‘Horizontal convection’ (HC) is the generic name for the flow resulting from a buoyancy variation imposed along a horizontal boundary of a fluid. We study the effects of rotation on three-dimensional HC numerically in two stages: first, when baroclinic instability is suppressed and, second, when it ensues and baroclinic eddies are formed. We concentrate on changes to the thickness of the near-surface boundary layer, the stratification at depth, the overturning circulation and the flow energetics during each of these stages. Our results show that, for moderate flux Rayleigh numbers ($O(1{0}^{11} )$), rapid rotation greatly alters the steady-state solution of HC. When the flow is constrained to be uniform in the transverse direction, rapidly rotating solutions do not support a boundary layer, exhibit weaker overturning circulation and greater stratification at all depths. In this case, diffusion is the dominant mechanism for lateral buoyancy flux and the consequent buildup of available potential energy leads to baroclinically unstable solutions. When these rapidly rotating flows are perturbed, baroclinic instability develops and baroclinic eddies dominate both the lateral and vertical buoyancy fluxes. The resulting statistically steady solution supports a boundary layer, larger values of deep stratification and multiple overturning cells compared with non-rotating HC. A transformed Eulerian-mean approach shows that the residual circulation is dominated by the quasi-geostrophic eddy streamfunction and that the eddy buoyancy flux has a non-negligible interior diabatic component. The kinetic and available potential energies are greater than in the non-rotating case and the mixing efficiency drops from ${\sim }0. 7$ to ${\sim }0. 17$. The eddies play an important role in the formation of the thermal boundary layer and, together with the negatively buoyant plume, help establish deep stratification. These baroclinically active solutions have characteristics of geostrophic turbulence.

Direct numerical simulations of turbulent channels with artificially prescribed velocity profiles are discussed, using both natural and purposely incorrect profiles. It is found that turbulence develops correctly when natural profiles are prescribed, but that even slightly incorrect ones modify the Reynolds stresses substantially. That is used to study the dynamics of the energy-containing velocity fluctuations. The stronger (weaker) structures generated by locally stronger (weaker) mean shears have essentially correct isotropy coefficients but they are out of energy equilibrium, with the energy imbalance compensated by turbulent diffusion. The velocity scale in smooth profiles changes with the distance to the wall, and is best described by a friction velocity derived from the local total tangential stress. The behaviour across sharper shear jumps is more consistent with non-equilibrium eddies that relax over wall-normal distances of the order of the distance to the wall, suggesting that the energy equilibrium in the logarithmic layer is not local to a given height, but applies to extended layers homogenized by wall-normal fluxes. Examples of that non-local character are the large-scale inactive fluctuations near the wall, whose velocities do not scale with the local shear stress, but with that of their active ‘cores’ farther away from the wall.

A series of wave basin experiments is reported that investigates the flexural response of one or two floating thin elastic discs to monochromatic waves. The work is motivated by numerical model validation. Innovative techniques are used to ensure the experimental configuration is consistent with the model. This demands linear motions, time-harmonic conditions, homogeneity of the plate and the restriction of horizontal motions of the disc or discs. An optical remote sensing device is employed to record the deflection of the discs accurately. Tests involving a single disc and two discs are conducted for a range of disc thicknesses, incident wave steepnesses, frequencies and, in the case of two discs, geometrical arrangements. A data processing technique is used to decompose the raw data into its spectral harmonics and filter the higher-order components. Pointwise comparisons of the linear first-order component of the experimental deflection with numerical predictions are presented. Satisfying agreement is found, although the model consistently over predicts the deflection. Disc–disc interactions are observed in the two-disc tests. A brief discussion of the shortcomings of the pointwise analysis, with associated possible sources of discrepancy, provides a link to the study reported in Part 2 (Montiel et al. J. Fluid Mech., vol. 723, 2013, pp. 629–652).

