The selective withdrawal of a layered fluid from a reservoir has interesting properties which are not present if the stratification is continuous. The problem investigated is that of finding the ratios of the discharges from each flowing layer when the total discharge from all layers is regulated at the outlet. It is shown that the ratios are determined by the requirement that the flow be smooth at critical points in the flow, each critical point being a point at which the long-wave velocity on one of the interfaces is zero. If there are too many critical points, the flow is over-determined and becomes unsteady. It follows therefore that if the external conditions change slowly, such as in the slow draining of a reservoir, the flow alternates between intervals in which the flow ratios also change slowly and intervals in which the flow ratios oscillate unsteadily. An experiment is described in which the theoretical conclusions were tested for two-layer flow. Good agreement was obtained between theory and experiment.

Liquid film flow along an inclined plane featuring a slit, normal to the main direction of flow, creates a second gas–liquid interface connecting the two side walls of the slit. This inner interface forms two three-phase contact lines and supports a widely varying amount of liquid under different physical and geometrical conditions. The exact liquid configuration is determined by employing the Galerkin/finite element method to solve the two-dimensional Navier–Stokes equations at steady state. The interplay of inertia, viscous, gravity and capillary forces along with the substrate wettability and orientation with respect to gravity and the width of the slit determine the extent of liquid penetration and free-surface deformation. Finite wetting lengths are predicted in hydrophilic and hydrophobic substrates for inclination angles more or less than the vertical, respectively. Multiple steady solutions, connected by turning points forming a hysteresis loop, are revealed by pseudo-arclength continuation. Under these conditions, small changes in certain parameter values leads to an abrupt change in the wetting length and the deformation amplitude of the outer film surface. In hydrophilic substrates the wetting lengths exhibit a local minimum for small values of the Reynolds number and a very small range of Bond numbers; when inertia increases, they exhibit the hysteresis loop with the second limit point in a very short range of Weber numbers. Simple force balances determine the proper rescaling in each case, so that critical points in families of solutions for different liquids or contact angles collapse. The flow inside the slit is quite slow in general because of viscous dissipation and includes counter-rotating vortices often resembling those reported by Moffatt (J. Fluid Mech., vol. 18, 1964, pp. 1–18). In hydrophobic substrates, the wetting lengths decrease monotonically until the first limit point of the hysteresis loop, which occurs in a limited range of Bond numbers when the Kapitza number is less than 300 and in a limited range of Weber numbers otherwise. Here additional solution families are possible as well, where one or both contact points (Cassie state) coincide with the slit corners.

This paper investigates the boundary-layer characteristics on a helical blade of large chord length, enclosed in an annulus and rotating in a fluid otherwise at rest. The three-dimensional form of momentum integral equations is derived, and used to predict the boundary-layer growth and limiting streamline angles on the blade surface. The measurements are in general agreement with the predictions. The wall shear stress correlation, which includes both Reynolds number and rotation parameters, valid for a rotating blade operating at zero pressure gradient, is derived. Radial and tangential velocity profiles, the tangential component of turbulence intensity and blade static pressures are measured at several locations on the blade surface. The nature of flow near the blade tip is discussed. An expression for the radial velocity profile, valid in the outer region of the boundary layer, is derived theoretically.

To clarify the function of gravity in the shallow-water theory of the interfacial instability in aluminium reduction cells, we analyse the existing long-wave theory of the instability in the limit of vanishing gravitational acceleration $g$. The flow then has an inner and outer structure, with gravity remaining essential within thin layers coating the cell walls. In those thin wall layers, the growing disturbance takes the form of a trapped magnetogravity wave propagating horizontally on the internal interface, and the growth rate $\sigma$ is determined by the coupling of that edge wave to the large-scale flow in the core of the cell; that coupling is expressed as an oblique-derivative problem for the core flow. Although $\sigma$ is asymptotically independent of $g$, gravity is essential to the long-wave instability because a correlation imposed by the magneto-gravity waves is essential for the disturbance to extract power from the mean state.

The effect of a small surface-tension coefficient on the classical theory of free-streamline separation from a sharp trailing edge is studied. The classical solution fails in a small region surrounding the edge, where it predicts singular behaviour, and an inner solution, satisfying linear boundary conditions, is required to obtain a uniformly valid first approximation. The solution valid near the edge removes the curvature and pressure-gradient singularities of the classical solution and predicts a standing capillary wave along the free streamline.

