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Dynamics of the Wavy Film Down an Inclined Plane. V. Newtonian Orr-Sommerfeld Problem Revisited
Wein, Ondřej ; Tihon, Jaroslav
Linear stability of the Newtonian film flow along an inclined plate is studied by treating the related 2D linearized equations of fluctuating motion (Orr-Sommerfeld). Quadratic character of the Orr-Sommerfeld (OS) equations and the corresponding multiplicity of solution (the regular and singular branch) is demonstrated. Region of validity of both the long-wave (20 terms) and short-wave (4 terms) analytic asymptotes to the regular branch is estimated by comparing it with numerical solution to the full Orr-Sommerfeld problem. In particular, the c1-stability criterion of the regular long-wave asymptote is confirmed. In addition to the basic wave characteristics of the inception region (celerity, growth rate, wavelength), the characteristics of the fluctuating wall shear rate are given.
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Stability of the Newtonian Film Flow Down an Oscillating Inclined Plane. II. The Related Orr-Sommerfeld-Floquet Problems in a Long-Wave Asymptote up to 3rd Order
Wein, Ondřej ; Tihon, Jaroslav
Linear stability of the film flow along an oscillating inclined plate is analyzed. Following the previous analyses by Yih (1968), Bajkov et al. (1982), Bauer and Kerczek (1991), Lin et al. (1996), the 2D linearized equations (Orr-Sommerfeld-Floquet) of fluctuating motion for a New-tonian fluid are treated. The long-wave (small wave number a ) asymptotic expansion for the complex-valued celerity coefficient, k = k(a) = R c(a), is solved analytically up to the or-der O(a3). In addition to well-known sufficient conditions for the wavy instability of the film under forced oscillation, the basic wave characteristics of the inception region are given: celer-ity, growth rate, and wavelength.
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