The wetting kinetics of a drop on a solid surface is measured by observing the movement of the contact line, which is often seen to be unstable, showing a scalloped profile. Many factors have been cited, which, although they can cause instability, can also be eliminated from the experiments, but still the instabilities appear. The basic shape of a spreading drop has a large curvature localized in the vicinity of the contact line as determined by microscopy. It is shown here using linear stability analysis that this curvature can destabilize the contact line region. When the drop profile is disturbed from a basic thickness of h to h + h′, there are two contributions from h′ in the form of added Laplace pressure. One of these is commonly accounted for in the stability analyses. The other is not, and occurs only if the basic shape has a curvature, and the drop has a large curvature near the apparent dynamic contact line, but only for a wetting liquid. This is why instability is not reported in the case of spreading of drops of non-wetting liquids. It also explains why instability gives rise to the changed spreading kinetics of drops that are sometimes reported in the literature, and suggests that as larger curvatures are expected in forced spreading those cases are probably accompanied quite frequently by unstable contact lines.

Convection effects in the melt of a vertical Bridgman furnace, used for solidifying a dilute binary alloy, are known to cause significant, and undesirable, non-uniformity in the alloy. We have found previously that non-axisymmetry significantly degrades the performance of the furnace at large Rayleigh number, Ra, and small Biot number, [Bscr ]. There have been a number of studies on improvement of the alloy quality by the introduction of additional forces into the melt flow – magnetic forces or d'Alembert forces due to various sorts of acceleration of the ampoule. In this paper, we explore the effects on the radial segregation generated by rotating the ampoule about its vertical axis. We determine that the magnitude of segregation is proportional to the product of [Bscr ] and the thickness of the thermal layer on the crystal–melt interface. As the rotation, as measured by a Taylor number, [Tscr ], increases beyond O(Ra1/3), the thermal layer thickens and so the segregation increases. Finally, at [Tscr ] = O(Ra1/2), the thermal adjustment occurs on outer scales, and hence the solutal concentration increases to O([Bscr ]). Hence rotation about the vertical axis actually degrades performance!

A linear stability analysis of a doubly diffusive system, with rotation and shear, shows that overstable oscillations can occur in stratifications typical of the equatorial ocean, that internal waves encountering an equatorial current can exchange energy with the current, and that the wave-induced fluxes of salt and heat can lead to layer formation in the salinity, temperature and velocity fields.

The inertial waves excited in a uniformly rotating fluid passing through a long circular tube are studied numerically. The waves are excited either by a local deformation of the tube wall or by an obstacle located on the tube axis. When the flow is subcritical, i.e. when the phase and group velocity of the fastest wave mode in their long-wave limit are larger than the incoming axial flow velocity, the excited waves propagate upstream of the excited position. The non-resonant waves have many linear aspects, including the upstream-advancing speed of the wave and the coexisting lee wavelength. When the flow is critical (resonant), i.e. when the long-wave velocity is nearly equal to the axial flow velocity, the large-amplitude waves are resonantly excited. The time development of these waves is described well by the equation derived by Grimshaw & Yi (1993). The integro-differential equation, which describes the strongly nonlinear waves until the axial flow reversal occurs, can predict the onset time and position of the recirculation eddies observed in the solutions of the Navier-Stokes equations. The numerical results and the theory both show that the flow reversal most probably occurs on the tube axis and also when the waves are excited by a contraction of the tube wall. The structure of the recirculation eddies obtained in the solutions of the Navier-Stokes equations at Re = 105 is similar to the axisymmetric or ‘bubble-type’ breakdown observed in the experiments of the vortex-breakdown which used a different non-uniform (Burgers-type) rotation. In uniformly rotating fluids the formation of the recirculation eddies has not been observed in the previous numerical studies of vortex breakdown where a straight tube was used and thus the inertial waves were not excited. This shows that the generation of the recirculation eddies in this study is genuinely explained by the topographically excited large-amplitude inertial ‘waves’ and not by other ‘instability’ mechanisms. Since the wave cannot be excited in a straight tube even in the non-uniformly rotating flows, the generation mechanism of the recirculation eddies in this study is different from the previous numerical studies for the vortex breakdown. The occurrence of the recirculation eddies depends not only on the Froude number and the strength of the excitation source but also on the Reynolds number since the wave amplitude generally decreases by the viscous effects. Some relations to the experiments of vortex breakdown, which have been exclusively done for non-uniformly rotating fluids but done in a ‘non-uniform tube’, are discussed. The flow states, which are classified as supercritical, subcritical or critical in hydraulic terminology, changes along the flow when the upstream flow is near resonant conditions and a non-uniform tube is used.

