Numerical studies of the fingering phenomena for the thin film equation |
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Authors: | Yibao Li Hyun Geun Lee Daeki Yoon Woonjae Hwang Suyeon Shin Youngsoo Ha Junseok Kim |
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Institution: | 1. Department of Mathematics, Korea University, Seoul 136‐701, Republic of Korea;2. Department of Information and Mathematics, Korea University, Jochiwon 339‐700, Republic of Korea;3. National Institute for Mathematical Science, Daejeon 305‐340, Republic of Korea |
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Abstract: | We present a new interpretation of the fingering phenomena of the thin liquid film layer through numerical investigations. The governing partial differential equation is ht + (h2?h3)x = ??·(h3?Δh), which arises in the context of thin liquid films driven by a thermal gradient with a counteracting gravitational force, where h = h(x, y, t) is the liquid film height. A robust and accurate finite difference method is developed for the thin liquid film equation. For the advection part (h2?h3)x, we use an implicit essentially non‐oscillatory (ENO)‐type scheme and get a good stability property. For the diffusion part ??·(h3?Δh), we use an implicit Euler's method. The resulting nonlinear discrete system is solved by an efficient nonlinear multigrid method. Numerical experiments indicate that higher the film thickness, the faster the film front evolves. The concave front has higher film thickness than the convex front. Therefore, the concave front has higher speed than the convex front and this leads to the fingering phenomena. Copyright © 2010 John Wiley & Sons, Ltd. |
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Keywords: | nonlinear diffusion equation Marangoni stress fingering instability thin film nonlinear multigrid method finite difference |
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