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Mathematical Problems in Engineering
Volume 2009, Article ID 948672, 15 pages
Research Article

Weakly Nonlinear Hydrodynamic Stability of the Thin Newtonian Fluid Flowing on a Rotating Circular Disk

Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan

Received 19 December 2008; Accepted 5 March 2009

Academic Editor: Mehrdad Massoudi

Copyright © 2009 Cha'o-Kuang Chen and Ming-Che Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The main object of this paper is to study the weakly nonlinear hydrodynamic stability of the thin Newtonian fluid flowing on a rotating circular disk. A long-wave perturbation method is used to derive the nonlinear evolution equation for the film flow. The linear behaviors of the spreading wave are investigated by normal mode approach, and its weakly nonlinear behaviors are explored by the method of multiple scales. The Ginzburg-Landau equation is determined to discuss the necessary condition for the existence of such flow pattern. The results indicate that the superctitical instability region increases, and the subcritical stability region decreases with the increase of the rotation number or the radius of circular disk. It is found that the rotation number and the radius of circular disk not only play the significant roles in destabilizing the flow in the linear stability analysis but also shrink the area of supercritical stability region at high Reynolds number in the weakly nonlinear stability analysis.