Rayleigh-Plateau instability of an elasto-viscoplastic filament

TL;DR

This work addresses the Rayleigh-Plateau instability in an elasto-viscoplastic filament by developing a slender-thread model based on Saramito rheology that captures elastic deformations below yield and plastic yielding above yield. Linear stability reveals a critical Weissenberg number $\mathcal{W}_c=\frac{6}{1-k^2}$ governing elastic RP growth, while nonlinear evolution shows yielding at the thinnest sections can trigger pinch-off, forming beads-on-a-string structures whose final shapes depend on the plastocapillarity $\mathcal{J}$ and the viscosity ratio $\beta$. The beads exhibit rich elasto-plastic anatomy, with fully plastic beads possible at small $\mathcal{J}$ and partially elastic beads at larger $\mathcal{J}$, and the dynamics can be strongly influenced by elastic transients for low solvent viscosity. Overall, the study demonstrates that allowing sub-yield elastic deformation reactivates the classical RP instability in yield-stress filaments and reveals how yielding history controls pinch-off and bead morphology, offering insights for applications in printing and atomization of complex fluids, while highlighting limitations of slender-thread theory and the need for full 3D simulations.

Abstract

A slender-thread model is derived to explore the Rayleigh-Plateau instability of a filament of elasto-viscoplastic fluid. Without elasticity, a finite yield stress suppresses any linear instability for a filament of constant radius. Including sub-yield elastic deformation permits an elastic Rayleigh-Plateau instability above a critical Weissenberg number. If stresses over the thinner sections of the thread breach the yield threshold, viscoplastic deformations then drive the filament towards pinch-off. The thread consequently evolves to a beads-on-a-string structure. The elasto-plastic anatomy of the beads is explored and categorized.

Figures (8)

• Figure 1: Sketch of the geometry of a thin thread, showing the main model variables.
• Figure 2: Sample solution for $\{\mathop{\mathrm{\mathcal{W}}}\nolimits,\mathop{\mathrm{\mathcal{L}}}\nolimits,\mathop{\mathrm{\mathcal{J}}}\nolimits,\beta,\mathop{\mathrm{\mathcal{R}}}\nolimits\}=\{10,20,0.2,1,0.1\}$. (a) Snapshots of the thread at the times indicated. The elastic regions are shaded grey when never previously yielded, and blue otherwise. (b) Radius plotted as a density on a space-time diagram, with superposed contours (black) showing the yield surfaces. The white lines indicate the paths taken by a selection of fluid elements; the red lines indicate the two fluid elements that border the region that passes through the yielded thinned section of the thread, but then falls below the yield stress and becomes elastic for later times. (c,d,e) Time series of $h_{max}-h_{min}$, $h_{min}$ and $|\tau_{rr}-\tau_{zz}|$ at $z=0$ (blue) and $z=\frac{1}{2}\mathop{\mathrm{\mathcal{L}}}\nolimits$ (red). In (c), the triangle shows the expected linear growth rate; that in (d) shows the exponential decay $e^{-t/(3\mathop{\mathrm{\mathcal{W}}}\nolimits)}$ expected over the thinnest part of the thread. The star in (d) identifies the moment that the stress over thinnest part of the thread breaches the yield stress. The dashed lines in (a,d-e) show the profiles and corresponding time series for a thread that never yields ($\mathop{\mathrm{\mathcal{J}}}\nolimits\gg1$).
• Figure 3: Numerical solutions showing examples of (a) case A ($\mathop{\mathrm{\mathcal{J}}}\nolimits=0.1$), (b) case B ($\mathop{\mathrm{\mathcal{J}}}\nolimits=0.2$) and (c) case C ($\mathop{\mathrm{\mathcal{J}}}\nolimits=0.6$), all with $\{\mathop{\mathrm{\mathcal{W}}}\nolimits,\mathop{\mathrm{\mathcal{L}}}\nolimits,\beta,\mathop{\mathrm{\mathcal{R}}}\nolimits\}=\{10,20,1,0.01\}$. For each case, the upper rows of panels show the final shapes and stress differences, whilst the bottom row presents the yield surfaces (thicker black lines) and the trajectories of sample fluid elements (thinner blue lines) on space-time diagrams. In the upper rows, the red dashed lines show solutions of \ref{['eq:finprof']}, the magenta dashed lines in (b,c) show the elastic stress state in \ref{['elastotaudif']}, and the (red) dot-dashed lines indicate the yield condition $\tau_{rr}-\tau_{zz}=\sqrt3 \mathop{\mathrm{\mathcal{J}}}\nolimits$. The shaded regions correspond to the elastic regions, either in grey if never yielded, or blue if previously so. In the bottom row, the red lines bordering the blue-shaded area show the fluid elements buffering the region that yields en route to a final elastic state.
• Figure 4: Final maximum radius (red) and aspect ratio (blue) of the beads formed for varying $\mathop{\mathrm{\mathcal{J}}}\nolimits$, all with $\{\mathop{\mathrm{\mathcal{W}}}\nolimits,\mathop{\mathrm{\mathcal{L}}}\nolimits,\beta,\mathop{\mathrm{\mathcal{R}}}\nolimits\}=\{10,20,1,0.01\}$. A selection of profiles are also indicated. The different bead types are indicated (cases A, B and C; see fig \ref{['fig:abc']}); U refers to the regime wherein the thread never yields. Bead aspect ratio is defined as the ratio of the half-length of the bead to its maximum radius at the end of a simulation. The half-length is defined by approximating the location of the bead's edge as the point, $z$, where $h(z,t_\mathrm{end})=2\min(h)=0.04$. Dashed lines indicate predictions from solutions to \ref{['eq:finprof']} for $h_{\max}$ in the purely plastic case (red), aspect ratio in the purely plastic case (blue), and $h_{\max}$ in the purely elastic case (black).
• Figure 5: Final (a) minimum and (b,c) maximum radii from computations with a range of values of $\mathop{\mathrm{\mathcal{W}}}\nolimits$ and $\mathop{\mathrm{\mathcal{J}}}\nolimits$, and $\{\mathop{\mathrm{\mathcal{L}}}\nolimits,\beta,\mathop{\mathrm{\mathcal{R}}}\nolimits\} = \{20,1,0.01\}$. The anatomy of the thread is identified by symbol (case A, $\blacktriangle$; case B, $\blacksquare$; case C $\bullet$; unyielded, $\blacktriangledown$). The solid red line shows the locus along which $\max_z|\tau_{rr}-\tau_{zz}|=\sqrt{3}\mathop{\mathrm{\mathcal{J}}}\nolimits$ for purely elastic steady states computed from \ref{['eq:finprof']}. The threshold $\mathop{\mathrm{\mathcal{W}}}\nolimits=\mathop{\mathrm{\mathcal{W}}}\nolimits_c=6/(1-k^2)$ for linear instability \ref{['LSAcritW']} is indicated by the dashed line. The data in (c) correspond to the low-$\mathop{\mathrm{\mathcal{J}}}\nolimits$ region of parameter space in (b), but with more data points. The black lines show linearly spaced contours of $h_{\max}$.