A recently developed symmetry-based theory is extended to derive an algebraic model for compressible turbulent boundary layers (CTBL) – predicting mean profiles of velocity, temperature and density – valid from incompressible to hypersonic flow regimes, thus achieving a Mach number ($Ma$) invariant description. The theory leads to a multi-layer analytic form of a stress length function which yields a closure of the mean momentum equation. A generalized Reynolds analogy is then employed to predict the turbulent heat transfer. The mean profiles and the friction coefficient are compared with direct numerical simulations of CTBL for a range of $Ma$ from 0 (e.g. incompressible) to 6.0 (e.g. hypersonic), with an accuracy notably superior to popular current models such as Baldwin–Lomax and Spalart–Allmaras models. Further analysis shows that the modification is due to an improved eddy viscosity function compared to competing models. The results confirm the validity of our $Ma$-invariant stress length function and suggest the path for developing turbulent boundary layer models which incorporate the multi-layer structure.

Streamwise velocity profiles and their wall-normal derivatives were used to investigate the properties of turbulent channel flow in the low polymer drag reduction $(DR)$ regime ($DR=6.5\,\%$ to $26\,\%$), as realized via polymer injection at the channel surface. Streamwise velocity data were obtained over a friction Reynolds number ranging from $650$ to $1800$ using the single-velocity-component version of molecular tagging velocimetry (1c-MTV). This adaptation of the MTV technique has the ability to accurately capture instantaneous profiles at very high spatial resolution (${\gtrsim}850$ data points per wall-normal profile), and thus generate well-resolved derivative information as well. Owing to this ability, the present study is able to build upon and extend the recent numerical simulation analysis of White et al. (J. Fluid Mech., vol. 834, 2018, pp. 409–433) that examined the mean dynamical structure of polymer drag-reduced channel flow at friction Reynolds numbers up to $1000$. Consistently, the present mean velocity profiles indicate that the extent of the logarithmic region diminishes with increasing polymer concentration, while statistically significant increases in the logarithmic profile slope begin to occur for drag reductions less than $15\,\%$. Profiles of the r.m.s. streamwise velocity indicate that the maximum moves farther from the wall and increases in magnitude with reductions in drag. Similarly, with increasing drag reduction, the profile of the combined Reynolds and polymer shear stress exhibits a decrease in its maximum value that also moves farther from the wall. Correlations are presented that estimate the location and value of the maximum r.m.s. streamwise velocity and combined Reynolds and polymer shear stress. Over the range of $DR$ investigated, these effects consistently exhibit approximately linear trends as a function of $DR$. The present measurements allow reconstruction of the mean momentum balance (MMB) for channel flow, which provides further insights regarding the physics described in the study by White et al. In particular, the present findings support a physical scenario in which the self-similar properties on the inertial domain identified from the leading-order structure of the MMB begin to detectably and continuously vary for drag reductions less than $10\,\%$.

In countless biological and technological processes, the flow of Newtonian liquids with a non-Newtonian interface is a common occurrence, such as in monomolecular films in ‘solid’ phases atop of aqueous bulk fluid. There is a lack of models that can predict the flow under conditions different from those used to measure the rheological response of the interface. Here, we present a model which describes interfacial hydrodynamics, including two-way coupling to a bulk Newtonian fluid described by the Navier–Stokes equations, that allows for shear-thinning response of the interface. The model includes a constitutive equation for the interface under steady shear that takes the Newtonian functional form but where the surface shear viscosity is generalized to be a function of the local shear rate. In the limit of a highly viscous interface, the interfacial hydrodynamics is decoupled from the bulk flow and the model can be solved analytically. This provides not only insight into the flow but also a means to validate the numerical technique for solving the two-way coupled problem. The numerical results of the coupled problem shed new light on existing experimental results on steadily sheared monolayers of dipalmitoylphosphatidylcholine (DPPC), the primary constituent of lung surfactant and the bilayers of mammalian cell walls. For low packing density DPPC monolayers, a Newtonian shear-independent surface shear viscosity model can reproduce the interfacial flows, but at high packing density, the shear-thinning properties of the new model presented here are needed.

