Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 987891, 17 pages
Research Article

Weakly Nonlinear Stability Analysis of a Thin Magnetic Fluid during Spin Coating

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

Received 8 April 2010; Accepted 1 June 2010

Academic Editor: Cristian Toma

Copyright © 2010 Cha'o-Kuang Chen and Dong-Yu Lai. 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.


This paper investigates the stability of a thin electrically conductive fluid under an applied uniform magnetic filed during spin coating. A generalized nonlinear kinematic model is derived by the long-wave perturbation method to represent the physical system. After linearizing the nonlinear evolution equation, the method of normal mode is applied to study the linear stability. Weakly nonlinear dynamics of film flow is studied by the multiple scales method. The Ginzburg-Landau equation is determined to discuss the necessary conditions of the various critical flow states, namely, subcritical stability, subcritical instability, supercritical stability, and supercritical explosion. The study reveals that the rotation number and the radius of the rotating circular disk generate similar destabilizing effects but the Hartmann number gives a stabilizing effect. Moreover, the optimum conditions can be found to alter stability of the film flow by controlling the applied magnetic field.