A Theoretical Study of Instabilities at the Advancing Front of Thermally Driven Coating Films

Kataoka, Dawn E.; Troian, Sandra M.

doi:10.1006/jcis.1997.5018

Cited by 125 publications

(150 citation statements)

References 14 publications

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“…This mechanism is quite different, however, from the fingering behavior observed at the leading edge of other free surface spreading problems, like the flow down an inclined plane 30,31 or the thermocapillary driven spreading of a thin liquid film. 32,33 In these other spreading processes, the instability occurs right at the leading edge and causes the spreading front to separate into long narrow rivulets which never undergo spreading, shielding, or tip-splitting. Furthermore, when the amplitude of the advancing ridge approaches values on the order of the thickness of the pre-existing liquid film, the fingering disappears altogether, in sharp contrast to a spreading surfactant film.…”

Section: A Linearized Transient Responsementioning

confidence: 99%

Spreading of a surfactant monolayer on a thin liquid film: Onset and evolution of digitated structures

Matar

Troian

1999

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We describe the response of an insoluble surfactant monolayer spreading on the surface of a thin liquid film to small disturbances in the film thickness and surfactant concentration. The surface shear stress, which derives from variations in surfactant concentration at the air-liquid interface, rapidly drives liquid and surfactant from the source toward the distal region of higher surface tension. A previous linear stability analysis of a quasi-steady state solution describing the spreading of a finite strip of surfactant on a thin Newtonian film has predicted only stable modes. ͓Dynamics in Small Confining Systems III, Materials Research Society Symposium Proceedings, edited by J. M. Drake, J. Klafter, and E. R. Kopelman ͑Materials Research Society, Boston, 1996͒, Vol. 464, p. 237; Phys. Fluids A 9, 3645 ͑1997͒; O. K. Matar Ph.D. thesis, Princeton University, Princeton, NJ, 1998͔. A perturbation analysis of the transient behavior, however, has revealed the possibility of significant amplification of disturbances in the film thickness within an order one shear time after the onset of flow ͓Phys. Fluids A 10, 1234 ͑1998͒; ''Transient response of a surfactant monolayer spreading on a thin liquid film: Mechanism for amplification of disturbances,'' submitted to Phys. Fluids͔. In this paper we describe the linearized transient behavior and interpret which physical parameters most strongly affect the disturbance amplification ratio. We show how the disturbances localize behind the moving front and how the inclusion of van der Waals forces further enhances their growth and lifetime. We also present numerical solutions to the fully nonlinear 2D governing equations. As time evolves, the nonlinear system sustains disturbances of longer and longer wavelength, consistent with the quasi-steady state and transient linearized descriptions. In addition, for the parameter set investigated, disturbances consisting of several harmonics of a fundamental wavenumber do not couple significantly. The system eventually singles out the smallest wavenumber disturbance in the chosen set. The summary of results to date seems to suggest that the fingering process may be a transient response which nonetheless has a dramatic influence on the spreading process since the digitated structures redirect the flux of liquid and surfactant to produce nonuniform surface coverage. © 1999 American Institute of Physics. ͓S1054-1500͑99͒02001-7͔ Surfactant molecules play a vital role in many biological and industrial processes because of their native ability to lower surface tension in proportion to their concentration. Their presence greatly improves the wetting and spreading capability of commonplace substances like inks, paints, and herbicides. Lung surfactants produced in the alveoli are critical in maintaining lung compliance. A deficiency in lung surfactant production, for example, is known to cause pulmonary edema and other respiratory difficulties. When surfactant molecules contact a liquid surface of higher surface tension, a shear stress develops...

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Section: A Linearized Transient Responsementioning

confidence: 99%

Spreading of a surfactant monolayer on a thin liquid film: Onset and evolution of digitated structures

Matar

Troian

1999

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…This updated interpretation was shown 20 to give consistent results for prior studies. 31,32 While the relevance of this new interpretation to the present problem is less clear because the neutrally stable mode for q =0 is continuous and the unstable modes are discrete, using this enhanced interpretation does not qualitatively alter the results shown in Fig. 10 because the Ė n for the discrete modes is much larger in magnitude than for the mode corresponding to q =0.…”

Section: B Energy Analysismentioning

confidence: 78%

“…10. The original interpretation of the energy analysis for spreading films [30][31][32] was subsequently improved 20,33 by considering not the absolute energy production rates of various terms but instead the deviation of the energy gains relative to their values for the transversely invariant, neutrally stable mode, Ė n ͑q͒ − Ė n ͑q =0͒. This updated interpretation was shown 20 to give consistent results for prior studies.…”

Section: B Energy Analysismentioning

confidence: 93%

“…Instability can occur even for a Biot number ͑Bi͒ of zero, in contrast to the thermocapillary instability of a thin film flowing over a uniformly heated plate. 29 The energy analysis of the governing linearized operator, which was introduced for spreading liquid films, [30][31][32] was subsequently interpreted 20,33 not based on the absolute energy production rates of various terms but instead on the deviation of the energy gains relative to their values for the transversely invariant, neutrally stable mode. It was revealed from this revised interpretation that the dominant instability mechanisms for zero Biot number are streamwise flow from the capillary pressure gradient ͑induced by the fluid ridge͒ and from gravity ͑mobility differences between slightly perturbed regions of the film͒.…”

Section: Introductionmentioning

confidence: 99%

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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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“…[53][54][55][56][57] The oscillations then include the ridge, are largest above the heater, and decay as they are convected downstream of the heater as waves. Shown in Fig.…”

Section: G Bifurcation To Time-periodic Base Statementioning

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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A Theoretical Study of Instabilities at the Advancing Front of Thermally Driven Coating Films

Cited by 125 publications

References 14 publications

Spreading of a surfactant monolayer on a thin liquid film: Onset and evolution of digitated structures

Spreading of a surfactant monolayer on a thin liquid film: Onset and evolution of digitated structures

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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