On the YoungLaplace relation and the evolution of a perturbed ellipsoid
Author:
George Dassios
Journal:
Quart. Appl. Math. 72 (2014), 2132
MSC (2010):
Primary 35B35, 35Q35, 35J25
Published electronically:
November 20, 2013
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Abstract: The YoungLaplace relation states that the interface separating two fluids, develops in such a way that the difference between the outer and the inner pressure remains proportional to the mean curvature at every point of the interface. This relation guides the evolution of a free boundary. Considering the importance of the ellipsoidal surfaces as free boundaries in anisotropic evolutions, it is of great interest to have readytouse formulae for the mean curvature of a perturbed ellipsoidal surface. These formulae provide the basis for the stability analysis of free boundary value problems in Fluid Mechanics. The present work calculates the first order approximation of the local curvatures for a surface which is a perturbation of an ellipsoid.
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G.K. Batchelor.
An Introduction to Fluid Mechanics. Cambridge University Press, Cambridge, 1967.
 [2]
George
Dassios, Ellipsoidal harmonics, Encyclopedia of Mathematics
and its Applications, vol. 146, Cambridge University Press, Cambridge,
2012. Theory and applications. MR
2977792
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E.W. Hobson.
The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, U.K., Cambridge, first edition, 1931.
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E.
T. Whittaker and G.
N. Watson, A course of modern analysis, Cambridge Mathematical
Library, Cambridge University Press, Cambridge, 1996. An introduction to
the general theory of infinite processes and of analytic functions; with an
account of the principal transcendental functions; Reprint of the fourth
(1927) edition. MR 1424469
(97k:01072)
 [1]
 G.K. Batchelor.
An Introduction to Fluid Mechanics. Cambridge University Press, Cambridge, 1967.
 [2]
 George Dassios, Ellipsoidal harmonics. Theory and applications, Encyclopedia of Mathematics and its Applications, vol. 146, Cambridge University Press, Cambridge, 2012. MR 2977792
 [3]
 E.W. Hobson.
The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, U.K., Cambridge, first edition, 1931.
 [4]
 E.T. Whittaker and G.N. Watson.
A Course of Modern Analysis. Cambridge University Press, third edition, 1920. MR 1424469 (97k:01072)
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Additional Information
George Dassios
Affiliation:
Department of Chemical Engineering, University of Patras and ICEHT/FORTH, Patras, Greece
DOI:
http://dx.doi.org/10.1090/S0033569X2013013217
PII:
S 0033569X(2013)013217
Received by editor(s):
February 22, 2012
Published electronically:
November 20, 2013
Article copyright:
© Copyright 2013 Brown University
