Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-18T06:27:01.889Z Has data issue: false hasContentIssue false
  • English
  • Français

The free surface on a liquid filling a trench heated from its side

Published online by Cambridge University Press:  29 March 2006

Daniel D. Joseph  and
Leroy Sturges
Daniel D. Joseph
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis
Leroy Sturges
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we compute the motion and the shape of the free surface on a liquid in a trench heated from its side. The analysis is based on Joseph's Lagrangian theory of domain perturbations, which is developed in general and through simple examples, chosen so as to make the comparison of the Lagrangian method with Stokes's Eulerian theory very clear. The perturbation problems are resolved analytically by application of biorthogonality conditions to a powerful set of biharmonic eigenfunctions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 69 , Issue 3 , 10 June 1975 , pp. 565 - 589
Copyright
© 1975 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Hadamard, J. 1908 Mémoire sur le problème d'analyse relatif á l'équilibre des plaques élastiques encastrées. Mémoires des Savants Etrangers, 33.Google Scholar
Hillman, A. P. & Salzer, H. E. 1943 Roots of sin z = z. Phil. Mag. 34, 575.Google Scholar
Johnson, M. W. & Little, R. W. 1965 The semi-infinite elastic strip Quart. Appl. Math. 22, 335.Google Scholar
Joseph, D. D. 1967 Parameter and domain dependence of eigenvalues of elliptic partial differential equations Arch. Rat. Mech. Anal. 24, 325–351.Google Scholar
Joseph, D. D. 1973 Domain perturbations: the higher-order theory of infinitesmal water waves. Arch. Rat. Mech. Anal. 51, 295–303.Google Scholar
Joseph, D. D. 1974 Slow motion and viscometric motion. Stability and bifurcation of the rest state of a simple fluid Arch. Rat. Mech. Anal. 56, 99–157.Google Scholar
Joseph, D. D. & Fosdick, R. L. 1973 The free surface on a liquid between cylinders rotating at different speeds. Part I Arch. Rat. Mech. Anal. 49, 321–380.Google Scholar
LEVI-CIVITA, T. 1925 Détermination rigoureuse des ondes permanentes d'ampleur finie Math. Annln, 93, 264–314.Google Scholar
Robbins, C. I. & Smith, R. C. T. 1948 A table of roots of sin z = —Z. Phil. Mag. 39, 1004–1005.Google Scholar
Sattinger, D. 1975 On the free surface of viscous fluid motion. Proc. Roy. Soc. A (to appear).Google Scholar
Smith, R. C. T. 1952 The bending of a semi-infinite strip Aust. J. Scie. Res. 5, 227–237.Google Scholar
Stoker, J. J. 1957 Water Waves. Interscience.
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8 (suppl.). (See also Scientific Papers, vol. 1. Cambridge University Press.)Google Scholar
Struik, D. J. 1926 Détermination rigoureuse des ondes irrotationnelles périodiques dans un canal á profondeur finie Math. Annln, 95, 595–634.Google Scholar
Wehausen, J. V. & Laitone, E. V. 1960 Surface Waves. Handbuch der Physik, vol. 9 (ed. S. Flugge & C. Truesdell). Springer.