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Thin-film flow down a fibre exhibits rich dynamics and is relevant to applications such as desalination, fibre coating and fog harvesting. These flows are subject to instabilities that result in dynamic bead-on-fibre patterns. We perform an experimental study of shear-thinning flow down fibres using 20 different xanthan gum solutions as our working liquid. The bead-on-fibre morphology can be oriented either symmetrically or asymmetrically on the fibre, and this depends upon the surface tension, fibre diameter and liquid rheology, as defined by the Ostwald power-law index. For highly shear-thinning liquids, it is possible for the pattern to be complex and exhibit simultaneously both asymmetric large beads and symmetric small beads in the isolated and convective flow regimes. We quantify the transition between flow regimes and bead dynamics for the asymmetric morphology, and compare with Newtonian flow, as it depends upon the experimental parameters. Finally, the dimensionless bead frequency is shown to scale with the Bond number for all of our experimental data (symmetric and asymmetric).

 

During solid surface impact, a falling drop's energy is transformed into oscillations of its liquid/gas interface. We consider drop deposition during oblique impact in the capillary-ballistic regime characterized by high Reynolds number and moderate Weber number. We treat this as an inverse problem showing that post-impact observations of the frequency spectrum and modal partition of energy allow one to determine a drop's pre-impact characteristics and wetting properties. Our analysis is useful for quantifying contact-line dissipation during inertial spreading and can be used as a diagnostic technique for determining substrate wetting properties. 

We study the linear stability of a compressible sessile bubble in an ambient fluid that partially wets a planar solid support, where the gas is assumed to be an ideal gas that obeys the adiabatic law. The frequency spectrum is computed from an integrodifferential boundary value problem and depends upon the wetting conditions through the static contact angle $\alpha$ , the dimensionless equilibrium bubble pressure $\varPi$ , and the contact-line dynamics that we assume to be either (i) pinned or (ii) freely moving with fixed contact angle. Corresponding mode shapes are defined by the polar-azimuthal mode number pair $[k,\ell ]$ with $k+\ell =\mathbb {Z}^{+}_{even}$ . We report instabilities to the (i) $[0,0]$ breathing mode associated with volume change, and (ii) $[1,1]$ mode that is linked to horizontal centre-of-mass motion of the bubble. Stability diagrams and instability growth rates are computed, and the respective instability mechanisms are revealed through an energy analysis. The zonal $\ell =0$ modes are associated with volume change, and we show that there is a complex dependence between the classical volume and shape change modes for wetting conditions that differ from neutral wetting $\alpha =90^\circ$ . Finally, we show how the classical frequency degeneracy for the Rayleigh–Lamb modes of the free bubble splits for the azimuthal modes $\ell \neq 0,1$ . 

Drop-on-demand printing applications involve a drop connected to a fluid reservoir between which volume can be exchanged, a situation that can be idealized as a sessile drop with prescribed volume flux across the drop/reservoir boundary. Here we compute the frequency spectrum for these pressure disturbances, as it depends upon the static contact-angle $\alpha$ (CA) and an empirical constant $\chi$ relating the reservoir pressure to volume exchanged, for either (i) pinned or (ii) free contact-lines. Mode shapes are characterized by the mode number pair $[k,\ell ]$ with property $k+\ell =\mathbb {Z}^{+}_{odd}$ that can be associated with the symmetry properties of the Rayleigh drop modes for the free sphere. We report instabilities to the axisymmetric $[1,0]$ and non-axisymmetric rocking $[2,1]$ modes that are related to centre-of-mass motions, and show how the spectral degeneracy of the Rayleigh drop modes breaks with the model parameters. 

The energetics of drop deposition are considered in the capillary-ballistic regime characterized by high Reynolds number and moderate Weber number. Experiments are performed impacting water/glycol drops onto substrates with varying wettability and contact-angle hysteresis. The impacting event is decomposed into three regimes: (i) pre-impact, (ii) inertial spreading and (iii) post contact-line (CL) pinning, conveniently framed using the theory of Dussan & Davis ( J. Fluid Mech. , vol. 173, 1986, pp. 115–130). During fast-time-scale inertial spreading, the only form of dissipation is CL dissipation ( $\mathcal {D}_{CL}$ ). High-speed imaging is used to resolve the stick-slip dynamics of the CL with $\mathcal {D}_{CL}$ measured directly from experiment using the $\Delta \alpha$ - $R$ cyclic diagram of Xia & Steen ( J. Fluid Mech. , vol. 841, 2018, pp. 767–783), representing the contact-angle deviation against the CL radius. Energy loss occurs on slip legs, and this observation is used to derive a closed-form expression for the kinetic K and interfacial $\mathcal{A}$ post-pinning energy $\{K+\mathcal {A}\}_p/\mathcal {A}_o$ independent of viscosity, only depending on the rest angle $\alpha _p$ , equilibrium angle $\bar {\alpha }$ and hysteresis $\Delta \alpha$ , which agrees well with experimental observation over a large range of parameters, and can be used to evaluate contact-line dissipation during inertial spreading. The post-pinning energy is found to be independent of the pre-impact energy, and it is broken into modal components with corresponding energy partitioning approximately constant for low-hysteresis surfaces with fixed pinning angle $\alpha _p$ . During slow-time-scale post-pinning, the liquid/gas ( $lg$ ) interface is found to vibrate with the frequencies and mode shapes predicted by Bostwick & Steen ( J. Fluid Mech. , vol. 760, 2014, pp. 5–38), irrespective of the pre-impact energy. Resonant mode decay rates are determined experimentally from fast Fourier transforms of the interface dynamics.