Inertia and slip effects on the instability of a liquid film coated on a fibre

Abstract

To investigate the influence of inertia and slip on the instability of a liquid film on a fibre, a theoretical framework based on the axisymmetric Navier-Stokes equations is proposed via linear instability analysis. The model reveals that slip significantly enhances perturbation growth in viscous film flows, whereas it exerts minimal influence on flows dominated by inertia. Moreover, under no-slip boundary conditions, the dominant instability mode of thin films remains unaltered by inertia, closely aligning with predictions from a no-slip lubrication model. Conversely, when slip is introduced, the dominant wavenumber experiences a noticeable reduction as inertia decreases. This trend is captured by an introduced lubrication model with giant slip. Direct numerical simulations of the Navier-Stokes equations are then performed to further confirm the theoretical findings at the linear stage. For the nonlinear dynamics, no-slip simulations show complex vortical structures within films, driven by fluid inertia near surfaces. Additionally, in scenarios with weak inertia, a reduction in the volume of satellite droplets is observed due to slip, following a power-law relationship.

Figures (15)

• Figure 1: Schematic of a liquid film on a slippery fibre
• Figure 2: The dispersion relation between the growth rate $\omega$ and the wavenumber $k$ for the limiting cases of inviscid and viscous fluids. (a) The inviscid liquid film ($Oh=10^{-3}$) on fibres with different radii $\alpha=0.8$ (green dash-dotted line), $0.5$ (blue dashed line), $0.2$ (red dotted line), $0.01$ (black solid line); (b) The viscous liquid film ($Oh=0.5$) on extremely thin fibres ($\alpha=0.01$) with different slip lengths $l_s=0.0$ (green dash-dotted line), $0.1$ (blue dashed line), $1.0$ (red dotted line), $10$ (black solid line). The circles are the predictions of rayleigh1878instability and goldin1969breakup. The lines are predictions from the NS dispersion relation (\ref{['eq_full_dispersion_relation']}).
• Figure 3: The dispersion relation between the growth rate $\omega$ and the wavenumber $k$ for the limiting cases of thin-film flows ($\alpha = 0.8$). (a) No-slip cases with different inertial effects, $Oh=2 \times 10^{-3}$ (green dash-dotted line), $8 \times 10^{-3}$ (blue dashed line), $3.2 \times 10^{-2}$ (red dotted line), $0.128$ (black solid line). (b) Slip cases ($l_s=10$) with different inertial effects, $Oh=0.01$ (green dash-dotted line), $0.1$ (blue dashed line), $1$ (red dotted line), $10$ (black solid line). The circles are the predictions from the slip-modified Stokes model zhao2023slip and the lines are predictions from the NS dispersion relation (\ref{['eq_full_dispersion_relation']}).
• Figure 4: The dispersion relation between the growth rate $\omega$ and the wavenumber $k$ on different boundary conditions of various fibre radii: (a,b,c) $\alpha=0.8$, (d,e,f) $\alpha=0.5$, (g,h,i) $\alpha=0.2$. For the inertial effects: (a,d,g) $Oh=10^{-3}$, (b,e,h) $Oh=0.1$, (c,f,i) $Oh=10$. Line types represent different values of the slip length: $l_s=0$ (green dash-dotted line), $0.1$ (blue dashed line), $1$ (red dotted line), $10$ (black solid line).
• Figure 5: Influence of inertia (different values of $Oh$) on the dominant wavenumber $k_{max}$ on fibres of two radii: (a) $\alpha=0.8$, (b) $\alpha=0.2$. The solid lines are the predictions of the NS dispersion relation (\ref{['eq_full_dispersion_relation']}) for different slip lengths: $l_s=0$ (green), $0.1$ (purple), $1.0$ (blue), $10$ (black), $100$ (red). The dotted lines and dashed lines represent the predictions from the no-slip lubrication model (\ref{['eq_LE_noslip']}) and the giant-slip one \ref{['eq_giant_LE']}