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On asymptotic methods for bifurcation and stability: confined Marangoni convection.

On asymptotic methods for bifurcation and stability: confined Marangoni convection.
On asymptotic methods for bifurcation and stability: confined Marangoni convection.
In 1979 Rosenblat developed a spectral method for studying bifurcation and stability problems. Drawing on an example from ordinary differential equations, he showed quite elegantly that, although the method bore striking similarities to the Lyapunov-Schmidt procedure, the range of validity of his method was significantly greater. In the early eighties Rosenblat, Homsy and Davis developed these pioneering ideas and extended them to partial differential equations, using Marangoni flow as an example. Several workers have subsequently employed the method to analyse other important problems.
In this paper we caution against following their precise implementation and suggest a modified procedure. We reconsider their Marangoni problem and show that the approximations they used to represent temperature and velocity are inadequate. In particular, the approximation provides a poor representation of the vertical component of the velocity as the number of members in the basis is increased. We remedy this by constructing a new set of basis functions which we show represents the solution well and, unlike the previous work, provides results in agreement with weakly nonlinear theory. The shortcomings lead to quantitative, rather than qualitative, changes in the results for the Marangoni problem considered here.
0033-5614
379-405
Shipp, D.J.
a981d0a5-4e07-4c1f-af81-3396e9f705af
Riley, D.S.
02306a8f-b410-456c-9c6b-1d083ee9010a
Wheeler, A.A.
eb831100-6e51-4674-878a-a2936ad04d73
Shipp, D.J.
a981d0a5-4e07-4c1f-af81-3396e9f705af
Riley, D.S.
02306a8f-b410-456c-9c6b-1d083ee9010a
Wheeler, A.A.
eb831100-6e51-4674-878a-a2936ad04d73

Shipp, D.J., Riley, D.S. and Wheeler, A.A. (1997) On asymptotic methods for bifurcation and stability: confined Marangoni convection. Quarterly Journal of Mechanics and Applied Mathematics, 50 (3), 379-405. (doi:10.1093/qjmam/50.3.379).

Record type: Article

Abstract

In 1979 Rosenblat developed a spectral method for studying bifurcation and stability problems. Drawing on an example from ordinary differential equations, he showed quite elegantly that, although the method bore striking similarities to the Lyapunov-Schmidt procedure, the range of validity of his method was significantly greater. In the early eighties Rosenblat, Homsy and Davis developed these pioneering ideas and extended them to partial differential equations, using Marangoni flow as an example. Several workers have subsequently employed the method to analyse other important problems.
In this paper we caution against following their precise implementation and suggest a modified procedure. We reconsider their Marangoni problem and show that the approximations they used to represent temperature and velocity are inadequate. In particular, the approximation provides a poor representation of the vertical component of the velocity as the number of members in the basis is increased. We remedy this by constructing a new set of basis functions which we show represents the solution well and, unlike the previous work, provides results in agreement with weakly nonlinear theory. The shortcomings lead to quantitative, rather than qualitative, changes in the results for the Marangoni problem considered here.

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Published date: 1997
Organisations: Applied Mathematics

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Local EPrints ID: 29092
URI: http://eprints.soton.ac.uk/id/eprint/29092
ISSN: 0033-5614
PURE UUID: e5f628af-f04f-4be5-b3bc-699159b45f44

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Date deposited: 09 Jan 2007
Last modified: 15 Mar 2024 07:28

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Contributors

Author: D.J. Shipp
Author: D.S. Riley
Author: A.A. Wheeler

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