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Dynamics of the Wavy Film Flow Down an Inclined Plane. II. Extended Similarity Profile for Newtonian Liquids
Wein, Ondřej ; Tihon, Jaroslav
Macroscopic balance method is a classic approximation in the theory of wavy film flows, see Kapitza (1949), Prokopiou et al. (1991), Wein (1993), Hwang (1994). Due to the mathemati-cal complexity of the related Orr-Sommerfeld problem for non-Newtonian liquids, the macro-scopic balance could be the only accessible method of attacking the problem at present. In the present report, we try to improve the macroscopic balance (MB) method for the Newto-nian fluids by (i) including the effect of normal stress changes into the x-moment balance, and (ii) using more realistic estimate of the time-dependent velocity field. Within the 3rd-order long-wave asymptotic expansion, O(a3), the improved MB method provides the results (celerity, criti-cal Reynolds number, wave number, identical with the solution of the related Orr-Sommerfeld problem. The new results indicate that (i) the improved x-momentum macroscopic balance is a correctly formulated functional property of the velocity field, (ii) the similarity velocity fie ld assumption is too simplified to provide a reasonable estimates of the linear stability parameters within the 3rd-order analysis, (iii) the extended similarity profile, correct up to O(a2), guarantees the estimate of the three basic wave characteristics, c, Recrit, a, within the accuracy including O(a3) terms. Macroscopic balance method is a classic approximation in the theory of wavy film flows, see Kapitza (1949), Prokopiou et al. (1991), Wein (1993), Hwang (1994). Due to the mathemati-cal complexity of the related Orr-Sommerfeld problem for non-Newtonian liquids, the macro-scopic balance could be the only accessible method of attacking the problem at present. In the present report, we try to improve the macroscopic balance (MB) method for the Newto-nian fluids by (i) including the effect of normal stress changes into the x-moment balance, and (ii) using more realistic estimate of the time-dependent velocity field. Within the 3rd-order long-wave asymptotic expansion, O(a3), the improved MB method prov ides the results (celerity, criti-cal Reynolds number, wave number, identical with the solution of the related Orr-Sommerfeld problem. The new results indicate that (i) the improved x-momentum macroscopic balance is a correctly formulated functional property of the velocity field, (ii) the similarity velocity field assumption is too simplified to provide a reasonable estimates of the linear stability parameters within the 3rd-order analysis, (iii) the extended similarity profile, correct up to O(a2), guarantees the estimate of the three basic wave characteristics, c, Recrit, a, within the accuracy including O(a3) terms. Macroscopic balance method is a classic approximation in the theory of wavy film flows, see Kapitza (1949), Prokopiou et al. (1991), Wein (1993), Hwang (1994). Due to the mathemati-cal complexity of the related Orr-Sommerfeld problem for non-Newtonian liquids, the macro-scopic balance could be the only accessible method of attacking the problem at present. In the present report, we try to improve the macroscopic balance (MB) method for the Newto-nian fluids by (i) including the effect of normal stress changes into the x-moment balance, and (ii) using more realistic estimate of the time-dependent velocity field. Within the 3rd-order long-wave asymptotic expansion, O(a3), the improved MB method provides the results (celerity, criti-cal Reynolds number, wave number, identical with the solution of the related Orr-Sommerfeld problem. The new results indicate that (i) the improved x-momentum macroscopic balance is a correctly formulated functional property of the velocity field, (ii) the similarity velocity field assumption is too simplified to provide a reasonable estimates of the linear stability parameters within the 3rd-order analysis, (iii) the extended similarity profile, correct up to O(a2), guarantees the estimate of the three basic wave characteristics, c, Recrit, a, within the accuracy including O(a3) terms.
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