Viscoplastic rimming flow inside a rotating cylinder

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Viscoplastic rimming flow inside a rotating cylinder

Thomasina V. Ball and Neil J. Balmforth

Phys. Rev. Fluids 9, 023304 – Published 27 February 2024

Abstract

A theoretical analysis is presented for the flow of a Herschel-Bulkley fluid around the inside surface of a rotating cylinder, at rotation speeds for which the fluid largely collects in a prominent pool in the lower part of the cylinder. The analysis, based on lubrication theory, predicts the steady states typically reached after a small number of rotations. The analysis is also modified to consider the drainage of the film around a stationary cylinder, which allows an exploration of the dynamics when a rotating drum is suddenly stopped. The predictions of the theory are compared with experiments in which a Carbopol suspension is rotated inside an acrylic drum.

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Received 23 August 2023
Accepted 16 January 2024

DOI:https://doi.org/10.1103/PhysRevFluids.9.023304

Physics Subject Headings (PhySH)

Lubrication theory Non-Newtonian fluids

Fluid Dynamics

Authors & Affiliations

Thomasina V. Ball ¹ and Neil J. Balmforth ²

¹Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
²Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada

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Issue

Vol. 9, Iss. 2 — February 2024

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Images

Figure 1
A sketch showing the geometry of the rimming flow. The red and blue stars mark the ends of the “pool” over which most of the fluid collects and the depth $\hat{h}$ varies significantly; beyond these points there is a residual, potentially plugged up film. The pool has a typical length $L ≪ R$ . Lubrication theory predicts that the flow profile takes a distinctive form in which a strongly sheared layer intervenes between the drum and a plug-like region, the “pseudoplug,” bordering the free surface; the strongly sheared layer has thickness $\hat{Y}$ . Beyond the pool, one anticipates a residual, plugged-up film to coat the remainder of the cylinder (with $\hat{Y} = 0$ ).
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Figure 2
Steady solutions with varying $Bi$ (as indicated) for $(Λ, f, n, S) = (10, 0.01, 1, 10^{- 4})$ . Shown are the positions of the free surface $η$ and yield surface $Y$ , with the pseudoplug and plug shaded according to the local value of plug speed, $u_{p} \equiv 1 - \frac{3}{2} Λ Γ Y^{2}$ . The vertical dot-dashed lines show the yield points. In panel (a), the dashed line shows the corresponding Newtonian solution, and the left-hand inset replots all the surface profiles (from blue to green) using the scaled variables $η / η_{max}$ and $(θ - θ_{max}) / \sqrt{η_{max}}$ , and comparing with the approximation in Eq. (38) (red dashed line). The right-hand insets show the profiles of some of the solutions as they would appear inside the drum.
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Figure 3
(a) Flux, $q$ , and (b), (c) peak height and position, $η_{max}$ and $θ_{max}$ , plotted against $Bi$ for $(Λ, f, S) = (10, 0.01, 10^{- 4})$ . The solid line in panel (a) shows the prediction from Eq. (40). The solid lines in panels (b) and (c) show the results from the analysis of the Appendix in which surface tension is neglected, and the dashed lines show the predictions from Eqs. (44) and (46).
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Figure 4
Steady solutions with increasing $Λ$ (from green to blue) for $(f, Bi, n, S) = (0.0838, 0.158, 0.38, 10^{- 4})$ . In panel (a) the free surface profile is shown for the values of $Λ$ indicated. The upper left inset shows a collapse of these profiles over the pool, along with the prediction in Eq. (38). In panels (b)–(d), the flux $q$ , and the peak height and position, $η_{max}$ and $θ_{max}$ , are plotted against $Λ$ for a wider set of solutions (stars). The solid lines in panel (b) shows the predictions from Eq. (41), and the solid lines in panels (c), (d) indicate the results from the $S = 0$ analysis of the Appendix. The dashed lines in panels (b), (c), (d) show the predictions from Eqs. (42), (44), and (46).
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Figure 5