Flows of incompressible inviscid heavy fluids with free or rigid boundary surfaces are considered. For slender streams of fluid, the flow and the free boundaries are represented by a number of different asymptotic expansions in powers of the slenderness ratio. There are three kinds of outer expansions representing respectively jets, which have two free boundaries, wall flows, which have one free and one rigid boundary, and channel flows, which have two rigid boundaries. The flow at the junction of two or more outer flows is represented by an inner expansion. Previously we constructed the three outer expansions and the inner expansion at the junction of a wall flow and a jet (Keller & Geer 1973). Now we construct the inner expansions at the junctions of a channel flow and a jet, a channel flow and a wall flow, and a jet and the two wall flows into which it splits upon hitting a wall. We also match each inner expansion to the adjacent outer expansions. These seven expansions can be combined to solve many problems involving flows of slender streams.

Recent experimental research on rotating cylinder wakes has found that a previously numerically predicted subharmonic instability mode, mode C, occurs for considerably lower rotation rates than predicted through stability analysis, yet other mode transitions occur closer to the predicted onset. One difference between the theoretical and experimental set-ups is the use of a small-diameter hydrogen bubble visualisation wire placed upstream of the rotating cylinder. The current paper tests the hypothesis that a wire, of only $1/100$th of the cylinder diameter, placed five diameters upstream of the cylinder, sufficiently perturbs the flow to substantially affect certain wake transitions, including the onset of mode C. This is achieved using stability analysis of a flow that includes the upstream wire. The results indeed show that the wire of a tiny diameter induces a non-negligible asymmetry in the flow, triggering the subharmonic mode at substantially lower rotation rates. Furthermore, at higher rotation rates, the onset of two other three-dimensional modes are delayed to higher Reynolds numbers. These results make the point that even seemingly minute perturbations caused by minimally intrusive methods may result in substantially altered experimental flow behaviour.

Convection plays a major role in a variety of natural hydrodynamic systems. Those in which convection drives exchange flows through a lateral contraction and/or over a sill form a special class with typical examples being the Red and Mediterranean Seas, the Persian Gulf, and the fjords that indent many coastlines. The present work focuses on the spatial distribution and scaling of the density difference between the inflowing and outflowing fluid layers. Using a long water-filled channel, fitted with buoyancy sources at its upper surface, experiments were conducted to investigate the influence of the geometry of the strait and the channel as well as the magnitude of the buoyancy flux. Two different scaling laws, one by Phillips (1966), and one by Maxworthy (1994, 1997) were compared with the experimental results. It has been shown that a scaling law for which g′ = kB02/3x/h4/3 best describes the distribution of the observed density difference along the channel, where B0 is the buoyancy flux, x the distance from the closed end of the channel, h its height at the open end (sill) and k a constant that depends on the details of the channel geometry and flow conditions. This result holds for the experimental results and appears to be valid for a number of natural systems as well.

The convergence rate and the structures of the proper orthogonal decomposition (POD) reconstruction are re-examined. The relatively slow convergence rate of the wall-normal velocity and the over-prediction of the Reynolds shear stress from POD-based representations of near-wall turbulence suggest that coherent structures are too well correlated. This is a consequence of the directional preference of the eigenfunctions toward the most energetic data (the streamwise motion at $x^+_2 \,{=}\, 13$) which may be relaxed by extracting the POD eigenfunctions from the streak modes and the streamwise vortex modes separately. The rate of convergence and representation of structures may be improved by performing the POD in Squire's coordinate system (PODS). The statistical reconstruction of the localized ($x^+_2 \,{\leq}\, 84$) PODS eigen-functions from the large-eddy simulation (LES) database suggests that the convergence rate of component energy, Reynolds shear stress and the correlation coefficient of streamwise and wall-normal velocity improve significantly. The PODS eigenvalue spectra and the spatial structures of the eigenfunctions indicate the presence of observed streaks, streamwise vortices, and ejection/sweep events.

Dynamical models of the full channel turbulence are constructed by projecting the Navier–Stokes equations onto the PODS eigenfunctions. The unresolved modes are taken into account by closure models adapted from a constant eddy-viscosity model, dynamic Smagorinsky, and dynamic one-coefficient mixed subgrid-scale (SGS) model. Statistical moments obtained from PODS-based simulations are much more accurate than their counterparts from POD-based simulations. This is particularly true for the mean velocity, streamwise turbulent intensity, Reynolds shear stress and the correlation coefficient of streamwise and wall-normal velocity.

