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.

A static rivulet is subject to disturbances in shape, velocity and pressure fields. Disturbances to interfacial shape accommodate a contact line that is either (i) fixed (pinned) or (ii) fully mobile (free) and preserves the static contact angle. The governing hydrodynamic equations for this inviscid, incompressible fluid are derived and then reduced to a functional eigenvalue problem on linear operators, which are parametrized by axial wavenumber and base-state volume. Solutions are decomposed according to their symmetry (varicose) or anti-symmetry (sinuous) about the vertical mid-plane. Dispersion relations are then computed. Static stability is obtained by setting growth rate to zero and recovers existing literature results. Critical growth rates and wavenumbers for the varicose and sinuous modes are reported. For the varicose mode, typical capillary break-up persists and the role of the liquid/solid interaction on the critical disturbance is illustrated. There exists a range of parameters for which the sinuous mode is the dominant instability mode. The sinuous instability mechanism is shown to correlate with horizontal centre-of-mass motion and illustrated using a toy model.

High-speed images of driven sessile water drops recorded under frequency scans are analysed for resonance peaks, resonance bands and hysteresis of characteristic modes. Visual mode recognition using back-lit surface distortion enables modes to be associated with frequencies, aided by the identifications in Part 1 (Bostwick & Steen, J. Fluid Mech., vol. 760, 2014, pp. 5–38). Part 1 is a linear stability analysis that predicts how inviscid drop spectra depend on base state geometry. Theoretically, the base states are spherical caps characterized by their ‘flatness’ or fraction of the full sphere. Experimentally, quiescent shapes are controlled by pinning the drop at a circular contact line on the flat substrate and varying the drop volume. The response frequencies of the resonating drop are compared with Part 1 predictions. Agreement with theory is generally good but does deteriorate for flatter drops and higher modes. The measured frequency bands agree better with an extended model, introduced here, that accounts for forcing and weak viscous effects using viscous potential flow. As the flatness varies, regions are predicted where modal frequencies cross and where the spectra crowd. Frequency crossings and spectral crowding favour interaction of modes. Modal interactions of two kinds are documented, called ‘mixing’ and ‘competing’. Mixed modes are two pure modes superposed with little evidence of hysteresis. In contrast, modal competition involves hysteresis whereby one or the other mode disappears depending on the scan direction. Perhaps surprisingly, a linear inviscid irrotational theory provides a useful framework for understanding observations of forced sessile drop oscillations.

A sessile droplet partially wets a planar solid support. We study the linear stability of this spherical-cap base state to disturbances whose three-phase contact line is (i) pinned, (ii) moves with fixed contact angle and (iii) moves with a contact angle that is a smooth function of the contact-line speed. The governing hydrodynamic equations for inviscid motions are reduced to a functional eigenvalue problem on linear operators, which are parameterized by the base-state volume through the static contact angle and contact-line mobility via a spreading parameter. A solution is facilitated using inverse operators for disturbances (i) and (ii) to report frequencies and modal shapes identified by a polar $k$ and azimuthal $l$ wavenumber. For the dynamic contact-line condition (iii), we show that the disturbance energy balance takes the form of a damped-harmonic oscillator with ‘Davis dissipation’ that encompasses the dynamic effects associated with (iii). The effect of the contact-line motion on the dissipation mechanism is illustrated. We report an instability of the super-hemispherical base states with mobile contact lines (ii) that correlates with horizontal motion of the centre-of-mass, called the ‘walking’ instability. Davis dissipation from the dynamic contact-line condition (iii) can suppress the instability. The remainder of the spectrum exhibits oscillatory behaviour. For the hemispherical base state with mobile contact line (ii), the spectrum is degenerate with respect to the azimuthal wavenumber. We show that varying either the base-state volume or contact-line mobility lifts this degeneracy. For most values of these symmetry-breaking parameters, a certain spectral ordering of frequencies is maintained. However, because certain modes are more strongly influenced by the support than others, there are instances of additional modal degeneracies. We explain the physical reason for these and show how to locate them.

A spherical drop is constrained by a solid support arranged as a latitudinal belt. This belt support splits the drop into two deformable spherical caps. The edges of the support are given by lower and upper latitudes yielding a ‘spherical belt’ of prescribed extent and position: a two-parameter family of constraints. This is a belt-constrained Rayleigh drop. In this paper we study the linear oscillations of the two coupled spherical-cap surfaces in the inviscid case, and the viscous case is studied in Part 2 (Bostwick & Steen, J. Fluid Mech., vol. 714, 2013, pp. 336–360), restricting to deformations symmetric about the axis of constraint symmetry. The integro-differential boundary-value problem governing the interface deformation is formulated as a functional eigenvalue problem on linear operators and reduced to a truncated set of algebraic equations using a Rayleigh–Ritz procedure on a constrained function space. This formalism allows mode shapes with different contact angles at the edges of the solid support, as observed in experiment, and readily generalizes to accommodate viscous motions (Part 2). Eigenvalues are mapped in the plane of constraints to reveal where near-multiplicities occur. The full problem is then approximated as two coupled harmonic oscillators by introducing a volume-exchange constraint. The approximation yields eigenvalue crossings and allows post-identification of mass and spring constants for the oscillators.

