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Travelling-wave similarity solutions for a steadily translating slender dry patch in a thin fluid film
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10.1063/1.4803906
/content/aip/journal/pof2/25/5/10.1063/1.4803906
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/5/10.1063/1.4803906
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Sketch of the geometry of the problem: a steadily translating dry patch in a thin fluid film on a planar substrate inclined at an angle α to the horizontal.

Image of FIG. 2.
FIG. 2.

Sketch of the possible physical forms of the steadily translating dry patch (shown unshaded) in a thin film (shown shaded): (a) and (b) show sessile cases, and (c) and (d) show pendent cases; in (a) and (c), with τ > 0, the dry patch is moving down the substrate, and in (b) and (d), with τ < 0, it is moving up the substrate.

Image of FIG. 3.
FIG. 3.

Plots of as a function of η for various values of τ.

Image of FIG. 4.
FIG. 4.

Enlargement of Fig. 3 , showing as a function of η for τ = −2/3, −11/16, −17/24, −35/48, −3/4, −37/48, −19/24 (upper group) and τ = −11/8, −67/48, −17/12, −3/2, −19/12, τ ≃ −1.6504, −5/3, −7/4, −11/6, −23/12, −2 (lower group). The “bounding curve” = 1 + τ is shown with a dashed line, and the asymptote of this curve at large η, namely = −1/3, is shown with a dotted line.

Image of FIG. 5.
FIG. 5.

Enlargement of Fig. 4 , showing as a function of η for τ = −0.7920, −0.7930, −0.7940, τ ≃ −0.7941, −0.7942, −0.7944, −0.7946, and τ ≃ −0.7947.

Image of FIG. 6.
FIG. 6.

Enlargement of Fig. 4 , showing as a function of η for various values of τ, including τ = τ ≃ −1.6547, τ = τ ≃ −1.6504, and τ = τ ≃ −1.6502. The bounding curve = 1 + τ is shown with a dashed line.

Image of FIG. 7.
FIG. 7.

Contour plot of in the τ-η plane; the contour values increase from = −5/2 (leftmost) to = 7/2 (rightmost) in steps of 1/2. There is no solution in the shaded region, given by 1 + τ < < 0, the left boundary of which, = 1 + τ, is a contour of .

Image of FIG. 8.
FIG. 8.

Cross-sectional profiles (η) for τ = 5, 0, −2/3, −5/3, and −5, with η = 1, for which ≃ 2.9611, 0.4113, 0.0494, −0.5560, and −2.2820, respectively.

Image of FIG. 9.
FIG. 9.

Plots of (a) and (b) as functions of η for various values of τ. The asymptotic forms in the limit η → ∞ given later in (55) , namely, ∼ η and ∼ (1/3 + τ/2)η, are shown with dashed lines.

Image of FIG. 10.
FIG. 10.

Free surface profiles and contour plots of the velocity for (a) τ = 2, (b) τ = −0.45, (c) τ = −2/3, and (d) τ = −2, with η = 1/2 in each case. Regions of downflow ( ⩾ 0) and of upflow ( < 0) are shown unshaded and shaded, respectively. The curves on which ∂/∂ = 0, namely, = + τ, are shown with dashed lines.

Image of FIG. 11.
FIG. 11.

Streamlines of the depth-integrated flow in a frame of reference moving with the dry patch, in the case η = 1/2 and τ = 2 (for which ≃ 1.6012), plotted for (a) a sessile case and (b) a pendent case. In both (a) and (b) the direction of flow in this moving reference frame is from bottom to top.

Image of FIG. 12.
FIG. 12.

Numerical solutions for of (21) in the cases (a) η = 0.2 and (b) η = 10, with τ = 0 (shown with solid lines), together with the leading-order asymptotic solution in the limit η → 0 obtained by solving (44) subject to (45) and (47) (shown with dashed lines). In part (b) the dashed line is virtually indistinguishable from the solid line.

Image of FIG. 13.
FIG. 13.

Numerical solutions for of (21) in the cases (a) η = 1 and (b) η = 5, with τ = 0 (shown with solid lines), together with the leading-order asymptotic solution (52) in the limit η → ∞ (shown with dashed lines). In part (b) the dashed line is virtually indistinguishable from the solid line.

Image of FIG. 14.
FIG. 14.

Plot of as a function of η for purely gravity-driven flow (τ = 0), together with the asymptotic value = = 1/3 in the limit η → ∞ (shown with a dashed line). The inset shows an enlargement of the behaviour near η = 0; the point = ≃ 0.6169 at η = 0 is shown as a dot, as are the (local) minimum = ≃ 0.6167 at η ≃ 0.0050 and the (global) maximum = ≃ 0.6225 at η ≃ 0.0630.

Image of FIG. 15.
FIG. 15.

Cross-sectional profiles (η) for purely gravity-driven flow (τ = 0) for the cases η = 0.1, 1, 3, 5, and 10, for which ≃ 0.6191, 0.4113, 0.3488, 0.3395, and 0.3350, respectively.

Image of FIG. 16.
FIG. 16.

Three-dimensional plots of the free-surface profiles for purely gravity-driven flow (τ = 0) in a sessile case with η = 1 (for which ≃ 0.4113) at times = 0, 5, and 10.

Image of FIG. 17.
FIG. 17.

Plot of (upper curve) and (= ) (lower curve) as functions of η for purely gravity-driven flow (τ = 0), together with the values ≃ 0.7449 and ≃ 0.4595 at η = 0 (shown as dots) and the asymptotic forms ∼ η and ∼ η/3 in the limit η → ∞ (shown with dashed lines).

Image of FIG. 18.
FIG. 18.

Plot of as a function of η for purely surface-shear-stress-driven flow, together with the asymptotic value = = 1/2 in the limit η → ∞ (shown with a dashed line). The inset shows an enlargement of the behaviour near η = 0; the point = ≃ 0.7712 at η = 0 is shown as a dot, as are the (local) minimum = ≃ 0.7711 at η ≃ 0.0050 and the (global) maximum = ≃ 0.7751 at η ≃ 0.0550.

Image of FIG. 19.
FIG. 19.

Cross-sectional profiles (η) for purely surface-shear-stress-driven flow for the cases η = 0.1, 1, 3, 5, and 10, for which ≃ 0.7706, 0.5745, 0.5151, 0.5061, and 0.5016, respectively.

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/content/aip/journal/pof2/25/5/10.1063/1.4803906
2013-05-21
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Travelling-wave similarity solutions for a steadily translating slender dry patch in a thin fluid film
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/5/10.1063/1.4803906
10.1063/1.4803906
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