Bifurcation Analysis of Cellular Interfaces in Unidirectional Solidification of a Dilute Binary Mixture

Impey, M. D.; Riley, D. S.; Wheeler, A. A.

doi:10.1137/0153006

Cited by 5 publications

(2 citation statements)

References 21 publications

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“…To overcome this difficulty at M c , the aspect ratio of the rectangular solution domain is used to unfold the multiple bifurcation point (see Impey et al [10]). To overcome this difficulty at M c , the aspect ratio of the rectangular solution domain is used to unfold the multiple bifurcation point (see Impey et al [10]).…”

Section: Methodsmentioning

confidence: 99%

On long-wave morphological instabilities in directional solidification

Skeldon

McFadden

Impey³

et al. 1995

Eur. J. Appl. Math

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A binary liquid that undergoes directional solidification is susceptible to morphological instabilities which cause the solid/liquid interface to change from a planar to a cellular state. This paper presents a numerical study of a class of long-wave equations that describe the evolution of interface morphology. We find new bifurcation points, new solution branches, and the existence of inverted hexagonal nodes and cells.

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Section: Methodsmentioning

confidence: 99%

On long-wave morphological instabilities in directional solidification

Skeldon

McFadden

Impey³

et al. 1995

Eur. J. Appl. Math

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“…Moreover, those PDEs are often invariant under some group of Euclidean transformations-often the whole Euclidean group. As well as convection (see Riley & Winters, 1989;Impey et al, 1990;Neveling & Dangelmayr, 1988;Gomes, 1992), examples of such systems include: the Couette-Taylor experiment (Golubitsky & Stewart, 1986 and references therein); the Faraday crispation experiment (Crawford, 199 1 a,b;Crawford et al, 1993); the Kuramoto-Sivashinsky equation (Ashwin, 1991); elastic buckling (Healey & Kielhofer, 199 la,b, 1992); and solidification of a binary fluid (Impey et al, 1993). Downloaded by [Virginia Tech Libraries] at 03:40 15 March 2015 'Hidden' symmetries arise as follows.…”

Section: Introductionmentioning

confidence: 98%

Hidden symmetries and pattern formation in Lapwood convection

Impey¹,

Roberts

Stewart

1996

Dynamics and Stability of Systems

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We study Lapwood convection (convection of afluid in aporous medium) on a two-dimensional rectangular domain. The linearized eigenmodes are symmetric p x q cellular patterns, which we call (p, q) modes. Numerical calculations of the branching structure near mode interaction points have derived bifurcation diagrams for the (3, 1)/(1, 1) and (3, 1)/(2, 2) mode interactions which are non-generic, even when the rectangular symmety of the domain is taken into account. This has raised questions about the accuracy of the numerical method used, a finite-element Galerkin approximation implemented using Harwell's E N T W F E code. We show that this apparent lack of genericity is partly a consequence of 'hidden' translational symmetries, which arise when the problem is extended to one with periodic bounda y conditions. This extension procedure has become standard for partial diferential equations (PDEs) with Neumann or Dirichlet bounday conditions, and it reveals restrictions on the Liapunou-Schmidt reduced bifurcation equations and the resulting singulan'ty-theoretic normal forms. Its application to Lapwood convection is unusual in that the PDE involves a mixture of both Neumann and Dirichlet bounday conditions. Specifically, on the vertical sidewalls the stream function satisfies Dirichlet bounda y conditions (is zero), but the temperature satisfies Neumann (no-flux) bounda y conditions. Nevertheless, we show that for abstract group-theoretical reasons the same symmety constraints that occur for purely Neumann bounda y conditions are imposed on the LiapunowSchmidt reduced bzfurcation equations, and therefore the same list of normal forms is valid. The hidden symmetries force certain terms in the reduced bzfurcation equations to be zero and change the generic branching geomety. With the aid of M A C S Y M A , we determine a small number of low-order coeficients of the reduced bzfurcation equations which are needed to find the correct normal form. We show that in some cases the normal form is more degenerate than might be anticipated, but that when these degeneracies are taken into account the resulting branching geomety reproduces that found in the earlier numerical approach. In particular, we obtain an analytic vindication of the numerical method. 0268-1 110/96/030155-38 Q 1996 Journals Oxford Ltd Downloaded by [Virginia Tech Libraries] at 03:40 15 March 2015 156 M. IMPEY ET AL.4

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