Three-Dimensional Long Waves on a Liquid Film

Roskes, G. J.

doi:10.1063/1.1693099

Cited by 65 publications

(35 citation statements)

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“…An extension to three dimensions of the Benney equation was calculated by Roskes. 8 In the very strong surface tension approximation, Sivashinsky and Michelson 9 demonstrated that this equation reduces to the Kuramoto-Sivashinsky equation, which has been shown to present chaotic behavior. Different approximations to the nonlinear surface wave on thin liquid films have been done, for example, by Alekseenko et al, 10 Alekseenko et al, 11 Trifonov,12 and Chang.…”

Section: Introductionmentioning

confidence: 98%

Nonlinear instability of a thin film flowing down a smoothly deformed surface

Dávalos‐Orozco

2007

Physics of Fluids

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Here, the nonlinear flow instability of a thin film flowing down an inclined wall with smooth deformations is investigated. A nonlinear evolution equation is obtained under the small wavenumber approximation. This equation describes the film free surface deformation including the effects of inertia, viscosity, surface tension, and deformation of the wall. The equation has a forcing term that corresponds to periodic time-dependent perturbations hitting on the free surface. These time-dependent perturbations are tested against the presence of the free surface waves due to wall deformations. In particular, a wall sinusoidal profile is investigated. It is found that when the wall wavelength is small enough, the valleys of the surface response to the wall profile become very deep, showing a kind of resonance, which has an important stabilizing influence in the evolution of the superposed time-dependent perturbations.

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Section: Introductionmentioning

confidence: 98%

Nonlinear instability of a thin film flowing down a smoothly deformed surface

Dávalos‐Orozco

2007

Physics of Fluids

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“…This leads to a single evolution equation for the film thickness h governing the dynamics of the flow at the onset of the instability. Several one-equation models have therefore been proposed to investigate the 3D dynamics of film flows (Roskes 1969;Atherton & Homsy 1976;Roy et al 2002;Saprykin et al 2005). However, for the range of Reynolds numbers where 3D wavy regimes have been reported by Liu et al (1995); Park & Nosoko (2003), one-equation models have been shown to fail, leading either to an underestimation of the wave speeds and heights, or exhibiting unphysical behaviours (Pumir et al 1983;Ooshida 1999;Scheid et al 2005b).…”

Section: Introductionmentioning

confidence: 99%

Wave patterns in film flows: modelling and three-dimensional waves

2006

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In a previous work, two-dimensional film flows were modelled using a weighted-residual approach that led to a four-equation model consistent at order 2 . A two-equation model resulted from a subsequent simplification but at the cost of lowering the degree of the approximation to order only (Ruyer-Quil & Manneville 2000). A Padé approximant technique is applied here to derive a refined two-equation model consistent at order 2 . This model, formulated in terms of coupled evolution equations for the film thickness h and the flow rate q, accounts for inertia effects due to the deviations of the velocity profile from the parabolic shape, and closely sticks to the asymptotic long-wave expansion in the appropriate limit. Comparisons of two-dimensional wave properties with experiments and direct numerical simulations show good agreement for the range of parameters where a two-dimensional wavy motion is reported in experiments.The stability of two-dimensional travelling waves against three-dimensional perturbations is investigated based on the extension of the models to include spanwise dependence. The secondary instability is found to be not much selective, which explains the widespread presence of the synchronous instability observed in the experiments by Liu et al. (1995) whereas Floquet analysis predicts a subharmonic scenario in most cases. Three-dimensional wave patterns are next computed assuming periodic boundary conditions. Transition from 2D to 3D flows is shown to be strongly dependent on initial conditions. The herringbone patterns, the synchronously deformed fronts and the threedimensional solitary waves observed in experiments (Liu et al. 1995;Park & Nosoko 2003;Alekseenko et al. 1994) are recovered using our regularised model, which is found to be an excellent compromise between the complete model, which has seven equations, and the simplified model, which does not include the second-order inertia corrections. Those corrections are found to play a role in the selection of the type of secondary instability as well as of the spanwise wavelength of the emerging pattern.

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“…Mathematical models and numerical simulations for the flow of a thin film of fluid have important applications in industrial and natural processes (Roskes 1969;Ruschak 1985;Moriarty, Schwartz & Tuck 1991;Chang 1994;Grotberg 1994;Schwartz, Weidner & Eley 1995;Decré & Baret 2003). The dynamics of a thin fluid film spreading or retracting from the surface of a supporting liquid or solid substrate has long been an active area of research because of its impact on many technological fields: for example, applications of coating flows (Ruschak 1985) range from a single decorative layer on packaging, to multiplelayer coatings on photographic film; coated products include adhesive tape, surgical dressings, magnetic and optical recording media, lithographic plates, paper and fabrics.…”

Section: Introductionmentioning

confidence: 99%

An accurate and comprehensive model of thin fluid flows with inertia on curved substrates

Roberts¹,

Li²

2006

J. Fluid Mech.

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Consider the three-dimensional flow of a viscous Newtonian fluid upon a curved two-dimensional substrate when the fluid film is thin, as occurs in many draining, coating and biological flows. We derive a comprehensive model of the dynamics of the film, the model being expressed in terms of the film thickness η and the average lateral velocityū. Centre manifold theory assures us that the model accurately and systematically includes the effects of the curvature of substrate, gravitational body force, fluid inertia and dissipation. The model resolves wavelike phenomena in the dynamics of viscous fluid flows over arbitrarily curved substrates such as cylinders, tubes and spheres. We briefly illustrate its use in simulating drop formation on cylindrical fibres, wave transitions, three-dimensional instabilities, Faraday waves, viscous hydraulic jumps, flow vortices in a compound channel and flow down and up a step. These models are the most complete models for thin-film flow of a Newtonian fluid; many other thin-film models can be obtained by different restrictions and truncations of the model derived here.

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Three-Dimensional Long Waves on a Liquid Film

Abstract: The equations of motion for three-dimensional long waves on a liquid film are formulated, and the equation for the surface height is derived. The interaction between two- and three-dimensional waves is considered.

Cited by 65 publications

References 1 publication

Nonlinear instability of a thin film flowing down a smoothly deformed surface

Nonlinear instability of a thin film flowing down a smoothly deformed surface

Wave patterns in film flows: modelling and three-dimensional waves

An accurate and comprehensive model of thin fluid flows with inertia on curved substrates

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