Nonlinear evolution of nonuniformly heated falling liquid films

Scheid, Benoît; Oron, Alexander; Colinet, Pierre; Thiele, Uwe; Legros, Jean Claude

doi:10.1063/1.1515270

Cited by 88 publications

(56 citation statements)

References 29 publications

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“…We believe, this ensures that the Benney equation will remain a helpful model to study thin film flows, especially to identify new phenomena but also in a limited parameter range for quantitative predictions. Actually, many other effects may be added to the Benney equation like evaporation (Joo et al 1991), Van der Waals force (Tan et al 1966), chemical reaction (Trevelyan et al 2002), topolographical effects (Kalliadasis et al 2000), non-uniform heating (Miladinova et al 2002;Kabov et al 2001;Scheid et al 2002;Skotheim et al 2002;Kalliadasis et al 2003b), etc. The validity of the Benney equation should in the future also be addressed including these additional effects.…”

Section: Discussionmentioning

confidence: 99%

“…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%

“…In contrast with several previous works (Joo et al 1991Scheid et al 2002), the value of the aspect ratio is not assumed a priori. However, doing so results merely in a rescaling of λ, as for instance in Joo et al (1991) where the wavenumber is defined by (2000) elaborated a model that predicts accurately and economically the linear and nonlinear properties of film flows up to relatively high Reynolds numbers.…”

Section: Travelling-wave Solutionsmentioning

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: 99%

Section: Frame and Objectives Of This Workmentioning

confidence: 99%

Section: Travelling-wave Solutionsmentioning

confidence: 99%

See 1 more Smart Citation

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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show abstract

“…[19][20][21][22][23][24][25][26] In particular, for films flowing under the influence of gravity over an inclined surface containing a rectangular heater, a thermocapillary ridge develops at the upstream edge of the heater where the Marangoni stress opposes the bulk flow. This ridge breaks up into an array of regular, spanwise rivulets above a critical value of the temperature increase at the heater 27,28 ͑characterized by a dimensionless Marangoni parameter that quantifies the gradient in surface tension͒.…”

Section: Introductionmentioning

confidence: 99%

Stabilization of thin liquid films flowing over locally heated surfaces via substrate topography

Tiwari¹,

Davis²

2010

Physics of Fluids

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A long-wave lubrication analysis is used to study the influence of topographical features on the linear stability of noninertial coating flows over a locally heated surface. Thin liquid films flowing over surfaces with localized heating develop a pronounced ridge at the upstream edge of the heater. This ridge becomes unstable to transverse perturbations above a critical Marangoni number and evolves into an array of rivulets even in the limit of noninertial flow. Similar fluid ridges form near topographical variations on isothermal surfaces, but these ridges are stable to perturbations. The influence of basic topographical features on the stability of the locally heated film is analyzed. In contrast to its destabilizing influence on liquid films resting on heated, horizontal walls, even such nonoptimized topography is found to be effective at stabilizing the flowing film with respect to rivulet formation and subsequent rupture. Optimal topographical features that suppress variations in the free-surface shape are also determined. The critical Marangoni number at the instability threshold increases substantially with appropriate topography even for nonzero Biot numbers. An energy analysis is used to provide insight into the mechanism by which the topography stabilizes the flow. Because the stabilizing effect of the topographical features is only weakly sensitive to the governing parameters and particular temperature profile, the use of such features could be a simple alternative in applications to more complicated methods of stabilization.

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“…[7][8][9][10] In addition, gravity-driven flow down inclined surfaces with nonuniform heating has been investigated recently. [11][12][13][14][15] While most of these studies were focused on the interaction of the inertial surface-wave instability with thermocapillary effects, there has also been much recent theoretical work on pattern formation in liquid films flowing over locally heated surfaces, [16][17][18][19][20][21][22][23] in which interesting dynamics occur even in the noninertial limit. These studies were motivated by a series of experiments in which a thin liquid film flows over an inclined plane wall bearing a rectangular heater.…”

Section: Introductionmentioning

confidence: 99%

Nonmodal and nonlinear dynamics of a volatile liquid film flowing over a locally heated surface

Tiwari

Davis

2009

Physics of Fluids

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The stability of a thin, volatile liquid film falling under the influence of gravity over a locally heated, vertical plate is analyzed in the noninertial regime using a model based on long-wave theory. The model is formulated to account for evaporation that is either governed by thermodynamic considerations at the interface in the one-sided limit or limited by the rate of mass transfer of the vapor from the interface. The temperature gradient near the upstream edge of the heater induces a gradient in surface tension that opposes the gravity-driven flow, and a pronounced thermocapillary ridge develops in the streamwise direction. Recent theoretical analyses predict that the ridge becomes unstable above a critical value of the Marangoni parameter, leading to the experimentally observed rivulet structure that is periodic in the direction transverse to the bulk flow. An oscillatory, thermocapillary instability in the streamwise direction above the heater is also predicted for films with sufficiently large heat loss at the free surface due to either evaporation or strong convection in the adjoining gas. This present work extends the recent linear stability analysis of such flows by Tiwari and Davis ͓Phys. Fluids 21, 022105 ͑2009͔͒ to a nonmodal analysis of the governing non-self-adjoint operator and computations of the nonlinear dynamics. The nonmodal analysis identifies the most destabilizing perturbations to the film and their maximum amplification. Computations of the nonlinear dynamics reveal that small perturbations can be sufficient to destabilize a linearly stable film for a narrow band of wave numbers predicted by the nonmodal, linearized analysis. This destabilization is linked to the presence of stable, discrete modes that appear as the Marangoni parameter approaches the critical value at which the film becomes linearly unstable. Furthermore, the thermocapillary instability leads to a new, time-periodic base state. This transition corresponds to a Hopf bifurcation with increasing Marangoni parameter. A linear stability analysis of this time-periodic state reveals further instability to transverse perturbations, with the wave number of the most unstable mode about 50% smaller than for the rivulet instability of the steady base state and exponential growth rate about three times larger. The resulting film behavior is reminiscent of inertial waves on locally heated films, although the wave amplitude is larger in the present case near the heater and decays downstream where the Marangoni stress vanishes. The film's heat transfer coefficient is found to increase significantly upon the transition to the time-periodic flow.

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Nonlinear evolution of nonuniformly heated falling liquid films

Cited by 88 publications

References 29 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

Stabilization of thin liquid films flowing over locally heated surfaces via substrate topography

Nonmodal and nonlinear dynamics of a volatile liquid film flowing over a locally heated surface

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