Viscoplasticity can stabilise liquid collar motion on vertical cylinders

Abstract

Liquid films coating vertical cylinders can form annular liquid collars which translate downwards under gravity. We investigate the dynamics of a thin viscoplastic liquid film coating the interior or exterior of a vertical cylindrical tube, quantifying how the yield stress modifies both the Rayleigh-Plateau instability leading to collar formation and the translation of collars down the tube. We use thin-film theory to derive an evolution equation for the layer thickness, which we solve numerically to examine the nonlinear dynamics. Instability and collar formation occur when gravity is sufficiently strong to make the fluid yield initially. We use matched asymptotics to derive a model describing the quasi-steady translation of a slender liquid collar when the Bond number is small. The structure of the asymptotic solution for a viscoplastic collar shares some features with the Newtonian version, but there are several novel asymptotic regions that emerge at the two ends of the collar. The global force balance, which determines the collar's speed, is modified by a leading-order contribution from viscous drag in the collar when the liquid is viscoplastic. We use the asymptotic model to describe slow changes in collar volume when the film thicknesses ahead of, and behind, the collar are unequal. When the film thickness ahead of the collar is less than a critical value that we determine, viscoplastic collars adjust their volume and reach a steadily-translating state. This contrasts with the Newtonian problem, where the only state in which steady translation occurs is unstable to small changes in the film thickness.

Figures (9)

• Figure 1: (a) Sketch of the model geometry. (b) Typical flow structure within the thin liquid film. In dimensionless coordinates, $y=h$ is the free surface, $y=Y$ is the boundary between regions of shear-dominated flow and weakly-yielded pseudo-plugs, and $w$ is the axial velocity.
• Figure 2: Numerical solution of \ref{['TFevoleqn']} with $B=0.1$, $J=0.05$, in a periodic domain of length $\mathcal{L}=2\pi/k_\mathrm{m}=2\sqrt{2}\pi$. (a) Snapshots of the evolving layer. Lines show $h(z,t)$ (blue) and $Y(z,t)$ (red); colour map illustrates the axial velocity. Arrow in upper-left panel indicates the direction of gravity. (b) Time evolution of $\max_z(h)-1$ (black) with the predicted early-time exponential growth rate from \ref{['LSAdispersion']} (dashed magenta), and $\min_z(h)$ (blue). (c) Location, $z_\mathrm{max}$, of the maximum of $h(z,t)$. The leading-order asymptotic prediction \ref{['res:steady_U']} for the steady collar speed, $B^{3/2}|\mathcal{U}|$, is also illustrated in (c), where we have used the approximation $V=\mathcal{L}$ to calculate $\mathcal{U}$ in \ref{['res:steady_U']}.
• Figure 3: Sketch of the structure of the small-$B$ asymptotic solution. (a) View of the whole translating collar, showing $h$ (black), $Y$ (red), and the length scales of various asymptotic regions and the sizes of $h$ and $Y$ in several regions. (b) Close view of the front end of the collar, showing the Bretherton region, the unyielded uniform film ahead of the collar, the rigidification region where $Y\rightarrow0^+$, and the inner regions within the Bretherton region. (c) Close view of the rear end of the collar, showing the Bretherton region, the rigidification region and the unyielded uniform film behind the collar.
• Figure 4: Dependence of collar volume and speed on Bingham number, $\mathcal{J}$, for fixed values of $H_\infty^-$. Variation of (a) $V$, and (b) $\mathcal{U}$, for $0<\mathcal{J}<1$, according to \ref{['res:steady_HinfV']} and \ref{['res:steady_U']}. Each solid line corresponds to one value of $H_\infty^-$, with the values shown being $H_\infty^-=\{0.4,0.7,1,1.3,1.6\}$. The large arrows indicate increasing values of $H_\infty^-$. The dashed line in (a) is $V=2\pi\mathcal{J}$, which corresponds to the minimum volume required for the collar to translate.
• Figure 5: Data from numerical solutions of \ref{['TFevoleqn']} with domain length $\mathcal{L}=2\pi/k_\mathrm{m}$, for various values of the Bond number, $B$, and Bingham number, $\mathcal{J}=J/B$. (a) Late-time collar speed, $\mathcal{U}=B^{-3/2}U$, evaluated as the gradient of $z_{\max}(t)$ at the end of the simulation, where $z_{\max}$ is the location of the maximum in $h$. (b) $B^{-1}h(z_{\max}\pm \mathcal{L}/2,t_{\mathrm{end}})$, which approximates the deposited film thickness, $H_\infty^-=B^{-1}h_\infty^-$, where $t_{\mathrm{end}}$ is the simulation end time. Solid lines show leading-order asymptotic predictions \ref{['res:steady_U']} for $\mathcal{U}$ and \ref{['res:steady_HinfV']} for $H_\infty^-$, evaluated by setting $V=\mathcal{L}$. The dashed line in (a) is $\mathcal{U}$ evaluated using \ref{['app_GSBcomp']}, the global force balance that includes $O(B^{5/4}\log{B})$ corrections to the wall shear stress, for $B=0.025$. The dotted line shows $\mathcal{U}$ determined via \ref{['GSB_J32']} and \ref{['tildeF89']}, the global force balance in the case that $J=O(B^{3/2})$, again for $B=0.025$. For each $B$, $t_{\mathrm{end}}$ is chosen to be large enough to achieve adequate convergence, ranging from $t_{\mathrm{end}}=8\times10^3$ for $B=0.1$ to $t_{\mathrm{end}}=4\times10^4$ for $B=0.0125$. The initial conditions \ref{['hIC']} are used for simulations with $\mathcal{J}=0$. For each $B$, $\mathcal{J}$ is then increased in increments of $0.1$ and the final solution from the previous value of $\mathcal{J}$ is used as the initial conditions; this enables faster convergence to the late-time solutions.