A spherical drop is constrained by a solid support arranged as a latitudinal belt. The spherical belt splits the drop into two deformable spherical caps. The edges of the belt support are given by lower and upper latitudes, yielding a support of prescribed extent and position: a two-parameter family of geometrical constraints. In this paper we study the linear oscillations of the two coupled surfaces in the viscous case, the inviscid case having been dealt with in Part 1 (Bostwick & Steen, J. Fluid Mech., vol. 714, 2013, pp. 312–335), restricting to axisymmetric disturbances. For the viscous case, limiting geometries are the spherical-bowl constraint of Strani & Sabetta (J. Fluid Mech., vol. 189, 1988, pp. 397–421) and free viscous drop of Prosperetti (J. Méc., vol. 19, 1980b, pp. 149–182). In this paper, a boundary-integral approach leads to an integro-differential boundary-value problem governing the interface disturbances, where the constraint is incorporated into the function space. Viscous effects arise due to relative internal motions and to the no-slip boundary condition on the support surface. No-slip is incorporated using a modified set of shear boundary conditions. The eigenvalue problem is then reduced to a truncated set of algebraic equations using a spectral method in the standard way. Limiting cases recover literature results to validate the proposed modification. Complex frequencies, as they depend upon the viscosity parameter and the support geometry, are reported for both the drop and bubble cases. Finally, for the drop, an approximate boundary between over- and under-damped motions is mapped over the constraint parameter plane.

A drop on a circular support spontaneously spreads upon contact with a substrate. The motion is driven by a loss of surface energy. The loss of recoverable energy can be expressed alternatively as work done at the liquid–gas interface or dissipation through viscosity and sliding friction. In this paper we require consistency with the energy lost by dissipation in order to infer details of the contact-line region through simulations. Simulations with the boundary integral method are used to compute the flow field of a corresponding experiment where polydimethylsiloxane spreads on a relatively hydrophobic surface. The flow field is used to calculate the energy dissipation, from which slip lengths for local slip and Navier slip boundary conditions are found. Velocities, shear rates and pressures along the interface as well as interface shapes in the microscopic region of the contact line are also reported. Angles, slip length and viscous bending length scale allow a test of the Voinov–Hocking–Cox model without free parameters.

In this paper we give a progress report on the Herschel observations of planetary nebulae that are carried out as part of the MESS guaranteed time key program.

Surface tension acting at a cylindrical interface holds an underlying liquid in motionless equilibrium. This static base state is subject to dynamic capillary instability, including Plateau–Rayleigh breakup. If the interface is partially supported by a cylindrical cup-like solid, the extent of the wetting contact can significantly influence the dynamics and the stability of the configuration. The equation for the motion of small disturbances is formulated as an eigenvalue equation on linear operators. A solution is constructed on a constrained function space using a Rayleigh–Ritz procedure. The influence of the extent-of-constraint on the dispersion relation and on modal structures is reported. In the extreme, the support reduces to a wire, aligned axially, and just touching the interface. From prior work, this constraint is known to stabilize the Plateau–Rayleigh limit by some 13%. We report the wavenumber of maximum growth and estimate the time to breakup. The constraint is then bent in-plane to add a weak secondary curvature to the now nearly cylindrical base state. This is referred to as the torus lift of the cylinder. The static stability of these toroidal equilibria, calculated using a perturbation approach, shows that the position of constraint is crucial – constraint can stabilize (outside) or destabilize (inside). The combined influence of secondary curvature and wire constraint on the Plateau–Rayleigh limit is tracked. Finally, attention is restricted to constraints that yield a lens-like cylindrical meniscus. For these lenses, the torus lift is used as apparatus along with a symmetrization procedure to prove a large-amplitude static stability result. Our study is conveniently framed by a classic paper on rivulets by Davis (J. Fluid Mech., vol. 98, 1980, p. 225).

Centre-of-mass motions of two coupled spherical-cap droplets are considered. A model with surface tension and inertia that accounts for finite-amplitude deformations is derived in closed form. Total droplet volume λ and half-length L of the tube that connects the droplets are the control parameters. The model dynamics reside in the phase-plane. For lens-like droplets λ < 1, and for any L there is a single steady state about which the droplets vibrate with limit-cycle behaviour. For λ>1, the symmetric state loses stability (saddle point) and new antisymmetric steady states arise about which limit-cycle oscillations occur. These mirror states – big-droplet up or big-droplet down – are also stable. In addition, there are large finite-amplitude ‘looping’ oscillations corresponding to limit cycles that enclose both steady states in the phase-plane. All three kinds of oscillations are documented in an experiment that sets the system into motion by ‘kicking’ one of the droplets with a prescribed pressure-pulse. Model predictions for frequencies are consistent with observations. Small-amplitude predictions are placed in the wider context of constrained Rayleigh vibrations. A model extension to account for the small but non-negligible influence of viscosity is also presented.

