Thin liquid films on a slightly inclined heated plate

Thiele, Uwe; Knobloch, Edgar

doi:10.1016/j.physd.2003.09.048

Cited by 113 publications

(130 citation statements)

References 61 publications

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“…Then equation (2.1) is clearly not longer applicable. However, the possible stationary two-dimensional solutions of equation (2.1) with Π = 0 can be determined directly [6] independently of the fact that they cannot be reached from the initial condition of a flat film by integration in time. The stationary solutions consist of a vast family of drop solutions separated by dry regions of different lengths.…”

Section: Thin Film Equationmentioning

confidence: 99%

“…Oron and Rosenau studied the effects of a quadratic dependence of surface tension on temperature [4] and extended the study towards an inclined substrate [5]. In the latter case the evolution equation looses its variational structure allowing for a richer bifurcation structure, as studied recently in some detail by Thiele and Knobloch [6]. All the mentioned work focused on the structure formation in two spatial dimensions, i.e.…”

Section: Introductionmentioning

confidence: 99%

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3D Large scale Marangoni convection in liquid films

Bestehorn

Pototsky

Thiele

2003

The European Physical Journal B - Condensed Matter

101

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Abstract.We study large scale surface deformations of a liquid film unstable due to the Marangoni effect caused by external heating on a smooth and solid substrate. The work is based on the thin film equation which can be derived from the basic hydrodynamic equations. To prevent rupture, a repelling disjoining pressure is included which accounts for the stabilization of a thin precursor film and so prevents the occurrence of completely dry regions. Linear stability analysis, nonlinear stationary solutions, as well as three-dimensional time dependent numerical solutions for horizontal and inclined substrates reveal a rich scenario of possible structures for several realistic fluid parameters.PACS. 68.15.+e Liquid thin films -47.20.Dr Surface-tension-driven instability -68.55.-a Thin film structure and morphology

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Section: Thin Film Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

3D Large scale Marangoni convection in liquid films

Bestehorn

Pototsky

Thiele

2003

The European Physical Journal B - Condensed Matter

101

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“…(i) The evolution of thin viscous films in the presence of gravity and thermo-capillary effects is modeled by u t + {u n (u xxx + u m−n u x − Au M−n u x )} x = 0, (1.2) where m, n, M, A are constants such that A 0, n > 0, m < M, as well as the more accurate variant of (1.2) given by u t + {u n (u xxx + h (u)u x )} x = 0, (1.3) where h (u) = −νG + B 1 /(u(1 + B 2 u) 2 ), G, B 1 , B 2 are positive constants, ν = ±1, and where ν = +1 [−1] represents stabilizing [destabilizing] gravitational forces [41,49,42]. Equation (1.2) with A = 0, u t + {u n (u xxx + u m−n u x )} x = 0, models thin films with thermo-capillary effects but without gravitational effects when m − n = −1 [21], or with destabilizing gravitational effects but without capillary effects when m − n = 0.…”

Section: Introductionmentioning

confidence: 99%

The thin film equation with backwards second order diffusion

Novick-Cohen¹,

Shishkov²

2011

Interfaces Free Bound.

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We focus on the thin film equation with lower order "backwards" diffusion which can describe, for example, the evolution of thin viscous films in the presence of gravity and thermo-capillary effects, or the thin film equation with a "porous media cutoff" of van der Waals forces. We treat in detail the equationwhere ν = ±1, n > 0, M > m, and A 0. Global existence of weak nonnegative solutions is proven when m − n > −2 and A > 0 or ν = −1, and when −2 < m − n < 2, A = 0, ν = 1. From the weak solutions, we get strong entropy solutions under the additional constraint that m−n > −3/2 if ν = 1. A local energy estimate is obtained when 2 n < 3 under some additional restrictions. Finite speed of propagation is proven when m > n/2, for the case of "strong slippage", 0 < n < 2, when ν = 1 based on local entropy estimates, and for the case of "weak slippage", 2 n < 3, when ν = ±1 based on local entropy and energy estimates.

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“…1 There has been considerable theoretical work focused on thermocapillary instabilities and wave formation in thin liquid films flowing over uniformly heated surfaces. 14,15 While inertial effects are significant in these studies, instabilities also lead to interesting behavior in the lubrication 16,17 and Stokes flow 18 regimes.…”

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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Thin liquid films on a slightly inclined heated plate

Cited by 113 publications

References 61 publications

3D Large scale Marangoni convection in liquid films

3D Large scale Marangoni convection in liquid films

The thin film equation with backwards second order diffusion

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

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