Abstract
We develop the long-wave theory for a film falling down a heated wall. The theory is based on a gradient expansion of the governing equations and wall and free-surface boundary conditions and leads to the Benney equation (BE—Chap. 1) for the evolution in time and space of the film thickness. A weakly nonlinear expansion of the equation leads to either the Kuramoto–Sivashinsky or the Kawahara equation depending on the distance from criticality, the orders-of-magnitude assignments of the different parameters and whether the film is inclined or vertical. BE fully resolves the behavior of the film close to the instability threshold, but it blows up in finite time at δ≃1, i.e., precisely where the transition between the drag-gravity and drag-inertia regimes takes place, as shown in Chap. 4. This in turn suggests that δ is the natural parameter for validation purposes/assessment of the validity of a model that aims to describe the dynamics of the film. The blow up behavior is strongly connected with the nonexistence of solitary waves of BE for δ≳1. A regularization procedure based on an extension of the Padé approximants technique to differential operators provides an evolution equation free of singularities which nevertheless seriously underestimates the amplitude and phase speed of the solitary waves at moderate Reynolds numbers. This is a direct consequence of slaving the dynamics of the film to its kinematics. It then seems that in the drag-inertia regime it is not possible to describe the dynamics of the flow with a single evolution equation for the film thickness.
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Notes
- 1.
- 2.
The form refers to an arrangement of an equation or system of equations in the form
$$\partial _t \mathbf{H} + \boldsymbol{\nabla}\cdot\mathbf{Q}(\mathbf{H}) = \mathbf{R}(\mathbf{H}),$$where R(H) is a “source” term and Q(H) is the “flux” associated with the quantity H. The right hand side of the above form should not involve any first-order spatial derivatives; these should be contained in the flux term of the left hand side.
- 3.
- 4.
The implicit assumption here is that a spatially periodic wavetrain results from a time periodic forcing at the inlet. However, this is not always the case, e.g., Fig. 7.39 shows that for x≲2m we have a wavetrain that is periodic in time but modulated in space so that the wavelength changes locally. As a consequence, we cannot relate the period in space with that in time.
- 5.
Indeed, as the wavenumber decreases, higher harmonics become linearly unstable at k n =k c/n with n=2,3,…. The resulting families \(\gamma^{(n)}_{1,2}\) for n>1 correspond therefore to trains of n identical negative- or n identical positive-hump traveling wave solutions. Their maximum heights \(h_{\max}^{(n)}(k)\) are not displayed in Fig. 5.2 because they are homothetic in k, i.e., given that \(h_{\max}^{(n)}(k_{n})=h_{\max}(k_{\mathrm{c}})\) it follows that \(h_{\max}^{(n)}(k/n)=h_{\max}(k)\). The individual solutions correspond simply to n identical solutions of the n=1 family placed in a domain of size 2πn/k.
- 6.
In general, the local Reynolds number can be defined by assuming that locally the flow is a Nusselt one, i.e., by replacing in the Reynolds number based on the Nusselt flat film thickness (2.35) \(\bar{h}_{\mathrm {N}}\) with \(\bar{h}\) or Reh 3. But the local flow rate is \(q = \bar{q}/[(\bar{h}_{\mathrm{N}}^{2}/(t_{\nu}l_{\nu})) \bar{h}_{\mathrm{N}}] = \bar{u}\bar{h}/[(\bar{h}_{\mathrm {N}}^{2}/(t_{\nu}l_{\nu})) \bar{h}_{\mathrm{N}}] = h^{3}/3\). The local Reynolds number then is 3qRe. Hence, the local Reynolds number based on the substrate thickness, say h s , is \(\sim\mathit{Re}h_{s}^{3}\). For a single soliton, the substrate thickness is almost the same with the inlet one, h s ∼h N. For many solitons, h s <h N. A physical explanation is given at the beginning of Sect. 7.2.3 (the reduction is not related to the arguments given at the end of Sect. 5.3.1 on the time-average film thickness). With coalescence, the number of waves goes down, which then leads to h s increasing and hence the local Reynolds number increases.
- 7.
As already pointed out, intriguingly, the KS and Kawahara equations obtained from a weakly nonlinear expansion of the first- and second-order BE, respectively, remain bounded for sufficiently smooth and small-amplitude initial conditions. As a matter of fact, they predict solitary wave solutions past the limit values Re ∗. Disappointingly, the advantage of the more complex BE over the KS and Kawahara equations is rather limited.
- 8.
For traveling waves the saddle node bifurcation corresponds precisely to the turning point of the solution branches. For time-dependent computations things are slightly different; even for slightly smaller values than Re ∗ we can have blow up (see also Sect. 5.4).
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Kalliadasis, S., Ruyer-Quil, C., Scheid, B., Velarde, M.G. (2012). Methodologies for Low-Reynolds Number Flows. In: Falling Liquid Films. Applied Mathematical Sciences, vol 176. Springer, London. https://doi.org/10.1007/978-1-84882-367-9_5
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