Nonlinear dynamics of temporally excited falling liquid films

Oron, Alexander; Gottlieb, O.

doi:10.1063/1.1485766

Cited by 52 publications

(54 citation statements)

References 30 publications

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“…We have established in this paper the equivalence between the boundary beyond which the dynamical system associated to the Benney equation has no travelling-wave solutions and the boundary for finite-time blow-up obtained in simulations by Oron & Gottlieb (2002). The solutions that tend to blow up first, i.e.…”

Section: Discussionmentioning

confidence: 80%

“…from neutrally unstable modes to infinite wavelength solitary waves. This boundary will be called absence-ofsolution boundary and will be linked to the blow-up boundary found with time-dependent simulations by Oron & Gottlieb (2002). In §4 we track the absence-of-solution boundary through the parameter space, investigating successively the influence of surface tension, inclination and thermocapillarity.…”

Section: Frame and Objectives Of This Workmentioning

confidence: 95%

“…Chang (1994) already evocated this condition speaking about constant-average thickness or constant-flux formulation. Many authors, as for instance (Joo et al 1991;Salamon et al 1994;Oron & Gottlieb 2002;Scheid et al 2002), implicitly prescribed the closed flow condition. This is due to the fact that in time-dependent simulations, using periodic boundary conditions, the amount of liquid leaving the domain downstream is reinjected upstream.…”

Section: Frame and Objectives Of This Workmentioning

confidence: 99%

“…‡ Some authors we refer to (Salamon et al 1994;Oron & Gottlieb 2002) based the Reynolds number on the surface velocity rather than on the mean velocity, which yields R = 3Re/2. Therefore, Re = 2.0667 corresponds to R = 3.1 and Ka = 3375 to We = 1000 with hN = 1.8371. ingly one-hump and faster as k decreases.…”

Section: Families Of Stationary Solutionsmentioning

confidence: 99%

“…For k > k * , two stationary solution branches coexist. Pumir et al (1983) and Oron & Gottlieb (2002) have shown that only the lower branch of small amplitude solution corresponds to bounded solutions (note that the WIBL model does not possess the larger amplitude branch at all). The bifurcation at k c is supercritical for Re ≤ 5.…”

Section: Blow-up Versus Wavenumbermentioning

confidence: 99%

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Validity domain of the Benney equation including the Marangoni effect for closed and open flows

Scheid¹,

Ruyer-Quil²,

Thiele³

et al. 2005

J. Fluid Mech.

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The Benney equation including thermocapillary effects is considered to study a liquid film flowing down a homogeneously heated inclined wall. The link between finite-time blow-up of the Benney equation and absence of one-hump travelling-wave solution of the associated dynamical system is accurately demonstrated in the whole range of linearly unstable wavenumbers. Then the blow-up boundary is tracked in the whole space of parameters accounting for flow rate, surface tension, inclination and thermocapillarity. Especially, the latter two effects can strongly reduce the validity range of the Benney equation. It is also shown that the subcritical bifurcation found for falling films with the Benney equation is related to the blow-up of solutions and is unphysical in any case, even with the thermocapillary effect, in contrast with horizontally heated films. The accuracy of bounded solutions of the Benney equation is also analysed by comparison with a reference weighted integral boundary layer model. A distinction is made between closed and open flow conditions, when calculating travelling-wave solutions; the former corresponding to the conservation of mass and the latter to the conservation of flow rate. The open flow condition matches experimental conditions more closely and is explored for the first time through the associated dynamical system. It yields bounded solutions for larger Reynolds numbers than the closed flow condition. Finally, solutions that are conditionally bounded are found to be unstable to disturbances of larger periodicity. In this case, coalescence is the pathway yielding finite-time blow-up.

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Section: Discussionmentioning

confidence: 80%

Section: Frame and Objectives Of This Workmentioning

confidence: 95%

Section: Frame and Objectives Of This Workmentioning

confidence: 99%

Section: Families Of Stationary Solutionsmentioning

confidence: 99%

Section: Blow-up Versus Wavenumbermentioning

confidence: 99%

See 3 more Smart Citations

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

Scheid¹,

Ruyer-Quil²,

Thiele³

et al. 2005

J. Fluid Mech.

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Methodologies for Low-Reynolds Number Flows

Kalliadasis

Ruyer-Quil

Scheid

et al. 2012

Applied Mathematical Sciences

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Isothermal Case: Two-Dimensional Flow

Kalliadasis

Ruyer-Quil

Scheid

et al. 2012

Applied Mathematical Sciences

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A falling liquid film can serve as a paradigm for the study of open flow hydrodynamic systems. This is because: (i) the flow is nearly parallel, unlike other hydrodynamic systems, e.g., jets that break up into drops; (ii) it can be studied experimentally relatively easily; (iii) the Reynolds number is small-to-moderate, which makes the problem amenable to theoretical analysis.In fact, falling liquid films have several similarities with many other hydrodynamic systems. The analogy with boundary layer flows has already been emphasized in the modeling approaches described in Chaps. 4 and 6, while similarities with the three-dimensional instabilities developed in boundary layer flows will be discussed in Chap. 8. Similarities can also be found with the propagation of bores in rivers (in the "torrential regime"), a topic that will be discussed in this chapter.More importantly, falling film flows offer an excellent opportunity for the theoretical study of the route toward spatio-temporal disorder and the specific events characterizing its development, not only in open flow hydrodynamic systems but other nonlinear systems as well. The wide variety of phenomena that can be investigated with falling film flows are: (i) development of convective instabilities; (ii) spatial response to external perturbations; (iii) development of traveling waves; (iv) competition/interplay between different instability mechanisms, e.g., for the problem of a heated film; (v) "condensation" phenomena such as formation of bound states, i.e., well-defined and robust groups of coherent structures. Figure 7.1 shows a snapshot of the thickness of the film at the end of a simulation of a naturally excited wavy motion. The initial growth of the waves at the inlet is rapidly followed by a wavy regime where localized structures are separated by relatively large portions of nearly flat films. Subsequently, the dynamics on the film is dominated by these dissipative structures, which seem to organize the flow. The dynamics is therefore "weakly disordered" and the spatial evolution of the film is an example of weak/dissipative turbulence in the Manneville sense [177]. Isolated waves look like tear drops made of a large-amplitude hump preceded by small capillary ripples, also referred to as radiation, as noted in previous chapters. These waves resemble the infinite-domain solitary waves we have already encountered at several places in this monograph. The evolution of the film is therefore dominated by solitary-like coherent structures, which are stable and robust and interact indefinitely with each other as "quasi-particles."Increasing the Reynolds number in the simulation of Fig. 7.1 makes the interface appear more complicated, but despite the apparent complexity one can still identify solitary-like coherent structures in what appears to be a randomly disturbed surface. It is then essential that in order to understand the spatio-temporal evolution of the film, we fully understand the properties of individual solitary waves, which in turn can help us unde...

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Nonlinear dynamics of temporally excited falling liquid films

Cited by 52 publications

References 30 publications

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

Methodologies for Low-Reynolds Number Flows

Isothermal Case: Two-Dimensional Flow

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