Two-dimensional wave dynamics in thin films. I. Stationary solitary pulses

Saprykin, Sergey; Demekhin, E. A.; Kalliadasis, Serafim

doi:10.1063/1.2128607

Cited by 40 publications

(25 citation statements)

References 27 publications

(20 reference statements)

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“…The main difference between the 1-D and 2-D KS equations lies in effects due to dispersion ͑from odd derivative terms͒. 48 When dispersion is weak, the dynamics of the 2-D case are qualitatively similar to those of the 1-D case. 45 If the effect is strong, the extra 2-D dispersion term can be important even at large tension ͑large ␥ 1 ͒, leading to a change of wave patterns or complicated wave interactions in the 2-D interface's dynamics.…”

Section: ͑55͒mentioning

confidence: 67%

Shear-flow and thermocapillary interfacial instabilities in a two-layer viscous flow

Wei

2006

Physics of Fluids

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Combined effects of shear-flow and thermocapillary instabilities in a two-layer Couette flow are asymptotically examined in the thin-layer limit. The basic features of the system instability are revealed by first analyzing the two-dimensional stability problem. A scaling analysis is devised to identify dominant mechanisms in various parameter regimes. With an appropriate scaling, the leading order linear stability is reduced to a one-dimensional evolution equation containing a nonlocal contribution from viscosity stratification. Viscosity stratification destabilizes ͑stabilizes͒ the system with a more ͑less͒ viscous film, but the effect can be compromised by thermocapillary stabilization ͑destabilization͒ as the film is cooled ͑heated͒. Thermocapillary effects dominate over viscosity stratification effects for short-wave perturbations albeit the latter could be stronger than the former for long waves. The competition between these two effects gives rise to the critical Reynolds number for the onset of stability/instability. A nontrivial interplay is found within a window in the weak interfacial-tension regime. It demonstrates a possibility of the existence of two neutral states in the wavenumber space. The three-dimensional problem is also examined. For the first time, a two-dimensional film evolution equation with the inclusion of a nonlocal term is systematically derived for the corresponding stability. It can be shown analytically that three-dimensional perturbations can be more unstable than two-dimensional ones due to thermocapillarity in line with the nonexistence of Squires' theorem. The three-dimensional problem has the critical Reynolds number larger than the two-dimensional problem, but an instability in the latter does not necessarily suggest an instability in the former. An extension of each problem to the weakly nonlinear regime is also discussed in the context of the Kuramoto-Sivashinsky equation.

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Section: ͑55͒mentioning

confidence: 67%

Shear-flow and thermocapillary interfacial instabilities in a two-layer viscous flow

Wei

2006

Physics of Fluids

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“…Note also the wave front is curved backwards on the sides as also depicted by the oblique dot-dashed line in Fig. 8b, reminiscent of tails in 3D solitons [41]. Figure 9a shows cross-sections for the type II m at three different streamwise positions.…”

Section: Mechanism For the Transition Ii-iiimentioning

confidence: 99%

Interaction of three-dimensional hydrodynamic and thermocapillary instabilities in film flows

et al. 2008

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We study three-dimensional wave patterns on the surface of a film flowing down a uniformly heated wall. Our starting point is a model of four evolution equations for the film thickness h, the interfacial temperature θ and the streamwise and spanwise flow rates, q and p, respectively, obtained by combining a gradient expansion with a weighted residual projection. This model is shown to be robust and accurate in describing the competition between hydrodynamic waves and thermocapillary Marangoni effects for a wide range of parameters. For small Reynolds numbers, i.e. in the 'drag-gravity regime', we observe regularly spaced rivulets aligned with the flow and preventing the development of hydrodynamic waves. The wavelength of the developed rivulet structures is found to closely match the one of the most amplified mode predicted by linear theory. For larger Reynolds numbers, i.e. in the 'drag-inertia regime', the situation is similar to the isothermal case and no rivulets are observed. Between these two regimes we observe a complex behavior for the hydrodynamic and thermocapillary modes with the presence of rivulets channeling quasi-two-dimensional waves of larger amplitude and phase speed than those observed in isothermal conditions, leading possibly to solitary-like waves. Two subregions are identified depending on the topology of the rivulet structures that can be either 'ridgelike' or 'groovelike'. A regime map is further proposed that highlights the influence of the Reynolds and the Marangoni numbers on the rivulet structures. Interestingly, this map is found to be related to the variations of amplitude and speed of the twodimensional solitary-wave solutions of the model. Finally, the heat transfer enhancement due to the increase of interfacial area in the presence of rivulet structures is shown to be significant.

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“…It can be used to describe long waves on a viscous fluid flowing down along an inclined plane [21], unstable drift waves in plasma [22] and stress waves in fragmented porous media. It can be applied to stationary solitary pulses in a falling film [23]. For β = 0, Eq.…”

Section: Introductionmentioning

confidence: 99%

Lattice Boltzmann method for the generalized Kuramoto–Sivashinsky equation

Lai

2009

Physica A: Statistical Mechanics and its Applications

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Two-dimensional wave dynamics in thin films. I. Stationary solitary pulses

Cited by 40 publications

References 27 publications

Shear-flow and thermocapillary interfacial instabilities in a two-layer viscous flow

Shear-flow and thermocapillary interfacial instabilities in a two-layer viscous flow

Interaction of three-dimensional hydrodynamic and thermocapillary instabilities in film flows

Lattice Boltzmann method for the generalized Kuramoto–Sivashinsky equation

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