Steady solutions with $S = 10^{- 4}, 10^{- 5}$ , and $10^{- 6}$ (solid, from blue to green), for $(Λ, Bi, n, f) = (10, 0.1, 1, 0.01)$ . The dashed line shows the construction of the Appendix for $S = 0$ . The dotted line shows the approximation (38). The main panel shows the pool solutions; the insets show the entire solutions and magnifications near the borders of the pool (as indicated by the shading and arrows).
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Figure 6
Anatomy of the solution with $(Λ, f, Bi, n, S) = (10, 0.01, 0.1, 1, 10^{- 4})$ . Plotted are $η$ (blue) and $Y$ (red), along with a selection of streamlines (light gray solid lines). In panel (b), the residual film is shown in more detail. The dashed gray lines show the root of the cubic $q = c o n s t a n t$ with $Γ = sin θ$ , to which the solution converges over the film; the gray dotted line shows a second root, which intersects the first at the green circle at $θ = \frac{1}{2} π$ . A magnification of the pool is shown in panel (c). Details of where the film reenters the pool are plotted in panel (d), along with $η_{θ θ}$ .
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Figure 7
A similar set of plots to Fig. 6, but for $Bi = 0.4$ .
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Figure 8
(a) Solutions in the plastic limit for $f = 0.055$ , 0.064, 0.0747, 0.0838, and 0.1, with $S = 0$ and $(n, Bi) = (0.38, 0.158)$ ; the stars indicate the yield points. The solutions are replotted as they would appear in the drum in panel (b), with the dashed lines indicating the approximation of the pool as a segment given by Eqs. (A2) and (48). (c) Peak depth $η_{max}$ and its position $θ_{max}$ plotted against $f$ . The dotted lines show Eqs. (43) and (48); the improvements in Eqs. (44) and (A2) are shown by dashed and dot-dashed lines, respectively. The stars indicate the values of $f$ shown in panel (a). (d), (e) Peak depth $η_{max}$ and its position $θ_{max}$ plotted against $Λ^{- n}$ for the same values of $f$ as in panel (a). The limits for $Λ \to \infty$ , from panel (b), are indicated by stars.
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Figure 9
Two viscous drainage solutions for $f = 0.0838$ and $S = 10^{- 4}$ ( $Λ = 1$ ). The initial conditions are (a) $η (θ, 0) = \frac{1}{2}$ and (b) a steady rotating solution with $Λ = 1$ . The residual film is scaled by $η (π, t)$ in panel (c), and the time series of $η (π, t)$ is plotted in panel (d). The times of the snapshots in panels (a), (b), which increase from blue to red, are plotted as dots in panel (d); the circles show the slightly fewer snapshots plotted in panel (c). The dots in panel (c) and gray dashed line in panel (d) show the prediction from Eq. (50) with $n = 1$ .
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Figure 10
Drainage solutions for $(f, Bi, n, S) = (0.0838, 0.158, 0.38, 10^{- 4})$ . In panel (a), a flat pool at the bottom of the cylinder is suddenly rotated clockwise by 90 degrees; shown are snapshots of $η (θ, t)$ as the fluid slumps back to rest (from blue to red). The black dashed line shows $η \sim Bi / | sin θ |$ , and the dots plot (38). In panel (b), the final state is plotted, along with three others generated by successive rotate-and-slump events (see main text, with the progression plotted from green to blue). In panels (c), (d), slumps from steady states in rotating drums are used (respectively, $Λ = 684$ and 68.4), simulating sudden stops. For panel (e), the slump occurs from a uniform coating $[η (θ, 0) = \frac{1}{2}]$ . The right-hand insets show the solutions as they would appear in the cylinder, and the insets on the left show the time series of $η (0, t)$ , with the stars showing the times of the snapshots in the main panels. For the solutions in panels (a) and (b), to avoid any issues with contact lines, a thin prewetted film of thickness $10^{- 2}$ is included to ensure that $η (θ, 0) > 0$ everywhere. The black dots in panels (c) and (e) plot the results predicted in the plastic limit for $S = 0$ [i.e., from integrating Eq. (47)] subject to either $η = Bi$ at the left of the pool and $θ = \frac{1}{2} π$ (c), or symmetry about $θ = 0$ (e).
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Figure 11