Stratified flow in nocturnal boundary layers is studied using direct numerical simulation (DNS) of the Ekman layer, a model problem that is useful to understand atmospheric boundary-layer (ABL) turbulence. A stabilizing buoyancy flux is applied for a finite time to a neutral Ekman layer. Based on previous studies and the simulations conducted here, the choice of $L_{\mathit{cri}}^{+}=Lu_{\ast }/\unicode[STIX]{x1D708}\approx 700$ ($L$ is the Obukhov length scale and $u_{\ast }$ is the friction velocity) provides a cooling flux that is sufficiently strong to cause the initial collapse of turbulence. The turbulent kinetic energy decays over a time scale of $4.06L/u_{\ast }$ during the collapse. The simulations suggest that imposing $L_{\mathit{cri}}^{+}\approx 700$ on the neutral Ekman layer results in turbulence collapse during the initial transient, independent of Reynolds number, $Re_{\ast }$. However, the long-time state of the flow, i.e. turbulent with spatial intermittency or non-turbulent, is found to depend on the initial value of $Re_{\ast }$ since the cooling flux and resultant stratification increase with $Re_{\ast }$ for a given $L^{+}$. The lower-$Re_{\ast }$ cases have sustained turbulence with shear and stratification profiles that evolve in a manner such that the gradient Richardson number, $Ri_{g}$, in the near-surface layer, including the low-level jet, remains subcritical. The highest $Re_{\ast }$ case has supercritical $Ri_{g}$ in the low-level jet and turbulence does not recover. A theoretical discussion is performed to infer that the bulk Richardson number, $Ri_{b}$, is more suitable than $L^{+}$ to determine the fate of stratified Ekman layers at late time. DNS results support the implications of $Ri_{b}$ for the effect of initial $Re_{\ast }$ and $L^{+}$ on the flow.

A liquid-metal flow driven by a rotating magnetic field in a finite-length cylinder is studied numerically as a function of the field frequency. In the high-frequency case, the magnetic field is expelled from the liquid-metal except in a skin-depth layer along the side and top walls of the cylinder. In the corner region, where the skin-depth layers intersect, the body force exhibits a large positive and negative azimuthal component as well as inward radial and axial components which are rotational. The flows for various frequencies are compared to the low-frequency flow.

Wave propagation across the continental shelf is studied by analogy with transmission-line theory. Fourier transformation along the contours of constant depth, which are assumed parallel to a straight coastline, yields a Sturm-Liouville equation for a prescribed depth profile h(x). The modal spectrum of the profile, which comprises a finite, discrete spectrum of trapped modes and a continuous spectrum of radiated modes, is established. The Green's function for a point source on the coastline is constructed by Fourier superposition over this spectrum. Detailed results are calculated for a two-step model (level shelf separated from level abyss by vertical cliff) and for a gradually sloping shelf that merges smoothly into a level abyss. The radiation impedance of a harbour is calculated, and the effects of the continental shelf on the resonant response of the harbour to a tsunami are discussed.

A model developed by Wilmott et al. (J. Fluid Mech., vol. 841, 2018, pp. 310–350) for the advance of a charged oil droplet along a charged capillary pore is considered. The oil droplet is surrounded by an aqueous phase filling the pore, and the model considers a uniformly curved capillary static droplet front plus an aqueous thin film separating the body of the oil droplet from the capillary wall, with these two regions being joined by a transition region. The methodology follows a classical asymptotic approach proposed by Bretherton (J. Fluid Mech., vol. 10, 1961, pp. 166–188) but incorporates additional electro-osmotic effects (specifically an electro-osmotic disjoining tension) due to the charged surfaces. A number of dimensionless parameters control the model’s behaviour, of which the most important is denoted $\unicode[STIX]{x1D712}^{\prime }$ and represents the ratio between the ‘nominal’ thickness of the aqueous film (as determined neglecting any electrostatic effects) and the Debye length within the film, which is sensitive to ion concentrations and hence to salinity. When $\unicode[STIX]{x1D712}^{\prime }$ is large, electro-osmotic effects are screened and Bretherton’s classical results are recovered. However as $\unicode[STIX]{x1D712}^{\prime }$ decreases, electro-osmotic effects come into play and the film becomes much thicker than Bretherton’s prediction to ensure that screening effects are not altogether lost, and also there is a noticeable increase in the pressure needed to drive the droplet front along. These results apply with minor variations in the case of singly charged surfaces (charge on either oil or on the capillary wall), oil and wall surfaces with like charges, or oil and wall surfaces with opposite but unequal charges. However in the case of opposite and equal charges, the system’s behaviour changes dramatically. There is now a conjoining electro-osmotic pressure rather than a disjoining tension, the film becomes thinner than the analogous Bretherton film, and the pressure needed to drive the droplet front along decreases. Surprisingly in this case, for sufficiently small $\unicode[STIX]{x1D712}^{\prime }$, the work done by the conjoining pressure can exceed the work done against viscous dissipation, meaning the pressure required to drive the droplet front is not just smaller than in Bretherton’s predictions but also slightly less than would be estimated based on capillary forces alone. Although the main effect of reducing salinity is to increase Debye length and hence reduce $\unicode[STIX]{x1D712}^{\prime }$, salinity also affects surface charges. A situation is explored whereby reducing salinity affects charges, producing a switch from disjoining tensions to conjoining pressures and back again: this leads to a non-monotonic response in film thickness and pressure required to drive the droplet front along.

