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Surface Instabilities on a Thin Power-Law Fluid During Spin Coating

Published online by Cambridge University Press:  22 May 2014

C.-I. Chen ,
M.-C. Lin  and
C.-K. Chen
C.-I. Chen
Affiliation:
Department of Industrial Management, I-Shou University, Kaohsiung County, Taiwan 84041, R.O.C.
M.-C. Lin*
Affiliation:
Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan 80778, R.O.C.
C.-K. Chen
Affiliation:
Department of Mechanical Engineering, National Cheng Kung University Tainan, Taiwan 70101, R.O.C.
*
*[email protected]
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Abstract

The phenomena of surface instabilities in a thin power-law fluid during spin coating are investigated. The set of Navier-Stokes equations with non-Newtonian behavior in the region of low Reynolds number serves as a mathematical description of the physical systems. Long-wave perturbation analysis is proposed to derive an evolution equation of the Ostwald de-Waele type fluid to govern the propagation of surface waves. Weakly nonlinear dynamics of film flow is studied by the multiple scales method. The amplitude of instability is determined by a Ginzburg-Landau equation. The study reveals that the degree of power-law index plays a vital role in stabilizing the film flow. The shear-thinning fluid is more unstable than the shear-thickening fluid in the stability analysis. Further, the nonlinear wave speed in the supercritical stability region decreases with increasing values of power-law index.

Keywords

Nonlinear stabilityPower-law fluidSpin coating
Type
Research Article
Information
Journal of Mechanics , Volume 30 , Issue 5 , October 2014 , pp. 505 - 513
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2014 

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References

1.Kitamura, A., Hasegawa, E. and Yoshizawa, M., “Asymptotic Analysis of the Formation of Thin Liquid Film in Spin Coating,” Fluid Dynamic Research, 30, pp. 107–125 (2002).CrossRefGoogle Scholar
2.Sisoev, G. M., Matar, O. K. and Lawrence, C. J., “Stabilizing Effect of the Coriolis Forces on a Viscous Liquid Film Flowing over a Spinning Disc,” Comptes Rendus Mechanique, 332, pp. 203–207 (2004).CrossRefGoogle Scholar
3.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
4.Li, Y., Ma, M., Wang, X. and Li, Z., “Preparation of Cerium-Doped Titania Macroporus Films by a SolGel Spin Coating Using Polypropylene Glycol (PPG) as Pore-Creating Agent: Effects of Ce Ions, PPG and Calculation on Photocatalytic Activity,” Surface & Coating Technology, 204, pp. 1353–1358 (2010).Google Scholar
5.Lai, C. Y., Rajagopal, K. R. and Szeri, A. Z., “Asymmetric Flow Above a Rotating Disk,” Journal of Fluid Mechanics, 157, pp. 471–492 (1985).CrossRefGoogle Scholar
6.Rajagopal, K. R., “Flow of Viscoelastic Fluids Between Rotating Disks,” Theoretical and Computational Fluid Dynamics, 3, pp. 185–206 (1992).CrossRefGoogle Scholar
7.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).Google Scholar
8.Higgins, B. G., “Film Flow on a Rotating Disk,” Physics Fluids, 29, pp. 3522–3529 (1986).Google Scholar
9.Acrivos, A., Shah, M. J. and Petersen, E. E., “On the Flow of a Non-Newtonian Liquid on a Rotating Disk,” Journal of Applied Physics, 31, pp. 963–968 (1960).CrossRefGoogle Scholar
10.Burgess, S. L. and Wilson, S. D. R., “Spin-Coating of a Viscoplastic Material,” Physics Fluids, 8, pp. 2291–2297 (1996).CrossRefGoogle Scholar
11.Hayat, T., Javed, T. and Sajid, M., “Analytic Solution for MHD Rotating Flow of a Second Grade Fluid over a Shrinking Surface,” Physics Letters A, 372, pp. 3264–3273 (2008).CrossRefGoogle Scholar
12.Chhabra, R. P. and Richardson, J. F., Non-Newtonian Flow and Applied Rheology: Engineering Applications, 2nd Edition, Butterworth-Heinemann, U.K. (2008).Google Scholar
13.Benney, D. J., “Long Waves on Liquid Films,” Journal of Mathematics Physics, 45, pp. 150–155 (1966).CrossRefGoogle Scholar
14.Lin, S. P., “Finite Amplitude Sine-Band Stability of a Viscous Film,” Journal of Fluid Mechanics, 63, pp. 417–429 (1974).CrossRefGoogle Scholar
15.Chen, C. I., Chen, C. K. and Yang, Y. T., “Perturbation Analysis to the Nonlinear Stability Characterization of Thin Condensate Falling Film on the Outer Surface of a Rotating Vertical Cylinder,” International Journal of Heat and Mass Transfer, 47, pp. 1937–1951 (2004).Google Scholar
16.Chen, C. K. and Lin, M. C., “Weakly Nonlinear Hy-drodynamic Stability of Thin Newtonian Fluid Flowing on a Rotating Circular Disk,” Mathematical Problems in Engineering, DOI:10.1155/2009/948672 (2009).Google Scholar
17.Lin, M. C. and Chen, C. K., “Finite Amplitude Long-Wave Instability of a Thin Viscoelastic Fluid During Spin Coating,” Applied Mathematical Modelling, 36, pp. 2536–2549 (2012).CrossRefGoogle Scholar
18.Cheng, P. J. and Liu, K. C., “Hydromagnetic Instability of a Power-Law Liquid Film Flowing Down a Vertical Cylinder Using Numerical Approximation Approach Techniques,” Applied Mathematical Modeling, 33, pp. 1904–1914 (2009).CrossRefGoogle Scholar
19.Eckhaus, W., Studies in Nonlinear Stability Theory, Springer, Berlin (1965).CrossRefGoogle Scholar
20.Krishna, M. V. G. and Lin, S. P., “Nonlinear Stability of a Viscous Film with Respect to Three-Dimensional Side-Band Disturbance,” Physics of Fluids, 20, pp. 1039–1044 (1977).CrossRefGoogle Scholar
21.Ginzburg, V. L. and Landau, L. D., “Theory of Superconductivity,” Journal of Experiential Theoretic Physics (USSR), 20, pp. 1064–1082 (1950).Google Scholar
22.Anderson, H. I. and Dahi, E. N., “Gravity-Driven Flow of a Viscoelastic Liquid Film Along a Vertical Wall,” Journal Physical D: Applied Physics, 32, pp. 1557–1562 (1999).CrossRefGoogle Scholar
23.Uma, B. and Usha, R., “Interfacial Phase Change Effects on the Stability Characteristics of Thin Viscoelastic Liquid Film Down a Vertical,” International Journal of Engineering Science, 42, pp. 1381–1406 (2004).CrossRefGoogle Scholar
24.Dandapat, B. S. and Mukhopadhyay, A., “Waves on the Surface of a Falling Power-Law Fluid Film,” International Journal of Non-linear Mechanics, 38, pp. 21–38 (2003).Google Scholar