The stability of open-channel flows (or film flows) has been extensively investigated for the case of impermeable smooth walls. In contrast, despite its relevance in many geophysical and industrial flows, the case that considers a permeable rather than an impermeable wall is almost unexplored. In the present work, a linear stability analysis of a film falling over a permeable and inclined wall is developed and discussed. The focus is on the mutual interaction between three modes of instability, namely, the well-known free-surface and hydrodynamic (i.e. shear) modes, which are commonly observed in open-channel flows over impermeable walls, plus a new one associated with the flow within the permeable wall (i.e. the porous mode). The flow in this porous region is modelled by the volume-averaged Navier–Stokes equations and, at the wall interface, the surface and subsurface flow are coupled through a stress-jump condition, which allows one to obtain a continuous velocity profile throughout the whole flow domain. The generalized eigenvalue problem is then solved via a novel spectral Galerkin method, and the whole spectrum of eigenvalues is presented and physically interpreted. The results show that, in order to perform an analysis with a full coupling between surface and subsurface flow, the convective terms in the volume-averaged equations have to be retained. In previous studies, this aspect has never been considered. For each kind of instability, the critical Reynolds number (${\mathit{Re}}_{c} $) is reported for a wide range of bed slopes ($\theta $) and permeabilities ($\sigma $). The results show that the free-surface mode follows the behaviour that was theoretically predicted by Benjamin and Yih for impermeable walls and is independent of wall permeability. In contrast, the shear mode shows a high dependence on $\sigma $: at $\sigma = 0$ the behaviour of ${\mathit{Re}}_{c} (\theta )$ recovers the well-known non-monotonic behaviour of the impermeable-wall case, with a minimum at $\theta \sim 0. 05\textdegree $. However, with an increase in wall permeability, ${\mathit{Re}}_{c} $ gradually decreases and eventually recovers a monotonic decreasing behaviour. At high values of $\sigma $, the porous mode of instability also occurs. A physical interpretation of the results is presented on the basis of the interplay between the free-surface-induced perturbation of pressure, the increment of straining due to shear with the increase in slope, and the shear stress condition at the free surface. Finally, the paper investigates the extent to which Squire’s theorem is applicable to the problem presented herein.

Convection in a closed domain driven by a dense buoyancy source along the upper boundary soon starts to wane owing to the increase of the average interior density. In this paper, theoretical and numerical models are developed of the subsequent long period of shutdown of convection in a two-dimensional porous medium at high Rayleigh number $\mathit{Ra}$. The aims of this paper are twofold. Firstly, the relationship between this slowly evolving ‘one-sided’ shutdown system and the statistically steady ‘two-sided’ Rayleigh–Bénard (RB) cell is investigated. Numerical measurements of the Nusselt number $\mathit{Nu}$ from an RB cell (Hewitt et al., Phys. Rev. Lett., vol. 108, 2012, 224503) are very well described by the simple parametrization $\mathit{Nu}= 2. 75+ 0. 0069\mathit{Ra}$. This parametrization is used in theoretical box models of the one-sided shutdown system and found to give excellent agreement with high-resolution numerical simulations of this system. The dynamical structure of shutdown can also be accurately predicted by measurements from an RB cell. Results are presented for a general power-law equation of state. Secondly, these ideas are extended to model more complex physical systems, which comprise two fluid layers with an equation of state such that the solution that forms at the (moving) interface is more dense than either layer. The two fluids are either immiscible or miscible. Theoretical box models compare well with numerical simulations in the case of a flat interface between the fluids. Experimental results from a Hele-Shaw cell and numerical simulations both show that interfacial deformation can dramatically enhance the convective flux. The applicability of these results to the convective dissolution of geologically sequestered ${\mathrm{CO} }_{2} $ in a saline aquifer is discussed.

