On nonlinear convection in mushy layers. Part 2. Mixed oscillatory and stationary modes of convection

Riahi, Daniel N.

doi:10.1017/s0022112004000849

Cited by 13 publications

(6 citation statements)

References 43 publications

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“…A number of linear-stability analyses (Worster 1992;Emms & Fowler 1994;Anderson & Worster 1996), weakly nonlinear analyses (Amberg & Homsy 1993;Anderson & Worster 1995;Chung & Chen 2000;Riahi 2002Riahi , 2004Guba & Worster 2006;Roper, Davis & Voorhees 2008) and nonlinear computations (Schulze & Worster 1998, 1999Chung & Worster 2002) of convection in mushy layers have been carried out. Among these, the linear-stability analysis of Anderson & Worster (1996) and the weakly nonlinear analyses of Anderson & Worster (1995) and Guba & Worster (2006) are closely related to the present study.…”

Section: Introductionmentioning

confidence: 99%

Interactions between steady and oscillatory convection in mushy layers

Guba

Worster

2010

J. Fluid Mech.

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We study nonlinear, two-dimensional convection in a mushy layer during solidification of a binary mixture. We consider a particular limit in which the onset of oscillatory convection just precedes the onset of steady overturning convection, at a prescribed aspect ratio of convection patterns. This asymptotic limit allows us to determine nonlinear solutions analytically. The results provide a complete description of the stability of and transitions between steady and oscillatory convection as functions of the Rayleigh number and the compositional ratio. Of particular focus are the effects of the basic-state asymmetries and non-uniformity in the permeability of the mushy layer, which give rise to abrupt (hysteretic) transitions in the system. We find that the transition between travelling and standing waves, as well as that between standing waves and steady convection, can be hysteretic. The relevance of our theoretical predictions to recent experiments on directionally solidifying mushy layers is also discussed.

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Section: Introductionmentioning

confidence: 99%

Interactions between steady and oscillatory convection in mushy layers

Guba

Worster

2010

J. Fluid Mech.

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“…We noted that, at the relevant scaling established by Anderson & Worster (1995, 1996, the linear system of the problem exhibits both stationary and oscillatory modes of convection at comparable critical values of the Rayleigh number for the onset of convection. Hence, to determine valid analytical results, which could also be compared with the available experimental results (Tait, Jahrling & Jaupart 1992), a nonlinear theory needed to be developed and analysed, which could account for both stationary and oscillatory modes of convection (Riahi 2002). However, prior to the present study, no nonlinear investigation of oscillatory convection alone in mushy layers had been carried out, and, thus, to understand its qualitative features the present investigation was necessary.…”

Section: Introductionmentioning

confidence: 99%

On nonlinear convection in mushy layers Part 1. Oscillatory modes of convection

Riahi

2002

J. Fluid Mech.

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We consider the problem of nonlinear convection in horizontal mushy layers during the solidification of binary alloys. We analyse the oscillatory modes of convection in the form of two-and three-dimensional travelling and standing waves. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the solutions to the nonlinear problem by using a perturbation technique, and the stability of two-and three-dimensional solutions in the form of simple travelling waves, general travelling waves and standing waves is investigated. The results of the stability and the nonlinear analyses indicate that supercritical simple travelling rolls are stable over most of the studied range of parameter values, while supercritical standing rolls can be stable only over some small range of parameter values, where the simple travelling rolls are unstable. The results of the investigation of the onset of plume convection and chimney formation leading to the occurrence of freckles in the alloy crystal indicate that the chimney of the plume can be generated internally or near the lower boundary of the mushy layer. The roles of a Stefan number, a permeability parameter and a concentration ratio on the flow instability in both linear and nonlinear regimes are also determined.

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“…The oscillatory instability identified by Anderson and Worster (1996) is caused by physical interactions internal to the mush itself, in contrast to an earlier oscillatory instability studied by Chen et al (1994) which owed its origin to double-diffusive convection in the solution above the mush. The theory of these oscillatory modes has been developed elsewhere (Riahi 2002, 2004, Guba and Worster 2006 and we shall discuss them no further here.…”

Section: Introductionmentioning

confidence: 99%

Patterns of convection in solidifying binary solutions

Keating

Spiegel

Worster

2011

Geophysical & Astrophysical Fluid Dynamics

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During the solidification of two-component solutions a two-phase mushy layer often forms consisting of solid dendritic crystals and solution in thermal equilibrium. Here, we extend previous weakly nonlinear analyses of convection in mushy layers to the derivation and study of a pattern equation by including a continuous spectrum of horizontal wave vectors in the development. The resulting equation is of the Swift-Hohenberg form with an additional quadratic term that destroys the up-down symmetry of the pattern as in other studies of nonBoussinesq convective pattern formation. In this case, the loss of symmetry is rooted in a nonBoussinesq dependence of the permeability on the solid-fraction of the mushy layer. We also study the motion of localized chimney structures that results from their interactions in a simplified one-dimensional approximation of the full pattern equation.

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On nonlinear convection in mushy layers. Part 2. Mixed oscillatory and stationary modes of convection

Abstract: This paper presents part 2 of the problem of nonlinear convection in horizontal mushy layers during the solidification of binary alloys. Part 1 of the problem dealt with only the oscillatory modes of convection, which was investigated very recently (Riahi2002). In

Cited by 13 publications

References 43 publications

Interactions between steady and oscillatory convection in mushy layers

Interactions between steady and oscillatory convection in mushy layers

On nonlinear convection in mushy layers Part 1. Oscillatory modes of convection

Patterns of convection in solidifying binary solutions

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