Images of the rotating drum, with fits to the surface profile of an enclosed Carbopol suspension, dyed blue with food colouring. In panels (b), (c), with rotation rates are $Ω = 0.024 rad / s$ and $0.14 rad / s$ , respectively, two fits are shown. The solid line shows a fit to the profile along the front face; the dashed line is a fit with a weaker threshold that captures more of the profile further inside the drum as well as the residual film outside the pool. The two fits are mostly the same in panel (b), but at the faster rotation speed in panel (c), the three-dimensionality of the front edge of the pool becomes more pronounced; the second fit takes some account of this to furnish a more averaged profile. An oblique view of a three-dimensional surface profile is shown in panel (d), along with some surface wrinkles generated by the flow dynamics. $f = 0.0838$ .
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Figure 12
(a) Experimental syrup profiles at the dimensionless rotation rates, $Λ^{- 1} = 3 μ Ω / (ρ g R f^{2}) = (0.21, 0.43, 0.67)$ (from green to blue), with $f = 0.0687$ . (b) Maximum depths ${\hat{h}}_{max}$ and (c) angular positions $θ_{max}$ plotted against $Λ^{- 1}$ . The open stars correspond to the drum with inner radius and width, $(R, W) = (27.8, 13.1) cm$ ; the filled stars correspond to a wider drum with $(R, W) = (28.8, 20) cm$ . The solid and dashed lines in panels (b), (c) show equivalent theoretical results with $S = 10^{- 5}$ and $S = 0$ , respectively. The red circle in panel (b) shows the depth expected for $Ω = 0$ . The fits of the surface in the experiments are all to the syrup profile as observed at the front face of the drum.
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Figure 13
(a) Experimental Carbopol profiles for $f = 0.0838$ at the rotation rates, $Ω = (0.012, 0.036, 0.097, 0.24, 0.44)$ rad/s, corresponding to $Λ = (457, 152, 57, 23, 12)$ (from blue to green) and $(n, Bi) = (0.38, 0.158)$ . The dots show the profiles fitted for the front face; the dash-dotted lines show the fits including more of the interior (see Fig. 11). (b) Maximum depths ${\hat{h}}_{max}$ and (c) angular positions $θ_{max}$ plotted against rotation rate $Ω$ . The solid and dashed lines in panels (b), (c) translate the theoretical results (with $S = 10^{- 4}$ and $S = 0$ , respectively) from Fig. 4; the stars and triangles show the two fits (front face and including interior, respectively).
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Figure 14
Experimental results for Carbopol with varying fill fraction ( $f = 0.055$ , 0.064, 0.0747, 0.0838 and 0.1, from green to blue), showing (a) maximum depth $h_{max} = {\hat{h}}_{max} / R \equiv f η_{max}$ and (b) position $θ_{max}$ plotted against $Ω^{n}$ (with $Ω$ in rad/s). The dotted lines show linear fits, and the insets plot the implied intercepts (for $Ω \to 0$ ) against $f$ . The solid lines in the insets show Eqs. (44) and (48) (with $q = Bi$ ).
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Figure 15
Experimental drainage tests with Carbopol; $f = 0.0838$ and $(n, Bi) = (0.38, 0.158)$ . In panel (a), a mound of material emplaced at the bottom of the cylinder was rotated clockwise by about $90^{\circ}$ to force an adjustment to rest (green curve). The new mound was then rotated again twice (by a similar angle) to prompt two more adjustments (each leaving a longer residual coating behind; solid lines, from green to blue). The cylinder was then rotated three more times, with the fluid now encountering the coating from the previous rotate-and-slump events (dashed lines, from green to blue). In panels (b)–(d), steady-state cylinders rotating with rates of $Ω = 0.008$ rad/s, 0.08 rad/s and 2 rad/s were abruptly stopped, leaving the fluid to drain (outlines from blue to red). In panel (d), the dots show the outcome of a repetition of the experiment. In all panels, the shaded region indicates the uniform film expected with a fill fraction of $f = 0.0838$ , and the units are in cm.
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Figure 16
Construction of steady solutions without surface tension for (a) Newtonian and (b), (c) Bingham fluid (parameter settings indicated). Beginning from the peak, we integrate left (blue solid) and right (red solid) until the solution converges to one of the residual film solutions (gray dashed) or a yield point (red stars). Over any plugs, $η = q$ . The thin black line shows the constructed solutions. The dotted line shows the approximation for the pool in panel (38). For panel (c), the steady solution is given approximately by Eq. (47) and the film solution is just $η \approx Bi / | sin θ |$ or $η \approx Bi$ .
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