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Buckling of a thin, viscous film in an axisymmetric geometry
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10.1063/1.4798825
/content/aip/journal/pof2/25/4/10.1063/1.4798825
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/4/10.1063/1.4798825

Figures

Image of FIG. 1.
FIG. 1.

Plan view of shear-driven buckling of a thin viscous film in (a) a rectilinear channel and (b) an axisymmetric channel. Troughs and crests of the associated wave field are indicated by the dashed lines. Panel (c) shows a side view of the thin film and its (sinuous) deflection from the horizontal where ρ denotes the film fluid density.

Image of FIG. 2.
FIG. 2.

Out-of-plane displacement field associated with the most unstable azimuthal mode of (2.12) for (a) ( = 16), (b) ( = 23), (c) ( = 27), and (d) ( = 27). Other parameter values are as follows: ɛ = 0.0207, Ω = 0.365, Υ = 14.2, and Υ = 4.53 × 10. The inner cylinder of Figure 1(b) occupies the white region in the center of each image and rotates in the counter-clockwise direction.

Image of FIG. 3.
FIG. 3.

Variation of with ɛ and β as predicted by the numerical solution of (2.12) . Note that the minimum value of ɛ is 6.9 × 10, not 0. Parameter values are as follows: (a) Ω = 0.365, Υ = 14.2, and Υ = 4.53 × 10; (b) Ω = 0.365, Υ = 5.93, and Υ = 1.89 × 10; (c) Ω = 0.632, Υ = 8.22, and Υ = 7.85 × 10; and (d) Ω = 0.632, Υ = 3.42, and Υ = 3.27 × 10. Thus moving from the left-hand side panels to their right-hand side counterparts, we increase μ by a factor of 2.4 (see Sec. III ). Likewise in moving from the upper panels to their lower counterparts, we decrease by a factor of 3.

Image of FIG. 4.
FIG. 4.

Variation of (solid symbols) and σ (open symbols) with Ω for β = 0.655, Υ = 14.2, and Υ = 4.53 × 10 as predicted by the numerical solution of (2.12) . Symbols are as follows – diamonds (⋄): ɛ = 6.90 × 10, left-facing triangles (◁): ɛ = 0.0138, right-facing triangles (▷): ɛ = 0.0207, squares ( ): ɛ = 0.0276, and stars (*): ɛ = 0.0345.

Image of FIG. 5.
FIG. 5.

Curves of constant (non-dimensional) growth rate with σ = 0, 0.030, 0.061, 0.122, 0.304, 0.608. The dashed curve connects the respective local minima and the arrow indicates the direction of increasing σ. Parameter values are as follows: β = 0.655, ɛ = 0.0207, Υ = 14.2, and Υ = 4.53 × 10.

Image of FIG. 6.
FIG. 6.

(Left) Schematic of the experimental set-up (camera elevation and light source position not to scale). The beaker rests on an adjustable frame of T-slotted aluminum bars, i.e., 80-20 Inc., product 1010. (Right) Detailed schematic of the disk assembly.

Image of FIG. 7.
FIG. 7.

Snapshot image from an experiment with Υ = 14.5, Υ = 4.50 × 10, i.e., (ν = 12 500 cSt), β = 0.667, ɛ = 0.0193, and Ω = 0.239. (Top) Raw image and (middle) subtracted image. In the latter case, the arrows and numbers denote individual wave crests. For the sake of comparison, the bottom panel shows the associated numerical solution.

Image of FIG. 8.
FIG. 8.

(a) Comparison between the solution of (2.12) and experimental measurements for ν = 12 500 cSt. Black, red, and blue curves correspond, respectively, to β = 0.497, 0.667, and 0.844; in all cases, Υ = 14.5 and Υ = 4.50 × 10. (b) Measured angular phase speeds for ν = 12 500 cSt. Data points for panels (a) and (b) are summarized in Table II . (c) Comparison between the solution of (2.12) and experimental measurements for ν = 30 000 cSt. Red and blue curves correspond, respectively, to β = 0.667 and 0.844; in both cases, Υ = 6.05 and Υ = 1.88 × 10. (d) Measured angular phase speeds for ν = 30 000 cSt. Data points for panels (c) and (d) are summarized in Table III . Representative vertical error bars are as indicated.

Image of FIG. 9.
FIG. 9.

Wave amplitude (normalized by , the film thickness) vs. Ω. (a) Υ = 14.5 and Υ = 4.50 × 10 (i.e., ν = 12 500 cSt); the open and closed circles correspond, respectively, to ɛ = 0.0214 and ɛ = 0.0269. (b) Υ = 6.05 and Υ = 1.88 × 10 (i.e., ν = 30 000 cSt); the open and closed circles correspond, respectively, to ɛ = 0.0139 and ɛ = 0.0205. In all cases, β = 0.667; the thin and thick solid vertical lines show representative vertical error bars for the open and closed circles, respectively. Note that wave amplitudes are too small to be reliably measured for small Ω, e.g., for Ω ≲ 0.23 when considering the open circles of (a). Conversely the wave pattern becomes irregular for large Ω, e.g., for Ω ≳ 0.14 when considering the closed circles of (b).

Tables

Generic image for table
Table I.

Geometric details for the disk assemblies. Measurements of are considered accurate to within 1 mm.

Generic image for table
Table II.

Data points descriptions – Figures 8(a) and 8(b) . In all cases, Υ = 14.5 and Υ = 4.50 × 10.

Generic image for table
Table III.

Data points descriptions – Figures 8(c) and 8(d) . In all cases, Υ = 6.05 and Υ = 1.88 × 10.

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/content/aip/journal/pof2/25/4/10.1063/1.4798825
2013-04-05
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Buckling of a thin, viscous film in an axisymmetric geometry
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/4/10.1063/1.4798825
10.1063/1.4798825
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