The equilibrium shape and stability of menisci formed at the contact line between two vertically aligned cylinders were investigated by developing a general bifurcation analysis from the classic equation of Young-Laplace. It was found that the maximum amount of liquid that can be held at the contact line is determined by the existence of a bifurcation of the equilibrium solutions. The onset of instability is characterized by a translationally symmetric bifurcation that always precedes the instability to asymmetric perturbations. The maximum stable liquid retention is a strong function of the ratio of gravitational to surface-tension forces, indicating that gravity acts as a destabilizing force. The effect of contact angle on the maximum liquid retention is more complex: when the gravitational effects are small, an increase in contact angle results in a decrease in liquid retention; on the other hand, when the gravitational effects are appreciable, a maximum value of the liquid retention is obtained for intermediate values of the contact angle.

The multi-layer exchange equations for gravitationally driven flows between two basins with stable Boussinesq type of stratification in discrete layers, specified far upstream on either side of a connecting strait, result in a hydraulic control condition that must be satisfied at the narrowest part of the contraction, the control point. If one stagnant layer is present at the control point, the control condition that applies to all layers collectively may be separated into two such conditions that apply independently to two groups of layers going in opposite directions separated by the stagnant layer. Such bidirectional flow regimes exist if the structure of the prespecified density profiles permits each of the opposing groups to vertically reduce their thickness by the ratio 2/3 relative to their upstream thicknesses, leaving space for the stagnant layer to protrude through the contraction. Under these restrictions, the bidirectional flow is controlled by the fastest propagating wave mode and the stationary solution then relies on the superposition of two previously known unidirectional self-similar flow regimes that are completely decoupled. Techniques for their numerical computation are presented. The transition into loosely coupled and fully coupled flow is discussed. The decoupling principle also applies when several non-adjacent stagnant layers are simultaneously present at control in which case multiple groups of decoupled layers flow in alternating directions.

The trajectories of surface and subsurface particles of nonlinear gravity–capillary waves are calculated. Surface tension is found to have a small effect on the trajectories and drift velocity of pure gravity waves (down to 20 cm in length). On the other hand, pure capillary wave trajectories can be considerably altered even when the influence of gravity is small (waves of up to 2 cm in length). When the restoring forces are of comparable magnitude, some remarkable trajectories are possible, containing one or more subloops. Overall, the influence of increased surface tension is to increase the relative horizontal distance travelled by a particle, as well as the magnitude of the time-averaged drift velocity ratio at the surface and, as far as short waves are concerned, its penetration depth.

These results can have implications for steep waves where parasitic capillaries are generated and for observations of the wind-drift current.

An approach is presented for studying Rossby wave interaction in a shear flow with both regular and singular modes (i.e. those possessing a critical level). The approach relies on a truncated normal mode expansion of the equations of motion. Such an expansion remains valid in the presence of singular modes, provided that these modes are not considered individually, but that complete packets are taken into account in the truncated system. Mathematically, this means that the interaction equations need to be integrated with respect to the phase velocity (or, equivalently, the critical level position) of the singular modes.

The action of two regular modes on a packet of singular modes is treated in detail; in particular, asymptotic results are deduced for the long-term behaviour of the packet. The case of a linear shear is considered as an illustration: analytical expressions are derived for the normal modes and their pseudomomentum, and they are used to present explicit results for the evolution of the packet of singular modes.

The unstable conclusion drawn by Ortloff & Ives of the cylinder motion in a viscous stream is shown to be incorrect. Furthermore, a region of stable solution is derived in the parameter space defined by the normal and tangential drag coefficients.

A method for calculating unsteady boundary-layer flows, based upon a vorticity–stream function formulation, has been supplemented by an appropriate viscous–inviscid interaction law to extend the calculations to large but finite Reynolds numbers. Two examples are considered in detail, namely the impulsive motion of a circular cylinder, and the flow induced when a line vortex is introduced into the neighbourhood of a circular cylinder. For the first of these a comparison of the results obtained with earlier boundary-layer and interactive calculations, Navier–Stokes solutions and experiment is made.

A rapidly scanning one-velocity-component directionally sensitive fringe-type laser-Doppler anemometer which scans the measurement volume perpendicular to the optical axis of the transmitting optics was used to investigate the flow structure of the steady freestream separated turbulent boundary layer of Simpson, Chew & Shivaprasad (1981a). Space–time correlations were obtained for the first time in a separated turbulent boundary layer and showed that the integral lengthscale Ly for the large eddies grows in size towards detachment, although the ratio of this lengthscale to the boundary-layer thickness remains constant. Results also indicate local dependence of the backflow on the middle and outer regions of the boundary layer at a given instant in time.