Mechanistic animal growth models can incorporate a description of the genotype as represented by underlying biological traits that aim to specify the animal's genetic potential for performance, independent from the environmental factors captured by the models. It can be argued that these traits may therefore be more closely associated to genetic potential, or components of genetic merit that are more robust across environments, than the environmentally dependent phenotypic traits currently used for genetic evaluation. The prediction of merit for underlying biological traits can be valuable for breeding and development of selection strategies across environments.

Model inversion has been identified as a valid method for obtaining estimates of phenotypic and genetic components of the biological traits representing the genotype in the mechanistic model. The present study shows how these estimates were obtained for two existing pig breeds based on genetic and phenotypic components of existing performance trait records. Some of the resulting parameter estimates associated with each breed differ substantially, implying that the genetic differences between the breeds are represented in the underlying biological traits. The estimated heritabilities for the genetic potentials for growth, carcass composition and feed efficiency as represented by biological traits exceed the heritability estimates of related phenotypic traits that are currently used in evaluation processes for both breeds. The estimated heritabilities for maintenance energy requirements are however relatively small, suggesting that traits associated with basic survival processes have low heritability, provided that maintenance processes are appropriately represented by the model.

The results of this study suggest that mechanistic animal growth models can be useful to animal breeding through the introduction of new biological traits that are less influenced by environmental factors than phenotypic traits currently used. Potential value comes from the estimation of underlying biological trait components and the explicit description of their expression across a range of environments as predicted by the model equations.

A cylindrical liquid bridge is unstable when its length is longer than its circumference, the Plateau–Rayleigh limit. This capillary instability is modified by fluid motions adjacent to the interface, which can be induced by thermocapillary stress, among other means. A simple flow model with symmetry that mimics the situation in encapsulated floating zones is analysed. The interfacial balance equation is formulated as a bifurcation problem, appropriate when the flows are nearly rectilinear. This balance captures the competition between capillary stress and the flow-induced pressure. The fluid motions are shown to have a stabilizing effect; bridges much longer than the classical limit are stabilized. Numerical branch-tracing and the Lyapunov–Schmidt reduction methods provide the bifurcation structures of branching solutions. A normal-form analysis predicts standing-wave patterns due to mode–mode interaction. The model is proposed as an explanation for the extra long float zones observed in various spacelab experiments.

High-sensitivity interferometry measurements of initial density distributions are reviewed for a wide range of gas-puff nozzles used in plasma radiation source (PRS) z-pinch experiments. Accurate gas distributions are required for determining experimental load parameters, modeling implosion dynamics, understanding the radiation properties of the stagnated pinch, and for predicting PRS performance in future experiments. For a number of these nozzles, a simple ballistic-gas-flow model (BFM) has been used to provide good physics-based analytic fits to the measured r, z density distributions. These BFM fits provide a convenient means to smoothly interpolate radial density distributions between discrete axial measurement locations for finer-zoned two-dimensional MHD calculations, and can be used to determine how changes in nozzle parameters and load geometry might alter implosion dynamics and radiation performance. These measurement and analysis techniques are demonstrated for a nested-shell nozzle used in Double Eagle and Saturn experiments. For this nozzle, the analysis suggests load modifications that may increase the K-shell yield.

An axisymmetric film bridge collapses under its own surface tension, disconnecting at a pair of pinchoff points that straddle a satellite bubble. The free-boundary problem for the motion of the film surface and adjacent inviscid fluid has a finite-time blowup (pinchoff). This problem is solved numerically using the vortex method in a boundary-integral formulation for the dipole strength distribution on the surface. Simulation is in good agreement with available experiments. Simulation of the trajectory up to pinchoff is carried out. The self-similar behaviour observed near pinchoff shows a ‘conical-wedge’ geometry whereby both principal curvatures of the surface are simultaneously singular – lengths scale with time as t2/3. The similarity equations are written down and key solution characteristics are reported. Prior to pinchoff, the following regimes are found. Near onset of the instability, the surface evolution follows a direction dictated by the associated static minimal surface problem. Later, the motion of the mid-circumference follows a t2/3 scaling. After this scaling ‘breaks’, a one-dimensional model is adequate and explains the second scaling regime. Closer to pinchoff, strong axial motions and a folding surface render the one-dimensional approximation invalid. The evolution ultimately recovers a t2/3 scaling and reveals its self-similar structure.