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Shooting Method for Good Estimates of the Eigenvalue in the Rayleigh-Bénard-Marangoni Convection Problem With General Boundary Conditions on Velocity and Temperature

[+] Author Affiliations
P. G. Siddheshwar

Bangalore University, Bangalore, India

B. R. Revathi

Nitte Meenakshi Institute of Technology, Bangalore, India

Paper No. IMECE2009-12761, pp. 915-924; 10 pages
  • ASME 2009 International Mechanical Engineering Congress and Exposition
  • Volume 9: Heat Transfer, Fluid Flows, and Thermal Systems, Parts A, B and C
  • Lake Buena Vista, Florida, USA, November 13–19, 2009
  • Conference Sponsors: ASME
  • ISBN: 978-0-7918-4382-6 | eISBN: 978-0-7918-3863-1
  • Copyright © 2009 by ASME


The paper presents a numerical study of Rayleigh-Bénard-Marangoni convection in a Newtonian liquid that is bounded on the underside by a permeable boundary and on the upperside by a free boundary. General boundary conditions arise in the study on both the velocity and temperature and hence the problem is not amenable to analytical treatment. First the single-term Galerkin technique is used to obtain an estimate of the critical value of the Marangoni and wave numbers as a function of Rayleigh, Darcy and Biot numbers. The estimates are then improved upon by the shooting method which is based on the Runge-Kutta-Fehlberg and Newton-Raphson methods. The results are documented in 4 figures and a table, and some conclusions are made on comparing the present results with those of the classical Rayleigh-Bénard, Bénard-Marangoni and Rayleigh-Bénard-Marangoni convection problems.

Copyright © 2009 by ASME



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