Abstract
We analyze the stability of the Nusselt flat film solution with respect to infinitesimal perturbations using the governing equations and boundary conditions presented in Chap. 2. We first present the governing equations for the primary instability (Orr–Sommerfeld eigenvalue problem) for both the specified temperature (ST) and the heat flux (HF) conditions. We then deal with transverse modulations triggered by the Marangoni effect (referred to in Chap. 1 as the S-mode) and, subsequently, with stream-wise waves triggered by both hydrodynamics (referred to in Chap. 1 as the H-mode) and the Marangoni effect. We obtain conditions for the onset of the instability in the presence of both modes. Via an energy method/energy balance for the disturbances we analyze the mechanism that triggers the H-mode in isothermal films.
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- 1.
Expression (3.15) now requires the evaluation of k⋅x where k is a vector with complex components. The dot product a⋅b for two vectors a and b with real components can be easily generalized to vectors with complex components (see e.g., [108]). Assume a=(a 1,a 2,…,a n ) and b=(b 1,b 2,…,b n ). Then \(\mathbf{a} \cdot\mathbf{b} =\sum_{i=1}^{n} a_{i} {\bar{b}}_{j}\) where the overbar denotes complex conjugation. Hence, in our case we simply have k⋅x=(k x r+ik x i)x+(k z r+ik z i)z.
- 2.
Some authors reserve the term “Orr–Sommerfeld eigenvalue problem” for the linear stability analysis of a parallel flow with respect to two-dimensional disturbances and use instead the term “generalized Orr–Sommerfeld eigenvalue problem” for three-dimensional disturbances—meaning the “generalization” of the Orr–Sommerfeld eigenvalue problem for two-dimensional disturbances to three-dimensional ones, e.g., [182]. Others use the term “Orr–Sommerfeld eigenvalue problem” for both two-dimensional and three-dimensional disturbances, e.g., [44].
- 3.
The S-mode is also a long-wave variety but has ω r=0 ∀k; on the other hand the P-mode is a short-wave variety but again with ω r=0 ∀k.
- 4.
For the horizontal case the film thickness is fixed by the amount of fluid; for the falling film the thickness is fixed through the flow rate.
- 5.
Though these works neglect the variation of surface tension with temperature in the normal stress boundary condition, it has been retained here. In the normal stress boundary condition (3.17g), the contribution of surface tension variation with temperature, k 2 MΘ(1), or equivalently the term k 2 M ⋆/B in the right hand sides of (3.18) and (3.19), can only be neglected compared to Ct and k 2 We when k is small and We is large, respectively—note that in experimental observations We is large for most liquids, the so called “strong surface tension effect” (this is a crucial point for the remaining of the monograph and will be discussed in detail in Chaps. 4 and 5). In the general case, however, i.e., for finite wavenumbers, \(k =\mathcal{O}(1 )\), MΘ(1), or equivalently, M ⋆/B should be retained.
- 6.
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Kalliadasis, S., Ruyer-Quil, C., Scheid, B., Velarde, M.G. (2012). Primary Instability. In: Falling Liquid Films. Applied Mathematical Sciences, vol 176. Springer, London. https://doi.org/10.1007/978-1-84882-367-9_3
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DOI: https://doi.org/10.1007/978-1-84882-367-9_3
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