Validation of a linear numerical model of wave interactions with floating compliant discs is sought using data obtained from the wave basin experiments reported in Part 1 (Montiel et al. J. Fluid Mech., vol. 723, 2013, pp. 604–628). Comparisons are made for both single-disc tests and the two-disc tests in which wave interactions between discs are observed. The deflection of the disc or discs is separated into the natural modes of vibration in vacuo. The decomposition allows the rigid-body motions and flexural motions to be analysed separately. Rigid-body motions are accurately replicated by the numerical model but, although passable agreement is found, the amplitudes of flexural modes are consistently overestimated. Extensions of the numerical model are used to discount the experimental configuration as a source of the discrepancies. An enhanced viscoelastic model for the discs is also proposed, which results in improved model/data agreement for the flexural motions but cannot account for all of the disagreement.

Villermaux & Pomeau (J. Fluid Mech., vol. 642, 2010, p. 147) analysed the motion of the interface of an inviscid liquid column released from rest in a vertical tube whose area expands gradually downwards, with application to an inverted conical container for which experimental measurements were carried out. An error in the analysis is found and corrected in the present investigation, which provides the new governing equation for the super-accelerated interface motion down gradually varying tubes in general, and integrated results for interface trajectories, velocities and accelerations down a conical tube in particular. Interestingly, the error does not affect any of the conclusions given in the 2010 paper. Further new results are reported here such as the equation governing the centre of mass and proof that the end point acceleration is exactly that of gravity.

The measurement of vortex flows with particle-image velocimetry (PIV) is particularly susceptible to error arising from the finite mass of the tracer particles, owing to the high velocities and accelerations typically experienced. A classical model of Stokes-flow particle transport is adopted, and an approximate solution for the case of particle transport within an axisymmetric, quasi-two-dimensional Batchelor $q$-vortex is presented. A generalized expression for the maximum particle tracking error is proposed for each of the velocity components, and the importance of finite particle size distributions is discussed. The results indicate that the tangential velocity component is significantly less sensitive to tracking error than the radial component, and that the conventional particle selection criterion (based on the particle Stokes number) may result in either over- or under-sized particles for a specified allowable error bound. Results were demonstrated by means of PIV measurements carried out in air and water using particles with very different properties.

We analyse two-dimensional clamped parallel elastic sheets which are partially immersed in liquid as a model for elasto-capillary coalescence. In the existing literature this problem is studied via minimal energy analysis of capillary and elastic energies of the post-coalescence state, yielding the maximal stable post-coalescence bundle size. Utilizing modal stability analysis and asymptotic analysis, we studied the stability of the configuration before the coalescence occurred. Our analysis revealed previously unreported relations between viscous forces, body forces, and the instability yielding the coalescence, thus undermining a common assumption that coalescence will occur as long as it will not create a bundle larger than the maximal stable post-coalesced size. A mathematical description of the process creating the hierarchical coalescence structure was obtained and yielded that the mean number of sheets per coalesced region is limited to the subset ${2}^{N} $ where $N$ is the set of natural numbers. Our theoretical results were illustrated by experiments and good agreement with the theoretical predictions was observed.

We consider flow in a finite-length channel, one wall of which contains a membrane under longitudinal tension. The upstream flux and downstream pressure are prescribed and an external linear pressure distribution is applied to the membrane such that the system admits uniform Poiseuille flow as a steady solution. The system is described using a one-dimensional model that accounts for viscous and inertial effects. A linear stability analysis reveals that the uniform state is unstable to static (or divergent) and oscillatory instabilities. Asymptotic analysis in the neighbourhood of a Takens–Bogdanov bifurcation point shows how, when the downstream rigid section of the channel is not substantially longer than the membrane, an oscillatory mode arises through an interaction between two static eigenmodes. Perturbations to the uniform state exhibit the dynamics of a weakly dissipative Hamiltonian system for which low-frequency self-excited oscillations are forced by the divergent instability of two nearby steady solutions, before ultimately growing to large amplitudes. Simulations show that the subsequent dynamics can involve slamming motion in which the membrane briefly comes into near-contact with the opposite rigid wall over short length scales.