelling-wave trains, and holes.…”

Section: Discussionmentioning

confidence: 99%

“…In surface tension-driven convection, VanHook et al (1995VanHook et al ( , 1997 have shown how the presence of a large-scale surface-deformational mode can lead to dry spots and high spots. In a two-layer viscous shear flow Charru 1998, Charru andBarthelet 1999), finite-amplitude interfacial waves become unstable through coupling to a large-scale mode which may be traced back to conservation of mass of each fluid. In liquid crystals (Hidaka et al 1997), complicated spatio-temporal convection patterns are traced to the presence of a large-scale mode arising in the degeneracy of the system.…”

Section: Introductionmentioning

confidence: 99%

Pattern formation with a conservation law

2000

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Abstract. Pattern formation in systems with a conserved quantity is considered by studying the appropriate amplitude equations. The conservation law leads to a large-scale neutral mode that must be included in the asymptotic analysis for pattern formation near onset. Near a stationary bifurcation, the usual Ginzburg-Landau equation for the amplitude of the pattern is then coupled to an equation for the large-scale mode. These amplitude equations show that for certain parameters all roll-type solutions are unstable. This new instability differs from the Eckhaus instability in that it is amplitude-driven and is supercritical. Beyond the stability boundary, there exist stable stationary solutions in the form of strongly modulated patterns. The envelope of these modulations is calculated in terms of Jacobi elliptic functions and, away from the onset of modulation, is closely approximated by a sech profile. Numerical simulations indicate that as the modulation becomes more pronounced, the envelope broadens. A number of applications are considered, including convection with fixed-flux boundaries and convection in a magnetic field, resulting in new instabilities for these systems.

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“…The question is then: at what order should the mean flow be introduced into the expansion? Regardless of the choice that is made, the amplitude equations contain terms of mixed asymptotic order (see, e.g., the equations of Bernoff 1994 and the analogous equations derived by Charru 1998 andBarthelet 1999 for interfacial waves). Ultimately, however, the point is moot because we merely use the amplitude equations to suggest other asymptotic scalings, then carry out calculations with these (consistent) scalings.…”

Section: The Small-angle Instabilitymentioning

confidence: 99%

Instability of rotating convection

Cox

Matthews

2000

J. Fluid Mech.

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Convection rolls in a rotating layer can become unstable to the Küppers–Lortz instability. When the horizontal boundaries are stress free and the Prandtl number is finite, this instability diverges in the limit where the perturbation rolls make a small angle with the original rolls. This divergence is resolved by taking full account of the resonant mode interactions that occur in this limit: it is necessary to include two roll modes and a large-scale mean flow in the perturbation. It is found that rolls of critical wavelength whose amplitude is of order ε are always unstable to rolls oriented at an angle of order ε2/5. However, these rolls are unstable to perturbations at an infinitesimal angle if the Taylor number is greater than 4π4. Unlike the Küppers–Lortz instability, this new instability at infinitesimal angles does not depend on the direction of rotation; it is driven by the flow along the axes of the rolls. It is this instability that dominates in the limit of rapid rotation. Numerical simulations confirm the analytical results and indicate that the instability is subcritical, leading to an attracting heteroclinic cycle. We show that the small-angle instability grows more rapidly than the skew-varicose instability.

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Benjamin-feir and eckhaus instabilities with galilean invariance: the case of interfacial waves in viscous shear flows

Cited by 15 publications

References 19 publications

Modelling the interfacial flow of two immiscible liquids in mixing processes

Modelling the interfacial flow of two immiscible liquids in mixing processes

Pattern formation with a conservation law

Instability of rotating convection

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