By focusing a laser pulse inside a capillary partially filled with liquid, a vapour bubble is created that emits a pressure wave. This pressure wave travels through the liquid and creates a fast, focused axisymmetric microjet when it is reflected at the meniscus. We numerically investigate the formation of this microjet using axisymmetric boundary integral simulations, where we model the pressure wave as a pressure pulse applied on the bubble. We find a good agreement between the simulations and experimental results in terms of the time evolution of the jet and on all parameters that can be compared directly. We present a simple analytical model that accurately predicts the velocity of the jet after the pressure pulse and its maximum velocity.

This paper examines the dynamic coupling between a sloshing fluid and the motion of the vessel containing the fluid. A mechanism is identified that leads to an energy exchange between the vessel dynamics and fluid motion. It is based on a 1:1 resonance in the linearized equations, but nonlinearity is essential for the energy transfer. For definiteness, the theory is developed for Cooker’s pendulous sloshing experiment. The vessel has a rectangular cross-section, is partially filled with a fluid and is suspended by two cables. A nonlinear normal form is derived close to an internal 1:1 resonance, with the energy transfer manifested by a heteroclinic connection, which connects the purely symmetric sloshing modes to the purely antisymmetric sloshing modes. Parameter values where this pure energy transfer occurs are identified. In practice, this energy transfer can lead to sloshing-induced destabilization of fluid-carrying vessels.

High-Reynolds-number data in turbulent boundary layers are analysed to examine statistical moments of streamwise velocity fluctuations ${u}^{\prime } $. Prior work has shown that the variance of ${u}^{\prime } $ exhibits logarithmic behaviour with distance to the surface, within an inertial sublayer. Here we extend these observations to even-order moments. We show that the $2p$-order moments, raised to the power $1/ p, $ also follow logarithmic behaviour according to $\langle \mathop{({u}^{\prime + } ){}^{2p} \rangle }\nolimits ^{1/ p} = {B}_{p} - {A}_{p} \ln (z/ \delta )$, where ${u}^{\prime + } $ is the velocity fluctuation normalized by the friction velocity, $\delta $ is an outer length scale and ${B}_{p} $ are non-universal constants. The slopes ${A}_{p} $ in the logarithmic region appear quite insensitive to Reynolds number, consistent with universal behaviour for wall-bounded flows. The slopes differ from predictions that assume Gaussian statistics, and instead are consistent with sub-Gaussian behaviour.

A Janus swimmer is any force-free, torque-free organism, particle or micro-robot operating at low Reynolds numbers and having contiguous regions of its boundary on which different boundary conditions are in play. In this paper we study a ‘slip–stick’ Janus swimmer theoretically within a two-dimensional model. The boundary of the swimmer is split into two zones: its motion is driven by an imposed tangential stress on a portion of the boundary with the complement taken to be a no-slip surface. The Stokes flow generated by the swimmer, and its swimming speed, are determined in closed analytical form as a function of the angle over which the stress actuation is active.

At finite Dukhin numbers, where Smoluchowski’s formula is inapplicable to thin-double-layer electrophoresis, the mobility of non-spherical particles is generally anisotropic. We consider bodies of revolution of otherwise arbitrary shape, where a uniformly applied electric field results in a rectilinear motion in the plane spanned by the field direction and the particle symmetry axis, as well as (for particles lacking fore–aft symmetry) rigid-body rotation about an axis perpendicular to that plane. Focusing upon slender particles, where the ratio $\epsilon $ of cross-sectional and longitudinal scales is asymptotically small, the translational and rotational mobilities are obtained as quadratures which depend upon the lengthwise distribution of the scaled cross-sectional width and the force densities associated with rigid-body motion. These mobility expressions approach finite limits as $\epsilon \rightarrow 0$, yielding closed-form expressions for specific particle geometries.

Simple stochastic models and direct nonlinear numerical simulations of three-dimensional internal waves are combined in order to understand the strong horizontal particle dispersion at second-order in wave amplitude that arises when small-amplitude internal waves are exposed to weak dissipation. This is contrasted with the well-known results for perfectly inviscid internal waves, in which such dispersion arises only at fourth-order in wave amplitude.