Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T17:48:46.749Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability Analysis of A Thin Micropolar Fluid Flowing on A Rotating Circular Disk

Published online by Cambridge University Press:  31 March 2011

C. K. Chen ,
M. C. Lin  and
C. I. Chen
C. K. Chen*
Affiliation:
Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
M. C. Lin
Affiliation:
Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences Kaohsiung, Taiwan 80778, R.O.C.
C. I. Chen
Affiliation:
Department of Industrial Engineering and Management, I-Shou UniversityKaohsiung County, Taiwan 84041, R.O.C.
*
* Professor, corresponding author
Get access

Abstract

The stability analysis of a thin micropolar fluid flowing on a rotating circular disk is investigated numerically. The target is restricted to some neighborhood of critical value in the linear stability analysis. First, a generalized nonlinear kinematic model is derived by the long wave perturbation method. The method of normal mode is applied to the linear stability. After the weakly nonlinear dynamics of a film flow is studied by using the method of multiple scales, the Ginzburg-Landau equation is determined to discuss the necessary condition in terms of the various states of subcritical stability, subcritical instability, supercritical stability, and supercritical explosion for the existence of such flow pattern. The modeling results indicate that the rotation number and the radius of circular disk play the significant roles in destabilizing the flow. Furthermore, the micropolar parameter K serves as the stabilizing factor in the thin film flow.

Keywords

Nonlinear stabilityMicropolar fluidGinzburg-Landau equation
Type
Articles
Information
Journal of Mechanics , Volume 27 , Issue 1 , March 2011 , pp. 95 - 105
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2011

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

REFERENCES

1.Yang, Y. K. and Chang, T. C., “Experimental Analysis and Optimization of a Photo Resist Coating Process for Photolithography in Wafer Fabrication,” Microelectronics Journal, 37, pp. 746–751 (2006).CrossRefGoogle Scholar
2.Kuo, Y. K. and Chao, C. G., “Control Ability of Spin Coating Planarization of Resist Film and Optimal of Developers,” Microelectronics Journal, 37, pp. 759–764 (2006).CrossRefGoogle Scholar
3.Emslie, A. G., Bonner, F. T. and Peck, L. G., “Flow of a Viscous Liquid on a Rotating Disk,” Journal of Applied Physics, 29, pp. 858–862 (1958).CrossRefGoogle Scholar
4.Higgins, B. G., “Film Flow on a Rotating Disk,” Physics of Fluids, 29, pp. 3522–3529 (1986).CrossRefGoogle Scholar
5.Kitamura, A., Hasegawa, E. and Yoshizawa, M., “Asymptotic Analysis of the Formation of Thin Liquid Film in Spin Coating,” Fluid Dynamics Research, 30, pp. 107–125 (2002).CrossRefGoogle Scholar
6.Lin, C. C., The Theory of Hydrodynamic Stability, Cambridge University Press. (1955).Google Scholar
7.Chandrasekhar, S.Hydrodynamic and Hydromagnetic Stability, Oxford University Press. (1961).Google Scholar
8.Kapitza, P. L., “Wave Flow of Thin Viscous Liquid Films,” Zhurnal Èksperimentalnoi I Teoreticheskoi Fiziki, 18, pp. 3–28 (1949).Google Scholar
9.Benney, D. J., “Long Waves on Liquid Film,” Journal of Mathematical Physics, 45, pp. 150–155 (1966).CrossRefGoogle Scholar
10.Landau, L. D., “On the Problem of Turbulence,” Comptes Rendus (Doklady) of Academy of sciences, 44, pp. 311–314 (1944).Google Scholar
11.Yih, C. S., “Stability of Liquid Flow Down an Inclined Plane,” Physics Fluids, 6, pp. 321–334 (1963).CrossRefGoogle Scholar
12.Stuart, J. T. “On the Role of Reynolds Stresses in Stability Theory,” Journal of the Aeronautical Sciences, 23, pp. 86–88 (1956).Google Scholar
13.Pumir, A., Manneville, P. and Pomeau, Y., “On Solitary Waves Running Down on Inclined Plane,” Journal of Fluid Mechanics, 135, pp. 27–50 (1983).CrossRefGoogle Scholar
14.Ruyer-Quil, C. and Manneville, P., “Modeling Film Flows Down an Inclined Planes,” European Physical Journal, B 6, pp. 277–292 (1998).CrossRefGoogle Scholar
15.Ruyer-Quil, C. and Manneville, P., “Improved Modeling of Flows Down Inclined Planes,” European Physical Journal, B 15, pp. 357–369 (2000).CrossRefGoogle Scholar
16.Ruyer-Quil, C. and Manneville, P., “Further Accuracy and Convergence Results on the Modeling of Flows Down Inclined Planes by Weighted Residual Approximations,” Physics of Fluids, 14, pp. 170–183 (2002).CrossRefGoogle Scholar
17.Amaouche, M., Mehidi, N. and Amatousse, N., “An Accurate Modeling of Thin Film Flows Down an Inclined for Inertia Dominated Regimes,” European Journal of Mechanics B/Fluids, 24, pp. 49–70 (2005).CrossRefGoogle Scholar
18.Samanta, A., “Stability of Liquid Film Down a Vertical Non-Uniformly Heated Wall,” Physica D: Nonlinear Phenomena, 237, pp. 2587–2598 (2008).CrossRefGoogle Scholar
19.Eringen, A. C., “Theory of Micropolar Fluids,” Journal of Mathematics Mechanics, 16, pp. 1–18 (1967).Google Scholar
20.Hung, C. I., Tsai, J. S. and Chen, C. K., “Nonlinear Stability of the Thin Micropolar Liquid Film Flowing Down on a Vertical Plate,” Journal of Fluids Engineering, ASME. 118, pp. 498–505 (1996).CrossRefGoogle Scholar
21.Dupuy, D. G., Panasenko, P. and Stavre, P., “Asymptotic Analysis for Micropolar Fluids,” Comptes Rendus Mecanique, 332, pp. 31–36 (2004).CrossRefGoogle Scholar
22.Chen, C. I., “Non-Linear Stability Characterization of the Thin Micropolar Liquid Film Flowing Down the Inner Surface of a Rotating Vertical Cylinder,” Communications in Nonlinear Science and Numerical Simulation, 12, pp. 760–775 (2007).CrossRefGoogle Scholar
23.Ahmadi, G., “Self-Similar Solution of Incompressible Micropolar Boundary Layer Flow Over a Semi-Infinite Plate,” International Journal of Engineering Science, 14, pp. 639–646 (1976).CrossRefGoogle Scholar
24.Jena, S. K. and Mathur, M. N., “Mixed Convenction Flow of a Micropolar Fluid from an Isothermal Vertical Plate,” Mathematics Applied, 10, pp. 291–304 (1984).Google Scholar
25.Ginzburg, V. L. and Landau, L. D., “Theory of Superconductivity,” Journal of Experimental and Theoretical Physics, 120, pp. 1064–1082 (1950).Google Scholar
26.Cheng, P-J. and Lai, H-Y., “Nonlinear Stability Analysis of Thin Film Flow from a Liquid Jet Impinging on a Circular Concentric Disk,” Journal of Mechanics, 22, pp. 115–124 (2006).CrossRefGoogle Scholar