Instabilities of a liquid film flowing down a slightly inclined plane

Floryan, J. M.; Davis, Stephen H.; Kelly, R. E.

doi:10.1063/1.866285

Cited by 113 publications

(115 citation statements)

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“…excluding near-horizontal configurations, for which instabilities that set in are typical of wall flows, involving Tollmien-Schlichting waves of shear-viscous origin, see e.g. Floryan et al (1987). The flow conditions can further be characterised by the dimensionless thickness of the flat film solution (Nusselt flow), h N , the inclination B = cot β and the Kapitza number Γ = σ ρν 4/3 (g sin β) 1/3 which compares the surface stress σ/l ν to the viscous stress µ/t ν .…”

Section: Primitive Equationsmentioning

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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Section: Primitive Equationsmentioning

confidence: 99%

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

2006

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“…These boundary conditions can be referred to in Ref. 5. At the free surface z = d + h, we have the kinematic condition…”

Section: ͑4͒mentioning

confidence: 99%

“…In the work of Floryan et al, 5 the surface tension number instead of the Weber number is introduced. is defined as…”

Section: ͑32͒mentioning

confidence: 99%

“…4 and investigated the instability of corrected system in the case of zero surface tension. Floryan et al 5 studied both the soft mode and the hard mode and paid particular attention to the effects of surface tension. Their results show that the characteristics of the hard mode depend on both the angle of inclination and the surface tension.…”

Section: Introductionmentioning

confidence: 99%

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Instabilities and transient behaviors of a liquid film flowing down a porous inclined plane

Liu

2010

Physics of Fluids

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The nonmodal linear stability of a falling film over a porous inclined plane has been investigated. The base flow is driven by gravity. We use Darcy's law to describe the flow in the porous medium. A simplified one-sided model is used to describe the fluid flow. In this model, the influence of the porous layer on the flow in the film can be identified by a parameter ␤. The instabilities of a falling film have traditionally been investigated by linearizing the governing equations and testing for unstable eigenvalues of the linearized problem. However, the results of eigenvalue analysis agree poorly in many cases with experiments, especially for shear flows. In the present paper, we have studied the linear stability of three-dimensional disturbances using the nonmodal stability theory. Particular attentions are paid to the transient behavior rather than the long time behavior of eigenmodes predicted by traditional normal mode analysis. The transient behaviors of the response to external excitations and the response to initial conditions are studied by examining the pseudospectral structures and the energy growth function G͑t͒. Before we study the nonmodal stability of the system, we extend the results of long-wave analysis in previous works by examining the linear stabilities for streamwise and spanwise disturbances. Results show that the critical conditions of both the surface mode and the shear mode instabilities are dependent on ␤ for streamwise disturbances. However, the spanwise disturbances have no unstable eigenvalue.

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“…Benjamin (1957) and Yih (1963) first performed a linear stability analysis of isothermal falling films and identified the surface wave instability through a long-wave asymptotic approach. DeBruin (1974), Floryan, Davis & Kelly (1987) and Woods & Lin (1996) revealed the short shear instability (Tollmien-Schlichting wave) occurring at a slightly inclined angle (<1 • ). Naturally, if the water film flows on a flat ice layer with a finite thickness, the traditional film model produces an excellent starting point for the analysis of the icing instability.…”

Section: Introductionmentioning

confidence: 99%

Linear spatio-temporal instability analysis of ice growth under a falling water film

Zhou

Yi-hong

et al. 2010

J. Fluid Mech.

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A linear spatio-temporal stability analysis is conducted for the ice growth under a falling water film along an inclined ice plane. The full system of linear stability equations is solved by using the Chebyshev collocation method. By plotting the boundary curve between the linear absolute and convective instabilities (AI/CI) of the ice mode in the parameter plane of the Reynolds number and incline angle, it is found that the linear absolute instability exists and occurs above a minimum Reynolds number and below a maximum inclined angle. Furthermore, by plotting the critical Reynolds number curves with respect to the inclined angle for the downstream and upstream branches, the convectively unstable region is determined and divided into three parts, one of which has both downstream and upstream convectively unstable wavepackets and the other two have only downstream or upstream convectively unstable wavepacket. Finally, the effect of the Stefan number and the thickness of the ice layer on the AI/CI boundary curve is investigated.

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Instabilities of a liquid film flowing down a slightly inclined plane

Cited by 113 publications

References 10 publications

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

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

Instabilities and transient behaviors of a liquid film flowing down a porous inclined plane

Linear spatio-temporal instability analysis of ice growth under a falling water film

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