^{1}, R. V. Craster

^{2}and M. R. Flynn

^{1}

### Abstract

By adapting the Föppl-von Kàrmàn equation, which describes the deformation of a thin elastic membrane, we present an analysis of the buckling pattern of a thin, very viscous fluid layer subject to shear in an axisymmetric geometry. A linear stability analysis yields a differential eigenvalue problem, whose solution, obtained using spectral techniques, yields the most unstable azimuthal wave-number, m ⋆. Contrary to the discussion of Slim et al. [J. Fluid Mech.694, 5–28 (Year: 2012)]10.1017/jfm.2011.437, it is argued that the axisymmetric problem shares the same degeneracy as its rectilinear counterpart, i.e., at the onset of instability, m ⋆ is indefinitely large. Away from this point, however, a comparison with analogue experimental results is both possible and generally favorable. In this vein, we describe the laboratory apparatus used to make new measurements of m ⋆, the phase speed and the wave amplitude; note that no prediction concerning the latter two quantities can be made using the present theory. Experiments reveal a limited range of angular velocities wherein waves of either small or large amplitude may be excited. Transition from one to the other regime does not appear to be associated with a notable change in m ⋆.

Financial support for this study was generously provided by Natural Sciences and Engineering Research Council (Canada) (NSERC) through the Discovery Grant and Research Tools and Instruments programs. The experiments of Sec. III benefitted from the helpful suggestions of J. A. Teichman and the fabrication assistance of the Technical Resources Group in the Department of Mechanical Engineering at the University of Alberta. The high speed camera used in the experiments was borrowed from the laboratory of Dr. David S. Nobes who provided additional experimental advice. INEOS Oligomers and the Kish Company kindly supplied free chemical samples. Finally, the thoughtful feedback provided by two anonymous referees helped to improve the clarity of our discussion.

I. INTRODUCTION

II. THEORY

A. Formulation

B. Theoretical predictions

III. EXPERIMENTAL PROCEDURE

IV. RESULTS AND DISCUSSION

A. Comparison between linear theory and laboratory measurements

B. Wave amplitudes

V. CONCLUSIONS

### Key Topics

- Viscosity
- 22.0
- Buckling
- 20.0
- Fluid equations
- 10.0
- Surface tension
- 9.0
- Shear rate dependent viscosity
- 8.0

## Figures

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.

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.

Out-of-plane displacement field associated with the most unstable azimuthal mode of (2.12) for (a) (m ⋆ = 16), (b) (m ⋆ = 23), (c) (m ⋆ = 27), and (d) (m ⋆ = 27). Other parameter values are as follows: ɛ = 0.0207, ΩT = 0.365, Υ1 = 14.2, and Υ2 = 4.53 × 10−3. The inner cylinder of Figure 1(b) occupies the white region in the center of each image and rotates in the counter-clockwise direction.

Out-of-plane displacement field associated with the most unstable azimuthal mode of (2.12) for (a) (m ⋆ = 16), (b) (m ⋆ = 23), (c) (m ⋆ = 27), and (d) (m ⋆ = 27). Other parameter values are as follows: ɛ = 0.0207, ΩT = 0.365, Υ1 = 14.2, and Υ2 = 4.53 × 10−3. The inner cylinder of Figure 1(b) occupies the white region in the center of each image and rotates in the counter-clockwise direction.

Variation of m ⋆ with ɛ and β as predicted by the numerical solution of (2.12) . Note that the minimum value of ɛ is 6.9 × 10−3, not 0. Parameter values are as follows: (a) ΩT = 0.365, Υ1 = 14.2, and Υ2 = 4.53 × 10−3; (b) ΩT = 0.365, Υ1 = 5.93, and Υ2 = 1.89 × 10−3; (c) ΩT = 0.632, Υ1 = 8.22, and Υ2 = 7.85 × 10−3; and (d) ΩT = 0.632, Υ1 = 3.42, and Υ2 = 3.27 × 10−3. 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 g by a factor of 3.

