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

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Title: On nonlinear convection in mushy layers: Part 2. Mixed oscillatory and stationary modes of convection
Author(s): Riahi, Daniel N.
Subject(s): nonlinear 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 were investigated very recently (2002). In the present paper we consider a particular range of the parameter values where the critical values of the scaled Rayleigh number R for the onset of oscillatory an dstationary convection are sufficiently close, and we develop and analyze a nonlinear theory in such a parameter regime which takes into account both stationary an doscillatory modes of convection. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the weakly nonlinear solutions due to mixed oscillatory and stationary modes. Twelve new classes of these solutions, which are referred to hearafter as mixed solutions, are detected. The oscillatory compomemt of any mixed solution depends on a parameter b, which lies in the range |b| less than or equal to 0.5. The standing wave component corresponds to b = 0, the simple travelling components correspond to |b| = 0.5, and the general travelling components correspond to 0 < |b| < 0.5. The preferred mixed solutions, which correspond to the lowest value of R, are found to be subcritical and composed of general travelling rectangular mode and stationary hexagonal mode, where the two wave number vectors of any oscillatory rectangular cell are along two wave number vectors of a corresponding stationary hexagonal cell. In contrast to the simple type of either oscillatory solution, where the subcritical three-diomensional pattern cannot be possible, or stationary solutions, where down-hexagons with down-flow at the cells' centers and up-flow at the cells' boundaris cannot be preferred or be predicted in the domain of the experimental observation for such pattern, the preferred mixed solutions with down-flow at the cells' centers and up-flow at the cells' boundaries are predicted in the domain of the experimental observation.
Issue Date: 2002-09
Publisher: Department of Theoretical and Applied Mechanics (UIUC)
Series/Report: TAM Reports 1007
Genre: Technical ReportArticle
Type: Text
Language: English
URI: http://hdl.handle.net/2142/275
ISSN: 0073-5264
Publication Status: published or submitted for publication
Peer Reviewed: is peer reviewed
Date Available in IDEALS: 2007-03-08
Is Version Of: Published as: Daniel N. Riahi. On nonlinear convection in mushy layers: Part 2. Mixed oscillatory and stationary modes of convection. Journal of Fluid Mechanics, Vol. 517, 2004, pp.71- 101. DOI 10.1017/S0022112004000849. Copyright 2004 Cambridge University Press. http://hdl.handle.net/2142/988.
 

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  • Theoretical and Applied Mechanics (TAM) Technical Reports
    TAM technical reports include manuscripts intended for publication, theses judged to have general interest, notes prepared for short courses, symposia compiled from outstanding undergraduate projects, and reports prepared for research-sponsoring agencies.

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