The extended mild-slope equation and the modified mild-slope equation have been used successfully to study refraction–diffraction of linear water waves by steep bottom roughness. Their consistency has been questioned. A systematic derivation of these model equations exposes and illuminates their rationale. Their good performance stems from an accurate representation of (Class I) Bragg resonance. As a benchmark test case, we consider scattering by a sloping bottom with random roughness. The rates of scattering found for the mean field in both of the approximate models agree exactly with the full theory for scattering by small roughness. This greatly improves the limited agreement which was found for the mild-slope equation, and establishes the validity of the above model equations. The study involves operator calculus, a powerful method for simplifying problems with variable coefficients. The augmented mild-slope equation serves to consistently derive accurate model equations.

Measurements of concentration fluctuations in a passive scalar plume released within a turbulent boundary layer are utilised to ascertain the scaling of concentration spectra. It is observed that the concentration spectra in a narrow meandering plume has a self-similar behaviour in both transverse ($y$) and vertical ($z$, i.e. wall-normal) directions. Experimental data reveal self-similarity when the magnitude of concentration spectra is scaled by the local concentration variance whereas frequency is suitably scaled utilising the integral length scale of the streamwise velocity or the boundary layer thickness and the source velocity as length and velocity scales, respectively. Furthermore, our data show that at each frequency, the concentration energy is distributed across the $y$ and $z$ directions that is proportional to concentration variance at that location. These results are consistent with our non-dimensional analysis. Based on these observations, if the mean plume statistics are known, a model is proposed with which concentration spectrum at any position within the plume can be calculated using the spectrum at any another location as the input. The model is tested extensively for point-source plumes released at various heights and streamwise distances in a turbulent boundary layer, and is found to predict spectra at different $y$ and $z$ locations in close agreement with measurements.

The equations for the hydrodynamic force and torque acting on a sphere in unsteady Stokes equations under different flow conditions are solved analytically by means of the singularity method. This analytical technique is based on the combination of suitable singularity solutions (also called fundamental solutions) such as primary Stokeslets, potential dipoles, or higher-order singularities, to construct the flow field. The different flows considered here include four examples: (1) a rotating sphere in a viscous flow, (2) a stationary sphere in a time-dependent shear flow, (3) a sphere with free rotation in a simple shear flow, as well as (4) a stationary sphere in a time-dependent axisymmetric parabolic flow. Our paradigm is to derive the fundamental solutions in unsteady Stokes flows and to express the solutions as a convolution integral in time using the time–space fundamental solutions. Next the Laplace transform is used to determine the strength of the distributed singularities that induce the velocity field around a stationary or rotating sphere. Then we use the computed strength of the singularities to derive hydrodynamic force and torque. In particular, for the problem of a stationary sphere in unsteady axisymmetric parabolic flow, our solution for the time-dependent force acting on the sphere consists of five force components – the well-known quasi-steady Stokes drag, the added mass term, the Basset historic (memory) force, and two additional memory forces. The first additional memory force due to the rate change of velocity, we find, is similar to the result obtained by Lawrence & Weinbaum (J. Fluid Mech., vol. 171, 1986, pp. 209–218) for the ostensibly unrelated setting of a slightly deformed translating spheroid. The second additional memory force comes from the effect of the rate change of acceleration and is found for the first time in this study to the best of our knowledge.