Variation of m ⋆ with ɛ and β as predicted by the numerical solution of (2.12) . Note that the minimum value of ɛ is 6.9 × 10−3, not 0. Parameter values are as follows: (a) ΩT = 0.365, Υ1 = 14.2, and Υ2 = 4.53 × 10−3; (b) ΩT = 0.365, Υ1 = 5.93, and Υ2 = 1.89 × 10−3; (c) ΩT = 0.632, Υ1 = 8.22, and Υ2 = 7.85 × 10−3; and (d) ΩT = 0.632, Υ1 = 3.42, and Υ2 = 3.27 × 10−3. 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 g by a factor of 3.

Variation of m ⋆ (solid symbols) and σ (open symbols) with ΩT for β = 0.655, Υ1 = 14.2, and Υ2 = 4.53 × 10−3 as predicted by the numerical solution of (2.12) . Symbols are as follows – diamonds (⋄): ɛ = 6.90 × 10−3, left-facing triangles (◁): ɛ = 0.0138, right-facing triangles (▷): ɛ = 0.0207, squares ( ): ɛ = 0.0276, and stars (*): ɛ = 0.0345.

Variation of m ⋆ (solid symbols) and σ (open symbols) with ΩT for β = 0.655, Υ1 = 14.2, and Υ2 = 4.53 × 10−3 as predicted by the numerical solution of (2.12) . Symbols are as follows – diamonds (⋄): ɛ = 6.90 × 10−3, left-facing triangles (◁): ɛ = 0.0138, right-facing triangles (▷): ɛ = 0.0207, squares ( ): ɛ = 0.0276, and stars (*): ɛ = 0.0345.

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, Υ1 = 14.2, and Υ2 = 4.53 × 10−3.

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, Υ1 = 14.2, and Υ2 = 4.53 × 10−3.

(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.

(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.

Snapshot image from an experiment with Υ1 = 14.5, Υ2 = 4.50 × 10−3, i.e., (ν = 12 500 cSt), β = 0.667, ɛ = 0.0193, and ΩT = 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.

Snapshot image from an experiment with Υ1 = 14.5, Υ2 = 4.50 × 10−3, i.e., (ν = 12 500 cSt), β = 0.667, ɛ = 0.0193, and ΩT = 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.

(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, Υ1 = 14.5 and Υ2 = 4.50 × 10−3. (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, Υ1 = 6.05 and Υ2 = 1.88 × 10−3. (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.

(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, Υ1 = 14.5 and Υ2 = 4.50 × 10−3. (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, Υ1 = 6.05 and Υ2 = 1.88 × 10−3. (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.

Wave amplitude (normalized by h, the film thickness) vs. ΩT. (a) Υ1 = 14.5 and Υ2 = 4.50 × 10−3 (i.e., ν = 12 500 cSt); the open and closed circles correspond, respectively, to ɛ = 0.0214 and ɛ = 0.0269. (b) Υ1 = 6.05 and Υ2 = 1.88 × 10−3 (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 ΩT, e.g., for ΩT ≲ 0.23 when considering the open circles of (a). Conversely the wave pattern becomes irregular for large ΩT, e.g., for ΩT ≳ 0.14 when considering the closed circles of (b).

Wave amplitude (normalized by h, the film thickness) vs. ΩT. (a) Υ1 = 14.5 and Υ2 = 4.50 × 10−3 (i.e., ν = 12 500 cSt); the open and closed circles correspond, respectively, to ɛ = 0.0214 and ɛ = 0.0269. (b) Υ1 = 6.05 and Υ2 = 1.88 × 10−3 (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 ΩT, e.g., for ΩT ≲ 0.23 when considering the open circles of (a). Conversely the wave pattern becomes irregular for large ΩT, e.g., for ΩT ≳ 0.14 when considering the closed circles of (b).

## Tables

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

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

Data points descriptions – Figures 8(a) and 8(b) . In all cases, Υ1 = 14.5 and Υ2 = 4.50 × 10−3.

Data points descriptions – Figures 8(a) and 8(b) . In all cases, Υ1 = 14.5 and Υ2 = 4.50 × 10−3.

Data points descriptions – Figures 8(c) and 8(d) . In all cases, Υ1 = 6.05 and Υ2 = 1.88 × 10−3.

Data points descriptions – Figures 8(c) and 8(d) . In all cases, Υ1 = 6.05 and Υ2 = 1.88 × 10−3.

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